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+@STRING{AP = {Academic Press}}
+
+@STRING{AREPS = {Annual Review of Earth Planetary Science}}
+
+@STRING{ARFM = {Annual Review of Fluid Mechanics}}
+
+@STRING{ASL = {Atmospheric Science Letters}}
+
+@STRING{AW = {Addison-Wesley}}
+
+@STRING{CD = {Clim. Dyn.}}
+
+@STRING{CP = {Clarendon Press}}
+
+@STRING{CUP = {Cambridge University Press}}
+
+@STRING{D = {Dover Publications}}
+
+@STRING{DAO = {Dyn. Atmos. Ocean}}
+
+@STRING{DSR = {Deep-Sea Res.}}
+
+@STRING{E = {Eyrolles}}
+
+@STRING{GRL = {Geophys. Res. Let.}}
+
+@STRING{I = {Interscience}}
+
+@STRING{JAOT = {J. Atmos. Ocean Tech.}}
+
+@STRING{JAS = {J. Atmos. Sc.}}
+
+@STRING{JC = {J. Climate}}
+
+@STRING{JCP = {J. Comput. Phys.}}
+
+@STRING{JGR = {J. Geophys. Res}}
+
+@STRING{JHUP = {The Johns Hopkins University Press}}
+
+@STRING{JMR = {J. Mar. Res.}}
+
+@STRING{JMS = {J. Mar. Sys.}}
+
+@STRING{JMSJ = {J. Met. Soc. Japan}}
+
+@STRING{JPO = {J. Phys. Oceanogr.}}
+
+@STRING{JWS = {John Wiley and Sons}}
+
+@STRING{M = {Macmillan}}
+
+@STRING{MGH = {McGraw-Hill}}
+
+@STRING{MWR = {Mon. Wea. Rev.}}
+
+@STRING{Nature = {Nat.}}
+
+@STRING{NH = {North-Holland}}
+
+@STRING{Ocean = {Oceanology}}
+
+@STRING{OS = {Ocean Science}}
+
+@STRING{OUP = {Oxford University Press}}
+
+@STRING{PH = {Prentice-Hall}}
+
+@STRING{PO = {Prog. Oceangr.}}
+
+@STRING{PP = {Pergamon Press}}
+
+@STRING{PRSL = {Proceedings of the Royal Society of London}}
+
+@STRING{QJRMS = {Quart J Roy Meteor Soc}}
+
+@STRING{Recherche = {La Recherche}}
+
+@STRING{Science = {Science}}
+
+@STRING{SV = {Springer-Verlag}}
+
+@STRING{Tellus = {Tellus}}
+
+@ARTICLE{Adcroft_Campin_OM04,
+  author = {A. Adcroft and J.-M. Campin},
+  title = {Re-scaled height coordinates for accurate representation of free-surface
+	flows in ocean circulation models},
+  journal = {Ocean Modelling},
+  year = {2004},
+  volume = {7},
+  owner = {gm},
+  timestamp = {2008.01.27}
+}
+
+@ARTICLE{Arakawa1966,
+  author = {A. Arakawa},
+  title = {Computational design for long term numerical integration of the equations
+	of fluid motion, two-dimensional incompressible flow, Part. I.},
+  journal = JCP,
+  year = {1966},
+  volume = {I},
+  pages = {119-149},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Arakawa1990,
+  author = {A. Arakawa and Y.-J. G. Hsu},
+  title = {Energy Conserving and Potential-Enstrophy Dissipating Schemes for
+	the Shallow Water Equations},
+  journal = MWR,
+  year = {1990},
+  volume = {118},
+  pages = {1960--1969
+	
+	},
+  number = {10},
+  abstract = {To incorporate potential enstrophy dissipation into discrete shallow
+	water equations with no or arbitrarily small energy dissipation,
+	a family of finite-difference schemes have been derived with which
+	potential enstrophy is guaranteed to decrease while energy is conserved
+	(when the mass flux is nondivergent and time is continuous). Among
+	this family of schemes, there is a member that minimizes the spurious
+	impact of infinite potential vorticities associated with infinitesimal
+	fluid depth. The scheme is, therefore, useful for problems in which
+	the free surface may intersect with the lower boundary.},
+  date = {October 01, 1990},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Arakawa1981,
+  author = {Arakawa, Akio and Lamb, Vivian R.},
+  title = {A Potential Enstrophy and Energy Conserving Scheme for the Shallow
+	Water Equations},
+  journal = MWR,
+  year = {1981},
+  volume = {109},
+  pages = {18--36
+	
+	},
+  number = {1},
+  abstract = {To improve the simulation of nonlinear aspects of the flow over steep
+	topography, a potential enstrophy and energy conserving scheme for
+	the shallow water equations is derived. It is pointed out that a
+	family of schemes can conserve total energy for general flow and
+	potential enstrophy for flow with no mass flux divergence. The newly
+	derived scheme is a unique member of this family, that conserves
+	both potential enstrophy and energy for general flow. Comparison
+	by means of numerical experiment with a scheme that conserves (potential)
+	enstrophy for purely horizontal nondivergent flow demonstrated the
+	considerable superiority of the newly derived potential enstrophy
+	and energy conserving scheme, not only in suppressing a spurious
+	energy cascade but also in determining the overall flow regime. The
+	potential enstrophy and energy conserving scheme for a spherical
+	grid is also presented.},
+  date = {January 01, 1981},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Arhan2006,
+  author = {M. Arhan and A.M. Treguier and B. Bourles and S. Michel},
+  title = {Diagnosing the annual cycle of the Equatorial Undercurrent in the
+	Atlantic Ocean from a general circulation model},
+  journal = JPO,
+  year = {2006},
+  volume = { 36},
+  pages = {1502-1522}
+}
+
+@ARTICLE{ASSELIN1972,
+  author = {R. Asselin},
+  title = {Frequency Filter for Time Integrations},
+  journal = MWR,
+  year = {1972},
+  volume = {100},
+  pages = {487-490},
+  number = {6},
+  abstract = {A simple filter for controlling high-frequency computational and physical
+	modes arising in time integrations is proposed. A linear analysis
+	of the filter with leapfrog, implicit, and semi-implicit, differences
+	is made. The filter very quickly removes the computational mode and
+	is also very useful in damping high-frequency physical waves. The
+	stability of the leapfrog scheme is adversely affected when a large
+	filter parameter is used, but the analysis shows that the use of
+	centered differences with frequency filter is still more advantageous
+	than the use of the Euler-backward method. An example of the use
+	of the filter in an actual forecast with the meteorological equations
+	is shown.},
+  date = {June 01, 1972},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Barnier_al_OD06,
+  author = {B. Barnier and G. Madec and T. Penduff and J.-M. Molines and A.-M.
+	Treguier and J. Le Sommer and A. Beckmann and A. Biastoch and C.
+	Boning and J. Dengg and C. Derval and E. Durand and S. Gulev and
+	E. Remy and C. Talandier and S. Theetten and M. Maltrud and J. McClean
+	and B. De Cuevas},
+  title = {Impact of partial steps and momentum advection schemes in a global
+	ocean circulation model at eddy-permitting resolution.},
+  journal = {Ocean Dyn.},
+  year = {2006},
+  pages = {doi: 10.1007/s10236-006-0082-1.},
+  owner = {gm},
+  timestamp = {2008.01.25}
+}
+
+@INCOLLECTION{Barnier1996,
+  author = {B. Barnier and P. Marchesiello and A.P. de Miranda},
+  title = {Modeling the ocean circulation in the South Atlantic: A strategy
+	for dealing with open boundaries},
+  booktitle = {The South Atlantic: Present and Past Circulation},
+  publisher = {Springer-Verlag, Berlin},
+  year = {1996},
+  editor = {G.Wefer and W.H. Berger and G Siedler and D. Webb},
+  pages = {289-304}
+}
+
+@ARTICLE{Barnier1998,
+  author = {B. Barnier and P. Marchesiello and A. P. de Miranda and J.M. Molines
+	and M. Coulibaly},
+  title = {A sigma-coordinate primitive equation model for studying the circulation
+	in the South Atlantic I, Model configuration with error estimates},
+  journal = {Deep Sea Res.},
+  year = {1998},
+  volume = {45},
+  pages = {543-572}
+}
+
+@ARTICLE{Beckmann1998,
+  author = {A. Beckmann},
+  title = {The representation of bottom boundary layer processes in numerical
+	ocean circulation models.},
+  journal = {Ocean modelling and parameterization, E. P. Chassignet and J. Verron
+	(eds.), NATO Science Series, Kluwer Academic Publishers},
+  year = {1998},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{BeckDos1998,
+  author = {A. Beckmann and R. D\"{o}scher},
+  title = {A method for improved representation of dense water spreading over
+	topography in geopotential-coordinate models},
+  journal = JPO,
+  year = {1998},
+  volume = {27},
+  pages = {581-591},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Beckmann1993,
+  author = {A. Beckmann and D. B. Haidvogel},
+  title = {Numerical Simulation of Flow around a Tall Isolated Seamount. Part
+	I - Problem Formulation and Model Accuracy},
+  journal = {Journal of Physical Oceanography},
+  year = {1993},
+  volume = {23},
+  pages = {1736--1753
+	
+	},
+  number = {8},
+  abstract = {A sigma coordinate ocean circulation model is employed to study flow
+	trapped to a tall seamount in a periodic f-plane channel. In Part
+	I, errors arising from the pressure gradient formulation in the steep
+	topography&#47;strong stratification limit are examined. To illustrate
+	the error properties, a linearized adiabatic version of the model
+	is considered, both with and without forcing, and starting from a
+	resting state with level isopycnals. The systematic discretization
+	errors from the horizontal pressure gradient terms are shown analytically
+	to increase with steeper topography (relative to a fixed horizontal
+	grid) and for stronger stratification (as measured by the Burger
+	number). For an initially quiescent unforced ocean, the pressure
+	gradient errors produce a spurious oscillating current that, at the
+	end of 10 days, is approximately 1 cm s&#8722;1 in amplitude. The
+	period of the spurious oscillation (about 0.5 days) is shown to be
+	a consequence of the particular form of the pressure gradient terms
+	in the sigma coordinate system. With the addition of an alongchannel
+	diurnal forcing, resonantly generated seamount-trapped waves are
+	observed to form. Error levels in these solutions are less than those
+	in the unforced cases; spurious time-mean currents are several orders
+	of magnitude less in amplitude than the resonant propagating waves.
+	However, numerical instability is encountered in a wider range of
+	parameter space. The properties of these resonantly generated waves
+	is explored in detail in Part II of this study. Several new formulations
+	of the pressure gradient terms are tested. Two of the formulations&#8212;constructed
+	to have additional conservation properties relative to the traditional
+	form of the pressure gradient terms (conservation of JEBAR and conservation
+	of energy)&#8212;are found to have error properties generally similar
+	to those of the traditional formulation. A corrected gradient algorithm,
+	based upon vertical interpolation of the pressure field, has a dramatically
+	reduced error level but a much more restrictive range of stable behavior.},
+  date = {August 01, 1993},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Blanke_al_JPO99,
+  author = {B. Blanke and M. Arhan and G. Madec and S. Roche},
+  title = {Warm Water Paths in the Equatorial Atlantic as Diagnosed with a General
+	Circulation Model},
+  journal = JPO,
+  year = {1999},
+  volume = {29, 11},
+  pages = {2753-2768},
+  owner = {gm},
+  timestamp = {2008.05.27}
+}
+
+@ARTICLE{Blanke1993,
+  author = {B. Blanke and P. Delecluse},
+  title = {Low frequency variability of the tropical Atlantic ocean simulated
+	by a general circulation model with mixed layer physics},
+  journal = JPO,
+  year = {1993},
+  volume = {23},
+  pages = {1363-1388}
+}
+
+@ARTICLE{blanketal97,
+  author = {B. Blanke and J. D. Neelin and D. Gutzler},
+  title = {Estimating the effect of stochastic wind forcing on ENSO irregularity},
+  journal = JC,
+  year = {1997},
+  volume = {10},
+  pages = {1473-1486},
+  abstract = {One open question in El NinoSouthern Oscillation (ENSO) simulation
+	and predictability is the role of random
+	
+	forcing by atmospheric variability with short correlation times, on
+	coupled variability with interannual timescales.
+	
+	The discussion of this question requires a quantitative assessment
+	of the stochastic component of the wind stress
+	
+	forcing. Self-consistent estimates of this noise (the stochastic forcing)
+	can be made quite naturally in an empirical
+	
+	atmospheric model that uses a statistical estimate of the relationship
+	between sea surface temperature (SST) and
+	
+	wind stress anomaly patterns as the deterministic feedback between
+	the ocean and the atmosphere. The authors
+	
+	use such an empirical model as the atmospheric component of a hybrid
+	coupled model, coupled to the GFDL
+	
+	ocean general circulation model. The authors define as residual the
+	fraction of the Florida State University wind
+	
+	stress not explained by the empirical atmosphere run from observed
+	SST, and a noise product is constructed by
+	
+	random picks among monthly maps of this residual.
+	
+	The impact of included or excluded noise is assessed with several
+	ensembles of simulations. The model is
+	
+	run in coupled regimes where, in the absence of noise, it is perfectly
+	periodic: in the presence of prescribed
+	
+	seasonal variability, the model is strongly frequency locked on a
+	2-yr period; in annual average conditions it
+	
+	has a somewhat longer inherent ENSO period (30 months). Addition of
+	noise brings an irregular behavior that
+	
+	is considerably richer in spatial patterns as well as in temporal
+	structures. The broadening of the model ENSO
+	
+	spectral peak is roughly comparable to observed. The tendency to frequency
+	lock to subharmonic resonances
+	
+	of the seasonal cycle tends to increase the broadening and to emphasize
+	lower frequencies. An inclination to
+	
+	phase lock to preferred seasons persists even in the presence of noise-induced
+	irregularity. Natural uncoupled
+	
+	atmospheric variability is thus a strong candidate for explaining
+	the observed aperiodicity in ENSO time series.
+	
+	Modelmodel hindcast experiments also suggest the importance of atmospheric
+	noise in setting limits to ENSO
+	
+	predictability.},
+  pdf = {Blanke_etal_JC97.pdf}
+}
+
+@ARTICLE{Blanke_Raynaud_JPO97,
+  author = {B. Blanke and S. Raynaud},
+  title = {Kinematics of the Pacific Equatorial Undercurrent: An Eulerian and
+	Lagrangian Approach from GCM Results},
+  journal = JPO,
+  year = {1997},
+  volume = {27, 6},
+  pages = {1038-1053},
+  owner = {gm},
+  timestamp = {2008.05.27}
+}
+
+@ARTICLE{Blayo2005,
+  author = {E. Blayo and L. Debreu},
+  title = {Revisiting open boundary conditions from the point of view of characteristic
+	variables},
+  journal = {Ocean Modelling},
+  year = {2005},
+  volume = {9},
+  pages = {231-252}
+}
+
+@ARTICLE{Bougeault1989,
+  author = {P. Bougeault and P. Lacarrere},
+  title = {Parameterization of Orography-Induced Turbulence in a Mesobeta--Scale
+	Model},
+  journal = MWR,
+  year = {1989},
+  volume = {117},
+  pages = {1872-1890},
+  number = {8},
+  abstract = {The possibility of extending existing techniques for turbulence parameterization
+	in the planetary boundary layer to attitude, orography-induced turbulence
+	events is examined. Starting from a well-tested scheme, we show that
+	it is possible to generalize the specification method of the length
+	scales, with no deterioration of the scheme performance in the boundary
+	layer. The new scheme is implemented in a two-dimensional version
+	of a limited-area, numerical model used for the simulation of mesobeta-scale
+	atmospheric flows. Three well-known cases of orographically induced
+	turbulence are studied. The comparison with observations and former
+	studies shows a satisfactory behavior of the new scheme.},
+  date = {August 01, 1989},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Brown1978,
+  author = {J. A. Brown and K. A. Campana},
+  title = {An Economical Time-Differencing System for Numerical Weather Prediction},
+  journal = MWR,
+  year = {1978},
+  volume = {106},
+  pages = {1125-1136},
+  number = {8},
+  month = aug,
+  abstract = {A simple method for integrating the primitive equations is presented
+	which allows for a timestep increment up to twice that of the conventional
+	leapfrog scheme. It consists of a time-averaging operator, which
+	incorporates three consecutive time levels, on the pressure gradient
+	terms in the equations of motion. An attractive feature of the method
+	is its case in programming, since the resulting finite-difference
+	equations can he solved explicitly.Presented here are linear analyses
+	of the method applied to the barotropic and two-layer baroclinic
+	gravity waves. Also presented is an analysis of the method with a
+	time-damping device incorporated, which is an alternative in controlling
+	linearly amplifying computational modes.},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Bryan1997,
+  author = {K. Bryan},
+  title = {A Numerical Method for the Study of the Circulation of the World
+	Ocean},
+  journal = JCP,
+  year = {1997},
+  volume = {135, 2},
+  owner = {gm},
+  timestamp = {2007.08.10}
+}
+
+@ARTICLE{Bryan1984,
+  author = {K. Bryan},
+  title = {Accelerating the convergence to equilibrium of ocean-climate models},
+  journal = JPO,
+  year = {1984},
+  volume = {14},
+  owner = {gm},
+  timestamp = {2007.08.10}
+}
+
+@ARTICLE{Bryden1973,
+  author = {H. L. Bryden},
+  title = {New polynomials for thermal expansion, adiabatic temperature gradient
+	
+	and potential temperature of sea water},
+  journal = DSR,
+  year = {1973},
+  volume = {20},
+  pages = {401-408},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Campin2004,
+  author = {J.-M. Campin and A. Adcroft and C. Hill and J. Marshall},
+  title = {Conservation of properties in a free-surface model},
+  journal = {Ocean Modelling},
+  year = {2004},
+  volume = {6, 3-4},
+  pages = {221-244},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Campin_Goosse_Tel99,
+  author = {J. M. Campin and H. Goosse},
+  title = {Parameterization of density-driven downsloping flow for a coarse-resolution
+	ocean model in z-coordinate},
+  journal = {Tellus},
+  year = {1999},
+  volume = {51},
+  pages = {412-430},
+  owner = {gm},
+  timestamp = {2008.01.20}
+}
+
+@ARTICLE{Cox1987,
+  author = {M. Cox},
+  title = {Isopycnal diffusion in a z-coordinate ocean model},
+  journal = {Ocean Modelling},
+  year = {1987},
+  volume = {74},
+  pages = {1-9},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Dorscher_Beckmann_JAOT00,
+  author = {R. D\"{o}scher and A. Beckmann},
+  title = {Effects of a Bottom Boundary Layer Parameterization in a Coarse-Resolution
+	Model of the North Atlantic Ocean},
+  journal = JAOT,
+  year = {2000},
+  volume = {17},
+  pages = {698-707},
+  owner = {gm},
+  timestamp = {2008.01.23}
+}
+
+@ARTICLE{Debreu_al_CG2008,
+  author = {L. Debreu and C. Vouland and E. Blayo},
+  title = {AGRIF: Adaptive Grid Refinement In Fortran},
+  journal = {Computers and Geosciences},
+  year = {2008},
+  volume = {34},
+  pages = {8-13},
+  owner = {gm},
+  timestamp = {2008.02.03}
+}
+
+@ARTICLE{Delecluse_Madec_Bk00,
+  author = {P. Delecluse and G. Madec},
+  title = {Ocean modelling and the role of the ocean in the climate system},
+  journal = {In \textit{Modeling the Earth's Climate and its Variability}, Les
+	Houches, Session, LXVII 1997,
+	
+	Eds. W. R. Holland, S. Joussaume and F. David, Elsevier Science,},
+  year = {2000},
+  pages = {237-313},
+  owner = {gm},
+  timestamp = {2008.02.03}
+}
+
+@ARTICLE{Dukowicz1994,
+  author = {J. K. Dukowicz and R. D. Smith},
+  title = {Implicit free-surface method for the Bryan-Cox-Semtner ocean model},
+  journal = JGR,
+  year = {1994},
+  volume = {99},
+  pages = {7991-8014},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@INCOLLECTION{Durran2001,
+  author = {D.R. Durran },
+  title = {Open boundary conditions: fact and fiction},
+  booktitle = {Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics},
+  publisher = {Kluwer Academic Publishers},
+  year = {2001},
+  editor = {P.F. Hodnett}
+}
+
+@ARTICLE{Dutay.J.C2004,
+  author = {J. -C. Dutay and P. J. -Baptiste and J. -M. Campin and A. Ishida
+	and E. M. -Reimer and R. J. Matear and A. Mouchet and I. J. Totterdell
+	and Y. Yamanaka and K. Rodgers and G. Madec and J.C. Orr},
+  title = {Evaluation of OCMIP-2 ocean models deep circulation
+	
+	with mantle helium-3},
+  journal = {Journal of Marine Systems},
+  year = {2004},
+  pages = {1-22},
+  abstract = {We compare simulations of the injection of mantle helium-3 into the
+	deep ocean from six global coarse resolution models which participated
+	in the Ocean Carbon Model Intercomparison Project (OCMIP). We also
+	discuss the results of a study carried out with one of the models,
+	which examines the effect of the subgrid-scale mixing parameterization.
+	These sensitivity tests provide useful information to interpret the
+	differences among the OCMIP models and between model simulations
+	and the data.
+	
+	We find that the OCMIP models, which parameterize subgrid-scale mixing
+	using an eddy-induced velocity, tend to
+	
+	underestimate the ventilation of the deep ocean, based on diagnostics
+	with d3He. In these models, this parameterization is implemented
+	with a constant thickness diffusivity coefficient. In future simulations,
+	we recommend using such a parameterization with spatially and temporally
+	varying coefficients in order to moderate its effect on stratification.
+	
+	The performance of the models with regard to the formation of AABW
+	confirms the conclusion from a previous evaluation with CFC-11. Models
+	coupled with a sea-ice model produce a substantial bottom water formation
+	in the Southern Ocean that tends to overestimate AABW ventilation,
+	while models that are not coupled with a sea-ice model systematically
+	underestimate the formation of AABW.
+	
+	We also analyze specific features of the deep 3He distribution (3He
+	plumes) that are particularly well depicted in the data and which
+	put severe constraints on the deep circulation. We show that all
+	the models fail to reproduce a correct propagation of these plumes
+	in the deep ocean. The resolution of the models may be too coarse
+	to reproduce the strong and narrow currents in the deep ocean, and
+	the models do not incorporate the geothermal heating that may also
+	contribute to the generation of these currents. We also use the context
+	of OCMIP-2 to explore the potential of mantle helium-3 as a tool
+	to compare and evaluate modeled deep-ocean circulations. Although
+	the source function of mantle helium is known with a rather large
+	uncertainty, we find that the parameterization used for the injection
+	of mantle helium-3 is sufficient to generate realistic results, even
+	in the Atlantic Ocean where a previous pioneering study [J. Geophys.
+	Res. 100 (1995) 3829] claimed this parameterization generates
+	
+	inadequate results. These results are supported by a multi-tracer
+	evaluation performed by considering the simulated distributions of
+	both helium-3 and natural 14C, and comparing the simulated tracer
+	fields with available data.},
+  owner = {sandra},
+  pdf = {Dutay_etal_OCMIP_JMS04.pdf},
+  timestamp = {2006.10.17}
+}
+
+@ARTICLE{Eiseman1980,
+  author = {P. R. Eiseman and A. P. Stone},
+  title = {Conservation lows of fluid dynamics -- A survey},
+  journal = {SIAM Review},
+  year = {1980},
+  volume = {22},
+  pages = {12-27},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@PHDTHESIS{Farge1987,
+  author = {M. Farge},
+  title = {Dynamique non lineaire des ondes et des tourbillons dans les equations
+	de Saint Venant},
+  school = {Doctorat es Mathematiques, Paris VI University, 401 pp.},
+  year = {1987},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Farrow1995,
+  author = {D. E. Farrow and D. P. Stevens},
+  title = {A new tracer advection scheme for Bryan--Cox type ocean general circulation
+	models},
+  journal = JPO,
+  year = {1995},
+  volume = {25},
+  pages = {1731-1741.},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Fujio1991,
+  author = {S. Fujio and N. Imasato},
+  title = {Diagnostic calculation for circulation and water mass movement in
+	the deep Pacific},
+  journal = JGR,
+  year = {1991},
+  volume = {96},
+  pages = {759-774},
+  month = jan,
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Gargett1984,
+  author = {A. E. Gargett},
+  title = {Vertical eddy diffusivity in the ocean interior},
+  journal = JMR,
+  year = {1984},
+  volume = {42},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Gaspar1990,
+  author = {P. Gaspar and Y. Gr{\'e}goris and J.-M. Lefevre},
+  title = {A simple eddy kinetic energy model for simulations of the oceanic
+	vertical mixing\: Tests at Station Papa and long-term upper ocean
+	study site},
+  journal = JGR,
+  year = {1990},
+  volume = {95(C9)},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Gent1990,
+  author = {P. R. Gent and J. C. Mcwilliams},
+  title = {Isopycnal Mixing in Ocean Circulation Models},
+  journal = JPO,
+  year = {1990},
+  volume = {20},
+  pages = {150-155},
+  number = {1},
+  abstract = {A subgrid-scale form for mesoscale eddy mixing on isopycnal surfaces
+	is proposed for use in non-eddy-resolving ocean circulation models.
+	The mixing is applied in isopycnal coordinates to isopycnal layer
+	thickness, or inverse density gradient, as well as to passive scalars,
+	temperature and salinity. The transformation of these mixing forms
+	to physical coordinates is also presented.},
+  date = {January 01, 1990},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Gerdes1993a,
+  author = {R. Gerdes},
+  title = {A primitive equation ocean circulation model using a general vertical
+	coordinate transformation 1. Description and testing of the model},
+  journal = JGR,
+  year = {1993},
+  volume = {98},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Gerdes1993b,
+  author = {R. Gerdes},
+  title = {A primitive equation ocean circulation model using a general vertical
+	coordinate transformation 2. Application to an overflow problem},
+  journal = JGR,
+  year = {1993},
+  volume = {98},
+  pages = {14703-14726},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@TECHREPORT{Gibson_TR86,
+  author = {J. K. Gibson},
+  title = {Standard software development and maintenance},
+  institution = {Operational Dep., ECMWF, Reading, UK.},
+  year = {1986},
+  owner = {gm},
+  timestamp = {2008.02.03}
+}
+
+@BOOK{Gill1982,
+  title = {Atmosphere-Ocean Dynamics},
+  publisher = {International Geophysics Series, Academic Press, New-York},
+  year = {1982},
+  author = {A. E. Gill}
+}
+
+@ARTICLE{Goosse_al_JGR99,
+  author = {H. Goosse and E. Deleersnijder and T. Fichefet and M. England},
+  title = {Sensitivity of a global coupled ocean-sea ice model to the parameterization
+	of vertical mixing},
+  journal = JGR,
+  year = {1999},
+  volume = {104},
+  pages = {13,681-13,695},
+  owner = {gm},
+  timestamp = {2008.05.27}
+}
+
+@ARTICLE{Griffes2005,
+  author = {S. M. Griffes and A. Gnanadesikan and K. W. Dixon and J. P. Dunne
+	and R. Gerdes and M. J. Harrison and A. Rosati and J. L. Russell
+	and B. L. Samuels and M. J. Spelman and M. Winton and R. Zhang},
+  title = {Formulation of an ocean model for global climate simulations},
+  journal = OS,
+  year = {2005},
+  pages = {165246},
+  abstract = {This paper summarizes the formulation of the ocean component to the
+	Geophysical
+	
+	Fluid Dynamics Laboratorys (GFDL) coupled climate model used for
+	the 4th IPCC As- Assessment
+	
+	(AR4) of global climate change. In particular, it reviews elements
+	of ocean
+	
+	sessment climate models and how they are pieced together for use in
+	a state-of-the-art coupled 5
+	
+	model. Novel issues are also highlighted, with particular attention
+	given to sensitivity of
+	
+	the coupled simulation to physical parameterizations and numerical
+	methods. Features
+	
+	of the model described here include the following: (1) tripolar grid
+	to resolve the Arctic
+	
+	Ocean without polar filtering, (2) partial bottom step representation
+	of topography to
+	
+	better represent topographically influenced advective and wave processes,
+	(3) more 10
+	
+	accurate equation of state, (4) three-dimensional flux limited tracer
+	advection to reduce
+	
+	overshoots and undershoots, (5) incorporation of regional climatological
+	variability in
+	
+	shortwave penetration, (6) neutral physics parameterization for representation
+	of the
+	
+	pathways of tracer transport, (7) staggered time stepping for tracer
+	conservation and
+	
+	numerical eciency, (8) anisotropic horizontal viscosities for representation
+	of equato- 15
+	
+	rial currents, (9) parameterization of exchange with marginal seas,
+	(10) incorporation
+	
+	of a free surface that accomodates a dynamic ice model and wave propagation,
+	(11)
+	
+	transport of water across the ocean free surface to eliminate unphysical
+	virtual tracer
+	
+	flux methods, (12) parameterization of tidal mixing on continental
+	shelves.},
+  owner = {sandra},
+  pdf = {Griffies_al_OSD05.pdf},
+  timestamp = {2007.01.25}
+}
+
+@BOOK{Griffies2004,
+  title = {Fundamentals of ocean climate models},
+  publisher = {Princeton University Press, 434pp},
+  year = {2004},
+  author = {S. M. Griffies},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Griffies1998,
+  author = {S. M. Griffies and A. Gnanadesikan and R. C. Pacanowski and V. D.
+	Larichev and J. K. Dukowicz and R. D. Smith},
+  title = {Isoneutral Diffusion in a z-Coordinate Ocean Model},
+  journal = JPO,
+  year = {1998},
+  volume = {28},
+  pages = {805-830},
+  number = {5},
+  abstract = {This paper considers the requirements that must be satisfied in order
+	to provide a stable and physically based isoneutral tracer diffusion
+	scheme in a z-coordinate ocean model. Two properties are emphasized:
+	1) downgradient orientation of the diffusive fluxes along the neutral
+	directions and 2) zero isoneutral diffusive flux of locally referenced
+	potential density. It is shown that the Cox diffusion scheme does
+	not respect either of these properties, which provides an explanation
+	for the necessity to add a nontrivial background horizontal diffusion
+	to that scheme. A new isoneutral diffusion scheme is proposed that
+	aims to satisfy the stated properties and is found to require no
+	horizontal background diffusion.},
+  date = {May 01, 1998},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Griffies2001,
+  author = {S. M. Griffies and R. C. Pacanowski and M. Schmidt and V. Balaji},
+  title = {Tracer Conservation with an Explicit Free Surface Method for z-Coordinate
+	Ocean Models},
+  journal = MWR,
+  year = {2001},
+  volume = {129},
+  pages = {1081-1098},
+  number = {5},
+  abstract = {This paper details a free surface method using an explicit time stepping
+	scheme for use in z-coordinate ocean models. One key property that
+	makes the method especially suitable for climate simulations is its
+	very stable numerical time stepping scheme, which allows for the
+	use of a long density time step, as commonly employed with coarse-resolution
+	rigid-lid models. Additionally, the effects of the undulating free
+	surface height are directly incorporated into the baroclinic momentum
+	and tracer equations. The novel issues related to local and global
+	tracer conservation when allowing for the top cell to undulate are
+	the focus of this work. The method presented here is quasi-conservative
+	locally and globally of tracer when the baroclinic and tracer time
+	steps are equal. Important issues relevant for using this method
+	in regional as well as large-scale climate models are discussed and
+	illustrated, and examples of scaling achieved on parallel computers
+	provided.},
+  date = {May 01, 2001},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Guily2001,
+  author = {E. Guilyardi and G. Madec and L. Terray},
+  title = {The role of lateral ocean physics in the upper ocean thermal balance
+	of a coupled ocean-atmosphere GCM},
+  journal = CD,
+  year = {2001},
+  volume = {17},
+  pages = {589-599},
+  number = {8},
+  pdf = {/home/ericg/TeX/Papers/Published_pdfs/Guilyardi_al_CD01.pdf}
+}
+
+@ARTICLE{Guyon_al_EP99,
+  author = {M. Guyon and G. Madec and F.-X. Roux and M. Imbard},
+  title = {A Parallel ocean model for high resolution studies},
+  journal = {Lecture Notes in Computer Science},
+  year = {1999},
+  volume = {Euro-Par'99},
+  pages = {603-607},
+  owner = {gm},
+  timestamp = {2008.05.27}
+}
+
+@ARTICLE{Guyon_al_CalPar99,
+  author = {M. Guyon and G. Madec and F.-X. Roux and M. Imbard and C. Herbaut
+	and P. Fronier},
+  title = {Parallelization of the OPA ocean model},
+  journal = {Calculateurs Paralleles},
+  year = {1999},
+  volume = {11, 4},
+  pages = {499-517},
+  owner = {gm},
+  timestamp = {2008.05.27}
+}
+
+@BOOK{Haltiner1980,
+  title = {Numerical prediction and dynamic meteorology},
+  publisher = {John Wiley {\&} Sons Eds., second edition, 477pp},
+  year = {1980},
+  author = {G. J. Haltiner and R. T. Williams},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Haney1991,
+  author = {R. L. Haney},
+  title = {On the Pressure Gradient Force over Steep Topography in Sigma Coordinate
+	Ocean Models},
+  journal = JPO,
+  year = {1991},
+  volume = {21},
+  pages = {610--619
+	
+	},
+  number = {4},
+  abstract = {The error in computing the pressure gradient force near steep topography
+	using terms following (&#963;) coordinates is investigated in an
+	ocean model using the family of vertical differencing schemes proposed
+	by Arakawa and Suarez. The truncation error is estimated by substituting
+	known buoyancy profiles into the finite difference hydrostatic and
+	pressure gradient terms. The error due to &#8220;hydrostatic inconsistency,&#8221;
+	which is not simply a space truncation error, is also documented.
+	The results show that the pressure gradient error is spread throughout
+	the water column, and it is sensitive to the vertical resolution
+	and to the placement of the grid points relative to the vertical
+	structure of the buoyancy field being modeled. Removing a reference
+	state, as suggested for the atmosphere by Gary, reduces the truncation
+	error associated with the two lowest vertical modes by a factor of
+	2 to 3. As an example, the error in computing the pressure gradient
+	using a standard 10-level primitive equation model applied to buoyancy
+	profiles and topographic slopes typical of the California Current
+	region corresponds to a false geostrophic current of the order of
+	10&#8211;12 cm s&#8722;1. The analogous error in a hydrostatically
+	consistent 30-level model with the reference state removed is about
+	an order of magnitude smaller.},
+  date = {April 01, 1991},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Hsu1990,
+  author = {Hsu, Yueh-Jiuan G. and Arakawa, Akio},
+  title = {Numerical Modeling of the Atmosphere with an Isentropic Vertical
+	Coordinate},
+  journal = MWR,
+  year = {1990},
+  volume = {118},
+  pages = {1933--1959
+	
+	},
+  number = {10},
+  abstract = {In constructing a numerical model of the atmosphere, we must choose
+	an appropriate vertical coordinate. Among the various possibilities,
+	isentropic vertical coordinates such as the &#952;-coordinate seem
+	to have the greatest potential, in spite of the technical difficulties
+	in treating the intersections of coordinate surfaces with the lower
+	boundary. The purpose of this paper is to describe the &#952;-coordinate
+	model we have developed and to demonstrate its potential through
+	simulating the nonlinear evolution of a baroclinic wave.In the model
+	we have developed, vertical discretization maintains important integral
+	constraints, such as conservation of the angular momentum and total
+	energy. In treating the intersections of coordinate surfaces with
+	the lower boundary, we have followed the massless-layer approach
+	in which the intersecting coordinate surfaces are extended along
+	the boundary by introducing massless layers. Although this approach
+	formally eliminates the intersection problem, it raises other computational
+	problems. Horizontal discretization of the continuity and momentum
+	equations in the model has been carefully designed to overcome these
+	problems.Selected results from a 10-day integration with the 25-layer,
+	&#946;-plane version of the model are presented. It seems that the
+	model can simulate the nonlinear evolution of a baroclinic wave and
+	associated dynamical processes without major computational difficulties.},
+  date = {October 01, 1990},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{JackMcD1995,
+  author = {D. R. Jackett and T. J. McDougall},
+  title = {Minimal adjustment of hydrographic data to achieve static stability},
+  journal = JAOT,
+  year = {1995},
+  volume = {12},
+  pages = {381-389},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@BOOK{Jerlov1968,
+  title = {Optical Oceanography},
+  publisher = {194pp},
+  year = {1968},
+  author = {N. G. Jerlov},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@INPROCEEDINGS{Killworth1989,
+  author = {P. D. Killworth},
+  title = {On the parameterization of deep convection in ocean models},
+  booktitle = {Parameterization of small-scale processes},
+  year = {1989},
+  editor = {Hawaiian winter workshop},
+  month = {January 17-20},
+  organization = {University of Hawaii at Manoa},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Killworth1992,
+  author = {P. D. Killworth},
+  title = {An equivalent-barotropic mode in the fine resolution Antarctic model},
+  journal = JPO,
+  year = {1992},
+  volume = {22},
+  pages = {1379-1387}
+}
+
+@ARTICLE{Killworth1991,
+  author = {Killworth, P. D. and Stainforth, D. and Webb, D. J. and Paterson,
+	S. M.},
+  title = {The Development of a Free-Surface Bryan-Cox-Semtner Ocean Model},
+  journal = JPO,
+  year = {1991},
+  volume = {21},
+  pages = {1333--1348},
+  number = {9},
+  abstract = {A version of the Bryan&#8211;Cox&#8211;Semtner numerical ocean general
+	circulation model, adapted to include a free surface, is described.
+	The model is designed for the following uses&#058; tidal studies
+	(a tidal option is explicitly included); assimilation of altimetric
+	data (since the surface elevation is now a prognostic variable);
+	and in situations where accurate relaxation to obtain the streamfunction
+	in the original model is too time consuming. Comparison is made between
+	a 300-year run of the original model and the free-surface version,
+	using a very coarse North Atlantic calculation as the basis. The
+	results are very similar, differing only in the streamfunction over
+	topography; this is to be expected, since the treatment of topographic
+	torques on the barotropic flow differs because of the nature of the
+	modifications.},
+  date = {September 01, 1991},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Kolmogorov1942,
+  author = {A. N. Kolmogorov},
+  title = {The equation of turbulent motion in an incompressible fluid},
+  journal = {Izv. Akad. Nauk SSSR, Ser. Fiz.},
+  year = {1942},
+  volume = {6},
+  pages = {56-58},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@PHDTHESIS{Levy1996,
+  author = {M. L\'{e}vy},
+  title = {Mod\'{e}lisation des processus biog\'{e}ochimiques en M\'{e}diterran\'{e}e
+	nord-occidentale. Cycle saisonnier et variabilit\'{e} m\'{e}so\'{e}chelle},
+  school = {Universit\'{e} Pierre et Marie Curie, Paris, France, 207pp},
+  year = {1996},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Levy2001,
+  author = {M. L\'{e}vy and A. Estubier and G Madec},
+  title = {Choice of an advection scheme for biogeochemical models},
+  journal = GRL,
+  year = {2001},
+  volume = {28},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Levy1998,
+  author = {M. L\'{e}vy and L. M\'{e}mery and G. Madec},
+  title = {The onset of a bloom after deep winter convection in the Northwestern
+	Mediterranean Sea: mesoscale 
+	
+	process study with a primitive equation model},
+  journal = JMS,
+  year = {1998},
+  volume = {16/1-2},
+  owner = {gm},
+  timestamp = {2007.08.10}
+}
+
+@BOOK{LargeYeager2004,
+  title = {Diurnal to decadal global forcing for ocean and sea-ice models: the
+	data sets and flux climatologies},
+  publisher = {NCAR Technical Note, NCAR/TN-460+STR, CGD Division of the National
+	Center for Atmospheric Research},
+  year = {2004},
+  author = {W. Large and S. Yeager},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{large1994,
+  author = {W. G. Large and J. C. McWilliams and S. C. Doney},
+  title = {Oceanic vertical mixing - a review and a model with a nonlocal boundary
+	layer parameterization},
+  journal = {Reviews of Geophysics},
+  year = {1994},
+  volume = {32},
+  pages = {363-404},
+  doi = {10.1029/94RG01872},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@PHDTHESIS{Lazar1997,
+  author = {A. Lazar},
+  title = {La branche froide de la circulation thermohaline - sensibilit\'{e}
+	\`{a} la diffusion turbulente dans un mod\`{e}le de circulation g\'{e}n\'{e}rale
+	id\'{e}alis\'{e}e},
+  school = {Universit\'{e} Pierre et Marie Curie, Paris, France, 200pp},
+  year = {1997},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Lazar1999,
+  author = {A. Lazar and G. Madec and P. Delecluse},
+  title = {The Deep Interior Downwelling, the Veronis Effect, and Mesoscale
+	Tracer Transport Parameterizations in an OGCM},
+  journal = JPO,
+  year = {1999},
+  volume = {29},
+  pages = {2945-2961},
+  number = {11},
+  abstract = {Numerous numerical simulations of basin-scale ocean circulation display
+	a vast interior downwelling and a companion intense western boundary
+	layer upwelling at midlatitude below the thermocline. These features,
+	related to the so-called Veronis effect, are poorly rationalized
+	and depart strongly from the classical vision of the deep circulation
+	where upwelling is considered to occur in the interior. Furthermore,
+	they significantly alter results of ocean general circulation models
+	(OGCMs) using horizontal Laplacian diffusion. Recently, some studies
+	showed that the parameterization for mesoscale eddy effects formulated
+	by Gent and McWilliams allows integral quantities like the streamfunction
+	and meridional heat transport to be free of these undesired effects.
+	In this paper, an idealized OGCM is used to validate an analytical
+	rationalization of the processes at work and help understand the
+	physics. The results show that the features associated with the Veronis
+	effect can be related quantitatively to three different width scales
+	that characterize the baroclinic structure of the deep western boundary
+	current. In addition, since one of these scales may be smaller than
+	the Munk barotropic layer, usually considered to determine the minimum
+	resolution and horizontal viscosity for numerical models, the authors
+	recommend that it be taken into account. Regarding the introduction
+	of the new parameterization, diagnostics in terms of heat balances
+	underline some interesting similarities between local heat fluxes
+	by eddy-induced velocities and horizontal diffusion at low and midlatitudes
+	when a common large diffusivity (here 2000 m2 s&#8722;1) is used.
+	The near-quasigeostrophic character of the flow explains these results.
+	As a consequence, the response of the Eulerian-mean circulation is
+	locally similar for runs using either of the two parameterizations.
+	However, it is shown that the advective nature of the eddy-induced
+	heat fluxes results in a very different effective circulation, which
+	is the one felt by tracers.},
+  date = {November 01, 1999},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Lengaigne_al_JGR03,
+  author = {M. Lengaigne and G. Madec and G. Alory and C. Menkes},
+  title = {Sensitivity of the tropical Pacific Ocean to isopycnal diffusion
+	on tracer and dynamics},
+  journal = JGR,
+  year = {2003},
+  volume = {108 (C11)},
+  pages = {3345, doi:10.1029/2002JC001704},
+  owner = {gm},
+  timestamp = {2008.01.26}
+}
+
+@ARTICLE{Leonard1991,
+  author = {B. P. Leonard},
+  title = {The ULTIMATE conservative difference scheme applied to unsteady one--dimensional
+	advection},
+  journal = {Computer Methods in Applied Mechanics and Engineering},
+  year = {1991},
+  pages = {17-74},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@TECHREPORT{Leonard1988,
+  author = {B. P. Leonard},
+  title = {Universal limiter for transient interpolation modelling of the advective
+	transport equations},
+  institution = {Technical Memorandum TM-100916 ICOMP-88-11, NASA},
+  year = {1988},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Leonard1979,
+  author = {B. P. Leonard},
+  title = {A stable and accurate convective modelling procedure based on quadratic
+	upstream interpolation},
+  journal = {Computer Methods in Applied Mechanics and Engineering},
+  year = {1979},
+  volume = {19},
+  pages = {59-98},
+  month = jun,
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@TECHREPORT{Levier2007,
+  author = {B. Levier and A.-M. Tr\'{e}guier and G. Madec and V. Garnier},
+  title = {Free surface and variable volume in the NEMO code},
+  institution = {MERSEA MERSEA IP report WP09-CNRS-STR-03-1A, 47pp, available on the
+	NEMO web site},
+  year = {2007},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@BOOK{levitus82,
+  title = {Climatological Atlas of the world ocean},
+  publisher = {NOAA professional paper No. 13, 174pp},
+  year = {1982},
+  author = {S Levitus },
+  note = {173 p.}
+}
+
+@TECHREPORT{Lott1989,
+  author = {F. Lott and G. Madec},
+  title = {Implementation of bottom topography in the Ocean General Circulation
+	Model OPA of the LODYC: formalism and experiments.},
+  institution = {LODYC, France, 36pp.},
+  year = {1989},
+  number = {3},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Lott1990,
+  author = {F. Lott and G. Madec and J. Verron},
+  title = {Topographic experiments in an Ocean General Circulation Model},
+  journal = {Ocean Modelling},
+  year = {1990},
+  volume = {88},
+  pages = {1-4},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@PHDTHESIS{Madec1990,
+  author = {G. Madec},
+  title = {La formation d'eau profonde et son impact sur la circulation r\'{e}gionale
+	en M\'{e}diterran\'{e}e Occidentale - une approche num\'{e}rique},
+  school = {Universit\'{e}Pierre et Marie Curie, Paris, France, 194pp.},
+  year = {1990},
+  owner = {gm},
+  timestamp = {2007.08.10}
+}
+
+@ARTICLE{Madec1991a,
+  author = {G. Madec and M. Chartier and M. Cr\'{e}pon},
+  title = {Effect of thermohaline forcing variability on deep water formation
+	in the Northwestern Mediterranean Sea - a high resulution three-dimensional
+	study},
+  journal = DAO,
+  year = {1991},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Madec1991b,
+  author = {G. Madec and M. Chartier and P. Delecluse and M. Cr\'{e}pon},
+  title = {A three-dimensional numerical study of deep water formation in the
+	
+	
+	Northwestern Mediterranean Sea .},
+  journal = JPO,
+  year = {1991},
+  volume = {21},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@INBOOK{Madec1991c,
+  chapter = {Thermohaline-driven deep water formation in the Northwestern Mediterranean
+	Sea},
+  title = {Deep convection and deep water formation in the oceans},
+  publisher = {Elsevier Oceanographic Series},
+  year = {1991},
+  author = {G. Madec and M. Cr\'{e}pon},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Madec1997,
+  author = {G. Madec and P. Delecluse},
+  title = {The OPA/ARPEGE and OPA/LMD Global Ocean-Atmosphere Coupled Model},
+  journal = {Int. WOCE Newsletter},
+  year = {1997},
+  volume = {26},
+  pages = {12-15},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@TECHREPORT{Madec1998,
+  author = {G. Madec and P. Delecluse and M. Imbard and C. Levy},
+  title = {OPA 8 Ocean General Circulation Model - Reference Manual},
+  institution = {LODYC/IPSL Note 11},
+  year = {1998}
+}
+
+@ARTICLE{MadecImb1996,
+  author = {G Madec and M Imbard},
+  title = {A global ocean mesh to overcome the north pole singularity},
+  journal = CD,
+  year = {1996},
+  volume = {12},
+  pages = {381-388}
+}
+
+@ARTICLE{Madec1996,
+  author = {G. Madec and F. Lott and P. Delecluse and M. Cr\'{e}pon},
+  title = {Large-Scale Preconditioning of Deep-Water Formation in the Northwestern
+	Mediterranean Sea},
+  journal = JPO,
+  year = {1996},
+  volume = {26},
+  pages = {1393-1408},
+  number = {8},
+  month = aug,
+  abstract = {The large-scale processes preconditioning the winter deep-water formation
+	in the northwestern Mediterranean Sea are investigated with a primitive
+	equation numerical model where convection is parameterized by a non-penetrative
+	convective adjustment algorithm. The ocean is forced by momentum
+	and buoyancy fluxes that have the gross features of mean winter forcing
+	found in the MEDOC area. The wind-driven barotropic circulation appears
+	to be a major ingredient of the preconditioning phase of deep-water
+	formation. After three months, the ocean response is dominated by
+	a strong barotropic cyclonic vortex located under the forcing area,
+	which fits the Sverdrup balance away from the northern coast. In
+	the vortex center, the whole water column remains trapped under the
+	forcing area all winter. This trapping enables the thermohaline forcing
+	to drive deep-water formation efficiently. Sensitivity studies show
+	that, &#946; effect and bottom topography play a paramount role and
+	confirm that deep convection occurs only in areas that combine a
+	strong surface thermohaline forcing and a weak barotropic advection
+	so that water masses are submitted to the negative buoyancy fluxes
+	for a much longer time. In particular, the impact of the Rh&#244;ne
+	Deep Sea Fan on the barotropic circulation dominates the &#946; effect:
+	the barotropic flow is constrained to follow the bathymetric contours
+	and the cyclonic vortex is shifted southward so that the fluid above
+	the fan remains quiescent. Hence, buoyancy fluxes trigger deep convection
+	above the fan in agreement with observations. The selection of the
+	area of deep-water formation through the defection of the barotropic
+	circulation by the topography seems a more efficient mechanism than
+	those associated with the wind- driven barotropic vortex. This is
+	due to its permanency, while the latter may be too sensitive to time
+	and space variations of the forcing.},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Madec1988,
+  author = {G. Madec and C. Rahier and M. Chartier},
+  title = {A comparison of two-dimensional elliptic solvers for the barotropic
+	streamfunction in a multilevel OGCM},
+  journal = {Ocean Modelling},
+  year = {1988},
+  volume = {78},
+  owner = {gm},
+  timestamp = {2007.08.10}
+}
+
+@ARTICLE{Maltrud1998,
+  author = {M. E. Maltrud and R. D. Smith and A. J. Semtner and R. C. Malone},
+  title = {Global eddy-resolving ocean simulations driven by 1985-1995 atmospheric
+	winds},
+  journal = JGR,
+  year = {1998},
+  volume = {103(C13)},
+  pages = {30,825-30,854},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Marchesiello2001,
+  author = { P. Marchesiello and J. Mc Williams and A. Shchepetkin },
+  title = {Open boundary conditions for long-term integrations of Regional Oceanic
+	Models},
+  journal = {Ocean Modelling},
+  year = {2001},
+  volume = {3},
+  pages = {1-20}
+}
+
+@PHDTHESIS{MartiTh1992,
+  author = {O. Marti},
+  title = {Etude de l'oc\'{e}an mondial : mod\'{e}lisation de la circulation
+	et du transport de traceurs anthropog\'{e}niques},
+  school = {Universit\'{e} Pierre et Marie Curie, Paris, France, 201pp},
+  year = {1992},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Marti1992,
+  author = {O. Marti and G. Madec and P. Delecluse},
+  title = {Comment on "Net diffusivity in ocean general circulation models with
+	nonuniform grids" by F. L. Yin and I. Y. Fung},
+  journal = JGR,
+  year = {1992},
+  volume = {97},
+  pages = {12763-12766},
+  month = aug,
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{McDougall1987,
+  author = {T. J. McDougall},
+  title = {Neutral Surfaces},
+  journal = {Journal of Physical Oceanography},
+  year = {1987},
+  volume = {17},
+  pages = {1950-1964},
+  number = {11},
+  abstract = {Scalar properties in the ocean are stirred (and subsequently mixed)
+	rather efficiently by mesoscale eddies and two-dimensional turbulence
+	along &#8220;neutral surfaces&#8221;, defined such that when water
+	parcels are moved small distances in the neutral surface, they experience
+	no buoyant restoring forces. By contrast, work would have to be done
+	on a moving fluid parcel in order to keep it on a potential density
+	surface. The differences between neutral surfaces and potential density
+	surfaces are due to the variation of &#945;&#47;&#946; with pressure
+	(where &#945; is the thermal expansion coefficient and &#946; is
+	the saline contraction coefficient). By regarding the equation of
+	state of seawater as a function of salinity, potential temperature,
+	and pressure, rather than in terms of salinity, temperature, and
+	pressure, it is possible to quantify the differences between neutral
+	surfaces and potential density surfaces. In particular, the spatial
+	gradients of scalar properties (e.g., S, &#952;, tritium or potential
+	vorticity) on a neutral surface can be quite different to the corresponding
+	gradients in a potential density surface. For example, at a potential
+	temperature of 4&#176;C and a pressure of 1000 db, the lateral gradient
+	of potential temperature in a potential density surface (referenced
+	to sea level) is too large by between 50&#37; and 350&#37; (depending
+	on the stability ratio Rp of the water column) compared with the
+	physically relevant gradient of potential temperature on the neutral
+	surface. Three-examples of neutral surfaces are presented, based
+	on the Levitus atlas of the North Atlantic.},
+  date = {November 01, 1987},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{McDougall_Taylor_JMR84,
+  author = {T. J. McDougall and J. R. Taylor},
+  title = {Flux measurements across a finger interface at low values of the
+	stability ratio},
+  journal = {Journal of Marine Research},
+  year = {1984},
+  volume = {42},
+  pages = {1-14},
+  owner = {gm},
+  timestamp = {2008.05.20}
+}
+
+@ARTICLE{Merryfield1999,
+  author = {W. J. Merryfield and G. Holloway and A. E. Gargett},
+  title = {A Global Ocean Model with Double-Diffusive Mixing},
+  journal = JPO,
+  year = {1999},
+  volume = {29},
+  pages = {1124-1142},
+  number = {6},
+  abstract = {A global ocean model is described in which parameterizations of diapycnal
+	mixing by double-diffusive fingering and layering are added to a
+	stability-dependent background turbulent diffusivity. Model runs
+	with and without double-diffusive mixing are compared for annual-mean
+	and seasonally varying surface forcing. Sensitivity to different
+	double-diffusive mixing parameterizations is considered. In all cases,
+	the locales and extent of salt fingering (as diagnosed from buoyancy
+	ratio R&#961;) are grossly comparable to climatology, although fingering
+	in the models tends to be less intense than observed. Double-diffusive
+	mixing leads to relatively minor changes in circulation but exerts
+	significant regional influences on temperature and salinity.},
+  date = {June 01, 1999},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@BOOK{Mesinger_Arakawa_Bk76,
+  title = {Numerical methods used in Atmospheric models},
+  publisher = {GARP Publication Series No 17},
+  year = {1976},
+  author = {F. Mesinger and A. Arakawa},
+  owner = {gm},
+  timestamp = {2008.02.09}
+}
+
+@ARTICLE{Murray1996,
+  author = {R. J. Murray},
+  title = {Explicit Generation of Orthogonal Grids for Ocean Models},
+  journal = JCP,
+  year = {1996},
+  volume = {126},
+  pages = {251-273},
+  number = {2},
+  month = {July},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@PHDTHESIS{OlivierPh2001,
+  author = {F. Olivier},
+  title = {Etude de l'activit\'{e} biologique et de la circulation oc\'{e}anique
+	dans un jet g\'{e}ostrophique: le front Alm\'{e}ria-Oran},
+  school = {Universit\'{e} Pierre et Marie Curie, Paris, France},
+  year = {2001},
+  owner = {gm},
+  timestamp = {2007.08.14}
+}
+
+@ARTICLE{PacPhil1981,
+  author = {R.C. Pacanowski and S.G.H. Philander},
+  title = {Parameterization of Vertical Mixing in Numerical Models of Tropical
+	Oceans},
+  journal = JPO,
+  year = {1981},
+  volume = {11},
+  pages = {1443-1451},
+  number = {11},
+  abstract = {Measurements indicate that mixing processes are intense in the surface
+	layers of the ocean but weak below the thermocline, except for the
+	region below the core of the Equatorial Undercurrent where vertical
+	temperature gradients are small and the shear is large. Parameterization
+	of these mixing processes by means of coefficients of eddy mixing
+	that are Richardson-number dependent, leads to realistic simulations
+	of the response of the equatorial oceans to different windstress
+	patterns. In the case of eastward winds results agree well with measurements
+	in the Indian Ocean. In the case of westward winds it is of paramount
+	importance that the nonzero heat flux into the ocean be taken into
+	account. This beat flux stabilizes the upper layers and reduces the
+	intensity of the mixing, especially in the cast. With an appropriate
+	surface boundary condition, the results are relatively insensitive
+	to values assigned to constants in the parameterization formula.},
+  date = {November 01, 1981},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Pacanowski_Gnanadesikan_MWR98,
+  author = {R. C. Pacanowski and A. Gnanadesikan},
+  title = {Transient response in a z-level ocean model that resolves topography
+	
+	
+	with partial-cells},
+  journal = MWR,
+  year = {1998},
+  volume = {126},
+  pages = {3248-3270},
+  owner = {gm},
+  timestamp = {2008.01.26}
+}
+
+@ARTICLE{Paulson1977,
+  author = {C. A. Paulson and J. J. Simpson},
+  title = {Irradiance Measurements in the Upper Ocean},
+  journal = JPO,
+  year = {1977},
+  volume = {7},
+  pages = {952-956},
+  number = {6},
+  abstract = {Observations were made of downward solar radiation as a function of
+	depth during an experiment in the North Pacific (35&#176;N, 155&#176;W).
+	The irradiance meter employed was sensitive to solar radiation of
+	wavelength 400&#8211;1000 nm arriving from above at a horizontal
+	surface. Because of selective absorption of the short and long wavelengths,
+	the irradiance decreases much faster than exponential in the upper
+	few meters, falling to one-third of the incident value between 2
+	and 3 m depth. Below 10 m the decrease was exponential at a rate
+	characteristic of moderately clear water of Type IA. Neglecting one
+	case having low sun altitude, the observations are well represented
+	by the expression I&#47;I0&#61;Rez&#47;&#950;1&#43;(1&#8722;R)ez&#950;2,
+	where I is the irradiance at depth &#8722;z, I0 is the irradiance
+	at the surface less reflected solar radiation, R&#61;0.62, &#950;1
+	and &#950;2 are attenuation lengths equal to 1.5 and 20 m, respectively,
+	and z is the vertical space coordinate, positive upward with the
+	origin at mean sea level. The depth at which the irradiance falls
+	to 10&#37; of its surface value is nearly the same as observations
+	of Secchi depth when cases with high wind speed or low solar altitude
+	are neglected. Parameters R, &#950;1, and &#950;2 are computed for
+	the entire range of oceanic water types.},
+  date = {November 01, 1977},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Penduff2000,
+  author = {T. Penduff and B. Barnier and A. Colin de Verdi\`{e}re},
+  title = { Self-adapting open boundaries for a regional model of the eastern
+	North Atlantic},
+  journal = JGR,
+  year = {2000},
+  volume = {105},
+  pages = {11,279-11,297}
+}
+
+@ARTICLE{Penduff2007,
+  author = {T. Penduff and J. Le Sommer and B. Barnier and A.M. Treguier and
+	J. Molines and G. Madec},
+  title = {Influence of numerical schemes on current-topography interactions
+	in 1/4$^{\circ}$ global ocean simulations},
+  journal = {Ocean Science},
+  year = {2007},
+  volume = {?},
+  pages = {in revision}
+}
+
+@ARTICLE{Phillips1959,
+  author = {R. S. Phillips},
+  title = {Dissipative Operators and Hyperbolic Systems of Partial Differential
+	Equations},
+  journal = {Transactions of the American Mathematical Society},
+  year = {1959},
+  volume = {90(2)},
+  pages = {193-254},
+  doi = {doi:10.2307/1993202},
+  owner = {gm},
+  timestamp = {2007.08.10}
+}
+
+@ARTICLE{Redi_JPO82,
+  author = {M. H. Redi},
+  title = {Oceanic isopycnal mixing by coordinate rotation},
+  journal = JPO,
+  year = {1982},
+  volume = {13},
+  pages = {1154-1158},
+  owner = {gm},
+  timestamp = {2008.02.02}
+}
+
+@ARTICLE{Reverdin1991,
+  author = {G. Reverdin and P. Delecluse and C. L\'{e}vy and P. Andrich and A.
+	Morli\`{e}re and J. M. Verstraete},
+  title = {The near surface tropical Atlantic in 1982-1984 : results from a
+	numerical simulation and a data analysis},
+  journal = PO,
+  year = {1991},
+  volume = {27},
+  pages = {273-340},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@BOOK{Richtmyer1967,
+  title = {Difference methods for initial-value problems},
+  publisher = {Interscience Publisher, Second Edition, 405pp},
+  year = {1967},
+  author = {R. D. Richtmyer and K. W. Morton},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Robert1966,
+  author = {A. J. Robert},
+  title = {The integration of a Low order spectral form of the primitive meteorological
+	equations},
+  journal = {J. Meteo. Soc. Japan},
+  year = {1966},
+  volume = {44, 2},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@INCOLLECTION{Roed1986,
+  author = {L.P. Roed and C.K. Cooper},
+  title = {Open boundary conditions in numerical ocean models},
+  booktitle = {Advanced Physical Oceanography Numerical Modelling},
+  publisher = { NATO ASI Series, vol. 186.},
+  year = {1986},
+  editor = {J.J. O'Brien}
+}
+
+@ARTICLE{Roullet2000,
+  author = {G. Roullet and G. Madec},
+  title = {salt conservation, free surface, and varying levels: a new formulation
+	for ocean general circulation models},
+  journal = JGR,
+  year = {2000},
+  volume = {105},
+  pages = {23,927-23,942},
+  owner = {sandra},
+  pdf = {Roullet_Madec_JGR00.pdf},
+  timestamp = {2007.03.22}
+}
+
+@ARTICLE{Sadourny1975,
+  author = {R. Sadourny},
+  title = {The Dynamics of Finite-Difference Models of the Shallow-Water Equations},
+  journal = JAS,
+  year = {1975},
+  volume = {32},
+  pages = {680-689},
+  number = {4},
+  abstract = {Two simple numerical models of the shallow-water equations identical
+	in all respects but for their con-servation properties have been
+	tested regarding their internal mixing processes. The experiments
+	show that violation of enstrophy conservation results in a spurious
+	accumulation of rotational energy in the smaller scales, reflected
+	by an unrealistic increase of enstrophy, which ultimately produces
+	a finite rate of energy dissipation in the zero viscosity limit,
+	thus violating the well-known dynamics of two-dimensional flow. Further,
+	the experiments show a tendency to equipartition of the kinetic energy
+	of the divergent part of the flow in the inviscid limit, suggesting
+	the possibility of a divergent energy cascade in the physical system,
+	as well as a possible influence of the energy mixing on the process
+	of adjustment toward balanced flow.},
+  date = {April 01, 1975},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Sarmiento1982,
+  author = {J. L. Sarmiento and K. Bryan},
+  title = {Ocean transport model for the North Atlantic},
+  journal = JGR,
+  year = {1982},
+  volume = {87},
+  pages = {394-409},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Sacha2005,
+  author = {A. F. Shchepetkin and J. C. McWilliams},
+  title = {The regional oceanic modeling system (ROMS) - a split-explicit, free-surface,
+	topography-following-coordinate oceanic modelr},
+  journal = {Ocean Modelling},
+  year = {2005},
+  volume = {9, 4},
+  pages = {347-404},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Sacha2003,
+  author = {A. F. Shchepetkin and J. C. McWilliams},
+  title = {A method for computing horizontal pressure-gradient force in an oceanic
+	model with a nonaligned 
+	
+	vertical coordinate},
+  journal = JGR,
+  year = {2003},
+  volume = {108(C3)},
+  pages = {3090, doi:10.1029/2001JC001047},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Shchepetkin1996,
+  author = {A. F. Shchepetkin and J. J. O'Brien},
+  title = {A Physically Consistent Formulation of Lateral Friction in Shallow-Water
+	Equation Ocean Models},
+  journal = MWR,
+  year = {1996},
+  volume = {124},
+  pages = {1285-1300},
+  number = {6},
+  abstract = {Dissipation in numerical ocean models has two purposes: to simulate
+	processes in which the friction is physically relevant and to prevent
+	numerical instability by suppressing accumulation of energy in the
+	smallest resolved scales. This study shows that even for the latter
+	case the form of the friction term should be chosen in a physically
+	consistent way. Violation of fundamental physical principles reduces
+	the fidelity of the numerical solution, even if the friction is small.
+	Several forms of the lateral friction, commonly used in numerical
+	ocean models, are discussed in the context of shallow-water equations
+	with nonuniform layer thickness. It is shown that in a numerical
+	model tuned for the minimal dissipation, the improper form of the
+	friction term creates finite artificial vorticity sources that do
+	not vanish with increased resolution, even if the viscous coefficient
+	is reduced consistently with resolution. An alternative numerical
+	implementation of the no-slip boundary conditions for an arbitrary
+	coast line is considered. It was found that the quality of the numerical
+	solution may be considerably improved by discretization of the viscous
+	stress tensor in such a way that the numerical boundary scheme approximates
+	not only the stress tensor to a certain order of accuracy but also
+	simulates the truncation error of the numerical scheme used in the
+	interior of the domain. This ensures error cancellation during subsequent
+	use of the elements of the tensor in the discrete version of the
+	momentum equations, allowing for approximation of them without decrease
+	in the order of accuracy near the boundary.},
+  date = {June 01, 1996},
+  owner = {gm},
+  timestamp = {2007.08.14}
+}
+
+@ARTICLE{Simmons2003,
+  author = {H. L. Simmons and S. R. Jayne and L. C. St. Laurent and A. J. Weaver},
+  title = {Tidally driven mixing in a numerical model of the
+	
+	ocean general circulation},
+  journal = OM,
+  year = {2003},
+  pages = {1-19},
+  abstract = {Astronomical data reveals that approximately 3.5 terawatts (TW) of
+	tidal energy is dissipated in the
+	
+	ocean. Tidal models and satellite altimetry suggest that 1 TW of this
+	energy is converted from the barotropic
+	
+	to internal tides in the deep ocean, predominantly around regions
+	of rough topography such as midocean
+	
+	ridges. Aglobal tidal model is used to compute turbulent energy levels
+	associated with the dissipation
+	
+	of internal tides, and the diapycnal mixing supported by this energy
+	?ux is computed using a simple parameterization.
+	
+	The mixing parameterization has been incorporated into a coarse resolution
+	numerical model of the
+	
+	global ocean. This parameterization o?ers an energetically consistent
+	and practical means of improving the
+	
+	representation of ocean mixing processes in climate models. Novel
+	features of this implementation are that
+	
+	the model explicitly accounts for the tidal energy source for mixing,
+	and that the mixing evolves both
+	
+	spatially and temporally with the model state. At equilibrium, the
+	globally averaged di?usivity pro?le
+	
+	ranges from 0.3 cm2 s1 at thermocline depths to 7.7 cm2 s1 in the
+	abyss with a depth average of 0.9
+	
+	cm2 s1, in close agreement with inferences from global balances.
+	Water properties are strongly in?uenced
+	
+	by the combination of weak mixing in the main thermocline and enhanced
+	mixing in the deep ocean.
+	
+	Climatological comparisons show that the parameterized mixing scheme
+	results in a substantial reduction},
+  owner = {sandra},
+  pdf = {Simmons_mixing_OM2003.pdf},
+  timestamp = {2007.03.22}
+}
+
+@ARTICLE{Song1994,
+  author = {Y. Song and D. Haidvogel},
+  title = {A Semi-implicit Ocean Circulation Model Using a Generalized Topography-Following
+	Coordinate System
+	
+	Authors:},
+  journal = JCP,
+  year = {1994},
+  volume = {115, 1},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Song1998,
+  author = {Y. T. Song},
+  title = {A General Pressure Gradient Formulation for Ocean Models. Part I:
+	Scheme Design and Diagnostic Analysis},
+  journal = MWR,
+  year = {1998},
+  volume = {126},
+  pages = {3213-3230},
+  number = {12},
+  abstract = {A Jacobian formulation of the pressure gradient force for use in models
+	with topography-following coordinates is proposed. It can be used
+	in conjunction with any vertical coordinate system and is easily
+	implemented. Vertical variations in the pressure gradient are expressed
+	in terms of a vertical integral of the Jacobian of density and depth
+	with respect to the vertical computational coordinate. Finite difference
+	approximations are made on the density field, consistent with piecewise
+	linear and continuous fields, and accurate pressure gradients are
+	obtained by vertically integrating the discrete Jacobian from sea
+	surface.Two discrete schemes are derived and examined in detail:
+	the first using standard centered differencing in the generalized
+	vertical coordinate and the second using a vertical weighting such
+	that the finite differences are centered with respect to the Cartesian
+	z coordinate. Both schemes achieve second-order accuracy for any
+	vertical coordinate system and are significantly more accurate than
+	conventional schemes based on estimating the pressure gradients by
+	finite differencing a previously determined pressure field.The standard
+	Jacobian formulation is constructed to give exact pressure gradient
+	results, independent of the bottom topography, if the buoyancy field
+	varies bilinearly with horizontal position, x, and the generalized
+	vertical coordinate, s, over each grid cell. Similarly, the weighted
+	Jacobian scheme is designed to achieve exact results, when the buoyancy
+	field varies linearly with z and arbitrarily with x, that is, b(x,z)
+	&#61; b0(x) &#43; b1(x)z.When horizontal resolution cannot be made
+	fine enough to avoid hydrostatic inconsistency, errors can be substantially
+	reduced by the choice of an appropriate vertical coordinate. Tests
+	with horizontally uniform, vertically varying, and with horizontally
+	and vertically varying buoyancy fields show that the standard Jacobian
+	formulation achieves superior results when the condition for hydrostatic
+	consistency is satisfied, but when coarse horizontal resolution causes
+	this condition to be strongly violated, the weighted Jacobian may
+	give superior results.},
+  date = {December 01, 1998},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{SongWright1998,
+  author = {Y. T. Song and D. G. Wright},
+  title = {A General Pressure Gradient Formulation for Ocean Models. Part II
+	- Energy, Momentum, and Bottom Torque Consistency},
+  journal = MWR,
+  year = {1998},
+  volume = {126},
+  pages = {3231-3247},
+  number = {12},
+  abstract = {A new formulation of the pressure gradient force for use in models
+	with topography-following coordinates is proposed and diagnostically
+	analyzed in Part I. Here, it is shown that important properties of
+	the continuous equations are retained by the resulting numerical
+	schemes, and their performance in prognostic simulations is examined.
+	Numerical consistency is investigated with respect to global energy
+	conservation, depth-integrated momentum changes, and the representation
+	of the bottom pressure torque. The performances of the numerical
+	schemes are tested in prognostic integrations of an ocean model to
+	demonstrate numerical accuracy and long-term integral stability.
+	Two typical geometries, an isolated tall seamount and an unforced
+	basin with sloping boundaries, are considered for the special case
+	of no external forcing and horizontal isopycnals to test numerical
+	accuracy. These test problems confirm that the proposed schemes yield
+	accurate approximations to the pressure gradient force. Integral
+	consistency conditions are verified and the energetics of the &#8220;advective
+	elimination&#8221; of the pressure gradient error (Mellor et al)
+	is considered.A large-scale wind-driven basin with and without topography
+	is used to test the model&#8217;s long-term integral performance
+	and the effects of bottom pressure torque on the transport in western
+	boundary currents. Integrations are carried out for 10 years in each
+	case and results show that the schemes are stable, and the steep
+	topography causes no obvious numerical problems. A realistic meandering
+	western boundary current is well developed with detached cold cyclonic
+	and warm anticyclonic eddies as it extends across the basin. In addition,
+	the results with topography show earlier separation and enhanced
+	transport in the western boundary currents due to the bottom pressure
+	torque.},
+  date = {December 01, 1998},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@PHDTHESIS{Speich1992,
+  author = {S. Speich},
+  title = {Etude du for\c{c}age de la circulation g\'{e}n\'{e}rale oc\'{e}anique
+	par les d\'{e}troits - cas de la mer d'Alboran},
+  school = {Universit\'{e} Pierre et Marie Curie, Paris, France},
+  year = {1992},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Speich1996,
+  author = {S. Speich and G. Madec and M. Cr\'{e}pon},
+  title = {The circulation in the Alboran Sea - a sensitivity study},
+  journal = JPO,
+  year = {1996},
+  volume = {26},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Steele2001,
+  author = {M. Steele and R. Morley and W. Ermold},
+  title = {PHC- A Global Ocean Hydrography with a High-Quality Arctic Ocean},
+  journal = {Journal of Climate},
+  year = {2001},
+  volume = {14},
+  pages = {2079--2087
+	
+	},
+  number = {9},
+  abstract = {A new gridded ocean climatology, the Polar Science Center Hydrographic
+	Climatology (PHC), has been created that merges the 1998 version
+	of the World Ocean Atlas with the new regional Arctic Ocean Atlas.
+	The result is a global climatology for temperature and salinity that
+	contains a good description of the Arctic Ocean and its environs.
+	Monthly, seasonal, and annual average products have been generated.
+	How the original datasets were prepared for merging, how the optimal
+	interpolation procedure was performed, and characteristics of the
+	resulting dataset are discussed, followed by a summary and discussion
+	of future plans.},
+  date = {May 01, 2001},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Stein1992,
+  author = {C. A. Stein and S. Stein},
+  title = {A model for the global variation in oceanic depth and heat flow with
+	lithospheric age},
+  journal = {Nature},
+  year = {1992},
+  volume = {359},
+  pages = {123-129},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Thiem2006,
+  author = {O. Thiem and J. Berntsen},
+  title = {Internal pressure errors in sigma-coordinate ocean models due to
+	anisotropy},
+  journal = {Ocean Modelling},
+  year = {2006},
+  volume = {12, 1-2},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Treguier1992,
+  author = {A.M. Tr\'{e}guier},
+  title = {Kinetic energy analysis of an eddy resolving, primitive equation
+	North Atlantic model},
+  journal = JGR,
+  year = {1992},
+  volume = {97},
+  pages = {687-701}
+}
+
+@ARTICLE{Treguier2001,
+  author = {A.M Tr\'{e}guier and B. Barnier and A.P. de Miranda and J.M. Molines
+	and N. Grima and M. Imbard and G. Madec and C. Messager and T. Reynaud
+	and S. Michel},
+  title = {An Eddy Permitting model of the Atlantic circulation: evaluating
+	open boundary conditions},
+  journal = JGR,
+  year = {2001},
+  volume = {106},
+  pages = {22115-22129}
+}
+
+@ARTICLE{Treguier1996,
+  author = {A.-M. Tr\'{e}guier and J. Dukowicz and K. Bryan},
+  title = {Properties of nonuniform grids used in ocean general circulation
+	models},
+  journal = JGR,
+  year = {1996},
+  volume = {101},
+  pages = {20877-20881},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@ARTICLE{Treguier1997,
+  author = {A. M. Tr\'{e}guier and I. M. Held and V. D. Larichev},
+  title = {Parameterization of Quasigeostrophic Eddies in Primitive Equation
+	Ocean Models},
+  journal = JPO,
+  year = {1997},
+  volume = {27},
+  pages = {567-580},
+  number = {4},
+  abstract = {A parameterization of mesoscale eddy fluxes in the ocean should be
+	consistent with the fact that the ocean interior is nearly adiabatic.
+	Gent and McWilliams have described a framework in which this can
+	be approximated in z-coordinate primitive equation models by incorporating
+	the effects of eddies on the buoyancy field through an eddy-induced
+	velocity. It is also natural to base a parameterization on the simple
+	picture of the mixing of potential vorticity in the interior and
+	the mixing of buoyancy at the surface. The authors discuss the various
+	constraints imposed by these two requirements and attempt to clarify
+	the appropriate boundary conditions on the eddy-induced velocities
+	at the surface. Quasigeostrophic theory is used as a guide to the
+	simplest way of satisfying these constraints.},
+  date = {April 01, 1997},
+  owner = {gm},
+  timestamp = {2007.08.03}
+}
+
+@BOOK{UNESCO1983,
+  title = {Algorithms for computation of fundamental property of sea water},
+  publisher = {Techn. Paper in Mar. Sci, 44, UNESCO},
+  year = {1983},
+  author = {UNESCO},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@TECHREPORT{OASIS2006,
+  author = {S. Valcke},
+  title = {OASIS3 User Guide (prism\_2-5)},
+  institution = {PRISM Support Initiative Report No 3, CERFACS, Toulouse, France,
+	64 pp},
+  year = {2006},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@TECHREPORT{valal00,
+  author = {S. Valcke and L. Terray and A. Piacentini },
+  title = {The OASIS Coupled User Guide Version 2.4},
+  institution = {CERFACS},
+  year = {2000},
+  number = {TR/CMGC/00-10}
+}
+
+@ARTICLE{Weatherly1984,
+  author = {G. L. Weatherly},
+  title = {An estimate of bottom frictional dissipation by Gulf Stream fluctuations},
+  journal = JMR,
+  year = {1984},
+  volume = {42, 2},
+  pages = {289-301},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Weaver1997,
+  author = {A. J. Weaver and M. Eby},
+  title = {On the numerical implementation of advection schemes for use in conjuction
+	with various mixing 
+	
+	parameterizations in the GFDL ocean model},
+  journal = JPO,
+  year = {1997},
+  volume = {27},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@ARTICLE{Webb1998,
+  author = {D. J. Webb and B. A. de Cuevas and C. S. Richmond},
+  title = {Improved Advection Schemes for Ocean Models},
+  journal = JAOT,
+  year = {1998},
+  volume = {15},
+  pages = {1171-1187},
+  number = {5},
+  abstract = {Leonard&#8217;s widely used QUICK advection scheme is, like the Bryan&#8211;Cox&#8211;Semtner
+	ocean model, based on a control volume form of the advection equation.
+	Unfortunately, in its normal form it cannot be used with the leapfrog&#8211;Euler
+	forward time-stepping schemes used by the ocean model. Farrow and
+	Stevens overcame the problem by implementing a predictor&#8211;corrector
+	time-stepping scheme, but this is computationally expensive to run.
+	The present paper shows that the problem can be overcome by splitting
+	the QUICK operator into an O(&#948;x2) advective term and a velocity
+	dependent biharmonic diffusion term. These can then be time-stepped
+	using the combined leapfrog and Euler forward schemes of the Bryan&#8211;Cox&#8211;Semtner
+	ocean model, leading to a significant increase in model efficiency.
+	A small change in the advection operator coefficients may also be
+	made leading to O(&#948;x4) accuracy. Tests of the improved schemes
+	are carried out making use of a global eddy-permitting ocean model.
+	Results are presented from cases where the schemes were applied to
+	only the tracer fields and also from cases where they were applied
+	to both the tracer and velocity fields. It is found that the new
+	schemes have the most effect in the western boundary current regions,
+	where, for example, the warm core of the Agulhas Current is no longer
+	broken up by numerical noise.},
+  date = {October 01, 1998},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Willebrand2001,
+  author = {J. Willebrand and B. Barnier and C. Boning and C. Dieterich and P.
+	D. Killworth and C. Le Provost and Y. Jia and J.-M. Molines and A.
+	L. New},
+  title = {Circulation characteristics in three eddy-permitting models of the
+	North Atlantic},
+  journal = {Progress in Oceanography},
+  year = {2001},
+  volume = {48, 2},
+  pages = {123-161},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Zalesak1979,
+  author = {S. T. Zalesak},
+  title = {Fully multidimensional flux corrected transport algorithms for fluids},
+  journal = JCP,
+  year = {1979},
+  volume = {31},
+  owner = {gm},
+  timestamp = {2007.08.04}
+}
+
+@ARTICLE{Zhang1992,
+  author = {Zhang, R.-H. and Endoh, M.},
+  title = {A free surface general circulation model for the tropical Pacific
+	Ocean},
+  journal = JGR,
+  year = {1992},
+  volume = {97},
+  pages = {11237-11255},
+  month = jul,
+  owner = {gm}
+}
+
+@comment{jabref-meta: groupsversion:3;}
+
+@comment{jabref-meta: groupstree:
+0 AllEntriesGroup:;
+1 ExplicitGroup:El Nino\;2\;blanketal97\;;
+2 ExplicitGroup:97/98 event\;0\;;
+2 ExplicitGroup:Forecast\;0\;;
+2 ExplicitGroup:GHG change\;0\;;
+2 ExplicitGroup:in GCMs\;0\;;
+2 ExplicitGroup:in MIPs\;0\;;
+2 ExplicitGroup:momentum balance\;0\;;
+2 ExplicitGroup:Obs analysis\;0\;;
+2 ExplicitGroup:Paleo\;0\;;
+2 ExplicitGroup:Previous events\;0\;;
+2 ExplicitGroup:Reviews\;0\;;
+2 ExplicitGroup:Simple models\;0\;Zhang1992\;;
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+2 ExplicitGroup:Global\;0\;;
+2 ExplicitGroup:Clouds\;0\;;
+2 ExplicitGroup:Scale interactions\;0\;;
+1 ExplicitGroup:Mechanisms\;2\;;
+2 ExplicitGroup:CRF\;0\;;
+2 ExplicitGroup:Water vapor\;0\;;
+2 ExplicitGroup:Atmos mechanisms\;0\;;
+1 ExplicitGroup:GCMs\;2\;;
+2 ExplicitGroup:Uncertainty\;0\;;
+2 ExplicitGroup:Momentum balance\;0\;;
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+1 KeywordGroup:EG publis\;0\;author\;guilyardi\;0\;0\;;
+}
+
Index: trunk/DOC/TexFiles/Chapters/0_page_de_garde_old.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/0_page_de_garde_old.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/0_page_de_garde_old.tex	(revision 994)
@@ -0,0 +1,317 @@
+%\begin{figure}[htbp]
+%\centerline{\includegraphics[width=1.64in,height=11.72in]{OPA9_Chap_0_Introduction1.eps}}
+%\label{fig1}
+%\end{figure}
+
+\textbf{Institut Pierre Simon Laplace}
+
+\textbf{des Sciences de l'Environnement Global}
+
+\textit{Notes du P\^{o}le de Mod\'{e}lisation}
+
+NEMO : the OPA ocean engine
+
+(NEMO version 2.3)
+
+Gurvan Madec, {\ldots} 
+
+Laboratoire d'Oc\'{e}anographie et du Climat: Exp\'{e}rimentation 
+
+et Approches Num\'{e}riques
+
+%\begin{figure}[htbp]
+%\centerline{\includegraphics[width=4.58in,height=4.35in]{OPA9_Chap_0_Introduction2.eps}}
+%\label{fig2}
+%\end{figure}
+
+\newpage 
+\begin{center}
+NEMO : the OPA ocean engine
+\end{center}
+
+Gurvan Madec
+
+Laboratoire d'Oc\'{e}anographie et du Climat: Exp\'{e}rimentation Approches 
+Num\'{e}riques
+
+CNRS/IRD/UPMC, UMR 7617
+
+\subsection{Abstract}
+\label{subsec:abstract}
+OPA is the ocean engine of NEMO, a framework of ocean related components 
+developed for both research and operational purposes. It is a primitive 
+equation model adapted to regional and global ocean circulation. It is 
+intended to be a flexible tool for studying the ocean and its interactions 
+with the others components of the earth climate system (atmosphere, sea-ice, 
+biogeochemical tracers, ...) over a wide range of space and time scales. 
+Prognostic variables are the three-dimensional velocity field, the sea 
+surface height, the temperature and the salinity. In the horizontal 
+direction, the model uses a curvilinear orthogonal grid and in the vertical 
+direction, a $z$ full or partial step coordinate, or $s$-coordinate, or a mixture 
+of the two. The distribution of variables is a three-dimensional Arakawa 
+C-type grid. Various physical choices are available to describe ocean 
+physics, including a 1.5 turbulent closure or a KPP scheme for the vertical 
+mixing. Within NEMO, OPA is interfaced with a sea-ice model (LIM), passive 
+tracer and biogeochemical models (TOP) and, via the OASIS coupler, with 
+several atmospheric general circulation models. In addition, it can be run 
+on many different computers, including shared and distributed memory 
+multiprocessor computers.
+
+\textsc{\textbf{R\'{e}sum\'{e}}}
+
+OPA est le moteur oc\'{e}anique de NEMO, un syst\`{e}me de mod\'{e}lisation 
+de l'oc\'{e}an d\'{e}velopp\'{e} \`{a} des fins de recherche et 
+d'oc\'{e}anographie op\'{e}rationnelle. Ce moteur est un mod\`{e}le aux 
+\'{e}quations primitives de la circulation oc\'{e}anique r\'{e}gionale et 
+globale. Il se veut un outil flexible pour \'{e}tudier sur un vaste spectre 
+spatiotemporel l'oc\'{e}an et ses interactions avec les autres composantes 
+du syst\`{e}me climatique terrestre (atmosph\`{e}re, glace de mer, traceurs 
+biog\'{e}ochimiques...). Les variables pronostiques sont le champ 
+tridimensionnel de vitesse, la hauteur de la mer , la temperature et la 
+salinit\'{e}. La distribution des variables se fait sur une grille $C$ 
+d'Arakawa tridimensionnelle utilisant une coordonn\'{e}e verticale $z$ \`{a} 
+niveaux entiers ou partiels, ou une coordonn\'{e}e$ s$, ou encore une 
+combinaison des deux. Diff\'{e}rents choix sont propos\'{e}s pour 
+d\'{e}crire la physique oc\'{e}anique, incluant notamment une fermeture 
+turbulente d'ordre 1.5 ou un sch\'{e}ma KPP pour le m\'{e}lange vertical. 
+Via l'infrastructure NEMO, OPA est interfac\'{e} avec un mod\`{e}le de glace 
+de mer, des mod\`{e}les biog\'{e}ochimiques et de traceur passif, et, via le 
+coupleur OASIS, \`{a} plusieurs mod\`{e}les de circulation g\'{e}n\'{e}rale 
+atmosph\'{e}rique. En outre, il peut \^{e}tre ex\'{e}cut\'{e} sur de 
+nombreux calculateurs, y compris des calculateurs multiprocesseurs \`{a} 
+m\'{e}moire partag\'{e}e ou distribu\'{e}e.
+
+\newpage 
+\textbf{PLAN :}
+
+Put it here
+
+\textsc{\textbf{Disclaimer 1}}
+
+\textsc{\textbf{Foreword 3}}
+
+\textsc{\textbf{Introduction 4}}
+
+\textsc{\textbf{Disclaimer}}
+
+OPA (an acronym for Ocean PArall\'{e}lis\'{e}) is the ocean component of 
+NEMO (Nucleus for European Modelling of the Ocean (\underline 
+{www.locean-ipsl.upmc.fr/NEMO}). Like all components of NEMO, it is 
+developed under the CECILL license, which is a french adaptation of the GNU 
+\textbf{GPL} (General Public license). Anyone may use OPA freely for 
+research purposes, and is encouraged to communicate back to the NEMO team 
+its own developments and improvements. The model and the present document 
+have been made available as a service to the community. We cannot certify 
+that the code and its manual are free of errors. Bugs are inevitable and 
+some have undoubtedly survived the testing phase. Users are encouraged to 
+bring them to our attention. The author assumes no responsibility for 
+problems, errors, or incorrect usage of OPA.
+
+The OPA OGCM reference in papers and other publications is as follows: 
+
+Madec, G., 2007: NEMO: the OPA ocean engine. Note du P\^{o}le de 
+mod\'{e}lisation, Institut Pierre-Simon Laplace (IPSL), France, N\r{ }XX, 
+YYpp.
+
+Gurvan Madec: gm@locean-ipsl.upmc.fr
+
+{\ldots}.
+
+\textsc{\textbf{Foreword}}
+
+OPA is the ocean engine of the Nucleus of European Model of the Ocean 
+(NEMO). This model, like all research tools, is in perpetual evolution. The 
+present document describes the OPA model include in the release 2.3 of NEMO. 
+This is the release 9 of OPA. This version differs significantly from 
+version 8, documented in Madec \textit{et al.} (1998). The major modifications are 
+
+(1) transition to full native Fortran 90, deep code restructuring and 
+drastic reduction of CPP keys, 
+
+(2) introduction of partial step representation of bottom topography 
+
+(3) reactivation of a terrain-following vertical coordinate 
+($s-$coordinates) with the addition of several options for pressure gradient 
+computation, 
+
+(4) mixed $z-s$ coordinate
+
+(5) more choices for the treatment of the free surface: full explicit, 
+split-explicit , filtered and rigid-lid 
+
+(6) non linear free surface option (variable level thickness distributed on 
+the whole water column) 
+
+(7) additional schemes for vector and flux forms of the momentum advection
+
+(8) addition of several advection schemes on tracers
+
+(9) implementation of the AGRIF package (Adaptative Grid Refinement in 
+Fortran)
+
+(10) online diagnostics : tracers trend in the mixed layer and vorticity 
+balance
+
+(11) rewriting of the I/O management
+
+(12) OASIS 3 and 4 couplers interfacing with atmospheric global circulation 
+models.
+
+In addition, several minor modifications in the coding have been introduced 
+with the constant concern of improving performance on both scalar and vector 
+computers. 
+
+At the time of this writing, the current release is NEMO 2.3. The new 
+surface module described in this document is not yet part of the current 
+distribution.
+
+\newpage 
+
+\textsc{\textbf{Introduction}}
+
+The Nucleus for European Modelling of the Ocean (NEMO) is a framework of 
+ocean related engines, namely OPA for the Ocean dynamics and thermodynamics, 
+LIM for the sea-ice dynamics and thermodynamics, TOP for the biogeochemistry 
+(both transport (TRP) and sources minus sinks (LOBSTER, PISCES). It is 
+intended to be a flexible tool for studying the ocean and its interactions 
+with the others components of the earth climate system (atmosphere, sea-ice, 
+biogeochemical tracers, ...) over a wide range of space and time scales. 
+This documentation provides information about the physics represented by the 
+OPA ocean model and the rationale for the choice of numerical schemes and 
+the model design. More specific information about running the model on 
+different computers, or how to set up a configuration, are found on the NEMO 
+web site (\underline {www.locean-ipsl.upmc.fr/NEMO}). 
+
+The ocean component of NEMO has been developed from the OPA8.2 model 
+described in Madec \textit{et al.} (1998). This model has been used for a wide range of 
+applications, either regional or global, as a forced ocean model or coupled 
+with the atmosphere. A complete list of references is found on the NEMO web 
+site. 
+
+This manual is organised in four parts. The first part presents the model 
+basics, i.e. the equations and their assumptions, the vertical coordinates 
+used, and the subgrid scale physics. This part deals with the continuous 
+equations of the model (primitive equations, with potential temperature, 
+salinity and an equation of state). The equations are written in a 
+curvilinear coordinate system, with a choice of vertical coordinates ($z$, $s$, 
+and variable volumes). Momentum equations are formulated in the vector 
+invariant form. The model equations are written in dimensional units in the 
+meter, kilogram, second (MKS) international system 
+
+The following chapters deal with the discrete equations. The second chapter 
+presents the space and time domain. The model is discretised on a staggered 
+grid (Arakawa C grid) with masking of land areas. Vertical discretisation 
+uses $z$ coordinates (including partial step), $s$ (terrain-following) coordinate 
+(fixed volume thickness and linear free surface), or$ s* $coordinate (variable 
+volume thickness and nonlinear free surface). The following chapters 
+describe the discretisation of the prognostic equations (momentum and 
+tracers). Explicit, split-explicit or implicit free surface formulations are 
+implemented as well as arid-lid approximation. A number of numerical schemes 
+are available for momentum advection, for the computation of the pressure 
+gradients, as well as for the advection of tracers (second or higher order 
+advection schemes, including positive ones).
+
+Physical parameterisations are described in chapters The model includes an 
+implicit treatment of vertical viscosity and diffusivity. The lateral 
+Laplacian and biharmonic viscosity and diffusion can be rotated following a 
+geopotential or neutral direction. There is an optional eddy induced 
+velocity (Gent and McWilliams 1992) with a space and time variable 
+coefficient, with options to compute the coefficients with Tr\'{e}guier et 
+al. (199X), or Visbeck et al. XXX) schemes. The model has vertical harmonic 
+viscosity and diffusion with a space and time variable coefficient, with 
+options to compute the coefficients with Blanke and Delecluse (1992), Large 
+et al. (1994), or Pacanowski and Philander (1981) mixing schemes.
+
+Optional tidal mixing parametrisation
+
+Other model characteristics (chapter {\ldots}) are the lateral boundary 
+conditions. Global configurations of the model make use of the ORCA tripolar 
+grid, with special north fold boundary condition. Free-slip or no-slip 
+boundary conditions are allowed at land boundaries. Closed basin geometries 
+as well as periodic domains and open boundary conditions are possible. 
+
+Surface boundary conditions can be implemented as prescribed fluxes, or bulk 
+formulations for the surface fluxes (wind stress, heat, freshwater). The 
+model allows penetration of solar radiation There is an optional geothermal 
+heating at the ocean bottom. Within the NEMO system the ocean model is 
+interactively coupled with a sea ice model (LIM) and with biogeochemistry 
+models (PISCES, LOBSTER).Interactive coupling to Atmospheric models is 
+possible via the OASIS (ref!!!) coupler. 
+
+Specific online diagnostics are available in the model: output of all the 
+tendencies of the momentum and tracers equations, output of tracers 
+tendencies averaged over the time evolving mixed layer.
+
+The model is implemented in Fortran 90, with preprocessing (C preprocessor). 
+It runs under UNIX. It is optimized for vector computers and parallelised by 
+domain decomposition with MPI. All input and output is done in NetCDF 
+(Network Common Data Format) with a optional direct access format for 
+output. To ensure the clarity and readability of the code it is necessary to 
+follow \textit{coding rules}. The coding rules for OPA include conventions for naming variables, 
+with different starting letters for different types of variables (real, 
+integer, parameter{\ldots}) Those rules are presented in a document 
+available on the NEMO web site..
+
+The model is organized with a high internal modularity based on physics. In 
+particular, each trend (e.g., a term in the rhs of the prognostic equation) 
+for momentum and tracers is computed in a dedicated module. To make it 
+easier for the user to find his way around the code, the module names follow 
+the \textit{three-letter rule}. Each module name is made of three-letter sequences. For example, 
+TRADMP.F90 is a module related to the TRAcers equation, computing the 
+DaMPing. The complete list of module names is presented in annex. 
+Furthermore, modules are organized in a few directories that correspond to 
+their category, as indicated by the first three letters of their name. 
+
+The manual follows this organization. After the presentation of the 
+continuous equations (chapter 1), the following chapters refer to specific 
+terms of the equations each associated with a group of modules .
+
+\begin{table}[htbp]
+\begin{center}
+\begin{tabular}{|p{143pt}|l|l|}
+\hline
+Chapter 2& 
+DOM& 
+Model DOMain \\
+\hline
+Chapter 3& 
+DYN& 
+DYNamic equations (momentum) \\
+\hline
+Chapter 4& 
+TRA& 
+TRAcer equations (potential temperature and salinity) \\
+\hline
+Chapter 5 & 
+SBC& 
+Surface Boundary Conditions \\
+\hline
+Chapter 6& 
+LDF& 
+Lateral DiFfusion (parameterisations) \\
+\hline
+Chapter 7& 
+ZDF& 
+Vertical DiFfusion  \\
+\hline
+Chapter 8& 
+OBC, lbclnk module& 
+Lateral boundary conditions  \\
+\hline
+Chapter 9& 
+miscellaneous& 
+ \\
+\hline
+\end{tabular}
+\label{tab1}
+\end{center}
+\end{table}
+
+\textbf{Nota Bene :}
+
+Red color : not in the current reference version (v2.3beta) but expected 
+soon
+
+Yellow : missing references, text to be updated{\ldots}
+
+\end{document}
Index: trunk/DOC/TexFiles/Chapters/Abstracts_Foreword - save.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Abstracts_Foreword - save.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Abstracts_Foreword - save.tex	(revision 994)
@@ -0,0 +1,52 @@
+
+% ================================================================
+% Abstract (English)
+% ================================================================
+
+\chapter*{Abstract}
+
+OPA is the ocean engine of NEMO, a framework of ocean related components developed for both research and operational purposes. It is a primitive equation model adapted to regional and global ocean circulation. It is intended to be a flexible tool for studying the ocean and its interactions with the others components of the earth climate system (atmosphere, sea-ice, biogeochemical tracers, ...) over a wide range of space and time scales. Prognostic variables are the three-dimensional velocity field, the sea surface height, the temperature and the salinity. In the horizontal direction, the model uses a curvilinear orthogonal grid and in the vertical direction, a z full or partial step coordinate, or s-coordinate, or a mixture of the two. The distribution of variables is a three-dimensional Arakawa C-type grid. Various physical choices are available to describe ocean physics, including a 1.5 turbulent closure  or a KPP scheme for the vertical mixing. Within NEMO, OPA is interfaced with a sea-ice model (LIM), passive tracer and biogeochemical models (TOP) and, via the OASIS coupler, with several atmospheric general circulation models. In addition, it can be run on many different computers, including shared and distributed memory multiprocessor computers.
+
+% ================================================================
+% Abstract (French)
+% ================================================================
+\chapter*{R\'{e}sum\'{e}}
+
+OPA est le moteur oc\'{e}anique de NEMO, un syst\`{e}me de mod\'{e}lisation de l'oc\'{e}an d\'{e}velopp\'{e} \`{a} des fins de recherche et d'oc\'{e}anographie op\'{e}rationnelle. Ce moteur est un mod\`{e}le aux \'{e}quations primitives de la circulation oc\'{e}anique r\'{e}gionale et globale. Il se veut un outil flexible pour \'{e}tudier sur un vaste spectre spatiotemporel l'oc\'{e}an et ses interactions avec les autres composantes du syst\`{e}me climatique terrestre (atmosph\`{e}re, glace de mer, traceurs biog\'{e}ochimiques...). Les variables pronostiques sont le champ tridimensionnel de vitesse, la hauteur de la mer , la temperature et la salinit\'{e}. La distribution des variables se fait sur une grille C d'Arakawa tridimensionnelle utilisant une coordonn\'{e}e verticale z \`{a} niveaux entiers ou partiels, ou une coordonn\'{e}e s, ou encore une combinaison des deux. Diff\'{e}rents choix sont propos\'{e}s pour d\'{e}crire la physique oc\'{e}anique, incluant notamment une fermeture turbulente d'ordre 1.5 ou un sch\'{e}ma KPP pour le m\'{e}lange vertical. Via l'infrastructure NEMO, OPA est interfac\'{e} avec un mod\`{e}le de glace de mer, des mod\`{e}les biog\'{e}ochimiques et de traceur passif, et, via le coupleur OASIS, \`{a} plusieurs mod\`{e}les de circulation g\'{e}n\'{e}rale atmosph\'{e}rique. En outre, il peut \^etre ex\'{e}cut\'{e} sur de nombreux calculateurs, y compris des calculateurs multiprocesseurs \`{a} m\'{e}moire partag\'{e}e ou distribu\'{e}e.
+
+
+% ================================================================
+% Disclaimer
+% ================================================================
+\chapter*{Disclaimer}
+
+OPA (an acronym for Ocean PArall\'{e}lis\'{e}) is the ocean component of NEMO (Nucleus for European Modelling of the Ocean (www.locean-ipsl.upmc.fr/NEMO). Like all components of NEMO, it is developed under the CECILL license, which is a french adaptation of the GNU GPL (General Public license). Anyone may use OPA freely for research purposes, and is encouraged to communicate back to the NEMO team its own developments and improvements. The model and the present document have been made available as a service to the community. We cannot certify that the code and its manual are free of errors. Bugs are inevitable and some have undoubtedly survived the testing phase. Users are encouraged to bring them to our attention. The author assumes no responsibility for problems, errors, or incorrect usage of OPA.
+
+ \vspace{1cm}
+The OPA OGCM reference in papers and other publications is as follows:
+ \vspace{0.5cm}
+
+Madec, G., 2007: NEMO ocean engine. \textit{Note du P\^ole de mod\'{e}lisation}, Institut Pierre-Simon Laplace (IPSL), France, NXX, \colorbox{yellow}{YYpp}.\\
+
+% ================================================================
+% Foreword
+% ================================================================
+\chapter*{Foreword}
+
+OPA is the ocean engine of the NEMO. This model, like all research tools, is in perpetual evolution. The present document describes the OPA model include in the release 2.3 of NEMO. This is the release 9 of OPA. This version differs significantly from version 8, documented in \citet{Madec1998}. The major modifications are :\\
+(1) transition to full native \textsc{Fortran} 90, deep code restructuring and drastic reduction of CPP keys, \\
+(2) introduction of partial step representation of bottom topography \\
+(3) partial reactivation of a terrain-following vertical coordinate ($s$- and hybrid $s$-$z$) with the addition of several options for pressure gradient computation \footnote{Partial support of $s$-coordinate: there is presently no support for neutral physics in $s$-coordinate and for the new options for horizontal pressure gradient computation with true equation of state.}, \\ 
+(4) more choices for the treatment of the free surface: full explicit, split-explicit , filtered and rigid-lid \\
+(5) non linear free surface option (variable level thickness distributed on the whole water column) \\
+(6) additional schemes for vector and flux forms of the momentum  advection \\
+(7) addition of several advection schemes on tracers \\
+ (8) implementation of the AGRIF package (Adaptative Grid Refinement in \textsc{Fortran} ) \\
+(9) online diagnostics : tracers trend in the mixed layer and vorticity balance \\
+(10) rewriting of the I/O management \\
+(11) OASIS 3 and 4 couplers interfacing with atmospheric global circulation models. 
+
+In addition, several minor modifications in the coding have been introduced with the constant concern of improving performance on both scalar and vector computers. 
+
+At the time of this writing, the current release is NEMO 2.3. The new surface module described in this document is not yet part of the current distribution.
+
Index: trunk/DOC/TexFiles/Chapters/Abstracts_Foreword.tex
===================================================================
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+++ trunk/DOC/TexFiles/Chapters/Abstracts_Foreword.tex	(revision 994)
@@ -0,0 +1,36 @@
+
+% ================================================================
+% Abstract (English / French)
+% ================================================================
+
+\chapter*{Abstract / R\'{e}sum\'{e}}
+
+\vspace{-40pt}
+
+\small{ 
+The ocean engine of NEMO (Nucleus for European Modelling of the Ocean) is a primitive equation model adapted to regional and global ocean circulation problems. It is intended to be a flexible tool for studying the ocean and its interactions with the others components of the earth climate system (atmosphere, sea-ice, biogeochemical tracers, ...) over a wide range of space and time scales. Prognostic variables are the three-dimensional velocity field, a linear or non-linear sea surface height, the temperature and the salinity. In the horizontal direction, the model uses a curvilinear orthogonal grid and in the vertical direction, a full or partial step $z$-coordinate, or $s$-coordinate, or a mixture of the two. The distribution of variables is a three-dimensional Arakawa C-type grid. Various physical choices are available to describe ocean physics, including TKE and KPP vertical physics. Within NEMO, the ocean is interfaced with a sea-ice model (LIM), passive tracer and biogeochemical models (TOP) and, via the OASIS coupler, with several atmospheric general circulation models. 
+
+% ================================================================
+ \vspace{0.5cm}
+
+Le moteur oc\'{e}anique de NEMO (Nucleus for European Modelling of the Ocean) est un mod\`{e}le aux \'{e}quations primitives de la circulation oc\'{e}anique r\'{e}gionale et globale. Il se veut un outil flexible pour \'{e}tudier sur un vaste spectre spatiotemporel l'oc\'{e}an et ses interactions avec les autres composantes du syst\`{e}me climatique terrestre (atmosph\`{e}re, glace de mer, traceurs biog\'{e}ochimiques...). Les variables pronostiques sont le champ tridimensionnel de vitesse, une hauteur de la mer lin\'{e}aire ou non, la temperature et la salinit\'{e}. La distribution des variables se fait sur une grille C d'Arakawa tridimensionnelle utilisant une coordonn\'{e}e verticale $z$ \`{a} niveaux entiers ou partiels, ou une coordonn\'{e}e s, ou encore une combinaison des deux. Diff\'{e}rents choix sont propos\'{e}s pour d\'{e}crire la physique oc\'{e}anique, incluant notamment des physiques verticales TKE et KPP . A travers l'infrastructure NEMO, l'oc\'{e}an est interfac\'{e} avec un mod\`{e}le de glace de mer, des mod\`{e}les biog\'{e}ochimiques et de traceur passif, et, via le coupleur OASIS, \`{a} plusieurs mod\`{e}les de circulation g\'{e}n\'{e}rale atmosph\'{e}rique. 
+} 
+
+% ================================================================
+% Disclaimer
+% ================================================================
+\chapter*{Disclaimer}
+
+Like all components of NEMO, the ocean component is developed under the CECILL license, which is a French adaptation of the GNU GPL (General Public License). Anyone may use it freely for research purposes, and is encouraged to communicate back to the NEMO team its own developments and improvements. The model and the present document have been made available as a service to the community. We cannot certify that the code and its manual are free of errors. Bugs are inevitable and some have undoubtedly survived the testing phase. Users are encouraged to bring them to our attention. The author assumes no responsibility for problems, errors, or incorrect usage of NEMO.
+
+ \vspace{1cm}
+NEMO reference in papers and other publications is as follows:
+ \vspace{0.5cm}
+
+Madec, G., 2008: NEMO ocean engine. \textit{Note du P\^ole de mod\'{e}lisation}, Institut Pierre-Simon Laplace (IPSL), France, No 27, ISSN No 1288-1619.\\
+
+
+ \vspace{0.5cm}
+Additional information can be found on www.locean-ipsl.upmc.fr/NEMO
+ \vspace{0.5cm}
+
Index: trunk/DOC/TexFiles/Chapters/Annex_A.tex
===================================================================
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+
+% ================================================================
+% Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations
+% ================================================================
+\chapter{Curvilinear $s$-Coordinate Equations}
+\label{Apdx_A}
+\minitoc
+
+In order to establish the set of Primitive Equation in curvilinear $s$-coordinates ($i.e.$ 
+orthogonal curvilinear coordinate in the horizontal and $s$-coordinate in the vertical), we 
+start from the set of equation established in \S\ref{PE_zco_Eq} for the special case 
+$k = z$ and thus $e_3 = 1$, and we introduce an arbitrary vertical coordinate 
+$s = s(i,j,z,t)$. Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ 
+(which now depends on $(i,j,z,t)$) and the horizontal slope of $s$-surfaces by :
+\begin{equation} \label{Apdx_A_s_slope}
+\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 
+\quad \text{and} \quad 
+\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s 
+\end{equation}
+
+The chain rule to establish the model equations in the curvilinear $s$-coordinate system 
+is:
+\begin{equation} \label{Apdx_A_s_chain_rule}
+\begin{aligned}
+&\left. {\frac{\partial \bullet }{\partial t}} \right|_z  =
+\left. {\frac{\partial \bullet }{\partial t}} \right|_s 
+	 -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\
+&\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =
+  \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
+  	  -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}=
+	  \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
+	  -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\
+&\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =
+\left. {\frac{\partial \bullet }{\partial j}} \right|_s 
+	- \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
+\left. {\frac{\partial \bullet }{\partial j}} \right|_s 
+	- \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\
+&\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\
+\end{aligned}
+\end{equation}
+
+In particular applying the time derivative chain rule to $z$ provide the expression of $w_s$,  the vertical velocity of the $s-$surfaces:
+\begin{equation} \label{Apdx_A_w_in_s}
+w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s 
+				= \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} 
+			    = e_3 \, \frac{\partial s}{\partial t} 
+\end{equation}
+
+% ================================================================
+% continuity equation
+% ================================================================
+\section{Continuity Equation}
+\label{Apdx_B_continuity}
+
+Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows:
+
+\begin{align*}
+\nabla \cdot {\rm {\bf U}} 
+&= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z 
+						+\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z  \right]
++ \frac{\partial w}{\partial z} 		\\
+\\
+&     = \frac{1}{e_1 \,e_2 }  \left[ 
+		  \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s       
+		  - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s}
+		+ \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s       
+		  - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s}	\right]
+	+ \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z}                        \\
+\\
+&     = \frac{1}{e_1 \,e_2 }   \left[ 
+		  \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s       
+		+ \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s       	\right]
+	+ \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s}
+						-  \sigma _1 \frac{\partial u}{\partial s}
+						-  \sigma _2 \frac{\partial v}{\partial s}      \right]          \\
+\\
+&     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[ 
+		  \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_s  
+		  -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s      
+		+ \left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s
+		  - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right]          \\
+& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+	+ \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s}
+						-  \sigma _1 \frac{\partial u}{\partial s}
+						-  \sigma _2 \frac{\partial v}{\partial s}      \right]      \\
+\\
+\end{align*}
+
+Noting that $\frac{1}{e_1 }\left. {\frac{\partial e_3 }{\partial i}} 
+\right|_s =\frac{1}{e_1 }\left. {\frac{\partial ^2z}{\partial i\,\partial 
+s}} \right|_s =\frac{\partial }{\partial s}\left( {\frac{1}{e_1 }\left. 
+{\frac{\partial z}{\partial i}} \right|_s } \right)=\frac{\partial \sigma _1 
+}{\partial s}$ and $\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} 
+\right|_s =\frac{\partial \sigma _2 }{\partial s}$, it becomes:
+
+\begin{align*}
+\nabla \cdot {\rm {\bf U}} 
+& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[   
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]        \\ 
+& \qquad \qquad \qquad \qquad \qquad \quad
+ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right] \\ 
+\\
+& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[   
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
+   + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma _1  - v\;\sigma _2  \right]
+ \end{align*} 
+ 
+Here, $w$ is the vertical velocity relative to the $z-$coordinate system. Introducing the dia-surface velocity component, $\omega $, defined as the velocity relative to the moving $s$-surfaces and normal to them:
+\begin{equation} \label{Apdx_A_w_s}
+\omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\
+\end{equation}
+with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression of the divergence of the velocity in the curvilinear $s$-coordinate system:
+\begin{align*} \label{Apdx_A_A4}
+\nabla \cdot {\rm {\bf U}} 
+&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
++ \frac{1}{e_3 } \frac{\partial w_s       }{\partial s}    \\
+\\
+&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
++ \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right)   \\
+\\
+&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
++ \frac{\partial}{\partial s} \frac{\partial s}{\partial t}
++ \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s}     \\
+\\
+&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
++ \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     \\
+\end{align*}
+
+As a result, the continuity equation \eqref{Eq_PE_continuity} in $s$-coordinates becomes:
+\begin{equation} \label{Apdx_A_A5}
+\frac{1}{e_3 } \frac{\partial e_3}{\partial t} 
++ \frac{1}{e_1 \,e_2 \,e_3 }\left[ 
+			{\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s 
+		    +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
+ +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0   
+\end{equation}
+
+% ================================================================
+% momentum equation
+% ================================================================
+\section{Momentum Equation}
+\label{Apdx_B_momentum}
+
+Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the 
+momentum equation in the vector invariant form (similar manipulations can be performed on the second one). Its non linear term can be transformed 
+as follows:
+
+\begin{align*}
+&+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z 
+- w \;\frac{\partial u}{\partial z} \\
+\\
+&\qquad=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} 
+\right|_z -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_z } 
+\right]\;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} 
+\right|_z -w\frac{\partial u}{\partial z}      \\
+\\
+&\qquad =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} 
+\right|_s -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_s }     \right. 
+ \left. {-\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (e_2 \,v)}{\partial s}+\frac{e_2 }{e_3 }\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right]\;v \\ 
+&\qquad \qquad \qquad \qquad \qquad
+{ -\frac{1}{2e_1 }\left( {\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}} \right)
+-\frac{w}{e_3 }\frac{\partial u}{\partial s} }    \\
+\end{align*}
+\begin{align*}
+\qquad  &= \left. \zeta \right|_s \;v
+	- \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 
+	- \frac{w}{e_3 }\frac{\partial u}{\partial s}
+	- \left[   {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s}
+	           - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v      \\
+\qquad&\qquad \qquad \qquad \qquad \qquad \qquad
+\qquad  \qquad \qquad \qquad \quad
+	+\frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s}      \\
+%\\
+\qquad &= \left. \zeta \right|_s \;v
+		- \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 
+\qquad&\qquad \qquad \qquad
+ -\frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s}
+	+\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s}
+	-\sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\
+\\
+\qquad &= \left. \zeta \right|_s \;v
+		- \frac{1}{2e_1 }\left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s    
+		  - \frac{1}{e_3} \left[  w - \sigma _2 v - \sigma _1 u  \right] 
+		  			 \; \frac{\partial u}{\partial s}   \\
+\\
+\qquad &= \left. \zeta \right|_s \;v
+		- \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s   
+		  - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s} 
+		  - \frac{\partial s}{\partial t}  \frac{\partial u}{\partial s} 
+\end{align*}
+
+Therefore, the non-linear terms of the momentum equation have the same form 
+in $z-$ and $s-$coordinates but with the addition of the time derivative of the velocity: 
+\begin{multline}  \label{Apdx_A_momentum_NL}
++\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z 
+- w \;\frac{\partial u}{\partial z}    \\
+= - \frac{\partial u}{\partial t} + \left. \zeta \right|_s \;v
+	- \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s   
+	- \frac{1}{e_3 }\omega \frac{\partial u}{\partial s} 
+\end{multline}
+
+The pressure gradient term can be transformed as follows:
+\begin{equation} \label{Apdx_A_grad_p}
+\begin{split}
+ -\frac{1}{\rho _o e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z& =-\frac{1}{\rho _o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\
+& =-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho _o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\
+&=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1
+\end{split}
+\end{equation}
+
+An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the tilt of model 
+levels.
+
+Introducing \eqref{Apdx_A_momentum_NL} and \eqref{Apdx_A_grad_p} in \eqref{Eq_PE_dyn_vect} and regrouping the time derivative terms in the left hand side, and performing the same manipulation on the second component, we obtain the vector invariant form of momentum equation in $s-$coordinate :
+\begin{subequations} \label{Apdx_A_dyn_vect}
+\begin{multline} \label{Apdx_A_PE_dyn_vect_u}
+ \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}=
+	+   \left( {\zeta +f} \right)\,v                                    
+	-   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
+	-   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
+	-   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)    
+	+  g\frac{\rho }{\rho _o}\sigma _1 
+	+   D_u^{\vect{U}}  +   F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Apdx_A_dyn_vect_v}
+ \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}=
+	-   \left( {\zeta +f} \right)\,u   
+	-   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)     	
+	-   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
+	-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right) 
+	 +  g\frac{\rho }{\rho _o }\sigma _2   
+	+  D_v^{\vect{U}}  +   F_v^{\vect{U}}
+\end{multline}
+\end{subequations}
+
+It has the same form as in $z-$coordinate but the vertical scale factor that has appeared inside the time derivative. The form of the vertical physics and forcing terms remain unchanged. The form of the lateral physics is discussed in appendix~\ref{Apdx_B}.  
+
+% ================================================================
+% Tracer equation
+% ================================================================
+\section{Tracer Equation}
+\label{Apdx_B_tracer}
+
+The tracer equation is obtained using the same calculation as for the 
+continuity equation and then regrouping the time derivative terms in the left hand side :
+
+\begin{multline} \label{Apdx_A_tracer}
+ \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t} 
+ 	= -\frac{1}{e_1 \,e_2 \,e_3 } 
+		\left[ {\frac{\partial }{\partial i}} \left( {e_2 \,e_3 \;Tu} \right) \right .
+		    +         \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)                \\
+		    + \left. \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] +D^{T} +F^{T} \; \;
+\end{multline}
+
+
+The expression of the advection term is a straight consequence of (A.4), the 
+expression of the 3D divergence in $s$-coordinates established above. 
+
Index: trunk/DOC/TexFiles/Chapters/Annex_B.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Annex_B.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Annex_B.tex	(revision 994)
@@ -0,0 +1,251 @@
+% ================================================================
+% Chapter Ñ Appendix B : Diffusive Operators
+% ================================================================
+\chapter{Appendix B : Diffusive Operators}
+\label{Apdx_B}
+\minitoc
+
+% ================================================================
+% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
+% ================================================================
+\section{Horizontal/Vertical 2nd Order Tracer Diffusive Operators}
+\label{Apdx_B_1}
+
+
+In $z$-coordinate, the horizontal/vertical second order tracer diffusive operator is given by:
+\begin{multline} \label{Apdx_B1}
+ D^T = \frac{1}{e_1 \, e_2}      \left[ 
+  \left. \frac{\partial}{\partial i} \left( 	\frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right.          \\
+						     \left.
++ \left. \frac{\partial}{\partial j} \left(  \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z   \right)   \right|_z      \right]         
++ \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
+\end{multline}
+
+In $s$-coordinate, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by (A.1) and the vertical/horizontal ratio of diffusive coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusive operator is given by:
+
+\begin{equation} \label{Apdx_B2}
+D^T = \left. \nabla \right|_s \cdot 
+           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
+\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\
+ 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\
+ {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1 
+^2+\sigma _2 ^2} \hfill \\
+\end{array} }} \right)
+\end{equation}
+or in expended form:
+\begin{multline} \label{Apdx_B3}
+D^T=\frac{1}{e_1\,e_2\,e_3 }\;\left[ {\quad \; \; e_2\,e_3\,A^{lT} \;\left. 
+{\frac{\partial }{\partial i}\left( {\frac{1}{e_1 }\;\left. {\frac{\partial 
+T}{\partial i}} \right|_s -\frac{\sigma _1 }{e_3 }\;\frac{\partial 
+T}{\partial s}} \right)} \right|_s } \right.  \\
++e_1\,e_3\,A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma _2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\
+ \;\;+e_1\,e_2\,A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma _1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. \; \\
+\shoveright{\;\;\left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;\,} \right]} 
+\end{multline}
+
+
+Equation \eqref{Apdx_B2} (or equivalently \eqref{Apdx_B3}) is obtained from \eqref{Apdx_B1} without any additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$, we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A} and use \eqref{Apdx_A_s_slope} and \eqref{Apdx_A_s_chain_rule}. Since no cross horizontal derivate $\partial _i \partial _j $ appears in \eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent. The demonstration can then be done for the ($i$,$z$)~$\to$~($j$,$s$) transformation without any loss of generality:
+
+\begin{equation*}
+D^T=\frac{1}{e_1\,e_2 }\left. {\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)        \qquad  \qquad  \qquad  \qquad \qquad  \qquad \qquad     \\
+\end{equation*}
+\begin{multline*}
+ =\frac{1}{e_1\,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ 
+ \left. { -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right]   
+\shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]} \\ 
+\end{multline*}
+\begin{multline*}
+ =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s -\left. {\frac{e_2 }{e_1 
+}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ 
+ \left. {-e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( 
+{\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\
+ \shoveright{ -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;} \right] }\\ 
+\end{multline*}
+
+Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:
+
+\begin{multline*}
+ =\frac{1}{e_1\,e_2\,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2\,e_3 }{e_1}\,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. 
+-\, {e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
+\qquad -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
+\shoveright{ \left. { +e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial z}} \right)\;\;\;} \right] }\\ 
+\end{multline*}
+\begin{multline*}
+ =\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-\frac{\partial }{\partial 
+i}\left( {e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
+ \left. {+\frac{e_2 \,\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}}  \right|_s -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ 
+-e_2 \,\sigma _1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ 
+\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s} \left( {\frac{\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s}} \right) + \frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
+\end{multline*}
+using the same remark as just above, it becomes:
+\begin{multline*}
+= \frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.\;\;\; \\ 
++\frac{e_1 \,e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma _1 }{\partial s} - \frac {\sigma _1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\ 
+-e_2 \left( {A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma _1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma _1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
+\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] }\\ 
+\end{multline*}
+
+Since the horizontal scale factor do not depend on the vertical coordinate, the last term of the first line and the first term of the last line cancel, while the second line reduces to a single vertical derivative, so it becomes:
+
+\begin{multline*}
+ =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma _1 \,A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
+ \shoveright{ \left. {+\frac{\partial }{\partial s}\left( {-e_2 \,\sigma _1 \,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s +A^{lT}\frac{e_1 \,e_2 }{e_3 }\;\left( {\varepsilon +\sigma _1 ^2} \right)\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
+\end{multline*}
+
+in other words, the horizontal Laplacian operator in the ($i$,$s$)plane takes the following expression :
+
+\begin{equation*}
+D^T = {\frac{1}{e_1\,e_2\,e_3}}
+\left( {{\begin{array}{*{30}c}
+{\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
+{\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
+\end{array}}}\right)
+\cdot \left[ {A^{lT}
+\left( {{\begin{array}{*{30}c}
+ {1} \hfill & {-\sigma_1 } \hfill \\
+ {-\sigma_1} \hfill & {\varepsilon_1^2} \hfill \\
+\end{array} }} \right)
+\cdot 
+\left( {{\begin{array}{*{30}c}
+{\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
+{\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
+\end{array}}}
+\right) \left( T \right)} \right]
+\end{equation*}
+ 
+
+% ================================================================
+% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
+% ================================================================
+\section{Iso/diapycnal 2nd Order Tracer Diffusive Operators}
+\label{Apdx_B_2}
+
+
+The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the  ($i$,$j$,$k$) curvilinear coordinate system in which the equations of the ocean circulation model are formulated, takes the following expression \citep{Redi_JPO82}:
+
+\begin{equation*}
+\textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
+\left[ {{\begin{array}{*{20}c}
+ {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
+ {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
+ {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
+\end{array} }} \right]
+\end{equation*}
+where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions:
+\begin{equation*}
+a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
+\qquad , \qquad
+a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
+\right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
+\end{equation*}
+
+In practice, the isopycnal slopes are generally less than $10^{-2}$ in the ocean, so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
+\begin{equation*}
+{\textbf{A}_{\textbf{I}}} \approx A^{lT}
+\left[ {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill & {-a_1 } \hfill \\
+ 0 \hfill & 1 \hfill & {-a_2 } \hfill \\
+ {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
+\end{array} }} \right]
+\end{equation*}
+
+The resulting isopycnal operator conserves the quantity and dissipates its square. The demonstration of the first property is trivial as \eqref{Apdx_B2} is the divergence of fluxes. Let us demonstrate the second one:
+\begin{equation*}
+\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
+\end{equation*}
+since
+\begin{align*}
+ \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} 
+\right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 
+\frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( 
+{\frac{\partial T}{\partial j}} \right)^2} \right. \\ 
+&\qquad \qquad \qquad \quad
+{ \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial 
+k}+\left( {a_1 ^2+a_2 ^2} \right)\left( {\frac{\partial T}{\partial k}} 
+\right)^2} \right]} \\
+&=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial j}-a_2 \frac{\partial T}{\partial k}} \right)^2} \right]\quad \geq 0
+\end{align*}
+the property becomes obvious. 
+
+The resulting diffusive operator in $z$-coordinates has the following 
+expression :
+\begin{multline*} \label{Apdx_B_ldfiso}
+ D^T=\frac{1}{e_1 e_2 }\left\{ {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2 }{e_1 }\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right.\;\; \\ 
+ \;\left. {\;\;\;+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1 }{e_2 }\frac{\partial T}{\partial j}-a_2 \frac{e_1 }{e_3 }\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\
+\shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]} \\ 
+\end{multline*}
+
+It has to be emphasised that the simplification introduced leads to a decoupling between ($i$,$z$) and ($j$,$z$) planes. The operator has therefore the same expression as \eqref{Apdx_B3}, the diffusive operator obtained for geopotential diffusion in $s$-coordinate. 
+
+% ================================================================
+% Lateral/Vertical Momentum Diffusive Operators
+% ================================================================
+\section{Lateral/Vertical Momentum Diffusive Operators}
+\label{Apdx_B_3}
+
+The second order momentum diffusive operator (Laplacian) in $z$-coordinate is found by applying \eqref{Eq_PE_lap_vector}, the expression of the Laplacian of a vector,  to the horizontal velocity vector : 
+\begin{align*}
+\Delta {\textbf{U}}_h 
+&=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
+\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)    \\
+\\
+&=\left( {{\begin{array}{*{20}c}
+ {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
+ {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
+ {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
+\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
+ {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 
+}\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial 
+u}{\partial k}} \right)} \hfill \\
+ {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 
+}\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta 
+}{\partial i}} \hfill \\
+ {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 
+}{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial 
+j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} 
+\hfill \\
+\end{array} }} \right)
+\\
+\\
+&=\left( {{\begin{array}{*{20}c}
+{\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
+{\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
+0 \\
+\end{array} }} \right)
++\frac{1}{e_3 }
+\left( {{\begin{array}{*{20}c}
+{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
+{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
+{\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\
+\end{array} }} \right)
+\end{align*}
+Using \eqref{Eq_PE_div}, the definition of the horizontal divergence, the third componant of the second vector is obviously zero and thus :
+\begin{equation*}
+\Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
+\end{equation*}
+
+Note that this operator ensures a full separation between the vorticity and horizontal divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian applied on each component in Cartesian coordinate, not on the sphere.
+
+The horizontal/vertical second order (Laplacian type) operator used to diffuse 
+horizontal momentum in $z$-coordinate takes therefore the following expression :
+\begin{equation} \label{Apdx_B_Lap_U}
+ {\textbf{D}}^{\textbf{U}} = 
+ 	  \nabla _h \left( {A^{lm}\;\chi } \right)
+	- \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
+	+ \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
+				\frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ 
+\end{equation}
+that is in expanded form:
+\begin{align*}
+D^{\textbf{U}}_u 
+& = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
+     -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
+     +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
+D^{\textbf{U}}_v   
+& = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
+     +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
+     +\frac{1}{e_3} \frac{\partial v}{\partial k}
+\end{align*}
+
+Note Bene: introducing a rotation in \eqref{Apdx_B_Lap_U} does not lead to any useful expression for the iso/diapycnal Laplacian operator in $z$-coordinate. Similarly, we did not found an expression of practical use for the geopotential horizontal/vertical Laplacian operator in $s$-coordinate. Generally, \eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate system, that is a Laplacian diffusion is applied on momentum along the coordinate directions.
Index: trunk/DOC/TexFiles/Chapters/Annex_C.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Annex_C.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Annex_C.tex	(revision 994)
@@ -0,0 +1,1298 @@
+% ================================================================
+% Chapter Ñ Appendix C : Discrete Invariants of the Equations
+% ================================================================
+\chapter{Discrete Invariants of the Equations}
+\label{Apdx_C}
+\minitoc
+
+%%%  Appendix put in gmcomment as it has not been updated for z* and s coordinate
+I'm writting this appendix. It will be available in a forthcoming release of the documentation
+
+%\gmcomment{
+
+% ================================================================
+% Conservation Properties on Ocean Dynamics
+% ================================================================
+\section{Conservation Properties on Ocean Dynamics}
+\label{Apdx_C.1}
+
+
+First, the boundary condition on the vertical velocity (no flux through the surface and the bottom) is established for the discrete set of momentum equations. Then, it is shown that the non linear terms of the momentum equation are written such that the potential enstrophy of a horizontally non divergent flow is preserved while all the other non-diffusive terms preserve the kinetic energy: the energy is also preserved in practice. In addition, an option is also offer for the vorticity term discretization which provides 
+a total kinetic energy conserving discretization for that term. 
+
+Nota Bene: these properties are established here in the rigid-lid case and for the 2nd order centered scheme. A forthcoming update will be their generalisation to the free surface case
+and higher order scheme.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Bottom Boundary Condition on Vertical Velocity Field
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Bottom Boundary Condition on Vertical Velocity Field}
+\label{Apdx_C.1.1}
+
+
+The discrete set of momentum equations used in rigid lid approximation 
+automatically satisfies the surface and bottom boundary conditions 
+(no flux through the surface and the bottom: $w_{surface} =w_{bottom} =~0$). 
+Indeed, taking the discrete horizontal divergence of the vertical sum of the 
+horizontal momentum equations (Eqs. (II.2.1) and (II.2.2)~) wheighted by the 
+vertical scale factors, it becomes:
+\begin{flalign*}
+\frac{\partial } {\partial t}  \left( \sum\limits_k    \chi    \right)
+\equiv 
+\frac{\partial } {\partial t}  \left( 	w_{surface} -w_{bottom}     \right)&&&\\
+\end{flalign*}
+\begin{flalign*}
+\equiv \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} 
+	\biggl\{ \quad
+	\delta_i 
+		&\left[ 
+		e_{2u}\,H_u 
+			\left( 
+			M_u - M_u - \frac{1} {H_u\,e_{2u}} \delta_j 
+				\left[ \partial_t\, \psi \right] 
+			\right) 
+		\right] &&
+	\biggr. \\
+	\biggl. 
+	+ \delta_j 
+		&\left[ 
+		e_{1v}\,H_v 
+			\left( M_v - M_v - \frac{1} {H_v\,e_{1v}} \delta_i 
+				\left[ \partial_i\, \psi \right] 
+			\right) 
+		\right] 
+	\biggr\}&& \\
+\end{flalign*}
+\begin{flalign*}
+\equiv \frac{1} {e_{1T} \,e_{2T} \,e_{3T}} \;
+	\biggl\{ 
+	- \delta_i 
+		\Bigl[ 
+		\delta_j 
+			\left[ \partial_t \psi  \right] 
+		\Bigr]
+	+ \delta_j 
+		\Bigl[ 
+		\delta_i 
+			\left[ \partial_t \psi  \right] 
+		\Bigr]\; 
+	\biggr\}\;
+	\equiv 0
+	&&&\\
+\end{flalign*}
+
+
+The surface boundary condition associated with the rigid lid approximation ($w_{surface} =0)$ is imposed in the computation of the vertical velocity (II.2.5). Therefore, it turns out to be:
+\begin{equation*}
+\frac{\partial } {\partial t}w_{bottom} \equiv 0
+\end{equation*}
+As the bottom velocity is initially set to zero, it remains zero all the time. Symmetrically, if $w_{bottom} =0$ is used in the computation of the vertical velocity (upward integral of the horizontal divergence), the same computation leads to $w_{surface} =0$ as soon as the surface vertical velocity is initially set to zero.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Coriolis and advection terms: vector invariant form
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Coriolis and advection terms: vector invariant form}
+\label{Apdx_C_vor_zad}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Vorticity Term
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Vorticity Term}
+\label{Apdx_C_vor} 
+
+Potential vorticity is located at $f$-points and defined as: $\zeta / e_{3f}$. The standard discrete formulation of the relative vorticity term obviously conserves potential vorticity (ENS scheme). It also conserves the potential enstrophy for a horizontally non-divergent flow (i.e. $\chi $=0) but not the total kinetic energy. Indeed, using the symmetry or skew symmetry properties of the operators (Eqs \eqref{DOM_mi_adj} and \eqref{DOM_di_adj}), it can be shown that:
+\begin{equation} \label{Apdx_C_1.1}
+\int_D {\zeta / e_3\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {\zeta \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0
+\end{equation}
+where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using 
+\eqref{Eq_dynvor_ens}, the discrete form of the right hand side of \eqref{Apdx_C_1.1} 
+can be transformed as follow:
+\begin{flalign*} 
+&\int_D \zeta / e_3\,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 
+	\left( 
+	\zeta \; \textbf{k} \times  \textbf{U}_h  
+	\right)\;
+	dv
+	&&& \displaybreak[0] \\
+%
+\equiv& \sum\limits_{i,j,k} 
+\frac{\zeta / e_{3f}} {e_{1f}\,e_{2f}\,e_{3f}} 
+	\biggl\{ \quad
+	\delta_{i+1/2} 
+		\left[ 
+		   - \overline {\left( {\zeta / e_{3f}} \right)}^{\,i}\;
+		      \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/ 2} 
+		 \right]  
+	&&  \\ & \qquad \qquad \qquad \;\;
+	- \delta_{j+1/2} 
+		\left[	\;\;\;
+		     \overline {\left( \zeta / e_{3f} \right)}^{\,j}\;
+		     \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 
+		\right] 
+	\;\;\biggr\} \;  e_{1f}\,e_{2f}\,e_{3f}       && \displaybreak[0] \\ 
+%
+\equiv& \sum\limits_{i,j,k} 
+	\biggl\{   \delta_i     \left[   \frac{\zeta} {e_{3f}}   \right] \;
+		     \overline{  \left(   \frac{\zeta} {e_{3f}}   \right)  }^{\,i}\; 
+		     \overline{  \overline{   \left( e_{1u}\,e_{3u}\,u \right)  }  }^{\,i,j+1/2} 
+			+ \delta_j   \left[   \frac{\zeta} {e_{3f}}   \right] \;
+        		\overline{   \left(  \frac{\zeta} {e_{3f}}    \right)  }^{\,j} \; 
+		\overline{\overline {\left( e_{2v}\,e_{3v}\,v \right)}}^{\,i+1/2,j}         \biggr\} 
+		&&&& \displaybreak[0] \\ 
+%
+\equiv& \frac{1} {2} \sum\limits_{i,j,k}  
+	\biggl\{	\delta_i    \Bigl[    \left(  \frac{\zeta} {e_{3f}} \right)^2   \Bigr]\;
+			\overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 
+		      + \delta_j 	\Bigl[    \left( \zeta / e_{3f} \right)^2     \Bigr]\; 
+			\overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 
+	\biggr\} 
+	&& \displaybreak[0] \\ 
+%
+\equiv& - \frac{1} {2} \sum\limits_{i,j,k} 	\left(  \frac{\zeta} {e_{3f}} \right)^2\;
+	\biggl\{    \delta_{i+1/2} 
+	      \left[   \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2}    \right]  
+	            + \delta_{j+1/2}
+		\left[   \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j}     \right]  
+	\biggr\}    && \\ 
+\end{flalign*}
+Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} 
+\left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, 
+and introducing the horizontal divergence $\chi $, it becomes:
+\begin{align*}
+\equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left(  \frac{\zeta} {e_{3f}} \right)^2 \; \overline{\overline{ e_{1T}\,e_{2T}\,e_{3T}\, \chi}}^{\,i+1/2,j+1/2} \;\;\equiv 0 
+\qquad \qquad \qquad \qquad \qquad \qquad \qquad &&&&\\
+\end{align*}
+
+Note that the demonstration is done here for the relative potential 
+vorticity but it still hold for the planetary ($f/e_3$) and the total 
+potential vorticity $((\zeta +f) /e_3 )$. Another formulation of 
+the two components of the vorticity term is optionally offered (ENE scheme) :
+\begin{equation*}
+	- \zeta \;{\textbf{k}}\times {\textbf {U}}_h 
+\equiv 
+	\left( {{
+	\begin{array} {*{20}c}
+		+ \frac{1} {e_{1u}} \; 
+		\overline {\left( \zeta / e_{3f} 	  \right)	
+		\overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} 
+		\hfill \\
+		- \frac{1} {e_{2v}} \; 
+		\overline {\left( \zeta / e_{3f}  	  \right)
+		\overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} 
+		\hfill \\
+	\end{array}} } 
+	\right)
+\end{equation*}
+
+This formulation does not conserve the enstrophy but the total kinetic 
+energy. It is also possible to mix the two formulations in order to conserve 
+enstrophy on the relative vorticity term and energy on the Coriolis term.
+\begin{flalign*}
+&\int\limits_D - \textbf{U}_h \cdot   \left(  \zeta \;\textbf{k} \times \textbf{U}_h  \right)  \;  dv &&  \\
+\equiv& \sum\limits_{i,j,k} 	\biggl\{ 	
+	   \overline {\left(  \frac{\zeta} {e_{3f}}      \right) 
+	     \overline {\left( e_{1v}e_{3v}v \right)}^{\,i+1/2}} ^{\,j} \, e_{2u}e_{3u}u 
+	- \overline {\left(  \frac{\zeta} {e_{3f}}       \right) 
+	    \overline {\left( e_{2u}e_{3u}u \right)}^{\,j+1/2}} ^{\,i} \, e_{1v}e_{3v}v \;
+	                                \biggr\}     
+\\
+\equiv& \sum\limits_{i,j,k}  \frac{\zeta} {e_{3f}}
+	\biggl\{  \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2}\;
+	    	    \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2}        
+		  - \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2}\;
+	            \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2}
+	\biggr\}
+	\equiv 0
+\end{flalign*}
+
+
+% -------------------------------------------------------------------------------------------------------------
+%       Gradient of Kinetic Energy / Vertical Advection
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Gradient of Kinetic Energy / Vertical Advection}
+\label{Apdx_C_zad} 
+
+The change of Kinetic Energy (KE) due to the vertical advection is exactly 
+balanced by the change of KE due to the horizontal gradient of KE~:
+\begin{equation*}
+      \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv
+ = - \int_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv
+\end{equation*}
+Indeed, using successively \eqref{DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator) and the incompressibility, then again \eqref{DOM_di_adj}, then 
+the commutativity of operators $\overline {\,\cdot \,}$ and $\delta$, and finally \eqref{DOM_mi_adj} ($i.e.$ the  symmetry property of the $\overline {\,\cdot \,}$ operator) applied in the horizontal and vertical direction, it becomes:
+\begin{flalign*}
+&\int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv	&&&\\
+\equiv& \frac{1}{2} \sum\limits_{i,j,k} 
+	\biggl\{	
+	\frac{1} {e_{1u}}  \delta_{i+1/2} 
+	\left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}    \right]  u\,e_{1u}e_{2u}e_{3u}  
+     + \frac{1} {e_{2v}}  \delta_{j+1/2} 
+	\left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right]  v\,e_{1v}e_{2v}e_{3v} 
+	\biggr\} 
+	&&& \displaybreak[0] \\ 
+%
+\equiv&  \frac{1}{2} \sum\limits_{i,j,k} 
+	\left(   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right)\;
+	\delta_k \left[  e_{1T}\,e_{2T} \,w   \right]
+%
+\;\; \equiv -\frac{1}{2} \sum\limits_{i,j,k}  \delta_{k+1/2} 
+	\left[ 
+	   \overline{ u^2}^{\,i} 
+	+ \overline{ v^2}^{\,j} 
+	\right] \;
+	e_{1v}\,e_{2v}\,w 
+	&&& \displaybreak[0]\\
+%
+\equiv &\frac{1} {2} \sum\limits_{i,j,k} 
+	\left(    \overline {\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 
+		+ \overline {\delta_{k+1/2} \left[ v^2 \right]}^{\,j}    \right) \; e_{1T}\,e_{2T} \,w 
+	&& \displaybreak[0] \\
+%	
+\equiv &\frac{1} {2} \sum\limits_{i,j,k} 
+	\biggl\{  \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\;2 	
+			\overline {u}^{\,k+1/2}\; \delta_{k+1/2}         \left[ u \right]     %&&&  \\
+	 + \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2}\;2 	\overline {v}^{\,k+1/2}\; \delta_{k+1/2} 			\left[ v \right]  \;
+	\biggr\} 
+	&&\displaybreak[0] \\ 
+%
+\equiv& -\sum\limits_{i,j,k}  
+	\biggl\{
+	\quad \frac{1} {b_u } \;
+	\overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\,\delta_{k+1/2}
+		\left[ u \right] 
+		       \Bigr\} }^{\,k} \;u\;e_{1u}\,e_{2u}\,e_{3u}  
+	&& \\
+	&\qquad \quad\; + \frac{1} {b_v } \;
+	\overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2} \delta_{k+1/2}
+		 \left[ v \right] 
+		 	\Bigr\} }^{\,k} \;v\;e_{1v}\,e_{2v}\,e_{3v}  \;
+	\biggr\} 
+	&& \\ 
+\equiv& -\int\limits_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv	&&&\\
+\end{flalign*}
+
+The main point here is that the satisfaction of this property links the choice of the discrete formulation of vertical advection and of horizontal gradient of KE. Choosing one imposes the other. For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. This leads to the following expression for the vertical advection:
+\begin{equation*}
+\frac{1} {e_3 }\; w\; \frac{\partial \textbf{U}_h } {\partial k}
+\equiv \left( {{\begin{array} {*{20}c}
+\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \;
+\overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2} 
+\left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k}  \hfill \\
+\frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \;
+\overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2}
+\left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\
+\end{array}} } \right)
+\end{equation*}
+a formulation that requires a additional horizontal mean compare to the one used in NEMO. Nine velocity points have to be used instead of 3. This is the reason why it has not been choosen.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Coriolis and advection terms: flux form
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Coriolis and advection terms: flux form}
+\label{Apdx_C.1.3}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Coriolis plus ``metric'' Term
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Coriolis plus ``metric'' Term}
+\label{Apdx_C.1.3.1} 
+
+In flux from the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the ``metric'' term. This altered Coriolis parameter is discretised at F-point. It is given by: 
+\begin{equation*}
+f+\frac{1} {e_1 e_2 } 
+\left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\;
+\equiv \;
+f+\frac{1} {e_{1f}\,e_{2f}} 
+\left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 
+-\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u}  \right] \right)
+\end{equation*}
+
+The ENE scheme is then applied to obtain the vorticity term in flux form. It therefore conserves the total KE. The demonstration is same as for the vorticity term in vector invariant form (\S\ref{Apdx_C_vor}).
+
+% -------------------------------------------------------------------------------------------------------------
+%       Flux form advection
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Flux form advection}
+\label{Apdx_C.1.3.2} 
+
+The flux form operator of the momentum advection is evaluated using a centered second order finite difference scheme. Because of the flux form, the discrete operator does not contribute to the global budget of linear momentum. Because of the centered second order scheme, it conserves the horizontal kinetic energy, that is :
+
+\begin{equation} \label{Apdx_C_I.3.10}
+\int_D \textbf{U}_h \cdot 
+\left( {{\begin{array} {*{20}c}
+\nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
+\nabla \cdot \left( \textbf{U}\,v \right) \hfill \\
+\end{array}} } \right)\;dv =\;0
+\end{equation}
+
+Let us demonstrate this property for the first term of the scalar product (i.e. considering just the the terms associated with the i-component of the advection):
+\begin{flalign*}
+&\int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv    &&&\\
+%
+\equiv& \sum\limits_{i,j,k} 
+\biggl\{    \frac{1} {e_{1u}\, e_{2u}\,e_{3u}}    \biggl(   
+      \delta_{i+1/2}  \left[   \overline {e_{2u}\,e_{3u}\,u}^{\,i}      \;\overline u^{\,i}          \right]   
+   + \delta_j           \left[   \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right] 
+		&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad
+   + \delta_k          \left[   \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;\overline u^{\,k+1/2} \right]
+			\biggr)   \;   \biggr\} \, e_{1u}\,e_{2u}\,e_{3u} \;u 
+		&&& \displaybreak[0] \\ 
+%
+\equiv& \sum\limits_{i,j,k} 
+	\biggl\{ 
+      \delta_{i+1/2} \left[   \overline {e_{2u}\,e_{3u}\,u}^{\,i}\;  \overline u^{\,i}  \right]
+   + \delta_j          \left[   \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right]
+		&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+   + \delta_k         \left[   \overline {e_{1w}\,e_{2w}\,w}^{\,i+12}\;\overline u^{\,k+1/2}  \right]
+   	\; \biggr\} \; u    &&& \displaybreak[0] \\
+%
+\equiv& - \sum\limits_{i,j,k}
+	\biggl\{ 
+	   \overline {e_{2u}\,e_{3u}\,u}^{\,i}\;			\overline u^{\,i} 		\delta_i 
+		\left[ u \right] 
+        + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;	\overline u^{\,j+1/2} 	\delta_{j+1/2} 
+		\left[ u \right] 
+		&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+       + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;	\overline u^{\,k+1/2}	\delta_{k+1/2} 	\left[ u \right]     \biggr\}     && \displaybreak[0] \\
+%
+\equiv& - \sum\limits_{i,j,k}
+	\biggl\{ 
+	     \overline {e_{2u}\,e_{3u}\,u}^{\,i} 			\delta_i 		\left[ u^2 \right] 
+	 + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2} 		\delta_{j+/2} 	\left[ u^2 \right]
+	 + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2} 	\delta_{k+1/2} 	\left[ u^2 \right] 
+	\biggr\} 
+	&& \displaybreak[0] \\
+%
+\equiv& \sum\limits_{i,j,k}
+	\bigg\{ 
+	   e_{2u}\,e_{3u}\,u\;		\delta_{i+1/2} 	   \left[ \overline {u^2}^{\,i} \right]
+        + e_{1u}\,e_{3u}\,v\;	\delta_{j+1/2} 	\; \left[ \overline {u^2}^{\,i} \right]
+	 + e_{1w}\,e_{2w}\,w\;	\delta_{k+1/2}  	   \left[ \overline {u^2}^{\,i} \right] 
+	\biggr\} 
+	&& \displaybreak[0] \\
+%
+\equiv& \sum\limits_{i,j,k}
+\overline {u^2}^{\,i} 
+	\biggl\{ 
+	   \delta_{i+1/2} 	\left[ e_{2u}\,e_{3u}\,u  \right]
+	+ \delta_{j+1/2} 	\left[ e_{1u}\,e_{3u}\,v  \right]
+	+ \delta_{k+1/2} 	\left[ e_{1w}\,e_{2w}\,w \right] 
+	\biggr\}  \;\;  \equiv 0
+	&&& \\
+\end{flalign*}
+
+When the UBS scheme is used to evaluate the flux form momentum advection, the discrete operator does not contribute to the global budget of linear momentum (flux form). The horizontal kinetic energy is not conserved, but forced to decrease (the scheme is diffusive). 
+
+% -------------------------------------------------------------------------------------------------------------
+%       Hydrostatic Pressure Gradient Term
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Hydrostatic Pressure Gradient Term}
+\label{Apdx_C.1.4}
+
+
+A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero. In $z$-coordinate, this properties is satisfied locally on a C-grid with 2nd order finite differences (property \eqref{Eq_DOM_curl_grad}). When the equation of state is linear ($i.e.$ when an advective-diffusive equation for density can be derived from those of temperature and salinity) the change of KE due to the work of pressure forces is balanced by the change of potential energy due to buoyancy forces: 
+\begin{equation*}
+\int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv 
+= \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+\end{equation*}
+
+This property can be satisfied in discrete sense for both $z$- and $s$-coordinates. Indeed, defining the depth of a $T$-point, $z_T$ defined as the sum of the vertical scale factors at $w$-points starting from the surface, the workof pressure forces can be written as:
+\begin{flalign*}
+&\int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv   &&& \\
+\equiv& \sum\limits_{i,j,k} \biggl\{ \;  - \frac{1} {\rho_o e_{1u}}   \Bigl( 
+\delta_{i+1/2} \left[ p^h \right] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] 
+		  			\Bigr)  \; u\;e_{1u}\,e_{2u}\,e_{3u}  
+	&&  \\ & \qquad \qquad
+						      	      - \frac{1} {\rho_o e_{2v}}    \Bigl( 
+\delta_{j+1/2} \left[ p^h \right] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2}  \left[ z_T \right] 
+					\Bigr)  \; v\;e_{1v}\,e_{2v}\,e_{3v} \;
+	\biggr\}   && \\ 
+\end{flalign*}
+
+Using  \eqref{DOM_di_adj}, $i.e.$ the skew symmetry property of the $\delta$ operator, \eqref{Eq_wzv}, the continuity equation), and \eqref{Eq_dynhpg_sco}, the hydrostatic 
+equation in $s$-coordinate, it turns out to be: 
+\begin{flalign*} 
+\equiv& \frac{1} {\rho_o} \sum\limits_{i,j,k}    \biggl\{ 
+		e_{2u}\,e_{3u} \;u\;g\;	\overline \rho^{\,i+1/2}\;\delta_{i+1/2}[ z_T]    
+	+ 	e_{1v}\,e_{3v} \;v\;g\;	\overline \rho^{\,j+1/2}\;\delta_{j+1/2}[ z_T]     
+&& \\  & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
+	+\Bigl(  \delta_i[ e_{2u}\,e_{3u}\,u] + \delta_j [ e_{1v}\,e_{3v}\,v]  \Bigr)\;p^h \biggr\}  &&\\
+%
+\equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k}
+	\biggl\{ 
+		 e_{2u}\,e_{3u} \;u\;g\;	\overline \rho^{\,i+1/2} \delta_{i+1/2} \left[ z_T \right]
+	+ 	e_{1v}\,e_{3v} \;v\;g\;	\overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right] 
+	&&&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
+	 - \delta_k \left[ e_{1w} e_{2w}\,w \right]\;p^h   \biggr\}   &&&\\ 
+%
+\equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k}
+	\biggl\{ 
+		e_{2u}\,e_{3u} \;u\;g\;	\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right]
+	+ 	e_{1v}\,e_{3v} \;v\;g\;	\overline \rho^{\,j+1/2} \;\delta_{j+1/2} \left[ z_T \right] 
+	&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
+	+	 e_{1w} e_{2w} \;w\;\delta_{k+1/2} \left[ p_h \right] 
+	\biggr\}  &&&\\ 
+%
+\equiv& \frac{g} {\rho_o}  \sum\limits_{i,j,k}
+	\biggl\{ 
+		e_{2u}\,e_{3u} \;u\;	\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right]
+	+	e_{1v}\,e_{3v} \;v\;	\overline \rho^{\,j+1/2}\;\delta_{j+1/2} \left[ z_T \right] 	
+	&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
+	- 	e_{1w} e_{2w} \;w\;e_{3w} \overline \rho^{\,k+1/2} 
+	\biggr\}   &&&\\ 
+\end{flalign*}
+noting that by definition of $z_T$, $\delta_{k+1/2} \left[ z_T \right] \equiv - e_{3w} $, thus:
+\begin{multline*}
+\equiv \frac{g} {\rho_o}  \sum\limits_{i,j,k}
+	\biggl\{ 
+		e_{2u}\,e_{3u} \;u\;	\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right]
+	+ 	e_{1v}\,e_{3v} \;v\;	\overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right]
+	\biggr. \\ 
+\shoveright{
+	\biggl. 
+	+ 	e_{1w} e_{2w} \;w\;	\overline \rho^{\,k+1/2}\;\delta_{k+1/2} \left[ z_T \right] 
+	\biggr\} } \\ 
+\end{multline*}
+Using \eqref{DOM_di_adj}, it becomes:
+\begin{flalign*}
+\equiv& - \frac{g} {\rho_o} \sum\limits_{i,j,k} z_T 
+	\biggl\{ 
+		\delta_i 	\left[ e_{2u}\,e_{3u}\,u\;	\overline \rho^{\,i+1/2} 	\right]
+	+ 	\delta_j 	\left[ e_{1v}\,e_{3v}\,v\;	\overline \rho^{\,j+1/2} 	\right]
+	+ 	\delta_k 	\left[ e_{1w} e_{2w}\,w\;	\overline \rho^{\,k+1/2} 	\right] 
+	\biggr\} 
+	&&& \\
+%
+\equiv& -\int_D \nabla \cdot \left( \rho \, \textbf{U} \right)\;g\;z\;\;dv 	&&& \\
+\end{flalign*}
+
+Note that this property strongly constraints the discrete expression of both 
+the depth of $T-$points and of the term added to the pressure gradient in 
+$s$-coordinate. Nevertheless, it is almost never satisfied as a linear equation of state 
+is rarely used.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Surface Pressure Gradient Term
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Surface Pressure Gradient Term}
+\label{Apdx_C.1.5}
+
+
+The surface pressure gradient has no contribution to the evolution of the vorticity. This property is trivially satisfied locally as the equation verified by $\psi $ has been derived from the discrete formulation of the momentum equation and of the curl. But it has to be noticed that since the elliptic equation verified by $\psi $ is solved numerically by an iterative solver (preconditioned conjugate gradient or successive over relaxation), the 
+property is only satisfied at the precision required on the solver used.
+
+With the rigid-lid approximation, the change of KE due to the work of surface pressure forces is exactly zero. This is satisfied in discrete form, at the precision required on the elliptic solver used to solve this equation. This can be demonstrated as follows:
+\begin{flalign*}
+\int\limits_D  - \frac{1} {\rho_o} \nabla_h \left( p_s \right) \cdot \textbf{U}_h \;dv%	&&& \\
+%
+&\equiv \sum\limits_{i,j,k}   \biggl\{ 	\;
+	 \left(  - M_u - \frac{1} {H_u \,e_{2u}}  \delta_j 	\left[ \partial_t \psi  \right]   \right)\;
+	 u\;e_{1u}\,e_{2u}\,e_{3u}  
+&&&\\&  \qquad \;\;\,
+      + \left(  - M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi \right]     \right)\;
+	  v\;e_{1v}\,e_{2v}\,e_{3v} 	\; \biggr\}      
+&&&\\
+\\
+%
+&\equiv \sum\limits_{i,j}  \Biggl\{   \;
+	\biggl( - M_u - \frac{1} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi  \right]   \biggr)
+	\biggl( \sum\limits_k  u\;e_{3u}   \biggr)\;  e_{1u}\,e_{2u} 
+&&&\\&  \qquad \;\;\,
+	+ \biggl( - M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi  \right]   \biggr)
+	   \biggl(   \sum\limits_k   v\;e_{3v}   \biggr)\;   e_{1v}\,e_{2v} \;   \Biggr\} 
+	&& \\ 
+%
+\intertext{using the relation between \textit{$\psi $} and the vertically sum of the velocity, it becomes:}
+%
+&\equiv \sum\limits_{i,j} 
+	\biggl\{  \;	
+		\left( \;\;\,
+		M_u + \frac{1} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi \right] 
+		\right)\;
+		e_{1u} \,\delta_j 
+			\left[ \partial_t \psi  \right] 
+	&& \\ &  \qquad \;\;\,
+		+ \left( 
+		- M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi \right] 
+		\right)\;
+		e_{2v} \,\delta_i \left[ \partial_t \psi \right] 	\;
+	\biggr\} 
+	&& \\ 
+%
+\intertext{applying the adjoint of the $\delta$ operator, it is now:}
+%
+&\equiv \sum\limits_{i,j}  - \partial_t \psi \;
+	\biggl\{ 	\;
+	  \delta_{j+1/2} \left[ e_{1u} M_u \right] 
+	  - \delta_{i+1/2} \left[ e_{1v} M_v \right] 
+	&& \\ &  \qquad \;\;\,
+	+ \delta_{i+1/2} 
+		\left[ \frac{e_{2v}} {H_v \,e_{2v}}  \delta_i \left[ \partial_t \psi \right] 
+		\right] 
+	+ \delta_{j+1/2} 
+	 	 \left[ \frac{e_{1u}} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi \right] 
+		 \right]  
+	\biggr\}   &&&\\
+	&\equiv 0                   && \\ 
+\end{flalign*}
+
+The last equality is obtained using \eqref{Eq_dynspg_rl}, the discrete barotropic streamfunction time evolution equation. By the way, this shows that \eqref{Eq_dynspg_rl} is the only way do compute the streamfunction, otherwise the surface pressure forces will work. Nevertheless, since the elliptic equation verified by $\psi $ is solved numerically by an iterative solver, the property is only satisfied at the precision required on the solver.
+
+% ================================================================
+% Conservation Properties on Tracers
+% ================================================================
+\section{Conservation Properties on Tracers}
+\label{Apdx_C.2}
+
+
+All the numerical schemes used in NEMO are written such that the tracer content is conserved by the internal dynamics and physics (equations in flux form). For advection, only the CEN2 scheme ($i.e.$ 2nd order finite different scheme) conserves the global variance of tracer. Nevertheless the other schemes ensure that the global variance decreases ($i.e.$ they are at least slightly diffusive). For diffusion, all the schemes ensure the decrease of the total tracer variance, but the iso-neutral operator. There is generally no strict conservation of mass, as the equation of state is non linear with respect to $T$ and $S$. In practice, the mass is conserved with a very good accuracy. 
+% -------------------------------------------------------------------------------------------------------------
+%       Advection Term
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Advection Term}
+\label{Apdx_C.2.1}
+
+Whatever the advection scheme considered it conserves of the tracer content as all the scheme are written in flux form. Let $\tau$ be the tracer interpolated at velocity point (whatever the interpolation is). The conservation of the tracer content is obtained as follows: 
+\begin{flalign*}
+&\int_D \nabla \cdot \left( T \textbf{U} \right)\;dv    &&&\\
+&\equiv  \sum\limits_{i,j,k}    \biggl\{
+	 \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} 
+	 \left(  \delta_i    \left[   e_{2u}\,e_{3u}\; u \;\tau_u   \right]
+	        + \delta_j    \left[   e_{1v}\,e_{3v}\; v  \;\tau_v   \right] \right) 
+&&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+	+ \frac{1} {e_{3T}} \delta_k \left[ w\;\tau \right]    \biggl\}  e_{1T}\,e_{2T}\,e_{3T} &&&\\
+%
+&\equiv  \sum\limits_{i,j,k}     \left\{
+	   \delta_i  \left[ e_{2u}\,e_{3u}  \,\overline T^{\,i+1/2}\,u \right]
+       	+ \delta_j  \left[ e_{1v}\,e_{3v}  \,\overline T^{\,j+1/2}\,v \right] 
+ 	+ \delta_k \left[ e_{1T}\,e_{2T} \,\overline T^{\,k+1/2}\,w \right] \right\} 
+ 	 && \\
+&\equiv 0 &&&
+\end{flalign*}
+
+The conservation of the variance of tracer can be achieved only with the CEN2 scheme. It can be demonstarted as follows:
+\begin{flalign*}
+&\int\limits_D T\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\
+\equiv& \sum\limits_{i,j,k} T\;
+	\left\{
+	   \delta_i  \left[ e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u \right]
+	+ \delta_j  \left[ e_{1v}\,e_{3v} \overline T^{\,j+1/2}\,v \right]
+	+ \delta_k \left[ e_{1T}\,e_{2T} \overline T^{\,k+1/2}w \right]
+	\right\} 
+	&& \\
+\equiv& \sum\limits_{i,j,k} 
+	\left\{
+	-           e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u\,\delta_{i+1/2}  \left[ T \right] \right.
+	-           e_{1v}\,e_{3v}  \overline T^{\,j+1/2}\,v\;\delta_{j+1/2}  \left[ T \right]
+&&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \quad \;
+	- \left. e_{1T}\,e_{2T} \overline T^{\,k+1/2}w\;\delta_{k+1/2} \left[ T \right]
+	\right\} 
+	&&\\
+\equiv&  -\frac{1} {2}  \sum\limits_{i,j,k}
+	\Bigl\{
+	   e_{2u}\,e_{3u} \;  u\;\delta_{i+1/2} \left[ T^2 \right]
+	+ e_{1v}\,e_{3v} \;  v\;\delta_{j+1/2}  \left[ T^2 \right]
+	+ e_{1T}\,e_{2T} \;w\;\delta_{k+1/2} \left[ T^2 \right]
+	\Bigr\} 
+	&& \\ 
+\equiv& \frac{1} {2}  \sum\limits_{i,j,k} T^2
+	\Bigl\{
+	   \delta_i  \left[ e_{2u}\,e_{3u}\,u \right]
+	+ \delta_j  \left[ e_{1v}\,e_{3v}\,v \right]
+	+ \delta_k \left[ e_{1T}\,e_{2T}\,w \right]
+	\Bigr\} 
+\quad \equiv 0 &&&
+\end{flalign*}
+
+
+% ================================================================
+% Conservation Properties on Lateral Momentum Physics
+% ================================================================
+\section{Conservation Properties on Lateral Momentum Physics}
+\label{Apdx_C.3}
+
+
+The discrete formulation of the horizontal diffusion of momentum ensures the 
+conservation of potential vorticity and horizontal divergence and the 
+dissipation of the square of these quantities (i.e. enstrophy and the 
+variance of the horizontal divergence) as well as the dissipation of the 
+horizontal kinetic energy. In particular, when the eddy coefficients are 
+horizontally uniform, it ensures a complete separation of vorticity and 
+horizontal divergence fields, so that diffusion (dissipation) of vorticity 
+(enstrophy) does not generate horizontal divergence (variance of the 
+horizontal divergence) and \textit{vice versa}. 
+
+These properties of the horizontal diffusive operator are a direct 
+consequence of properties \eqref{Eq_DOM_curl_grad} and \eqref{Eq_DOM_div_curl}. When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken, the term associated to the horizontal gradient of the divergence is zero locally. 
+
+% -------------------------------------------------------------------------------------------------------------
+%       Conservation of Potential Vorticity
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation of Potential Vorticity}
+\label{Apdx_C.3.1}
+
+The lateral momentum diffusion term conserves the potential vorticity :
+\begin{flalign*}
+&\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
+	\Bigl[ \nabla_h 
+		\left( A^{\,lm}\;\chi  \right)
+	- \nabla_h \times
+		\left( A^{\,lm}\;\zeta \; \textbf{k} \right)
+	\Bigr]\;dv  = 0
+\end{flalign*}
+%%%%%%%%%%  recheck here....  (gm)
+\begin{flalign*}
+= \int \limits_D  -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
+	\Bigl[ \nabla_h \times 
+		\left( A^{\,lm}\;\zeta \; \textbf{k} \right) 
+	\Bigr]\;dv &&& \\ 
+\end{flalign*}
+\begin{flalign*}
+\equiv& \sum\limits_{i,j}
+	\left\{
+	\delta_{i+1/2} 
+	\left[ 
+	\frac {e_{2v}} {e_{1v}\,e_{3v}}  \delta_i
+		\left[ A_f^{\,lm} e_{3f} \zeta  \right]
+	 \right]
+ 	+ \delta_{j+1/2} 
+	\left[ 
+	\frac {e_{1u}} {e_{2u}\,e_{3u}} \delta_j 
+		\left[ A_f^{\,lm} e_{3f} \zeta  \right]
+	\right]
+	\right\} 
+	&& \\ 
+%
+\intertext{Using \eqref{DOM_di_adj}, it follows:}
+%
+\equiv& \sum\limits_{i,j,k} 
+	-\,\left\{
+	   \frac{e_{2v}} {e_{1v}\,e_{3v}}  \delta_i
+		\left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_i \left[ 1\right]
+	+ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j 
+		\left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_j \left[ 1\right]
+	\right\} \quad \equiv 0 
+	&& \\ 
+\end{flalign*}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Dissipation of Horizontal Kinetic Energy
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Dissipation of Horizontal Kinetic Energy}
+\label{Apdx_C.3.2}
+
+
+The lateral momentum diffusion term dissipates the horizontal kinetic energy:
+%\begin{flalign*}
+\begin{equation*}
+\begin{split}
+\int_D \textbf{U}_h \cdot 
+	\left[ \nabla_h 		\right.   &     \left.       \left( A^{\,lm}\;\chi \right)      
+	- \nabla_h \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)     \right] \; dv    \\
+\\  %%%
+\equiv& \sum\limits_{i,j,k} 
+	\left\{
+	  \frac{1} {e_{1u}}               \delta_{i+1/2} \left[  A_T^{\,lm}          \chi     \right]
+	- \frac{1} {e_{2u}\,e_{3u}}  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta   \right]
+	\right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u     \\
+&\;\; + 	\left\{
+	   \frac{1} {e_{2u}}             \delta_{j+1/2}	\left[ A_T^{\,lm}          \chi    \right] 
+	+ \frac{1} {e_{1v}\,e_{3v}} \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right] 
+	\right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v     \qquad \\ 
+\\  %%%
+\equiv& \sum\limits_{i,j,k} 
+ 	\Bigl\{
+	  e_{2u}\,e_{3u} \;u\;  \delta_{i+1/2} \left[ A_T^{\,lm}           \chi    \right]
+	- e_{1u}             \;u\;  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta  \right]
+	 \Bigl\} 
+	 \\ 
+&\;\; + \Bigl\{
+	   e_{1v}\,e_{3v} \;v\;  \delta_{j+1/2}  \left[ A_T^{\,lm}           \chi    \right]
+	+ e_{2v}             \;v\;  \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right]
+	\Bigl\}      \\ 
+\\  %%%
+\equiv& \sum\limits_{i,j,k} 
+	- \Bigl(
+	  \delta_i   \left[  e_{2u}\,e_{3u} \;u  \right]
+	+ \delta_j  \left[  e_{1v}\,e_{3v}  \;v  \right] 
+	     \Bigr) \;  A_T^{\,lm} \chi   \\ 
+&\;\; - \Bigl(
+	  \delta_{i+1/2}  \left[  e_{2v}  \;v  \right]
+	- \delta_{j+1/2}  \left[  e_{1u} \;u  \right] 
+	     \Bigr)\;  A_f^{\,lm} e_{3f} \zeta      \\ 
+\\  %%%
+\equiv& \sum\limits_{i,j,k} 
+	- A_T^{\,lm}  \,\chi^2   \;e_{1T}\,e_{2T}\,e_{3T}
+	- A_f ^{\,lm}  \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f}  
+\quad \leq 0       \\
+\end{split}
+\end{equation*}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Dissipation of Enstrophy
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Dissipation of Enstrophy}
+\label{Apdx_C.3.3}
+
+
+The lateral momentum diffusion term dissipates the enstrophy when the eddy 
+coefficients are horizontally uniform:
+\begin{flalign*}
+&\int\limits_D  \zeta \; \textbf{k} \cdot \nabla \times 
+	\left[ 
+	  \nabla_h 
+		\left( A^{\,lm}\;\chi  \right)
+	-\nabla_h \times 
+		\left( A^{\,lm}\;\zeta \; \textbf{k} \right)
+	\right]\;dv &&&\\
+&= A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 
+ 	\left[ 
+	\nabla_h \times 
+		\left( \zeta \; \textbf{k} \right)
+	\right]\;dv &&&\displaybreak[0]\\
+&\equiv A^{\,lm} \sum\limits_{i,j,k}  \zeta \;e_{3f} 
+	\left\{
+	  \delta_{i+1/2} 
+	\left[ 
+	\frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i 
+		\left[ e_{3f} \zeta  \right]
+	\right]
+	+ \delta_{j+1/2} 
+	\left[
+	\frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j 
+		\left[ e_{3f} \zeta  \right]
+	\right]
+	\right\} 
+	&&&\\ 
+%
+\intertext{Using \eqref{DOM_di_adj}, it follows:}
+%
+&\equiv  - A^{\,lm} \sum\limits_{i,j,k} 
+	\left\{
+	  \left( 
+	  \frac{1} {e_{1v}\,e_{3v}}  \delta_i 
+		\left[ e_{3f} \zeta  \right] 
+	  \right)^2   e_{1v}\,e_{2v}\,e_{3v}
+	+ \left(
+	  \frac{1} {e_{2u}\,e_{3u}}  \delta_j 
+	  	\left[ e_{3f} \zeta  \right]
+	  \right)^2   e_{1u}\,e_{2u}\,e_{3u}
+	  \right\}      &&&\\
+& \leq \;0       &&&\\ 
+\end{flalign*}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Conservation of Horizontal Divergence
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation of Horizontal Divergence}
+\label{Apdx_C.3.4}
+
+When the horizontal divergence of the horizontal diffusion of momentum 
+(discrete sense) is taken, the term associated to the vertical curl of the 
+vorticity is zero locally, due to (II.1.8). The resulting term conserves the 
+$\chi$ and dissipates $\chi^2$ when the eddy coefficients are 
+horizontally uniform.
+\begin{flalign*}
+& \int\limits_D  \nabla_h \cdot 
+	\Bigl[ 
+	  \nabla_h 
+		\left( A^{\,lm}\;\chi \right)
+	- \nabla_h \times 
+		\left( A^{\,lm}\;\zeta \;\textbf{k} \right)
+	\Bigr]
+	dv
+= \int\limits_D  \nabla_h \cdot \nabla_h 
+	\left( A^{\,lm}\;\chi  \right)
+	dv 
+&&&\\
+&\equiv \sum\limits_{i,j,k} 
+	\left\{ 
+	  \delta_i
+		\left[ 
+		A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} 
+			\left[ \chi \right] 
+		\right]
+	+ \delta_j 
+		\left[ 
+		A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}}  \delta_{j+1/2} 
+			\left[ \chi \right] 
+		\right] 
+	\right\}
+	&&&\\ 
+%
+\intertext{Using \eqref{DOM_di_adj}, it follows:}
+%
+&\equiv \sum\limits_{i,j,k} 
+	- \left\{
+	\frac{e_{2u}\,e_{3u}} {e_{1u}}  A_u^{\,lm} \delta_{i+1/2} 
+		\left[ \chi \right]
+	\delta_{i+1/2} 
+		\left[ 1 \right] 
+	+ \frac{e_{1v}\,e_{3v}} {e_{2v}}  A_v^{\,lm} \delta_{j+1/2} 
+		\left[ \chi \right]
+	\delta_{j+1/2} 
+		\left[ 1 \right]
+	\right\} \;
+	\equiv 0 
+	&&& \\ 
+\end{flalign*}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Dissipation of Horizontal Divergence Variance
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Dissipation of Horizontal Divergence Variance}
+\label{Apdx_C.3.5}
+
+\begin{flalign*}
+&\int\limits_D \chi \;\nabla_h \cdot 
+	\left[    \nabla_h              \left( A^{\,lm}\;\chi                    \right)
+	        - \nabla_h   \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \right]\;  dv
+ = A^{\,lm}   \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\;  dv    &&&\\ 
+%
+&\equiv A^{\,lm}  \sum\limits_{i,j,k}  \frac{1} {e_{1T}\,e_{2T}\,e_{3T}}  \chi 
+	\left\{
+	   \delta_i  \left[   \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]   \right]
+	+ \delta_j  \left[   \frac{e_{1v}\,e_{3v}} {e_{2v}}   \delta_{j+1/2} \left[ \chi \right]   \right]
+	\right\} \;   e_{1T}\,e_{2T}\,e_{3T}    &&&\\ 
+%
+\intertext{Using \eqref{DOM_di_adj}, it turns out to be:}
+%
+&\equiv - A^{\,lm} \sum\limits_{i,j,k}
+	\left\{ 
+   \left(  \frac{1} {e_{1u}}  \delta_{i+1/2}  \left[ \chi \right]  \right)^2  e_{1u}\,e_{2u}\,e_{3u}
++ \left(  \frac{1} {e_{2v}}  \delta_{j+1/2}  \left[ \chi \right]  \right)^2  e_{1v}\,e_{2v}\,e_{3v}
+	\right\} \;    &&&\\
+&\leq 0              &&&\\
+\end{flalign*}
+
+% ================================================================
+% Conservation Properties on Vertical Momentum Physics
+% ================================================================
+\section{Conservation Properties on Vertical Momentum Physics}
+\label{Apdx_C_4}
+
+
+As for the lateral momentum physics, the continuous form of the vertical diffusion of momentum satisfies the several integral constraints. The first two are associated to the conservation of momentum and the dissipation of horizontal kinetic energy:
+\begin{align*}
+ \int\limits_D  
+	 \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+ 	\left( 
+	\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}
+	\right)\;
+	dv \qquad \quad &= \vec{\textbf{0}}
+	\\
+%
+\intertext{and}
+%
+\int\limits_D 
+	\textbf{U}_h \cdot 
+	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
+	\left(
+	\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}
+	\right)\;
+	dv \quad &\leq 0
+	\\
+\end{align*}
+The first property is obvious. The second results from:
+
+\begin{flalign*}
+\int\limits_D 
+	\textbf{U}_h \cdot 
+	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
+	\left( 
+	\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
+	\right)\;dv
+	&&&\\
+\end{flalign*}
+\begin{flalign*}
+&\equiv \sum\limits_{i,j,k} 
+	\left( 
+	  u\; \delta_k 
+		\left[ 
+		\frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
+			\left[ u \right] 
+		\right]\;
+		e_{1u}\,e_{2u} 
+	+ v\;\delta_k 
+		\left[
+		\frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
+			\left[ v \right] 
+		\right]\;
+		e_{1v}\,e_{2v} 
+	\right)
+	&&&\\ 
+%
+\intertext{as the horizontal scale factor do not depend on $k$, it follows:}
+%
+&\equiv - \sum\limits_{i,j,k} 
+	\left( 
+	   \frac{A_u^{\,vm}} {e_{3uw}}
+		\left( 
+		\delta_{k+1/2} 
+			\left[ u \right] 
+		\right)^2\;
+		e_{1u}\,e_{2u} 
+	+ \frac{A_v^{\,vm}} {e_{3vw}} 
+		\left( \delta_{k+1/2} 
+			\left[ v \right] 
+		\right)^2\;
+		e_{1v}\,e_{2v}
+	\right)
+	 \quad \leq 0
+	 &&&\\
+\end{flalign*}
+
+The vorticity is also conserved. Indeed:
+\begin{flalign*}
+\int \limits_D 
+	\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
+		\left( 
+		\frac{1} {e_3 }\; \frac{\partial } {\partial k}
+			\left( 
+			\frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} 
+			\right)
+		\right)\;
+		dv
+		&&&\\ 
+\end{flalign*}
+\begin{flalign*}
+\equiv \sum\limits_{i,j,k}  \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}}
+	\bigg\{    \biggr.   \quad
+	\delta_{i+1/2} 
+		&\left(
+		\frac{e_{2v}} {e_{3v}} \delta_k 
+			\left[
+			\frac{1} {e_{3vw}} \delta_{k+1/2} 
+				\left[ v \right] 
+			\right]
+		\right)
+	 &&\\
+	\biggl. 
+	- \delta_{j+1/2} 
+		&\left(
+		\frac{e_{1u}} {e_{3u}} \delta_k 
+			\left[
+			\frac{1} {e_{3uw}}\delta_{k+1/2} 
+				\left[ u \right]
+			\right]
+		\right)
+	\biggr\} \;
+	e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0
+	&& \\
+\end{flalign*}
+If the vertical diffusion coefficient is uniform over the whole domain, the 
+enstrophy is dissipated, i.e.
+\begin{flalign*}
+\int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 
+	\left( 
+	\frac{1} {e_3}\; \frac{\partial } {\partial k}
+		\left( 
+		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
+		\right) 
+	\right)\;
+	dv = 0
+	&&&\\
+\end{flalign*}
+This property is only satisfied in $z$-coordinates:
+
+\begin{flalign*}
+\int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 
+	\left( 
+	\frac{1} {e_3}\; \frac{\partial } {\partial k}
+		\left( 
+		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
+		\right)
+	\right)\;
+	dv
+	&&& \\ 
+\end{flalign*}
+\begin{flalign*}
+\equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \;
+	\biggl\{ 	\biggr.	\quad
+	\delta_{i+1/2} 
+		&\left( 
+		   \frac{e_{2v}} {e_{3v}} \delta_k 
+			\left[ 
+			\frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
+				\left[ v \right] 
+			\right]
+		   \right)
+		   &&\\
+	- \delta_{j+1/2} 
+		&\biggl.
+		\left( 	
+		   \frac{e_{1u}} {e_{3u}} \delta_k 
+			\left[ 
+			\frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
+				\left[ u \right]
+			 \right] 
+		   \right) 
+	\biggr\} 
+	&&\\ 
+\end{flalign*}
+\begin{flalign*}
+\equiv \sum\limits_{i,j,k} \zeta \;e_{3f} 
+	\biggl\{		\biggr.	\quad
+	\frac{1} {e_{3v}} \delta_k 
+		&\left[ 
+		\frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
+			\left[ \delta_{i+1/2} 
+				\left[ e_{2v}\,v \right]
+			 \right]
+		\right]
+		&&\\ 
+ 	 \biggl. 
+	- \frac{1} {e_{3u}} \delta_k 
+		&\left[ 
+		\frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
+			\left[ \delta_{j+1/2} 
+				\left[ e_{1u}\,u \right]
+			 \right]
+		\right]
+	\biggr\} 
+	&&\\ 
+\end{flalign*}
+Using the fact that the vertical diffusive coefficients are uniform and that in $z$-coordinates, the vertical scale factors do not depends on $i$ and $j$ so that: $e_{3f} =e_{3u} =e_{3v} =e_{3T} $ and $e_{3w} =e_{3uw} =e_{3vw} $, it follows:
+\begin{flalign*}
+\equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k 
+	\left[ 
+	\frac{1} {e_{3w}} \delta_{k+1/2} 
+		\Bigl[ 
+		\delta_{i+1/2} 
+			\left[ e_{2v}\,v \right]
+		- \delta_{j+1/ 2} 
+			\left[ e_{1u}\,u \right]
+		 \Bigr] 
+	\right] 
+	&&&\\
+\end{flalign*}
+\begin{flalign*}
+\equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}}
+	\left( 
+	\delta_{k+1/2} 
+		\left[ \zeta  \right]
+	 \right)^2 \;
+	 e_{1f}\,e_{2f} 
+	 \; \leq 0
+	 &&&\\
+\end{flalign*}
+Similarly, the horizontal divergence is obviously conserved:
+
+\begin{flalign*}
+\int\limits_D \nabla \cdot 
+	\left( 
+	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
+		\left( 
+		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
+		\right) 
+	\right)\;
+	dv = 0
+	&&&\\
+\end{flalign*}
+and the square of the horizontal divergence decreases (i.e. the horizontal divergence is dissipated) if vertical diffusion coefficient is uniform over the whole domain:
+
+\begin{flalign*}
+\int\limits_D \chi \;\nabla \cdot 
+	\left( 
+	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
+		\left( 
+		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
+		\right) 
+	\right)\;
+	dv = 0
+	&&&\\
+\end{flalign*}
+This property is only satisfied in $z$-coordinates:
+
+\begin{flalign*}
+\int\limits_D \chi \;\nabla \cdot 
+	\left( 
+	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
+		\left( 
+		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
+		\right) 
+	\right)\;
+	dv
+	&&&\\
+\end{flalign*}
+\begin{flalign*}
+\equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1T}\,e_{2T}}
+	\biggl\{ 	\Biggr.	\quad
+	\delta_{i+1/2} 
+		&\left( 
+		   \frac{e_{2u}} {e_{3u}} \delta_k 
+		   	\left[ 
+			\frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
+				\left[ u \right] 
+			\right]
+		 \right)
+		 &&\\ 
+ 	\Biggl. 
+	+ \delta_{j+1/2} 
+		&\left( 
+		\frac{e_{1v}} {e_{3v}} \delta_k 
+			\left[ 
+			\frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
+				\left[ v \right]
+			 \right]
+		\right)
+	\Biggr\} \;
+	e_{1T}\,e_{2T}\,e_{3T} 
+	&&\\ 
+\end{flalign*}
+
+\begin{flalign*}
+\equiv A^{\,vm} \sum\limits_{i,j,k}  \chi \,
+	\biggl\{	\biggr.	\quad
+	\delta_{i+1/2}
+		&\left( 
+		   \delta_k 
+			\left[ 
+			\frac{1} {e_{3uw}} \delta_{k+1/2} 
+				\left[ e_{2u}\,u \right]
+			 \right]
+		   \right) 
+		   && \\ 
+	\biggl. 
+	+ \delta_{j+1/2} 
+		&\left( 
+		   \delta_k 
+			\left[ 
+			\frac{1} {e_{3vw}} \delta_{k+1/2} 
+				\left[ e_{1v}\,v \right]
+			 \right] 
+		   \right)
+	\biggr\} 
+	&& \\ 
+\end{flalign*}
+
+\begin{flalign*}
+\equiv -A^{\,vm} \sum\limits_{i,j,k} 
+\frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\;
+	\biggl\{ 
+	\delta_{k+1/2} 
+		\Bigl[
+		   \delta_{i+1/2} 
+			\left[ e_{2u}\,u \right]
+		+ \delta_{j+1/2} 
+			\left[ e_{1v}\,v \right]
+		\Bigr]
+	\biggr\} 
+	&&&\\
+\end{flalign*}
+
+\begin{flalign*}
+\equiv -A^{\,vm} \sum\limits_{i,j,k}
+ \frac{1} {e_{3w}} 
+\delta_{k+1/2} 
+	\left[ \chi \right]\;
+\delta_{k+1/2} 
+	\left[ e_{1T}\,e_{2T} \;\chi \right]
+&&&\\
+\end{flalign*}
+
+\begin{flalign*}
+\equiv -A^{\,vm} \sum\limits_{i,j,k} 
+\frac{e_{1T}\,e_{2T}} {e_{3w}}\;
+	\left( 
+	\delta_{k+1/2} 
+		\left[ \chi \right]
+	\right)^2
+	\quad  \equiv 0
+&&&\\
+\end{flalign*}
+
+% ================================================================
+% Conservation Properties on Tracer Physics
+% ================================================================
+\section{Conservation Properties on Tracer Physics}
+\label{Apdx_C.5}
+
+
+
+The numerical schemes used for tracer subgridscale physics are written such that the heat and salt contents are conserved (equations in flux form, second order centered finite differences). As a form flux is used to compute the temperature and salinity, the quadratic form of these quantities (i.e. their variance) globally tends to diminish. As for the advection term, there is generally no strict conservation of mass even if, in practice, the mass is conserved with a very good accuracy. 
+
+% -------------------------------------------------------------------------------------------------------------
+%       Conservation of Tracers
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation of Tracers}
+\label{Apdx_C.5.1}
+
+constraint of conservation of tracers:
+\begin{flalign*}
+&\int\limits_D  \nabla  \cdot \left( A\;\nabla T \right)\;dv  &&&\\ 
+\\
+&\equiv \sum\limits_{i,j,k} 
+	\biggl\{ 	\biggr.
+	\delta_i 
+		\left[ 
+		A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 
+			\left[ T \right]
+		\right]
+	+ \delta_j 
+		\left[ 
+		A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 
+			\left[ T \right] 
+		\right]
+	&&\\  & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;
+	+ \delta_k 
+		\left[ 
+		A_w^{\,vT} \frac{e_{1T}\,e_{2T}} {e_{3T}} \delta_{k+1/2} 
+			\left[ T \right] 
+		\right]
+	\biggr\}   \quad  \equiv 0
+	&&\\ 
+\end{flalign*}
+
+In fact, this property is simply resulting from the flux form of the operator. 
+
+% -------------------------------------------------------------------------------------------------------------
+%       Dissipation of Tracer Variance
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Dissipation of Tracer Variance}
+\label{Apdx_C.5.2}
+
+constraint of dissipation of tracer variance:
+\begin{flalign*}
+\int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv	&&&\\ 
+&\equiv   \sum\limits_{i,j,k} \; T
+\biggl\{  \biggr.
+     \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right]
+& + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] 
+		\quad&& \\ 
+ \biggl. 
+&&+ \delta_k \left[A_w^{\,vT}\frac{e_{1T}\,e_{2T}} {e_{3T}}\delta_{k+1/2}\left[T\right]\right]
+\biggr\} && 
+\end{flalign*}
+\begin{flalign*}
+\equiv - \sum\limits_{i,j,k} 
+	\biggl\{ 	\biggr.	\quad
+	& A_u^{\,lT} 
+		\left( 
+		\frac{1} {e_{1u}} \delta_{i+1/2} 
+			\left[ T \right]
+		\right)^2
+		e_{1u}\,e_{2u}\,e_{3u}
+	&& \\
+	& + A_v^{\,lT} 
+		\left( 
+		\frac{1} {e_{2v}} \delta_{j+1/2} 
+			\left[ T \right] 
+		\right)^2
+		e_{1v}\,e_{2v}\,e_{3v}
+	&& \\ 
+ 	\biggl. 
+	& + A_w^{\,vT} 
+		\left( 
+		\frac{1} {e_{3w}} \delta_{k+1/2} 
+			\left[ T \right] 
+		\right)^2
+		e_{1w}\,e_{2w}\,e_{3w} 
+	\biggr\} 
+	\quad \leq 0 
+	&& \\ 
+\end{flalign*}
+
+
+%%%%  end of appendix in gm comment
+%}
Index: trunk/DOC/TexFiles/Chapters/Annex_D.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Annex_D.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Annex_D.tex	(revision 994)
@@ -0,0 +1,182 @@
+% ================================================================
+% Appendix D Ñ Coding Rules
+% ================================================================
+\chapter{Coding Rules}
+\label{Apdx_D}
+\minitoc
+
+
+A "model life" is more than ten years. Its software, composed by a few 
+hundred modules, is used by many people who are scientists or students and 
+do not necessary know very well all computer aspects. Moreover, a well 
+thought-out programme is easy to read and understand, less difficult to 
+modify, produces fewer bugs and is easier to maintain. Therefore, it is 
+essential that the model development follows some rules :
+
+- well planned and designed
+
+- well written
+
+- well documented (both on- and off-line)
+
+- maintainable
+
+- easily portable
+
+- flexible.
+
+To satisfy part of these aims, \NEMO is written with a coding standard which 
+is close to the ECMWF rules, named DOCTOR \citep{Gibson_TR86}. These rules present some advantages like :
+
+- to provide a well presented program
+
+- to use rules for variable names which allow recognition of their type 
+(integer, real, parameter, local or shared variables, etc. ) so that debugging is 
+facilitated.
+
+% ================================================================
+% The program structure
+% ================================================================
+\section{The program structure}
+\label{Apdx_D_structure}
+
+Each program begins with a set of headline comments containing :
+
+- the program title
+
+- the purpose of the routine
+
+- the method and algorithms used
+
+- the detail of input and output interfaces
+
+- the external routines and functions used (if exist)
+
+- references (if exist)
+
+- the author name (s), the date of creation and of updates.
+
+- Each program is splitted into several well separated sections and 
+sub-sections with a underlined title and specific labelled statements.
+
+- A program has not more than 200 to 300 lines.
+
+% ================================================================
+% Coding conventions
+% ================================================================
+\section{Coding conventions}
+\label{Apdx_D_coding}
+
+- Use of the universal language \textsc{Fortran} 5 ANSI 77, with non 
+standard extensions, NAMELIST and \textsc{Fortran} 90 (matrix resolution 
+algorithm), and some well-identified particular statements or functions for 
+weak and massive parallelism, and vectorization 
+
+- A comment line begins with a uppercase character C at the first column. A 
+space line must have a C. For the on-line documentation, comments are 
+classified into three levels :
+
+Overview, triggered by CCC in columns 1 to 3. Only the title and the purpose 
+of the program are identified like that. This overview documentation can be 
+extracted by the UNIX function : grep -e '\^{}CCC' *
+
+External, triggered by CC in columns 1 to 2, and which correspond to 
+headlines of each programme, extracted by : grep -e '\^{}CC' *
+
+Internal which are all the comments, extracted by : grep -e '\^{}C' *
+
+- Statements GO TO, EQUIVALENCE are forbidden.
+
+- A section is numbered with labels which are in agreement with the 
+paragraph label and increase from the begin to the end of routine. Labels of 
+a hundred ( 200, 201.. 220..) are reserved to a unique section. The 
+\textsc{Fortran} 90 extension syntax DO/ENDDO is used except for multitasked 
+do-loop. In this case labels 1000, 2000, ... are used. The FORMAT statement 
+are labelled with numbers in the range 9000 to 9999.
+
+- A continuation line begins with the character {\$} in column 6.
+
+- All statements begin in column 7 with the following gaps :
+
+2 spaces toward the right in a DO loop.
+
+4 spaces toward the right in IF, ELSEIF, ELSE and ENDIF statements, with 
+only 2 spaces for ELSE and ELSEIF lines. All IF statement must be followed 
+by a ELSE statement.
+
+Some spaces in the continuation line for alignment.
+
+- Use of different labels for each DO loop statement.
+
+- STOP must be well documented with the name of the subroutine or a number.
+
+% ================================================================
+% Naming Conventions
+% ================================================================
+\section{Naming Conventions}
+\label{Apdx_D_naming}
+
+The purpose of the naming conventions is to use prefix letters to classify 
+model variables. These conventions allow to know easily the variable type 
+and to identify them rapidly:
+
+
+%--------------------------------------------------TABLE--------------------------------------------------
+\begin{table}[htbp]
+\begin{center}
+\begin{tabular}{|p{50pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}|}
+\hline  Type \par / Status &   integer&   real&   logical &   character&   double \par precision&   complex \\  
+\hline
+public & 
+\textbf{m n} \par \textit{but not } \par \textbf{nam}& 
+\textbf{a b e f g h o} \textbf{q} \textit{to} \textbf{x} \par but not \par \textbf{sf}& 
+\textbf{l} \par \textit{but not} \par \textbf{lp ld ll}& 
+\textbf{c} \par \textit{but not} \par \textbf{cp cd cl } \par \textbf{com cim}& 
+\textbf{d} \par \textit{but not} \par \textbf{dp dd dl}& 
+\textbf{y} \par \textit{but not} \par \textbf{yp yd yl} \\
+\hline
+dummy \par argument& 
+\textbf{k} \par \textit{but not} \par \textbf{kf}& 
+\textbf{p} \par \textit{but not}  \par \textbf{pp pf}& 
+\textbf{ld}& 
+\textbf{cd}& 
+\textbf{dd}& 
+\textbf{yd} \\
+\hline
+local \par variable& 
+\textbf{i}& 
+\textbf{z}& 
+\textbf{ll}& 
+\textbf{cl}& 
+\textbf{cd}& 
+\textbf{yl} \\
+\hline
+loop \par control& 
+\textbf{j} \par \textit{but not } \par \textbf{jp}& 
+& 
+& 
+& 
+& 
+ \\
+\hline
+parameter& 
+\textbf{jp}& 
+\textbf{pp}& 
+\textbf{lp}& 
+\textbf{cp}& 
+\textbf{dp}& 
+\textbf{yp} \\
+\hline
+statement \par function& 
+\textbf{kf}& 
+\textbf{sf} \par & 
+& 
+& 
+& 
+ \\
+\hline
+\end{tabular}
+\label{tab1}
+\end{center}
+\end{table}
+%--------------------------------------------------------------------------------------------------------------
Index: trunk/DOC/TexFiles/Chapters/Annex_E.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Annex_E.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Annex_E.tex	(revision 994)
@@ -0,0 +1,210 @@
+% ================================================================
+% Appendix E : UBS scheme
+% ================================================================
+\chapter{UBS scheme}
+\label{AN_E}
+\minitoc
+
+
+% -------------------------------------------------------------------------------------------------------------
+%        UBS scheme  
+% -------------------------------------------------------------------------------------------------------------
+\section{Upstream Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=T)}
+\label{TRA_adv_ubs}
+
+The UBS advection scheme is an upstream biased third order scheme based on 
+an upstream-biased parabolic interpolation. It is also known as Cell 
+Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective 
+Kinematics). For example, in the $i$-direction :
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\tau _u^{ubs} = \left\{	 \begin{aligned}
+  & \tau _u^{cen4} + \frac{1}{12} \,\tau"_i	   & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
+  & \tau _u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2}       <       0
+  						 \end{aligned}    \right.
+\end{equation}
+or equivalently, the advective flux is
+\begin{equation} \label{Eq_tra_adv_ubs2}
+U_{i+1/2} \ \tau _u^{ubs} 
+=U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2}
+- \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+\end{equation}
+where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$. By choosing this expression for $\tau "$ we consider a fourth order approximation of $\partial_i^2$ with a constant i-grid spacing ($\Delta i=1$). 
+
+Alternative choice: introduce the scale factors:  $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta _{i+1/2}[\tau] \right]$.
+
+
+This results in a dissipatively dominant (i.e. hyper-diffusive) truncation 
+error \citep{Sacha2005}. The overall performance of the 
+advection scheme is similar to that reported in \cite{Farrow1995}. 
+It is a relatively good compromise between accuracy and smoothness. It is 
+not a \emph{positive} scheme meaning false extrema are permitted but the 
+amplitude of such are significantly reduced over the centred second order 
+method. Nevertheless it is not recommended to apply it to a passive tracer 
+that requires positivity. 
+
+The intrinsic diffusion of UBS makes its use risky in the vertical direction 
+where the control of artificial diapycnal fluxes is of paramount importance. 
+It has therefore been preferred to evaluate the vertical flux using the TVD 
+scheme when \np{ln\_traadv\_ubs}=T.
+
+For stability reasons, in \eqref{Eq_tra_adv_ubs}, the first term which corresponds to a 
+second order centred scheme is evaluated using the \textit{now} velocity (centred in 
+time) while the second term which is the diffusive part of the scheme, is 
+evaluated using the \textit{before} velocity (forward in time. This is discussed by \citet{Webb1998} in the context of the Quick advection scheme. UBS and QUICK 
+schemes only differ by one coefficient. Substituting 1/6 with 1/8 in 
+(\ref{Eq_tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. This 
+option is not available through a namelist parameter, since the 1/6 
+coefficient is hard coded. Nevertheless it is quite easy to make the 
+substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme
+
+NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can 
+be controlled by vertical advection (not vertical diffusion which is usually 
+solved using an implicit scheme). Computer time can be saved by using a 
+time-splitting technique on vertical advection. This possibility have been 
+implemented and validated in ORCA05-L301. It is not currently offered in the 
+current reference version. 
+
+NB 2 : In a forthcoming release four options will be proposed for the 
+vertical component used in the UBS scheme. $\tau _w^{ubs}$ will be 
+evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme , or  \textit{(b)} a TVD 
+scheme, or  \textit{(c)} an interpolation based on conservative parabolic splines 
+following \citet{Sacha2005} implementation of UBS in ROMS, 
+or  \textit{(d)} an UBS. The $3^{rd}$ case has dispersion properties similar to an 
+eight-order accurate conventional scheme.
+
+NB 3 : It is straight forward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\tau _u^{ubs} = \left\{	 \begin{aligned}
+	& \tau _u^{cen4} + \frac{1}{12} \tau"_i		& \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
+	& \tau _u^{cen4} - \frac{1}{12} \tau"_{i+1}	& \quad \text{if }\ u_{i+1/2}       <       0
+  						 \end{aligned}    \right.
+\end{equation}
+or equivalently 
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\begin{split}
+e_{2u} e_{3u}\,u_{i+1/2} \ \tau _u^{ubs} 
+&= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\
+& - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+\end{split}
+\end{equation}
+\eqref{Eq_tra_adv_ubs2} has several advantages. First it clearly evidence that the UBS scheme is based on the fourth order scheme to which is added an upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order part as stated above using \eqref{Eq_tra_adv_ubs}. Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient with is simply proportional to the velocity.
+
+
+laplacian diffusion:
+\begin{equation} \label{Eq_tra_ldf_lap}
+\begin{split}
+D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta _i 
+\left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} 
+\left[ T \right]} \right]} \right.
+\\
+&\ \left. {+\; \delta _j \left[ 
+{A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta _{j+1/2} \left[ T 
+\right]} \right)} \right]\quad } \right]
+\end{split}
+\end{equation}
+
+bilaplacian:
+\begin{equation} \label{Eq_tra_ldf_lap}
+\begin{split}
+D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\
+& \delta _i \left[  \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2} 
+		  \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}
+    \delta _i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2} 
+    		  [T] \right] \right] \right]
+\end{split}
+\end{equation}
+with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, 
+$i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$
+it comes :
+\begin{equation} \label{Eq_tra_ldf_lap}
+\begin{split}
+D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\
+& \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2} 
+		 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 
+    \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2} 
+    		[T] \right] \right] \right]
+\end{split}
+\end{equation}
+if the velocity is uniform ($i.e.$ $|u|=cst$) then the diffusive flux is
+\begin{equation} \label{Eq_tra_ldf_lap}
+\begin{split}
+F_u^{lT} = - \frac{1}{12}
+ e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta _{i+1/2} 
+		 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 
+    \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta _{i+1/2} 
+    		[T] \right] \right]
+\end{split}
+\end{equation}
+beurk....  reverte the logic: starting from the diffusive part of the advective flux it comes:
+
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\begin{split}
+F_u^{lT}
+&= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+\end{split}
+\end{equation}
+if the velocity is uniform ($i.e.$ $|u|=cst$) and choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right]$
+
+sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$):
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\begin{split}
+F_u^{lT}
+&= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right] \right]
+\end{split}
+\end{equation}
+which leads to ${A_T^{lT}}^2 = \frac{1}{12} {e_{1T}}^3\ \overline{|u|}^{\,i+1/2}$
+
+sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\begin{split}
+F_u^{lT}
+&= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta _i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right] \right] \\
+&= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta _i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta _{i+1/2}[\tau] \right] \right]
+\end{split}
+\end{equation}
+which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$
+
+
+% -------------------------------------------------------------------------------------------------------------
+%        Leap-Frog energetic  
+% -------------------------------------------------------------------------------------------------------------
+\section{Leap-Frog energetic }
+\label{LF}
+
+We adopt the following semi-discrete notation for time derivative. Given the values of a variable $q$ at successive time step, the time derivation and averaging operators at the mid time step are:
+\begin{subequations} \label{dt_mt}
+\begin{align}
+ \delta _{t+\Delta t/2} [q]     &=  \  \ \,   q^{t+\Delta t}  - q^{t}		\\
+ \overline q^{\,t+\Delta t/2} &= \left\{ q^{t+\Delta t} + q^{t} \right\} \; / \; 2
+\end{align}
+\end{subequations}
+As for space operator, the adjoint of the derivation and averaging time operators are $\delta_t^*=\delta_{t+\Delta t/2}$ and $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$, respectively. 
+
+The Leap-frog time stepping given by \eqref{Eq_DOM_nxt} can be defined as:
+\begin{equation} \label{LF}
+   \frac{\partial q}{\partial t} 
+   		\equiv \frac{1}{\Delta t} \overline{ \delta _{t+\Delta t/2}[q]}^{\,t} 
+		=         \frac{q^{t+\Delta t}-q^{t-\Delta t}}{2\Delta t}
+\end{equation} 
+Note that \eqref{LF} shows that the leapfrog time step is $\Delta t$, not $2\Delta t$ as it can be found sometime in literature. 
+The leap-Frog time stepping is a second order centered scheme. As such it respects the quadratic invariant in integral forms, $i.e.$ the following continuous property,
+\begin{equation} \label{Energy}
+\int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 
+	=\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} 
+	=  \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) ,
+\end{equation}
+is satisfied in discrete form. Indeed, 
+\begin{equation} \begin{split}
+\int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 
+	&\equiv \sum\limits_{0}^{N} 
+			{\frac{1}{\Delta t} q^t \ \overline{ \delta _{t+\Delta t/2}[q]}^{\,t} \ \Delta t} 
+	   \equiv \sum\limits_{0}^{N}  { q^t \ \overline{ \delta _{t+\Delta t/2}[q]}^{\,t} } \\
+	&\equiv \sum\limits_{0}^{N}  { \overline{q}^{\,t+\Delta/2}{ \delta _{t+\Delta t/2}[q]}}
+	   \equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta _{t+\Delta t/2}[q^2] }\\
+	&\equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta _{t+\Delta t/2}[q^2] }
+	   \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right)
+\end{split} \end{equation}
+NB here pb of boundary condition when applying the adjoin! In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition (equivalently of the boundary value of the integration by part). In time this boundary condition is not physical and \textbf{add something here!!!}
+
+
+
Index: trunk/DOC/TexFiles/Chapters/Chap_Conservation.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_Conservation.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_Conservation.tex	(revision 994)
@@ -0,0 +1,332 @@
+
+% ================================================================
+% Invariant of the Equations
+% ================================================================
+\chapter{Annex --- Invariants of the Primitive Equations}
+\label{Invariant}
+\minitoc
+
+The continuous equations of motion have many analytic properties. Many 
+quantities (total mass, energy, enstrophy, etc.) are strictly conserved in 
+the inviscid and unforced limit, while ocean physics conserve the total 
+quantities on which they act (momentum, temperature, salinity) but dissipate 
+their total variance (energy, enstrophy, etc.). Unfortunately, the finite 
+difference form of these equations is not guaranteed to retain all these 
+important properties. In constructing the finite differencing schemes, we 
+wish to ensure that certain integral constraints will be maintained. In 
+particular, it is desirable to construct the finite difference equations so 
+that horizontal kinetic energy and/or potential enstrophy of horizontally 
+non-divergent flow, and variance of temperature and salinity will be 
+conserved in the absence of dissipative effects and forcing. \citet{Arakawa1966} 
+has first pointed out the advantage of this approach. He showed that if 
+integral constraints on energy are maintained, the computation will be free 
+of the troublesome "non linear" instability originally pointed out by 
+\citet{Phillips1959}. A consistent formulation of the energetic properties is 
+also extremely important in carrying out long-term numerical simulations for 
+an oceanographic model. Such a formulation avoids systematic errors that 
+accumulate with time \citep{Bryan1997}.
+
+The general philosophy of OPA which has led to the discrete formulation 
+presented in {\S}II.2 and II.3 is to choose second order non-diffusive 
+scheme for advective terms for both dynamical and tracer equations. At this 
+level of complexity, the resulting schemes are dispersive schemes. 
+Therefore, they require the addition of a diffusive operator to be stable. 
+The alternative is to use diffusive schemes such as upstream or flux 
+corrected schemes. This last option was rejected because we prefer a 
+complete handling of the model diffusion, i.e. of the model physics rather 
+than letting the advective scheme produces its own implicit diffusion 
+without controlling the space and time structure of this implicit diffusion. 
+Note that in some very specific cases as passive tracer studies, the 
+positivity of the advective scheme is required. In that case, and in that 
+case only, the advective scheme used for passive tracer is a flux correction 
+scheme \citep{Marti1992, Levy1996, Levy1998}.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Conservation Properties on Ocean Dynamics
+% -------------------------------------------------------------------------------------------------------------
+\section{Conservation Properties on Ocean Dynamics}
+\label{Invariant_dyn}
+
+The non linear term of the momentum equations has been split into a 
+vorticity term, a gradient of horizontal kinetic energy and a vertical 
+advection term. Three schemes are available for the former (see {\S}~II.2) 
+according to the CPP variable defined (default option\textbf{ 
+}or \textbf{key{\_}vorenergy } or \textbf{key{\_}vorcombined 
+} defined). They differ in their conservative 
+properties (energy or enstrophy conserving scheme). The two latter terms 
+preserve the total kinetic energy: the large scale kinetic energy is also 
+preserved in practice. The remaining non-diffusive terms of the momentum 
+equation (namely the hydrostatic and surface pressure gradient terms) also 
+preserve the total kinetic energy and have no effect on the vorticity of the 
+flow.
+
+\textbf{* relative, planetary and total vorticity term:}
+
+Let us define as either the relative, planetary and total potential 
+vorticity, i.e. , , and , respectively. The continuous formulation of the 
+vorticity term satisfies following integral constraints:
+\begin{equation} \label{Eq_vor_vorticity}
+\int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma 
+\;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0
+\end{equation}
+
+\begin{equation} \label{Eq_vor_enstrophy}
+if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot 
+\frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv} 
+=0
+\end{equation}
+
+\begin{equation} \label{Eq_vor_energy}
+\int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0
+\end{equation}
+where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element. 
+(II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an integration by part. It means that $\varsigma^2$ is conserved for a horizontally non-divergent flow. 
+(II.4.1c) is even satisfied locally since the vorticity term is orthogonal 
+to the horizontal velocity. It means that the vorticity term has no 
+contribution to the evolution of the total kinetic energy. (II.4.1a) is 
+obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied 
+simultaneously with a second order scheme. Using the symmetry or 
+anti-symmetry properties of the operators (Eqs II.1.10 and 11), it can be 
+shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), while 
+scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). 
+Note that the enstrophy conserving scheme on total vorticity has been chosen 
+as the standard discrete form of the vorticity term.
+
+\textbf{* Gradient of kinetic energy / vertical advection}
+
+In continuous formulation, the gradient of horizontal kinetic energy has no 
+contribution to the evolution of the vorticity as the curl of a gradient is 
+zero. This property is satisfied locally with the discrete form of both the 
+gradient and the curl operator we have made (property (II.1.9)~). Another 
+continuous property is that the change of horizontal kinetic energy due to 
+vertical advection is exactly balanced by the change of horizontal kinetic 
+energy due to the horizontal gradient of horizontal kinetic energy:
+
+\begin{equation} \label{Eq_keg_zad}
+\int_D {{\textbf{U}}_h \cdot \nabla _h \left( {1/2\;{\textbf{U}}_h ^2} \right)\;dv} =-\int_D {{\textbf{U}}_h \cdot \frac{w}{e_3 }\;\frac{\partial 
+{\textbf{U}}_h }{\partial k}\;dv}
+\end{equation}
+
+Using the discrete form given in {\S}II.2-a and the symmetry or 
+anti-symmetry properties of the mean and difference operators, \eqref{Eq_keg_zad} is 
+demonstrated in the Appendix C. The main point here is that satisfying 
+\eqref{Eq_keg_zad} links the choice of the discrete forms of the vertical advection 
+and of the horizontal gradient of horizontal kinetic energy. Choosing one 
+imposes the other. The discrete form of the vertical advection given in 
+{\S}II.2-a is a direct consequence of formulating the horizontal kinetic 
+energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in the gradient term.
+
+\textbf{* hydrostatic pressure gradient term}
+
+In continuous formulation, a pressure gradient has no contribution to the 
+evolution of the vorticity as the curl of a gradient is zero. This 
+properties is satisfied locally with the choice of discretization we have 
+made (property (II.1.9)~). In addition, when the equation of state is linear 
+(i.e. when an advective-diffusive equation for density can be derived from 
+those of temperature and salinity) the change of horizontal kinetic energy 
+due to the work of pressure forces is balanced by the change of potential 
+energy due to buoyancy forces:
+
+\begin{equation} \label{Eq_hpg_pe}
+\int_D {-\frac{1}{\rho _o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv}
+\end{equation}
+
+Using the discrete form given in {\S}~II.2-a and the symmetry or 
+anti-symmetry properties of the mean and difference operators, (II.4.3) is 
+demonstrated in the Appendix C. The main point here is that satisfying 
+(II.4.3) strongly constraints the discrete expression of the depth of 
+$T$-points and of the term added to the pressure gradient in $s-$coordinates: the 
+depth of a $T$-point, $z_T$, is defined as the sum the vertical scale 
+factors at $w$-points starting from the surface.
+
+\textbf{* surface pressure gradient term}
+
+In continuous formulation, the surface pressure gradient has no contribution 
+to the evolution of vorticity. This properties is trivially satisfied 
+locally as (II.2.3) (the equation verified by $\psi$ has been 
+derived from the discrete formulation of the momentum equations, vertical 
+sum and curl. Nevertheless, the $\psi$-equation is solved numerically by an 
+iterative solver (see {\S}~III.5), thus the property is only satisfied with 
+the accuracy required on the solver. In addition, with the rigid-lid 
+approximation, the change of horizontal kinetic energy due to the work of 
+surface pressure forces is exactly zero:
+\begin{equation} \label{Eq_spg}
+\int_D {-\frac{1}{\rho _o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0
+\end{equation}
+
+(II.4.4) is satisfied in discrete form only if the discrete barotropic 
+streamfunction time evolution equation is given by (II.2.3) (see appendix 
+C). This shows that (II.2.3) is the only way to compute the streamfunction, 
+otherwise there is no guarantee that the surface pressure force work 
+vanishes.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Conservation Properties on Ocean Thermodynamics
+% -------------------------------------------------------------------------------------------------------------
+\section{Conservation Properties on Ocean Thermodynamics}
+\label{Invariant_tra}
+
+In continuous formulation, the advective terms of the tracer equations 
+conserve the tracer content and the quadratic form of the tracer, i.e.
+\begin{equation} \label{Eq_tra_tra2}
+\int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
+\;\text{and}
+\int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
+\end{equation}
+
+The numerical scheme used ({\S}II.2-b) (equations in flux form, second order 
+centred finite differences) satisfies (II.4.5) (see appendix C). Note that 
+in both continuous and discrete formulations, there is generally no strict 
+conservation of mass, since the equation of state is non linear with respect 
+to $T$ and $S$. In practice, the mass is conserved with a very good accuracy. 
+
+% -------------------------------------------------------------------------------------------------------------
+%       Conservation Properties on Momentum Physics
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation Properties on Momentum Physics}
+\label{Invariant_dyn_physics}
+
+\textbf{* lateral momentum diffusion term}
+
+The continuous formulation of the horizontal diffusion of momentum satisfies 
+the following integral constraints~:
+\begin{equation} \label{Eq_dynldf_dyn}
+\int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla 
+_h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta 
+\;{\rm {\bf k}}} \right)} \right]\;dv} =0
+\end{equation}
+
+\begin{equation} \label{Eq_dynldf_div}
+\int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
+\right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} 
+\right]\;dv} =0
+\end{equation}
+
+\begin{equation} \label{Eq_dynldf_curl}
+\int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
+\right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} 
+\right]\;dv} \leqslant 0
+\end{equation}
+
+\begin{equation} \label{Eq_dynldf_curl2}
+\mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot 
+\nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h 
+\times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} 
+\leqslant 0
+\end{equation}
+
+\begin{equation} \label{Eq_dynldf_div2}
+\mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ 
+{\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( 
+{A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0
+\end{equation}
+
+
+(II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum 
+conserve both the potential vorticity and the divergence of the flow, while 
+Eqs (II.4.6c) to (II.4.6e) mean that it dissipates the energy, the enstrophy 
+and the square of the divergence. The two latter properties are only 
+satisfied when the eddy coefficients are horizontally uniform.
+
+Using (II.1.8) and (II.1.9), and the symmetry or anti-symmetry properties of 
+the mean and difference operators, it is shown that the discrete form of the 
+lateral momentum diffusion given in {\S}II.2-c satisfies all the integral 
+constraints (II.4.6) (see appendix C). In particular, when the eddy 
+coefficients are horizontally uniform, a complete separation of vorticity 
+and horizontal divergence fields is ensured, so that diffusion (dissipation) 
+of vorticity (enstrophy) does not generate horizontal divergence (variance 
+of the horizontal divergence) and \textit{vice versa}. When the vertical curl of the horizontal 
+diffusion of momentum (discrete sense) is taken, the term associated to the 
+horizontal gradient of the divergence is zero locally. When the horizontal 
+divergence of the horizontal diffusion of momentum (discrete sense) is 
+taken, the term associated to the vertical curl of the vorticity is zero 
+locally. The resulting term conserves $\chi$ and dissipates 
+$\chi^2$ when the 
+eddy coefficient is horizontally uniform.
+
+\textbf{* vertical momentum diffusion term}
+
+As for the lateral momentum physics, the continuous form of the vertical 
+diffusion of momentum satisfies following integral constraints~:
+
+conservation of momentum, dissipation of horizontal kinetic energy
+
+\begin{equation} \label{Eq_dynzdf_dyn}
+\begin{aligned}
+& \int_D {\frac{1}{e_3 }}  \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\ 
+& \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\ 
+ \end{aligned} 
+ \end{equation}
+conservation of vorticity, dissipation of enstrophy
+\begin{equation} \label{Eq_dynzdf_vor}
+\begin{aligned}
+& \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 
+}\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm 
+{\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\ 
+& \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 
+}\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm 
+{\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\ 
+\end{aligned}
+\end{equation}
+conservation of horizontal divergence, dissipation of square of the 
+horizontal divergence
+\begin{equation} \label{Eq_dynzdf_div}
+\begin{aligned}
+ &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial 
+k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} 
+\right)} \right)\;dv} =0 \\ 
+& \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial 
+k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} 
+\right)} \right)\;dv} \leq 0 \\ 
+\end{aligned}
+\end{equation}
+
+In discrete form, all these properties are satisfied in $z$-coordinate (see 
+Appendix C). In $s$-coordinates, only first order properties can be 
+demonstrated, i.e. the vertical momentum physics conserve momentum, 
+potential vorticity, and horizontal divergence.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Conservation Properties on Tracer Physics
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation Properties on Tracer Physics}
+\label{Invariant_tra_physics}
+
+The numerical schemes used for tracer subgridscale physics are written in 
+such a way that the heat and salt contents are conserved (equations in flux 
+form, second order centred finite differences). As a form flux is used to 
+compute the temperature and salinity, the quadratic form of these quantities 
+(i.e. their variance) globally tends to diminish. As for the advective term, 
+there is generally no strict conservation of mass even if, in practice, the 
+mass is conserved with a very good accuracy. 
+
+\textbf{* lateral physics: }conservation of tracer, dissipation of tracer 
+variance, i.e.
+
+\begin{equation} \label{Eq_traldf_t_t2}
+\begin{aligned}
+&\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ 
+&\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\ 
+\end{aligned}
+\end{equation}
+
+\textbf{* vertical physics: }conservation of tracer, dissipation of tracer 
+variance, i.e.
+
+\begin{equation} \label{Eq_trazdf_t_t2}
+\begin{aligned}
+& \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv = 0 \\ 
+& \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv \leq 0 \\ 
+\end{aligned}
+\end{equation}
+
+Using the symmetry or anti-symmetry properties of the mean and difference 
+operators, it is shown that the discrete form of tracer physics given in 
+{\S}~II.2-c satisfies all the integral constraints (II.4.8) and (II.4.9) 
+except the dissipation of the square of the tracer when non-geopotential 
+diffusion is used (see appendix C). A discrete form of the lateral tracer 
+physics can be derived which satisfies these last properties. Nevertheless, 
+it requires a horizontal averaging of the vertical component of the lateral 
+physics that prevents the use of implicit resolution in the vertical. It has 
+not been implemented.
+
Index: trunk/DOC/TexFiles/Chapters/Chap_DOM.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_DOM.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_DOM.tex	(revision 994)
@@ -0,0 +1,1094 @@
+
+% ================================================================
+% Chapter 2 Ñ Space and Time Domain (DOM)
+% ================================================================
+\chapter{Space and Time Domain (DOM) }
+\label{DOM}
+\minitoc
+
+% Missing things:
+% 	- istate: description of the initial state   ==> this has to be put elsewhere..
+%                  perhaps in MISC ?  By the way the initialisation of T S and dynamics 
+%                  should be put outside of DOM routine (better with TRC staff and off-line
+%                  tracers)
+%	- daymod: definition of the time domain (nit000, nitend andd the calendar)
+% 	-geo2ocean:  how to switch from geographic to mesh coordinate
+%  - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
+
+\gmcomment{STEVEN :maybe a picture of the directory structure in the introduction which could be referred to here, would help  ==> to be added}
+%%%%
+
+
+\newpage
+$\ $\newline    % force a new ligne
+
+
+Having defined the continuous equations in Chap.~\ref{PE}, we need to choose a 
+discretization on a grid, and numerical algorithms. In the present chapter, we 
+provide a general description of the staggered grid used in \NEMO, and other 
+information relevant to the main directory routines (time stepping, main program) 
+as well as the DOM (DOMain) directory. 
+
+% ================================================================
+% Fundamentals of the Discretisation
+% ================================================================
+\section{Fundamentals of the Discretisation}
+\label{DOM_basics}
+
+% -------------------------------------------------------------------------------------------------------------
+%        Arrangement of Variables 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Arrangement of Variables}
+\label{DOM_cell}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \label{Fig_cell}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_cell.pdf}
+\caption{Arrangement of variables. $T$ indicates scalar points where temperature, 
+salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$) 
+indicates vector points, and $f$ indicates vorticity points where both relative and 
+planetary vorticities are defined}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The numerical techniques used to solve the Primitive Equations in this model are 
+based on the traditional, centred second-order finite difference approximation. 
+Special attention has been given to the homogeneity of the solution in the three 
+space directions. The arrangement of variables is the same in all directions. 
+It consists of cells centred on scalar points ($T$, $S$, $p$, $\rho$) with vector 
+points $(u, v, w)$ defined in the centre of each face of the cells (Fig. \ref{Fig_cell}). 
+This is the generalisation to three dimensions of the well-known ``C'' grid in 
+Arakawa's classification \citep{Mesinger_Arakawa_Bk76}. The relative and 
+planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge 
+and the barotropic stream function $\psi$ is defined at horizontal points overlying 
+the $\zeta$ and $f$-points.
+
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 
+by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$. 
+The grid-points are located at integer or integer and a half value of $(i,j,k)$ as 
+indicated on Table \ref{Tab_cell}. In all the following, subscripts $u$, $v$, $w$, 
+$f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where the scale 
+factors are defined. Each scale factor is defined as the local analytical value 
+provided by \eqref{Eq_scale_factors}. As a result, the mesh on which partial 
+derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and 
+$\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity. Discrete partial derivatives are formulated by the traditional, centred second order 
+finite difference approximation while the scale factors are chosen equal to their 
+local analytical value. An important point here is that the partial derivative of the 
+scale factors must be evaluated by centred finite difference approximation, not 
+from their analytical expression. This preserves the symmetry of the discrete set 
+of equations and therefore satisfies many of the continuous properties (see 
+Appendix~\ref{Apdx_C}). A similar, related remark can be made about the domain 
+size: when needed, an area, volume, or the total ocean depth must be evaluated 
+as the sum of the relevant scale factors (see \eqref{DOM_bar}) in the next section). 
+
+\begin{table}[!tb] \label{Tab_cell}
+\begin{center} \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|}
+\hline
+T	&$i$ 		& $j$		& $k$		 \\ \hline
+u	& $i+1/2$	& $j$		& $k$	 	\\ \hline
+v	& $i$	 	& $j+1/2$	& $k$	 	\\ \hline
+w	& $i$	 	& $j$		& $k+1/2$ 	\\ \hline
+f	& $i+1/2$ 	& $j+1/2$	& $k$	 	\\ \hline
+uw	& $i+1/2$ 	& $j$		& $k+1/2$ 	\\ \hline
+vw	& $i$	 	& $j+1/2$	& $k+1/2$ 	\\ \hline
+fw	& $i+1/2$ 	& $j+1/2$	& $k+1/2$ 	\\ \hline
+\end{tabular}
+\caption{Location of grid-points as a function of integer or integer and a half value 
+of the column, line or level. This indexing is only used for the writing of the semi-
+discrete equation. In the code, the indexing uses integer values only and has a 
+reverse direction in the vertical (see \S\ref{DOM_Num_Index})}
+\end{center}
+\end{table}
+
+% -------------------------------------------------------------------------------------------------------------
+%        Vector Invariant Formulation 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Discrete Operators}
+\label{DOM_operators}
+
+Given the values of a variable $q$ at adjacent points, the differencing and 
+averaging operators at the midpoint between them are:
+\begin{subequations} \label{Eq_di_mi}
+\begin{align}
+ \delta _i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)		\\
+ \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2
+\end{align}
+\end{subequations}
+
+Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and 
+$k+1/2$. Following \eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a 
+variable $q$ defined at a $T$-point has its three components defined at $u$-, $v$- 
+and $w$-points while its Laplacien is defined at $T$-point. These operators have 
+the following discrete forms in the curvilinear $s$-coordinate system:
+\begin{equation} \label{Eq_DOM_grad}
+\nabla q\equiv 	\frac{1}{e_{1u} }\delta _{i+1/2} \left[ q \right]\;\,{\rm {\bf i}}
+			+	\frac{1}{e_{2v} }\delta _{j+1/2} \left[ q \right]\;\,{\rm {\bf j}}
+			+	\frac{1}{e_{3w} }\delta _{k+1/2} \left[ q \right]\;\,{\rm {\bf k}}
+\end{equation}
+\begin{multline} \label{Eq_DOM_lap}
+\Delta q\equiv \frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}e_{3T} }\;\left( 
+{\delta _i \left[ {\frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} 
+\left[ q \right]} \right]
++\delta _j \left[ {\frac{e_{1v} e_{3v} }{e_{2v} 
+}\;\delta _{j+1/2} \left[ q \right]} \right]\;} \right)		\\
++\frac{1}{e_{3T} }\delta _k \left[ {\frac{1}{e_{3w} }\;\delta _{k+1/2} 
+\left[ q \right]} \right]
+\end{multline}
+
+Following \eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ defined at vector points $(u,v,w)$ has its three curl 
+components defined at $vw$-, $uw$, and $f$-points, and its divergence defined 
+at $T$-points:
+\begin{equation} \label{Eq_DOM_curl}
+\begin{split}
+ \nabla \times {\rm {\bf A}}\equiv \frac{1}{e_{2v} {\kern 1pt}e_{3vw} 
+}{\kern 1pt}\,\;\left( {\delta _{j+1/2} \left[ {e_{3w} a_3 } \right]-\delta 
+_{k+1/2} \left[ {e_{2v} a_2 } \right]} \right)	&\;\;{\rm {\bf i}} \\ 
+ +\frac{1}{e_{2u} {\kern 1pt}e_{3uw} }\;\left( {\delta _{k+1/2} \left[ {e_{1u} a_1 } 
+\right]-\delta _{i+1/2} \left[ {e_{3w} a_3 } \right]} \right) 	&\;\;{\rm{\bf j}} \\ 
+ +\frac{e_{3f} }{e_{1f} {\kern 1pt}e_{2f} }\,{\kern 1pt}\;\left( {\delta 
+_{i+1/2} \left[ {e_{2v} a_2 } \right]-\delta _{j+12} \left[ {e_{1u} a_1 } \right]} 
+\right)	&\;\;{\rm {\bf k}}
+ \end{split}
+\end{equation}
+\begin{equation} \label{Eq_DOM_div}
+\nabla \cdot {\rm {\bf A}}=\frac{1}{e_{1T} e_{2T} e_{3T} }\left( {\delta 
+_i \left[ {e_{2u} e_{3u} a_1 } \right]+\delta _j \left[ {e_{1v} e_{3v} a_2 } 
+\right]} \right)+\frac{1}{e_{3T} }\delta _k \left[ {a_3 } \right]
+\end{equation}
+
+In the special case of a pure $z$-coordinate system, \eqref{Eq_DOM_lap} and 
+\eqref{Eq_DOM_div} can be simplified. In this case, the vertical scale factor 
+becomes a function of the single variable $k$ and thus does not depend on the 
+horizontal location of a grid point. For example \eqref{Eq_DOM_div} reduces to: 
+\begin{equation*}
+\nabla \cdot {\rm {\bf A}}=\frac{1}{e_{1T} e_{2T} }\left( {\delta 
+_i \left[ {e_{2u} a_1 } \right]+\delta _j \left[ {e_{1v}  a_2 } 
+\right]} \right)+\frac{1}{e_{3T} }\delta _k \left[ {a_3 } \right]
+\end{equation*}
+
+The vertical average over the whole water column denoted by an overbar becomes 
+for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area):
+\begin{equation} \label{DOM_bar}
+\bar q 	= \frac{1}{H}\int_{k^b}^{k^o} {q\;e_{3q} \,dk} 
+		\equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} }
+\end{equation}
+where $H_q$  is the ocean depth, which is the masked sum of the vertical scale 
+factors at $q$ points, $k^b$ and $k^o$ are the bottom and surface $k$-indices, 
+and the symbol $k^o$ refers to a summation over all grid points of the same type 
+in the direction indicated by the subscript (here $k$). 
+
+In continuous form, the following properties are satisfied:
+\begin{equation} \label{Eq_DOM_curl_grad}
+\nabla \times \nabla q ={\rm {\bf {0}}}
+\end{equation}
+\begin{equation} \label{Eq_DOM_div_curl}
+\nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0
+\end{equation}
+
+It is straightforward to demonstrate that these properties are verified locally in 
+discrete form as soon as the scalar $q$ is taken at $T$-points and the vector 
+\textbf{A} has its components defined at vector points $(u,v,w)$.
+
+Let $a$ and $b$ be two fields defined on the mesh, with value zero inside 
+continental area. Using integration by parts it can be shown that the differencing 
+operators ($\delta_i$, $\delta_j$ and $\delta_k$) are anti-symmetric linear 
+operators, and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$, 
+$\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear 
+operators, $i.e.$
+\begin{align} 
+\label{DOM_di_adj}
+\sum\limits_i { a_i \;\delta _i \left[ b \right]} 
+	&\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\
+\label{DOM_mi_adj}
+\sum\limits_i { a_i \;\overline b^{\,i}} 
+	& \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } 
+\end{align}
+
+In other words, the adjoint of the differencing and averaging operators are 
+$\delta_i^*=\delta_{i+1/2}$ and 
+${(\overline{\,\cdot \,}^{\,i})}^*= \overline{\,\cdot\,}^{\,i+1/2}$, respectively. 
+These two properties will be used extensively in the Appendix~\ref{Apdx_C} to 
+demonstrate integral conservative properties of the discrete formulation chosen.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Numerical Indexing 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Numerical Indexing}
+\label{DOM_Num_Index}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \label{Fig_index_hor}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_index_hor.pdf}
+\caption{Horizontal integer indexing used in the \textsc{Fortran} code. The dashed 
+area indicates the cell in which variables contained in arrays have the same 
+$i$- and $j$-indices}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The array representation used in the \textsc{Fortran} code requires an integer 
+indexing while the analytical definition of the mesh (see \S\ref{DOM_cell}) is 
+associated with the use of integer values for $T$-points and both integer and 
+integer and a half values for all the other points. Therefore a specific integer 
+indexing must be defined for points other than $T$-points ($i.e.$ velocity and 
+vorticity grid-points). Furthermore, the direction of the vertical indexing has 
+been changed so that the surface level is at $k=1$.
+
+% -----------------------------------
+%        Horizontal Indexing 
+% -----------------------------------
+\subsubsection{Horizontal Indexing}
+\label{DOM_Num_Index_hor}
+
+The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}. For an increasing $i$ index ($j$ index), the $T$-point 
+and the eastward $u$-point (northward $v$-point) have the same index 
+(see the dashed area in Fig.\ref{Fig_index_hor}). A $T$-point and its 
+nearest northeast $f$-point have the same $i$-and $j$-indices.
+
+% -----------------------------------
+%        Vertical indexing 
+% -----------------------------------
+\subsubsection{Vertical Indexing}
+\label{DOM_Num_Index_vertical}
+
+In the vertical, the chosen indexing requires special attention since the 
+$k$-axis is re-orientated downward in the \textsc{Fortran} code compared 
+to the indexing used in the semi-discrete equations and given in \S\ref{DOM_cell}. 
+The sea surface corresponds to the $w$-level $k=1$ which is the same index 
+as $T$-level just below (Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$) 
+either corresponds to the ocean floor or is inside the bathymetry while the last 
+$T$-level is always inside the bathymetry (Fig.\ref{Fig_index_vert}). Note that 
+for an increasing $k$ index, a $w$-point and the $T$-point just below have the 
+same $k$ index, in opposition to what is done in the horizontal plane where 
+it is the $T$-point and the nearest velocity points in the direction of the horizontal 
+axis that have the same $i$ or $j$ index (compare the dashed area in Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are chosen 
+to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran} code 
+\emph{before all the vertical derivatives} of the discrete equations given in this 
+documentation.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!pt] \label{Fig_index_vert}  \begin{center}
+\includegraphics[width=.90\textwidth]{./Figures/Fig_index_vert.pdf}
+\caption{Vertical integer indexing used in the \textsc{Fortran } code. Note that 
+the $k$-axis is orientated downward. The dashed area indicates the cell in 
+which variables contained in arrays have the same $k$-index.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -----------------------------------
+%        Domain Size
+% -----------------------------------
+\subsubsection{Domain Size}
+\label{DOM_size}
+
+The total size of the computational domain is set by the parameters \jp{jpiglo}, 
+\jp{jpjglo} and \jp{jpk} in the $i$, $j$ and $k$ directions respectively. They are 
+given as parameters in the \mdl{par\_oce} module\footnote{When a specific 
+configuration is used (ORCA2 global ocean, etc...) the parameter are actually 
+defined in additional files introduced by \mdl{par\_oce} module via CPP 
+\textit{include} command. For example, ORCA2 parameters are set in 
+\textit{par\_ORCA\_R2.h90} file}. The use of parameters rather than variables 
+(together with dynamic allocation of arrays) was chosen because it ensured that 
+the compiler would optimize the executable code efficiently, especially on vector 
+machines (optimization may be less efficient when the problem size is unknown 
+at the time of compilation). Nevertheless, it is possible to set up the code with full 
+dynamical allocation by using the AGRIF packaged \citep{Debreu_al_CG2008}. 
+%
+\gmcomment{  add the following ref 
+\colorbox{yellow}{(ref part of the doc)} } 
+%
+Note that are other parameters in \mdl{par\_oce} that refer to the domain size. 
+The two parameters $jpidta$ and $jpjdta$ may be larger than $jpiglo$, $jpjglo$ 
+when the user wants to use only a sub-region of a given configuration. This is 
+the "zoom" capability described in \S\ref{MISC_zoom}. In most applications of 
+the model, $jpidta=jpiglo$, $jpjdta=jpjglo$, and $jpizoom=jpjzoom=1$. Parameters 
+$jpi$ and $jpj$ refer to the size of each processor subdomain when the code is 
+run in parallel using domain decomposition (\key{mpp\_mpi} defined, see 
+\S\ref{LBC_mpp}).
+
+% ================================================================
+% Domain: Horizontal Grid (mesh) 
+% ================================================================
+\section  [Domain: Horizontal Grid (mesh) (\textit{domhgr})]               
+		{Domain: Horizontal Grid (mesh) \small{(\mdl{domhgr} module)} }
+\label{DOM_hgr}
+
+% -------------------------------------------------------------------------------------------------------------
+%        Coordinates and scale factors 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Coordinates and scale factors}
+\label{DOM_hgr_coord_e}
+
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 
+by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 
+The grid-points are located at integer or integer and a half values of as indicated 
+in Table~\ref{Tab_cell}. The associated scale factors are defined using the 
+analytical first derivative of the transformation \eqref{Eq_scale_factors}. These 
+definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which 
+provide the horizontal and vertical meshes, respectively. This section deals with 
+the horizontal mesh parameters.
+
+In a horizontal plane, the location of all the model grid points is defined from the 
+analytical expressions of the longitude $\lambda$ and  latitude $\varphi$ as a 
+function of  $(i,j)$. The horizontal scale factors are calculated using 
+\eqref{Eq_scale_factors}. For example, when the longitude and latitude are 
+function of a single value ($i$ and $j$, respectively) (geographical configuration 
+of the mesh), the horizontal mesh definition reduces to define the wanted 
+$\lambda(i)$, $\varphi(j)$, and their derivatives $\lambda'(i)$ $\varphi'(j)$ in the 
+\mdl{domhgr} module. The model computes the grid-point positions and scale 
+factors in the horizontal plane as follows:
+\begin{flalign*}
+\lambda_T &\equiv \text{glamt}= \lambda(i)	  & \varphi_T &\equiv \text{gphit} = \varphi(j)\\
+\lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\
+\lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\
+\lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2) 
+\end{flalign*}
+\begin{flalign*}
+e_{1T} &\equiv \text{e1t} = r_a |\lambda'(i)		\; \cos\varphi(j)  |&
+e_{2T} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\
+e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)	\; \cos\varphi(j)  |&
+e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\
+e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)		\; \cos\varphi(j+1/2)  |&
+e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\
+e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |&
+e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|
+\end{flalign*}
+where the last letter of each computational name indicates the grid point 
+considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with 
+all universal constants). Note that the horizontal position of and scale factors 
+at $w$-points are exactly equal to those of $T$-points, thus no specific arrays 
+are defined at $w$-points. 
+
+Note that the definition of the scale factors ($i.e.$ as the analytical first derivative 
+of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) is 
+specific to the \NEMO model \citep{Marti1992}. As an example, $e_{1T}$ is defined 
+locally at a $T$-point, whereas many other models on a C grid choose to define 
+such a scale factor as the distance between the $U$-points on each side of the 
+$T$-point. Relying on an analytical transformation has two advantages: firstly, there 
+is no ambiguity in the scale factors appearing in the discrete equations, since they 
+are first introduced in the continuous equations; secondly, analytical transformations 
+encourage good practice by the definition of smoothly varying grids (rather than 
+allowing the user to set arbitrary jumps in thickness between adjacent layers) 
+\citep{Treguier1996}. An example of the effect of such a choice is shown in 
+Fig.~\ref{Fig_zgr_e3}.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_zgr_e3}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_zgr_e3.pdf}
+\caption{Comparison of (a) traditional definitions of grid-point position and grid-size 
+in the vertical, and (b) analytically derived grid-point position and scale factors. For 
+both grids here,  the same $w$-point depth has been chosen but in (a) the 
+$T$-points are set half way between $w$-points while in (b) they are defined from 
+an analytical function: $z(k)=5\,(i-1/2)^3 - 45\,(i-1/2)^2 + 140\,(i-1/2) - 150$. 
+Note the resulting difference between the value of the grid-size $\Delta_k$ and 
+those of the scale factor $e_k$. }
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+%        Choice of horizontal grid
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Choice of horizontal grid}
+\label{DOM_hgr_msh_choice}
+
+The user has three options available in defining a horizontal grid, which involve 
+the parameter $jphgr\_mesh$ of the \mdl{par\_oce} module. 
+\begin{description}
+\item[\jp{jphgr\_mesh}=0]  The most general curvilinear orthogonal grids.
+The coordinates and their first derivatives with respect to $i$ and $j$ are 
+provided in a file, read in \rou{hgr\_read} subroutine of the domhgr module.
+\item[\jp{jphgr\_mesh}=1 to 5] A few simple analytical grids are provided (see below). 
+For other analytical grids, the \mdl{domhgr} module must be modified by the user. 
+\end{description}
+
+There are two simple cases of geographical grids on the sphere. With 
+\jp{jphgr\_mesh}=1, the grid (expressed in degrees) is regular in space, 
+with grid sizes specified by parameters \pp{ppe1\_deg} and \pp{ppe2\_deg}, 
+respectively. Such a geographical grid can be very anisotropic at high latitudes 
+because of the convergence of meridians (the zonal scale factors $e_1$ 
+become much smaller than the meridional scale factors $e_2$). The Mercator 
+grid (\jp{jphgr\_mesh}=4) avoids this anisotropy by refining the meridional scale 
+factors in the same way as the zonal ones. In this case, meridional scale factors 
+and latitudes are calculated analytically using the formulae appropriate for 
+a Mercator projection, based on \pp{ppe1\_deg} which is a reference grid spacing 
+at the equator (this applies even when the geographical equator is situated outside 
+the model domain). 
+%%%
+\gmcomment{ give here the analytical expression of the Mercator mesh}
+%%%
+In these two cases (\jp{jphgr\_mesh}=1 or 4), the grid position is defined by the 
+longitude and latitude of the south-westernmost point (\pp{ppglamt0} 
+and \pp{ppgphi0}). Note that for the Mercator grid the user need only provide 
+an approximate starting latitude: the real latitude will be recalculated analytically, 
+in order to ensure that the equator corresponds to line passing through $T$- 
+and $u$-points.  
+
+Rectangular grids ignoring the spherical geometry are defined with 
+\jp{jphgr\_mesh} = 2, 3, 5. The domain is either an $f$-plane (\jp{jphgr\_mesh} = 2, 
+Coriolis factor is constant) or a beta-plane (\jp{jphgr\_mesh} = 3, the Coriolis factor 
+is linear in the $j$-direction). The grid size is uniform in meter in each direction, 
+and given by the parameters \pp{ppe1\_m} and \pp{ppe2\_m} respectively. 
+The zonal grid coordinate (\textit{glam} arrays) is in kilometers, starting at zero 
+with the first $T$-point. The meridional coordinate (gphi. arrays) is in kilometers, 
+and the second $T$-point corresponds to coordinate $gphit=0$. The input 
+parameter \pp{ppglam0} is ignored. \pp{ppgphi0} is used to set the reference 
+latitude for computation of the Coriolis parameter. In the case of the beta plane, 
+\pp{ppgphi0} corresponds to the center of the domain. Finally, the special case 
+\jp{jphgr\_mesh}=5 corresponds to a beta plane in a rotated domain for the 
+GYRE configuration, representing a classical mid-latitude double gyre system. 
+The rotation allows us to maximize the jet length relative to the gyre areas 
+(and the number of grid points). 
+
+The choice of the grid must be consistent with the boundary conditions specified 
+by the parameter \jp{jperio} (see {\S\ref{LBC}).
+
+% -------------------------------------------------------------------------------------------------------------
+%        Grid files
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Grid files}
+\label{DOM_hgr_files}
+
+All the arrays relating to a particular ocean model configuration (grid-point 
+position, scale factors, masks) can be saved in files if $\np{nmsh} \not= 0$ 
+(namelist parameter). This can be particularly useful for plots and off-line 
+diagnostics. In some cases, the user may choose to make a local modification 
+of a scale factor in the code. This is the case in global configurations when 
+restricting the width of a specific strait (usually a one-grid-point strait that 
+happens to be too wide due to insufficient model resolution). An example 
+is Gibraltar Strait in the ORCA2 configuration. When such modifications are done, 
+the output grid written when $\np{nmsh} \not=0$ is no more equal to the input grid.
+
+% ================================================================
+% Domain: Vertical Grid (domzgr)
+% ================================================================
+\section  [Domain: Vertical Grid (\textit{domzgr})]
+		{Domain: Vertical Grid \small{(\mdl{domzgr} module)} }
+\label{DOM_zgr}
+%-----------------------------------------nam_zgr & namdom-------------------------------------------
+\namdisplay{nam_zgr} 
+\namdisplay{namdom} 
+%-------------------------------------------------------------------------------------------------------------
+
+In the vertical, the model mesh is determined by four things: 
+(1) the bathymetry given in meters ; 
+(2) the number of levels of the model (\jp{jpk}) ; 
+(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors 
+(derivatives of the transformation) ; 
+and (4) the masking system, $i.e.$ the number of wet model levels at each 
+$(i,j)$ column of points.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \label{Fig_z_zps_s_sps}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./Figures/Fig_z_zps_s_sps.pdf}
+\caption{The ocean bottom as seen by the model: 
+(a) $z$-coordinate with full step, 
+(b) $z$-coordinate with partial step, 
+(c) $s$-coordinate: terrain following representation, 
+(d) hybrid $s-z$ coordinate, 
+(e) hybrid $s-z$ coordinate with partial step, and 
+(f) same as (e) but with variable volume associated with the non-linear free surface. 
+Note that the variable volume option (\key{vvl}) can be used with any of the 
+5 coordinates (a) to (e).}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The choice of a vertical coordinate, even if it is made through a namelist parameter, 
+must be done once of all at the beginning of an experiment. It is not intended as an 
+option which can be enabled or disabled in the middle of an experiment. Three main 
+choices are offered (Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step 
+bathymetry (\np{ln\_zco}=true), $z$-coordinate with partial step bathymetry 
+(\np{ln\_zps}=true), or generalized, $s$-coordinate (\np{ln\_sco}=true). 
+Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate 
+(Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). When using the variable 
+volume option \key{vvl}) ($i.e.$ non-linear free surface), the coordinate follow the 
+time-variation of the free surface so that the transformation is time dependent: 
+$z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). This option can be used with full step 
+bathymetry or $s$-coordinate (hybride and partial step coordinates have not 
+yet been tested in NEMO v2.3). 
+
+Contrary to the horizontal grid, the vertical grid is computed in the code and no 
+provision is made for reading it from a file. The only input file is the bathymetry 
+(in meters)\footnote{N.B. in full step $z$-coordinate, a \textit{bathy\_level} file can 
+replace the \textit{bathy\_meter} file, so that the computation of the number of 
+wet ocean point in each water column is by-passed}. After reading the bathymetry, 
+the algorithm for vertical grid definition differs between the different options:
+\begin{description}
+\item[\textit{zco}] set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
+\item[\textit{zps}] set a reference coordinate transformation $z_0 (k)$, and 
+calculate the thickness of the deepest level at each $(i,j)$ point using the 
+bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
+\item[\textit{sco}] smooth the bathymetry to fulfil the hydrostatic consistency 
+criteria and set the three-dimensional transformation.
+\item[\textit{s-z} and \textit{s-zps}] smooth the bathymetry to fulfil the hydrostatic 
+consistency criteria and set the three-dimensional transformation $z(i,j,k)$, and 
+possibly introduce masking of extra land points to better fit the original bathymetry file
+\end{description}
+%%%
+\gmcomment{   add the description of the smoothing:  envelop topography...}
+%%%
+
+Generally, the arrays describing the grid point depths and vertical scale factors 
+are three dimensional arrays $(i,j,k)$. In the special case of $z$-coordinates with 
+full step bottom topography, it is possible to define those arrays as one-dimensional, 
+in order to save memory. This is performed by defining the \key{zco} 
+C-Pre-Processor (CPP) key. To improve the code readability while providing this 
+flexibility, the vertical coordinate and scale factors are defined as functions of 
+$(i,j,k)$ with "fs" as prefix (examples: \textit{fsdeptht, fse3t,} etc) that can be either 
+three-dimensional arrays, or a one dimensional array when \key{zco} is defined. 
+These functions are defined in the file \hf{domzgr\_substitute} of the DOM directory. 
+They are used throughout the code, and replaced by the corresponding arrays at 
+the time of pre-processing (CPP capability).
+
+% -------------------------------------------------------------------------------------------------------------
+%        Meter Bathymetry
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Meter Bathymetry}
+\label{DOM_bathy}
+
+Three options are possible for defining the bathymetry, according to the 
+namelist variable \np{ntopo}: 
+\begin{description}
+\item[\np{ntopo} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ 
+is given by the coordinate transformation. The domain can either be a closed 
+basin or a periodic channel depending on the parameter \jp{jperio}. 
+\item[\np{ntopo} = -1] a domain with a bump of topography one third of the 
+domain width at the central latitude. This is meant for the "EEL-R5" configuration, 
+a periodic or open boundary channel with a seamount. 
+\item[\np{ntopo} = 1] read a bathymetry. The bathymetry file (Netcdf format) 
+provides the ocean depth (positive, in meters) at each grid point of the model grid. 
+The bathymetry is usually built by interpolating a standard bathymetry product 
+($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also 
+defines the coastline: where the bathymetry is zero, no model levels are defined 
+(all levels are masked).
+\end{description}
+
+When using the rigid lid approximation (\key{dynspg\_rl} is defined) isolated land 
+masses (islands) must be identified by negative integers in the input bathymetry file 
+(see \S\ref{MISC_solisl}).
+
+When a global ocean is coupled to an atmospheric model it is better to represent 
+all large water bodies (e.g, great lakes, Caspian sea...) even if the model 
+resolution does not allow their communication with the rest of the ocean. 
+This is unnecessary when the ocean is forced by fixed atmospheric conditions, 
+so these seas can be removed from the ocean domain. The user has the option 
+to set the bathymetry in closed seas to zero (see \S\ref{MISC_closea}), but the 
+code has to be adapted to the user's configuration. 
+
+% -------------------------------------------------------------------------------------------------------------
+%        z-coordinate  and reference coordinate transformation
+% -------------------------------------------------------------------------------------------------------------
+\subsection[$z$-coordinate (\np{ln\_zco} or \key{zco})]
+	  	  {$z$-coordinate (\np{ln\_zco}=.true. or \key{zco}) and reference coordinate}
+\label{DOM_zco}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \label{Fig_zgr}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_zgr.pdf}
+\caption{Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level 
+functions for (a) T-point depth and (b) the associated scale factor as computed 
+from \eqref{DOM_zgr_ana} using \eqref{DOM_zgr_coef} in $z$-coordinate.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ 
+and $gdepw_0$ for $T$- and $w$-points, respectively. As indicated on 
+Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the 
+ocean surface. There are at most \jp{jpk}-1 $T$-points inside the ocean, the 
+additional $T$-point at $jk=jpk$ is below the sea floor and is not used. 
+The vertical location of $w$- and $T$-levels is defined from the analytic expression 
+of the depth $z_0(k)$ whose analytical derivative with respect to $k$ provides the 
+vertical scale factors. The user must provide the analytical expression of both 
+$z_0$ and its first derivative with respect to $k$. This is done in routine \mdl{domzgr} 
+through statement functions, using parameters provided in the \textit{par\_oce.h90} file. 
+
+It is possible to define a simple regular vertical grid by giving zero stretching (\pp{ppacr=0}). In that case, the parameters \jp{jpk} (number of $w$-levels) and \pp{pphmax} (total ocean depth in meters) fully define the grid. 
+
+For climate-related studies it is often desirable to concentrate the vertical resolution 
+near the ocean surface. The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps): 
+\begin{equation} \label{DOM_zgr_ana}
+\begin{split}
+ z_0 (k) 	&= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ 
+ e_3^0 (k) 	&= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right| 
+\end{split}
+\end{equation}
+where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. Such an 
+expression allows us to define a nearly uniform vertical location of levels at the 
+ocean top and bottom with a smooth hyperbolic tangent transition in between 
+(Fig.~\ref{Fig_zgr}).
+
+The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the 
+surface (bottom) layers and a depth which varies from 0 at the sea surface to a 
+minimum of $-5000~m$. This leads to the following conditions:
+\begin{equation} \label{DOM_zgr_coef}
+\begin{split}
+ e_3 (1+1/2)		&=10. \\ 
+ e_3 (jpk-1/2)	&=500. \\ 
+ z(1)			&=0. \\ 
+ z(jpk)			&=-5000. \\ 
+\end{split}
+\end{equation}
+
+With the choice of the stretching $h_{cr} =3$ and the number of levels 
+\jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in 
+\eqref{DOM_zgr_ana} have been determined such that \eqref{DOM_zgr_coef} is 
+satisfied, through an optimisation procedure using a bisection method. For the first 
+standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$, 
+$h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and 
+scale factors as a function of the model levels are shown in Fig.~\ref{Fig_zgr} and 
+given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters 
+\pp{ppsur}, \pp{ppa0}, \pp{ppa1}, \pp{ppkth} in the parameter file \mdl{par\_oce}. 
+
+Rather than entering parameters $h_{sur}$, $h_{0}$, and $h_{1}$ directly, it is 
+possible to recalculate them. In that case the user sets 
+\pp{ppsur}=\pp{ppa0}=\pp{ppa1}=\pp{pp\_to\_be\_computed}, in \mdl{par\_oce}, 
+and specifies instead the four following parameters:
+\begin{itemize}
+\item 	\pp{ppacr}=$h_{cr} $: stretching factor (nondimensional). The larger 
+\pp{ppacr}, the smaller the stretching. Values from $3$ to $10$ are usual.
+\item 	\pp{ppkth}=$h_{th} $: is approximately the model level at which maximum 
+stretching occurs (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
+\item 	\pp{ppdzmin}: minimum thickness for the top layer (in meters)
+\item 	\pp{pphmax}: total depth of the ocean (meters).
+\end{itemize}
+As an example, for the $45$ layers used in the DRAKKAR configuration those 
+parameters are: \jp{jpk}=46, \pp{ppacr}=9, \pp{ppkth}=23.563, \pp{ppdzmin}=6m, 
+\pp{pphmax}=5750m.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{table} \label{Tab_orca_zgr}
+\begin{center} \begin{tabular}{c||r|r|r|r}
+\hline
+\textbf{LEVEL}& \textbf{GDEPT}& \textbf{GDEPW}& \textbf{E3T }& \textbf{E3W  } \\ \hline
+1	&	\textbf{  5.00}	&       0.00 &	\textbf{ 10.00} &	 10.00 \\	\hline
+2	&	\textbf{15.00}	& 	  10.00 &	\textbf{ 10.00} &	 10.00 \\	\hline
+3	&	\textbf{25.00}	&	  20.00 &	\textbf{ 10.00} & 	 10.00 \\	\hline
+4	&	\textbf{35.01}	&	  30.00 & 	\textbf{ 10.01} & 	 10.00 \\	\hline
+5	&	\textbf{45.01}	&	  40.01 &	\textbf{ 10.01} &	 10.01 \\	\hline
+6	&	\textbf{55.03}	&	  50.02 &	\textbf{ 10.02} & 	 10.02 \\	\hline
+7	&	\textbf{65.06}	&	  60.04 &	\textbf{ 10.04} &	 10.03 \\	\hline
+8	&	\textbf{75.13}	&	  70.09 &	\textbf{ 10.09} &	 10.06 \\	\hline
+9	&	\textbf{85.25}	&	  80.18 &	\textbf{ 10.17} &	 10.12 \\	\hline
+10	& 	\textbf{95.49}	& 	  90.35 &	\textbf{ 10.33} &	 10.24 \\	\hline
+11	& 	\textbf{105.97}	& 	 100.69 &	\textbf{ 10.65} &	 10.47 \\	\hline
+12	& 	\textbf{116.90}	& 	 111.36 &	\textbf{ 11.27} &	 10.91 \\	\hline
+13	& 	\textbf{128.70}	& 	 122.65 &	\textbf{ 12.47} &	 11.77 \\	\hline
+14	& 	\textbf{142.20}	& 	 135.16 &	\textbf{ 14.78} &	 13.43 \\	\hline
+15	& 	\textbf{158.96}	& 	 150.03 &	\textbf{ 19.23} &	 16.65 \\	\hline
+16	& 	\textbf{181.96}	& 	 169.42 &	\textbf{ 27.66} &	 22.78 \\	\hline
+17	& 	\textbf{216.65}	& 	 197.37 & 	\textbf{ 43.26} &	 34.30 \\ \hline
+18	& 	\textbf{272.48}	& 	 241.13 & 	\textbf{ 70.88} &	 55.21 \\ \hline
+19	& 	\textbf{364.30}	& 	 312.74 & 	\textbf{116.11} &	 90.99 \\ \hline
+20	& 	\textbf{511.53}	& 	 429.72 & 	\textbf{181.55} & 	146.43 \\ \hline
+21	& 	\textbf{732.20}	& 	 611.89 & 	\textbf{261.03} & 	220.35 \\ \hline
+22	& 	\textbf{1033.22}&	 872.87 & 	\textbf{339.39} & 	301.42 \\ \hline
+23	& 	\textbf{1405.70}&	1211.59 & \textbf{402.26} & 	373.31 \\ \hline
+24	& 	\textbf{1830.89}&	1612.98 & \textbf{444.87} & 	426.00 \\ \hline
+25	& 	\textbf{2289.77}&	2057.13 & \textbf{470.55} & 	459.47 \\ \hline
+26	& 	\textbf{2768.24}&	2527.22 & \textbf{484.95} & 	478.83 \\ \hline
+27	& 	\textbf{3257.48}&	3011.90 & \textbf{492.70} & 	489.44 \\ \hline
+28	& 	\textbf{3752.44}&	3504.46 & \textbf{496.78} & 	495.07 \\ \hline
+29	& 	\textbf{4250.40}&	4001.16 & \textbf{498.90} & 	498.02 \\ \hline
+30	& 	\textbf{4749.91}&	4500.02 & \textbf{500.00} &	499.54 \\ \hline
+31	& 	\textbf{5250.23}&	5000.00 &	\textbf{500.56} &	500.33 \\ \hline
+\end{tabular} \end{center} 
+\caption{Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration 
+as computed from \eqref{DOM_zgr_ana} using the coefficients given in 
+\eqref{DOM_zgr_coef}}
+\end{table}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+%        z-coordinate with partial step
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [$z$-coordinate with partial step (\np{ln\_zps})]
+			{$z$-coordinate with partial step (\np{ln\_zps}=.true.)}
+\label{DOM_zps}
+%--------------------------------------------namdom-------------------------------------------------------
+\namdisplay{namdom} 
+%--------------------------------------------------------------------------------------------------------------
+
+In $z$-coordinate partial step, the depths of the model levels are defined by the 
+reference analytical function $z_0 (k)$ as described in the previous 
+section, \emph{except} in the bottom layer. The thickness of the bottom layer is 
+allowed to vary as a function of geographical location $(\lambda,\varphi)$ to allow a 
+better representation of the bathymetry, especially in the case of small 
+slopes (where the bathymetry varies by less than one level thickness from 
+one grid point to the next). The reference layer thicknesses $e_{3t}^0$ have been 
+defined in the absence of bathymetry. With partial steps, layers from 1 to 
+\jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$. The model deepest layer (\jp{jpk}-1) is 
+allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: the 
+maximum thickness allowed is $2*e_{3t}(jpk-1)$. This has to be kept in mind when 
+specifying the maximum depth \pp{pphmax} in partial steps: for example, with 
+\pp{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean depth allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$). Two 
+variables in the namdom namelist are used to define the partial step 
+vertical grid. The mimimum water thickness (in meters) allowed for a cell 
+partially filled with bathymetry at level jk is the minimum of \np{e3zpsmin} 
+(thickness in meters, usually $20~m$) or $e_{3t}(jk)*\np{e3zps\_rat}$ (a fraction, 
+usually 10\%, of the default thickness $e_{3t}(jk)$).
+
+ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level }
+
+% -------------------------------------------------------------------------------------------------------------
+%        s-coordinate
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [$s$-coordinate (\np{ln\_sco})]
+			{$s$-coordinate (\np{ln\_sco}=true)}
+\label{DOM_sco}
+%------------------------------------------nam_zgr_sco---------------------------------------------------
+\namdisplay{nam_zgr_sco} 
+%--------------------------------------------------------------------------------------------------------------
+In $s$-coordinate (\key{sco} is defined), the depth and thickness of the model 
+levels are defined from the product of a depth field and either a stretching 
+function or its derivative, respectively:
+\begin{equation} \label{DOM_sco_ana}
+\begin{split}
+ z(k) 		&= h(i,j) \; z_0(k)	\\
+ e_3(k)	&= h(i,j) \; z_0'(k)
+\end{split}
+\end{equation}
+where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $T$-point 
+location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea 
+surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean 
+depth, since a mixed step-like and bottom-following representation of the 
+topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e). In the example provided (\hf{zgr\_s} file) $h$ is a smooth envelope bathymetry and steps are used to represent sharp bathymetric gradients.
+
+A new flexible stretching function, modified from \citet{Song1994} is provided as an example:
+\begin{equation} \label{DOM_sco_function}
+\begin{split}
+z	&= h_c +( h-h_c)\;c s)	\\
+c(s)	&=  \frac{ \left[	  \tanh{ \left( \theta \, (s+b) \right)} 
+	  	 			- \tanh{ \left(  \theta \, b      \right)}  \right]}
+		      {2\;\sinh \left( \theta \right)}
+\end{split}
+\end{equation}
+where $h_c$ is the thermocline depth and $\theta$ and $b$ are the surface and 
+bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 
+$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom increase of the vertical resolution (Fig.~\ref{Fig_sco_function}).
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \label{Fig_sco_function}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./Figures/Fig_sco_function.pdf}
+\caption{Examples of the stretching function applied to a sea mont; from left to right: surface, surface and bottom, and bottom intensified resolutions}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+%        z*- or s*-coordinate
+% -------------------------------------------------------------------------------------------------------------
+\subsection{$z^*$- or $s^*$-coordinate (add \key{vvl}) }
+\label{DOM_zgr_vvl}
+
+This option is described in the Report by Levier \textit{et al.} (2007), available on 
+the \NEMO web site. 
+
+%gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances
+
+% -------------------------------------------------------------------------------------------------------------
+%        level bathymetry and mask 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{level bathymetry and mask}
+\label{DOM_msk}
+
+Whatever the vertical coordinate used, the model offers the possibility of 
+representing the bottom topography with steps that follow the face of the 
+model cells (step like topography) \citep{Madec1996}. The distribution of 
+the steps in the horizontal is defined in a 2D integer array, mbathy, which 
+gives the number of ocean levels ($i.e.$ those that are not masked) at each 
+$T$-point. mbathy is computed from the meter bathymetry using the definiton of 
+gdept as the number of $T$-points which gdept $\leq$ bathy. Note that in version 
+NEMO v2.3, the user still has to provide the "level" bathymetry in a NetCDF
+file when using the full step option (\np{ln\_zco}), rather than the bathymetry 
+in meters: both will be allowed in future versions.
+
+Modifications of the model bathymetry are performed in the \textit{bat\_ctl} 
+routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points 
+that do not communicate with another ocean point at the same level are eliminated.
+
+In the case of the rigid-lid approximation when islands occur in the computational 
+domain (\np{ln\_dynspg\_rl}=.true. and \key{island} is defined), the \textit{mbathy} 
+array must be provided and takes values from $-N$ to \jp{jpk}-1. It provides the 
+following information: $mbathy(i,j) = -n, \ n \in \left] 0,N \right]$, $T$-points are 
+land points on the $n^{th}$ island ; $mbathy(i,j) =0$, $T$-points are land points 
+on the main land (continent) ; $mbathy(i,j) =k$, the first $k$ $T$- and $w$-points 
+are ocean points, the others are points below the ocean floor. 
+
+This is used to compute the island barotropic stream function used in the rigid lid 
+computation (see \S\ref{MISC_solisl}).
+
+From the \textit{mbathy} array, the mask fields are defined as follows:
+\begin{align*}
+tmask(i,j,k) &= \begin{cases}   \; 1&   \text{ if $k\leq mbathy(i,j)$  }    \\
+                                                \; 0&   \text{ if $k\leq mbathy(i,j)$  }    \end{cases}     \\
+umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)	\\
+vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k)	\\
+fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)	\\
+                   & \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k)
+\end{align*}
+
+\gmcomment{ STEVEN: are the dots multiplications?}      
+
+Note that \textit{wmask} is not defined as it is exactly equal to \textit{tmask} with 
+the numerical indexing used (\S~\ref{DOM_Num_Index}). Moreover, the 
+specification of closed lateral boundaries requires that at least the first and last 
+rows and columns of the \textit{mbathy} array are set to zero. In the particular 
+case of an east-west cyclical boundary condition, \textit{mbathy} has its last 
+column equal to the second one and its first column equal to the last but one 
+(and so too the mask arrays) (see \S~\ref{LBC_jperio}).
+
+%%%
+\gmcomment{   \colorbox{yellow}{Add one word on tricky trick !} mbathy in further modified in zdfbfr{\ldots}.  }
+%%%
+
+% ================================================================
+% Time Discretisation
+% ================================================================
+\section{Time Discretisation}
+\label{DOM_nxt}
+
+The time stepping used in \NEMO is a three level scheme that can be 
+represented as follows:
+\begin{equation} \label{Eq_DOM_nxt}
+   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t-\Delta t,t,t+\Delta t}
+\end{equation} 
+where $x$ stands for $u$, $v$, $T$ or $S$; RHS is the Right-Hand-Side of the 
+corresponding time evolution equation; $\Delta t$ is the time step; and the 
+superscripts indicate the time at which a quantity is evaluated. Each term of the 
+RHS is evaluated at a specific time step(s) depending on the physics with which 
+it is associated. 
+
+The choice of the time step used for this evaluation is discussed below as 
+well as the implications in terms of starting or restarting a model 
+simulation. Note that the time stepping is generally performed in a one step 
+operation. With such a complex and nonlinear system of equations it would be dangerous to let a prognostic variable evolve in time for each term separately.
+%%%
+\gmcomment{ STEVEN  suggest separately instead of successively...  wrong?}
+%%%
+
+The three level scheme requires three arrays for each prognostic variables. 
+For each variable $x$ there is $x_b$ (before) and $x_n$ (now). The third array, 
+although referred to as $x_a$ (after) in the code, is usually not the variable at 
+the next time step; but rather it is used to store the time derivative (RHS in 
+\eqref{Eq_DOM_nxt}) prior to time-stepping the equation. Generally, the time 
+stepping is performed once at each time step in \mdl{tranxt} and \mdl{dynnxt} 
+modules, except for implicit vertical diffusion or sea surface height when 
+time-splitting options are used.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Non-Diffusive Part---Leapfrog Scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Non-Diffusive Part --- Leapfrog Scheme}
+\label{DOM_nxt_leap_frog}
+
+The time stepping used for non-diffusive processes is the well-known 
+leapfrog scheme. It is a time centred scheme, i.e. the RHS is evaluated at 
+time step $t$, the now time step. It is only used for non-diffusive terms, 
+that is momentum and tracer advection, pressure gradient, and Coriolis 
+terms. This scheme is widely used for advective processes in low-viscosity 
+fluids. It is an efficient method that achieves second-order accuracy with 
+just one right hand side evaluation per time step. Moreover, it does not 
+artificially damp linear oscillatory motion nor does it produce instability 
+by amplifying the oscillations. These advantages are somewhat diminished by 
+the large phase-speed error of the leapfrog scheme, and the unsuitability of 
+leapfrog differencing for the representation of diffusive and Rayleigh 
+damping processes. However, the most serious problem associated with the 
+leapfrog scheme is a high-frequency computational noise called 
+"time-splitting" \citep{Haltiner1980} that develops when the method 
+is used to model non linear fluid dynamics: the even and odd time steps tend 
+to diverge into a physical and a computational mode. Time splitting can 
+be controlled through the use of an Asselin time filter (first designed by 
+\citep{Robert1966} and more comprehensively studied by \citet{Asselin1972}), or by 
+periodically reinitialising the leapfrog solution through a single 
+integration step with a two-level scheme. In \NEMO we follow the first 
+strategy:
+\begin{equation} \label{Eq_DOM_nxt_asselin}
+x_F^t  = x^t + \gamma \, \left[ x_f^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right]
+\end{equation} 
+where the subscript $f$ denotes filtered values and $\gamma$ is the Asselin 
+coefficient. $\gamma$ is initialized as \np{atfp} (namelist parameter). 
+Its default value is \np{atfp}=0.1.  This default value causes a significant dissipation 
+of high frequency motions. Recommended values in idealized studies of shallow 
+water turbulence are two orders of magnitude smaller (\citep{Farge1987}). 
+Both strategies do, nevertheless, degrade the accuracy of the calculation from 
+second to first order. The leapfrog scheme combined with a Robert-Asselin 
+time filter has been preferred to other time differencing schemes such as 
+predictor corrector or trapezoidal schemes, because the user has an explicit 
+and simple control of the magnitude of the time diffusion of the scheme. 
+In association with the 2nd order centred space discretisation of the 
+advective terms in the momentum and tracer equations, it avoids implicit 
+numerical diffusion in both the time and space discretisations of the 
+advective term: they are both set explicitly by the user through the Robert-Asselin 
+filter parameter and the viscous and diffusive coefficients.
+
+\gmcomment{
+%gm - reflexion about leapfrog: ongoing work with Matthieu Leclair
+% to be updated latter with addition of new time stepping strategy
+\colorbox{yellow}{Note}: 
+The Robert-Asselin time filter slightly departs from a simple second order time 
+diffusive operator computed with a forward time stepping due to the presence of 
+$x_f^{t-\Delta t}$ in the right hand side of  \ref{Eq_DOM_nxt_asselin}. The original 
+willing of Robert1966 and Asselin1972 was to design a time filter that allow much 
+larger parameter than 0.5.   is due to computer saving consideration. In the original 
+asselin filter, $x^{t-\Delta t}$ is used instead:
+ \begin{equation} \label{Eq_DOM_nxt_asselin_true}
+x_f^t  = x^t + \gamma \, \left[ x^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right]
+\end{equation} 
+Applying a "true" Asselin time filter is nothing more than adding a harmonic 
+diffusive operator in time. Indeed, equations \ref{Eq_DOM_nxt} and 
+\ref{Eq_DOM_nxt_asselin_true} can be rewritten together as:
+\begin{equation} \label{Eq_DOM_nxt2}
+\begin{split}
+  \frac{ x^{t+\Delta t} - x^{t-\Delta t} } { 2 \,\Delta t } 
+  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} + \frac{ x_f^t  - x^t }{2 \,\Delta t} \\
+  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} 
+    + \gamma\ \frac{  \, \left[ x^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right] }{2 \,\Delta t}  \\
+  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} 
+  + 2 \Delta t \ \gamma \ \frac{1}{{2 \Delta t}^2}  
+  	\,\delta_{t-1}\,\left[ \delta_{t+1/2}\left[ x^t \right] \right] 
+  \end{split}
+\end{equation} 
+expressing this again in a continuous form, one finds that the Asselin filter leads to :
+\begin{equation} \label{Eq_DOM_nxt3}
+  \frac{ \partial x} { \partial t } =  \text{RHS} + 2 \,\Delta t \ \gamma \ \frac{ {\partial}^2 x}{ \partial t ^2 }
+\end{equation} 
+
+Equations  \ref{Eq_DOM_nxt2} and \ref{Eq_DOM_nxt3} suggest several remarks. 
+First the Asselin filter is definitively a second order time diffusive operator which is 
+evaluated at centered time step. The magnitude of this diffusion is proportional to 
+the time step (with $\gamma$ usually taken between $10^{-1}$ to $10^{-3}$). 
+Second, this term has to be taken into account in all budgets of the equations 
+(mass, heat content, salt content, kinetic energy...). Nevertheless,we stress here 
+that it is small and does not create systematic biases. Indeed let us evaluate how 
+it contributes to the time evolution of $x$ between $t_o$ and $t_1$:
+\begin{equation} \label{Eq_DOM_nxt4}
+\begin{split}
+ t_1-t_o &= \int_{t_o}^{t_1} \frac{ \partial x} { \partial t } dt \\
+ 		&= \int_{t_o}^{t_1} \text{RHS} dt + 2 \,\Delta t \ \gamma \left(
+		  \left. \frac{ \partial x}{ \partial t } \right|_1 
+		- \left. \frac{ \partial x}{ \partial t } \right|_o  \right)
+ \end{split}
+\end{equation} 
+}
+
+Alternative time stepping schemes are currently under investigation.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Diffusive Part---Forward or Backward Scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Diffusive Part --- Forward or Backward Scheme}
+\label{DOM_nxt_forward_imp}
+
+The leapfrog differencing scheme is unsuitable for the representation of 
+diffusive and damping processes. For a tendancy $D_x$, representing a 
+diffusive term or a restoring term to a tracer climatology 
+(when present, see \S~\ref{TRA_dmp}), a forward time differencing scheme
+ is used :
+\begin{equation} \label{Eq_DOM_nxt_euler}
+   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \ {D_x}^{t-\Delta t}
+\end{equation} 
+
+This is diffusive in time and conditionally stable. For example, the 
+conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies2004}:
+\begin{equation} \label{Eq_DOM_nxt_euler_stability}
+A^h < \left\{
+\begin{aligned}
+                    &\frac{e^2}{  8 \; \Delta t } 	&&\quad \text{laplacian diffusion} 	\\
+                    &\frac{e^4}{64 \; \Delta t } 	&&\quad \text{bilaplacian diffusion} 
+            \end{aligned}
+\right.
+\end{equation}
+where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. The linear constraint \eqref{Eq_DOM_nxt_euler_stability} is a necessary condition, but not sufficient. If it is not satisfied, even mildly, then the model soon becomes wildly unstable. The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient.
+
+For the vertical diffusion terms, a forward time differencing scheme can be 
+used, but usually the numerical stability condition implies a strong 
+constraint on the time step. Two solutions are available in \NEMO to overcome 
+the stability constraint: $(a)$ a forward time differencing scheme using a 
+time splitting technique (\np{ln\_zdfexp}=.true.) or $(b)$ a backward (or implicit) 
+time differencing scheme by \np{ln\_zdfexp}=.false.). In $(a)$, the master 
+time step $\Delta $t is cut into $N$ fractional time steps so that the 
+stability criterion is reduced by a factor of $N$. The computation is done as 
+follows:
+\begin{equation} \label{Eq_DOM_nxt_ts}
+\begin{split}
+& u_\ast ^{t-\Delta t} = u^{t-\Delta t}   \\
+& u_\ast ^{t-\Delta t+L\frac{2\Delta t}{N}}=u_\ast ^{t-\Delta t+\left( {L-1} 
+\right)\frac{2\Delta t}{N}}+\frac{2\Delta t}{N}\;\text{DF}^{t-\Delta t+\left( {L-1} \right)\frac{2\Delta t}{N}}
+        \quad \text{for $L=1$ to $N$}      \\
+& u^{t+\Delta t} = u_\ast^{t+\Delta t}
+\end{split}
+\end{equation}
+with DF a vertical diffusion term. The number of fractional time steps, $N$, is given 
+by setting \np{n\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally 
+stable but diffusive. It can be written as follows:
+\begin{equation} \label{Eq_DOM_nxt_imp}
+   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t+\Delta t}
+\end{equation} 
+
+This scheme is rather time consuming since it requires a matrix inversion, 
+but it becomes attractive since a splitting factor of 3 or more is needed 
+for the forward time differencing scheme. For example, the finite difference 
+approximation of the temperature equation is:
+\begin{equation} \label{Eq_DOM_nxt_imp_zdf}
+\frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\Delta t}\equiv \text{RHS}+\frac{1}{e_{3T} }\delta 
+_k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 
+\right]
+\end{equation}
+where RHS is the right hand side of the equation except for the vertical diffusion term. We rewrite \eqref{Eq_DOM_nxt_imp} as:
+\begin{equation} \label{Eq_DOM_nxt_imp_mat}
+-c(k+1)\;u^{t+1}(k+1)+d(k)\;u^{t+1}(k)-\;c(k)\;u^{t+1}(k-1) \equiv b(k)
+\end{equation}
+where 
+\begin{align*} 
+ c(k) &= A_w^{vm} (k) \, / \, e_{3uw} (k)     \\
+ d(k) &= e_{3u} (k)       \, / \, (2\Delta t) + c_k + c_{k+1}    \\
+ b(k) &= e_{3u} (k) \; \left( u^{t-1}(k) \, / \, (2\Delta t) + \text{RHS} \right)    
+\end{align*}
+
+\eqref{Eq_DOM_nxt_imp_mat} is a linear system of equations which associated 
+matrix is tridiagonal. Moreover, $c(k)$ and $d(k)$ are positive and the diagonal 
+term is greater than the sum of the two extra-diagonal terms, therefore a special 
+adaptation of the Gauss elimination procedure is used to find the solution 
+(see for example \citet{Richtmyer1967}).
+
+% -------------------------------------------------------------------------------------------------------------
+%        Start/Restart strategy
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Start/Restart strategy}
+\label{DOM_nxt_rst}
+%--------------------------------------------namrun-------------------------------------------
+\namdisplay{namrun}         
+%--------------------------------------------------------------------------------------------------------------
+
+The first time step of this three level scheme when starting from initial conditions 
+is a forward step (Euler time integration): 
+\begin{equation} \label{Eq_DOM_euler}
+	x^1 = x^0 + \Delta t \ \text{RHS}^0
+\end{equation}
+
+It is also possible to restart from a previous computation, by using a 
+restart file. The restart strategy is designed to ensure perfect 
+restartability of the code: the user should obtain the same results to 
+machine precision either by running the model for $2N$ time steps in one go, 
+or by performing two consecutive experiments of $N$ steps with a restart. 
+This requires saving two time levels and many auxiliary data in the restart 
+files in machine precision. 
+
+Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure 
+gradient (see \S\ref{DYN_hpg_imp}), an extra three-dimensional field has to be 
+added in the restart file to ensure an exact restartability. This is done only optionally 
+via the namelist parameter \np{nn\_dynhpg\_rst}, so that a reduction of the size of restart file can be obtained when the restartability is not a key issue (operational oceanography or ensemble simulation for seasonal forcast).
+%%%
+\gmcomment{add here how to force the restart to contain only one time step for operational purposes}
+%%%
+
+\gmcomment{       % add a subsection here  
+
+%-------------------------------------------------------------------------------------------------------------
+%        Time Domain
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Time domain}
+\label{DOM_nxt_time}
+
+ \colorbox{yellow}{add here a few word on nit000 and nitend}
+
+ \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj}
+
+}     	%% end add
+
Index: trunk/DOC/TexFiles/Chapters/Chap_DYN.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_DYN.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_DYN.tex	(revision 994)
@@ -0,0 +1,1110 @@
+% ================================================================
+% Chapter Ñ Ocean Dynamics (DYN)
+% ================================================================
+\chapter{Ocean Dynamics (DYN)}
+\label{DYN}
+\minitoc
+
+% add a figure for  dynvor ens, ene latices
+
+
+$\ $\newline      %force an empty line
+
+Using the representation described in Chap.\ref{DOM}, several semi-discrete 
+space forms of the dynamical equations are available depending on the vertical 
+coordinate used and on the conservation properties of the vorticity term. In all 
+the equations presented here, the masking has been omitted for simplicity. 
+One must be aware that all the quantities are masked fields and that each time a 
+average or difference operator is used, the resulting field is multiplied by a mask.
+
+The prognostic ocean dynamics equation can be summarized as follows:
+\begin{equation*}
+\text{NXT} = \dbinom	{\text{VOR} + \text{KEG} + \text {ZAD} }
+						{\text{COR} + \text{ADV}                       }
+			+ \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
+\end{equation*}
+
+NXT stands for next, referring to the time-stepping. The first group of terms on 
+the rhs of the momentum equations corresponds to the Coriolis and advection 
+terms that are decomposed into a vorticity part (VOR), a kinetic energy part (KEG) 
+and, a vertical advection part (ZAD) in the vector invariant formulation or a Coriolis 
+and advection part(COR+ADV) in the flux formulation. The terms following these 
+are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient, 
+and SPG, Surface Pressure Gradient); and contributions from lateral diffusion 
+(LDF) and vertical diffusion (ZDF), which are added to the rhs in the \mdl{dynldf} 
+and \mdl{dynzdf} modules. The vertical diffusion term includes the surface and 
+bottom stresses. The external forcings and parameterisations require complex 
+inputs (surface wind stress calculation using bulk formulae, estimation of mixing 
+coefficients) that are carried out in modules SBC, LDF and ZDF and are described 
+in Chapters \ref{SBC}, \ref{LDF} and \ref{ZDF}, respectively. 
+
+In the present chapter we also describe the diagnostic equations used to compute 
+the horizontal divergence and curl of the velocities (\emph{divcur} module) as well 
+as the vertical velocity (\emph{wzvmod} module).
+
+The different options available to the user are managed by namelist variables. 
+For equation term \textit{ttt}, the logical namelist variables are \textit{ln\_dynttt\_xxx}, 
+where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 
+If a CPP key is used for this term its name is \textbf{key\_ttt}. The corresponding 
+code can be found in the \textit{dynttt\_xxx} module in the DYN directory, and it is 
+usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
+
+The user has the option of extracting each tendency term of both the rhs of the 
+3D momentum equation (\key{trddyn} defined) for output, as described in 
+Chap.\ref{MISC}.  Furthermore, the tendency terms associated to the 2D 
+barotropic vorticity balance (\key{trdvor} defined) can be derived on-line from the 
+3D terms.
+%%%
+\gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does MISC correspond to "extracting tendency terms" or "vorticity balance"?}
+
+% ================================================================
+% Coriolis and Advection terms: vector invariant form
+% ================================================================
+\section{Coriolis and Advection: vector invariant form}
+\label{DYN_adv_cor_vect}
+%-----------------------------------------nam_dynadv----------------------------------------------------
+\namdisplay{nam_dynadv} 
+%-------------------------------------------------------------------------------------------------------------
+
+The vector invariant form of the momentum equations is the one most 
+often used in applications of \NEMO ocean model. The flux form option 
+(see next section) has been recently introduced in version $2$. 
+Coriolis and momentum 
+advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity 
+appearing in these expressions is centred in time (\textit{now} velocity). 
+At the lateral boundaries either free slip, no slip or partial slip boundary 
+conditions are applied following Chap.\ref{LBC}.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Vorticity term 
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Vorticity term (\textit{dynvor}) ]
+			{Vorticity term (\mdl{dynvor})}
+\label{DYN_vor}
+%------------------------------------------nam_dynvor----------------------------------------------------
+\namdisplay{nam_dynvor} 
+%-------------------------------------------------------------------------------------------------------------
+
+Different discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=.true.) are 
+available: conserving potential enstrophy of horizontally non-divergent flow; 
+conserving horizontal kinetic energy; or conserving potential enstrophy for the 
+relative vorticity term and horizontal kinetic energy for the planetary vorticity term 
+(see  Appendix~\ref{Apdx_C}). The vorticity terms are given below for the general 
+case, but note that in the full step $z$-coordinate (\key{zco} is defined), 
+$e_{3u} =e_{3v} =e_{3f}$ so that the vertical scale factors disappear.
+
+%-------------------------------------------------------------
+%                 enstrophy conserving scheme
+%-------------------------------------------------------------
+\subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=.true.)}
+\label{DYN_vor_ens}
+
+In the enstrophy conserving case (ENS scheme), the discrete formulation of the 
+vorticity term provides a global conservation of the enstrophy 
+($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent 
+flow ($i.e.$ $\chi=0$), but does not conserve the total kinetic energy. It is given by:
+\begin{equation} \label{Eq_dynvor_ens}
+\left\{ 
+\begin{aligned}
+{+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} & {\overline{\overline {\left( {e_{1v} e_{3v} v} \right)}} }^{\,i, j+1/2}    \\
+{-\frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}  & {\overline{\overline {\left( {e_{2u} e_{3u} u} \right)}} }^{\,i+1/2, j}  
+\end{aligned} 
+ \right.
+\end{equation} 
+
+%-------------------------------------------------------------
+%                 energy conserving scheme
+%-------------------------------------------------------------
+\subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=.true.)}
+\label{DYN_vor_ene}
+
+The kinetic energy conserving scheme (ENE scheme) conserves the global 
+kinetic energy but not the global enstrophy. It is given by:
+\begin{equation} \label{Eq_dynvor_ene}
+\left\{ {
+\begin{aligned}
+{+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
+\;\overline {\left( {e_{1v} e_{3v} v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
+{-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
+\;\overline {\left( {e_{2u} e_{3u} u} \right)} ^{\,j+1/2}} }^{\,i} }
+\end{aligned} 
+} \right.
+\end{equation} 
+
+%-------------------------------------------------------------
+%                 mix energy/enstrophy conserving scheme
+%-------------------------------------------------------------
+\subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=.true.) }
+\label{DYN_vor_mix}
+
+The mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the 
+two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens}) 
+to the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied 
+to the planetary vorticity term.
+\begin{equation} \label{Eq_dynvor_mix}
+\left\{ {
+\begin{aligned}
+ {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} 
+ \; {\overline{\overline {\left( {e_{1v} \; e_{3v} \ v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
+ \; {\overline {\left( {\frac{f}{e_{3f} }} \right) 
+ \;\overline {\left( {e_{1v} \; e_{3v} \ v} \right)} ^{\,i+1/2}} }^{\,j} } \\
+{-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
+ \; {\overline{\overline {\left( {e_{2u} \; e_{3u} \ u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
+ \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
+ \;\overline {\left( {e_{2u}\; e_{3u} \ u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
+\end{aligned} 
+} \right.
+\end{equation} 
+
+%-------------------------------------------------------------
+%                 energy and enstrophy conserving scheme
+%-------------------------------------------------------------
+\subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=.true.) }
+\label{DYN_vor_een}
+
+In the energy and enstrophy conserving scheme (EEN scheme), the vorticity term 
+is  evaluated using the vorticity advection scheme of \citet{Arakawa1990}. 
+This scheme conserves both total energy and potential enstrophy in the limit of 
+horizontally nondivergent flow ($i.e. \ \chi=0$). While EEN is more complicated 
+than ENS or ENE and does not conserve potential enstrophy and total energy in 
+general flow, it tolerates arbitrarily thin layers. This feature is essential for 
+$z$-coordinate with partial step. 
+%%%
+\gmcomment{gm :   it actually conserve kinetic energy  !   show that in appendix C }
+%%%
+
+The \citet{Arakawa1990} vorticity advection scheme for a single layer is modified 
+for spherical coordinates as described by \citet{Arakawa1981} to obtain the EEN 
+scheme. The potential vorticity, defined at an $f$-point, is: 
+\begin{equation} \label{Eq_pot_vor}
+q_f  = \frac{\zeta +f} {e_{3f} }
+\end{equation}
+where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter 
+is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: 
+\begin{equation} \label{Eq_een_e3f}
+e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
+\end{equation}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \label{Fig_DYN_een_triad}
+\begin{center}
+\includegraphics[width=0.70\textwidth]{./Figures/Fig_DYN_een_triad.pdf}
+\caption{Triads used in the energy and enstrophy conserving scheme (een) for 
+$u$-component (upper panel) and $v$-component (lower panel).}
+\end{center}
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Note that a key point in \eqref{Eq_een_e3f} is that the averaging in \textbf{i}- and 
+\textbf{j}- directions uses the masked vertical scale factor but is always divided by 
+$4$, not by the sum of the mask at $T$-point. This preserves the continuity of 
+$e_{3f}$ when one or more of the neighbouring $e_{3T}$ tends to zero and 
+extends by continuity the value of $e_{3f}$ in the land areas. 
+%%%
+\gmcomment{this has to be further investigate in case of several step topography}
+%%%
+
+The vorticity terms are represented as: 
+\begin{equation} \label{Eq_dynvor_een}
+\left\{ {
+\begin{aligned}
+ +q\,e_3 \, v 	&\equiv +\frac{1}{e_{1u} }  \left[ 
+{{\begin{array}{*{20}c}
+    	{\,\ \ a_{j+1/2}^{i   } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i+1/2} } 
+  	{\,+\,b_{j+1/2}^{i   } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i-1/2}  } \\
+ \\
+     {  +\,c_{j-1/2}^{i   }  \left( {e_{1v} e_{3v} \ v} \right)_{j    }^{i+1/2}         } 
+ 	{\,+\,d_{j+1/2}^{i   } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i+1/2} } \\
+\end{array} }} \right] \\ 
+\\
+-q\,e_3 \,u  		&\equiv -\frac{1}{e_{2v} }  \left[ 
+{{\begin{array}{*{20}c}
+ 	{\,\ \ a_{j-1/2}^{i   }  \left( {e_{2u} e_{3u} \ u} \right)_{j+1}^{i+1/2} } 
+  	{\,+\,b_{j-1/2}^{i+1}  \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i+1} } \\
+ \\
+     	{  +\,c_{j+1/2}^{i+1} \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i+1} } 
+  	{\,+\,d_{j+1/2}^{i   }  \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i   } } \\
+\end{array} }} \right]
+\end{aligned} 
+} \right.
+\end{equation} 
+where $a$, $b$, $c$ and $d$ are triad combinations of the neighbouring 
+potential vorticities (Fig. \ref{Fig_DYN_een_triad}): 
+\begin{equation} \label{Eq_een_triads}
+\left\{ 
+\begin{aligned}
+ a_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j+1/2}^{i+1} + q_{j+1 /2}^i + q_{j-1/2}^i  \right)    \\ 
+ \\
+ b_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j+1/2}^{i-1} +q_{j+1/2}^i +q_{j-1/2}^i   \right)     \\ 
+\\
+ c_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j-1/2}^{i-1} +q_{j+1/2}^i +q_{j-1/2}^i   \right)     \\ 
+\\
+ d_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j-1/2}^{i+1} +q_{j+1/2}^i +q_{j-1/2}^i  \right)     \\ 
+\end{aligned} 
+\right.
+\end{equation}
+
+%--------------------------------------------------------------------------------------------------------------
+%           Kinetic Energy Gradient term
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Kinetic Energy Gradient term (\textit{dynkeg})]
+			{Kinetic Energy Gradient term (\mdl{dynkeg})}
+\label{DYN_keg}
+
+As demonstarted in Appendix~\ref{Apdx_C}, there is a single discrete formulation 
+of the kinetic energy gradient term that, together with the formulation chosen for 
+the vertical advection (see below), conserves the total kinetic energy:
+\begin{equation} \label{Eq_dynkeg}
+\left\{ \begin{aligned}
+ -\frac{1}{2 \; e_{1u} } 
+ & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
+ -\frac{1}{2 \; e_{2v} } 
+ & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]    
+\end{aligned} \right.
+\end{equation} 
+
+%--------------------------------------------------------------------------------------------------------------
+%           Vertical advection term
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Vertical advection term (\textit{dynzad}) ]
+			{Vertical advection term (\mdl{dynzad}) }
+\label{DYN_zad}
+
+The discrete formulation of the vertical advection, together with the formulation 
+chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic 
+energy. Indeed, the change of KE due to the vertical advection is exactly 
+balanced by the change of KE due to the gradient of KE (see Appendix~\ref{Apdx_C}).
+\begin{equation} \label{Eq_dynzad}
+\left\{ 		\begin{aligned}
+ -\frac{1}  { e_{1u}\,e_{2u}\,e_{3u} }  &  
+  \ {\overline {\overline{ e_{1T}\,e_{2T}\,w } ^{\,i+1/2}  \;\delta _{k+1/2} \left[ u \right]  }^{\,k}   } \\
+ -\frac{1}  { e_{1v}\,e_{2v}\,e_{3v} }  &
+  \ {\overline {\overline{ e_{1T}\,e_{2T}\,w } ^{\,j+1/2}  \;\delta _{k+1/2} \left[ u \right]  }^{\,k}   }
+\end{aligned} \right.
+\end{equation} 
+
+% ================================================================
+% Coriolis and Advection : flux form
+% ================================================================
+\section{Coriolis and Advection: flux form}
+\label{DYN_adv_cor_flux}
+%------------------------------------------nam_dynadv----------------------------------------------------
+\namdisplay{nam_dynadv} 
+%-------------------------------------------------------------------------------------------------------------
+
+In the flux form (as in the vector invariant form), the Coriolis and momentum 
+advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity 
+appearing in their expressions is centred in time (\textit{now} velocity). At the 
+lateral boundaries either free slip, no slip or partial slip boundary conditions 
+are applied following Chap.\ref{LBC}.
+
+
+%--------------------------------------------------------------------------------------------------------------
+%           Coriolis plus curvature metric terms
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Coriolis plus curvature metric terms (\textit{dynvor}) ]
+			{Coriolis plus curvature metric terms (\mdl{dynvor}) }
+\label{DYN_cor_flux}
+
+In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis 
+parameter has been modified to account for the "metric" term. This altered 
+Coriolis parameter is thus discretised at $f$-points. It is given by: 
+\begin{multline} \label{Eq_dyncor_metric}
+f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right)  \\
+   \equiv   f + \frac{1}{e_{1f} e_{2f} } 
+	\left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right]  
+	         -  \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
+\end{multline} 
+
+Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een}) 
+schemes can be used to compute the product of the Coriolis parameter and the 
+vorticity. However, the energy-conserving scheme  (\ref{Eq_dynvor_een}) has 
+exclusively been used to date. This term is evaluated using a leapfrog scheme, 
+$i.e.$ the velocity is centred in time (\textit{now} velocity).
+
+%--------------------------------------------------------------------------------------------------------------
+%           Flux form Advection term
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Flux form Advection term (\textit{dynadv}) ]
+			{Flux form Advection term (\mdl{dynadv}) }
+\label{DYN_adv_flux}
+
+The discrete expression of the advection term is given by :
+\begin{equation} \label{Eq_dynadv}
+\left\{ 
+\begin{aligned}
+\frac{1}{e_{1u}\,e_{2u}\,e_{3u}} 
+\left(      \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\ u }^{i       }  \ u_T      \right]    
+          + \delta _{j       } \left[ \overline{e_{1u}\,e_{3u}\ v }^{i+1/2}  \ u_F      \right] \right.  \ \;   \\
+\left.   + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w} w}^{i+1/2}  \ u_{uw} \right] \right)   \\
+\\
+\frac{1}{e_{1v}\,e_{2v}\,e_{3v}} 
+\left( 	\delta _{i      } \left[  \overline{e_{2u}\,e_{3u } \ u }^{j+1/2} \ v_F       \right] 
+         + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u } \ v }^{i       } \ v_T       \right] \right.  \ \, \\
+\left.  + \delta _{k     } \left[  \overline{e_{1w}\,e_{2w} \ w}^{j+1/2} \ v_{vw}  \right] \right) \\
+\end{aligned}
+\right.
+\end{equation}
+
+Two advection schemes are available: a $2^{nd}$ order centered finite 
+difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS. 
+The latter is described in \citet{Sacha2005}. The schemes are selected using 
+the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. In flux 
+form, the schemes differ by the choice of a space and time interpolation to 
+define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells, 
+$i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and 
+$vw$-points for $v$. 
+
+%-------------------------------------------------------------
+%                 2nd order centred scheme
+%-------------------------------------------------------------
+\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=.true.)}
+\label{DYN_adv_cen2}
+
+In the centered $2^{nd}$ order formulation, the velocity is evaluated as the 
+mean of the two neighbouring points :
+\begin{equation} \label{Eq_dynadv_cen2}
+\left\{ 		\begin{aligned}
+ u_T^{cen2} &=\overline u^{i }		\quad & 	
+  u_F^{cen2} &=\overline u^{j+1/2}		\quad &
+ u_{uw}^{cen2} &=\overline u^{k+1/2}      \\
+ v_F^{cen2} &=\overline v ^{i+1/2}		\quad &
+ v_F^{cen2} &=\overline v^j		\quad &
+ v_{vw}^{cen2} &=\overline v ^{k+1/2}      \\
+\end{aligned} \right.
+\end{equation} 
+
+The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive 
+($i.e.$ it may create false extrema). It is therefore notoriously noisy and must 
+be used in conjunction with an explicit diffusion operator to produce a sensible 
+solution. The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, so $u$ and $v$ are the \emph{now} 
+velocities.
+
+%-------------------------------------------------------------
+%                 UBS scheme
+%-------------------------------------------------------------
+\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=.true.)}
+\label{DYN_adv_ubs}
+
+The UBS advection scheme is an upstream biased third order scheme based on 
+an upstream-biased parabolic interpolation. For example, the evaluation of 
+$u_T^{ubs} $ is done as follows:
+\begin{equation} \label{Eq_dynadv_ubs}
+u_T^{ubs} =\overline u ^i-\;\frac{1}{6} 	\begin{cases}
+		u"_{i-1/2}& 	\text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ } 	\\
+		u"_{i+1/2}& 	\text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
+\end{cases}
+\end{equation}
+where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results 
+in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Sacha2005}. 
+The overall performance of the advection scheme is similar to that reported in 
+\citet{Farrow1995}. It is a relatively good compromise between accuracy and 
+smoothness. It is not a \emph{positive} scheme, meaning that false extrema are 
+permitted. But the amplitudes of the false extrema are significantly reduced over 
+those in the centred second order method. 
+
+The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 
+order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and 
+$u_{vw}^{ubs}$ in \eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is 
+associated with vertical mixing of momentum. \gmcomment{ gm  pursue the 
+sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
+
+For stability reasons,  the first term in (\ref{Eq_dynadv_ubs}), which corresponds 
+to a second order centred scheme, is evaluated using the \textit{now} velocity 
+(centred in time), while the second term, which is the diffusive part of the scheme, 
+is evaluated using the \textit{before} velocity (forward in time). This is discussed 
+by \citet{Webb1998} in the context of the Quick advection scheme.
+
+Note that the UBS and Quadratic Upstream Interpolation for Convective Kinematics 
+(QUICK) schemes only differ by one coefficient. Substituting $1/6$ with $1/8$ in 
+(\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. 
+This option is not available through a namelist parameter, since the $1/6$ coefficient 
+is hard coded. Nevertheless it is quite easy to make the substitution in 
+\mdl{dynadv\_ubs} module and obtain a QUICK scheme.
+
+Note also that in the current version of \mdl{dynadv\_ubs}, there is also the 
+possibility of using a $4^{th}$ order evaluation of the advective velocity as in 
+ROMS. This is an error and should be suppressed soon.
+%%%
+\gmcomment{action :  this have to be done}
+%%%
+
+% ================================================================
+%           Hydrostatic pressure gradient term
+% ================================================================
+\section  [Hydrostatic pressure gradient (\textit{dynhpg})]
+		{Hydrostatic pressure gradient (\mdl{dynhpg})}
+\label{DYN_hpg}
+%------------------------------------------nam_dynhpg---------------------------------------------------
+\namdisplay{nam_dynhpg} 
+\namdisplay{namflg} 
+%-------------------------------------------------------------------------------------------------------------
+%%%
+\gmcomment{Suppress the namflg namelist and incorporate it in the namhpg namelist}
+%%%
+
+The key distinction between the different algorithms used for the hydrostatic 
+pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal} 
+pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any 
+tilt of the surface of the computational levels will require a specific treatment to 
+compute the hydrostatic pressure gradient.
+
+The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, 
+$i.e.$ the density appearing in its expression is centred in time (\emph{now} rho), or 
+a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip 
+boundary conditions are applied.
+
+%--------------------------------------------------------------------------------------------------------------
+%           z-coordinate with full step
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ]
+			{$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=.true.)}
+\label{DYN_hpg_zco}
+
+The hydrostatic pressure can be obtained by integrating the hydrostatic equation 
+vertically from the surface. However, the pressure is large at great depth while its 
+horizontal gradient is several orders of magnitude smaller. This may lead to large 
+truncation errors in the pressure gradient terms. Thus, the two horizontal components 
+of the hydrostatic pressure gradient are computed directly as follows:
+
+for $k=km$ (surface layer, $jk=1$ in the code)
+\begin{equation} \label{Eq_dynhpg_zco_surf}
+\left\{ \begin{aligned}
+					\left. \delta _{i+1/2} \left[  p^h 			 \right] \right|_{k=km} 
+&= \frac{1}{2} g \ 	\left. \delta _{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
+     					\left. \delta _{j+1/2} \left[  p^h  			 \right] \right|_{k=km} 
+&= \frac{1}{2} g \ 	\left. \delta _{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
+\end{aligned} \right.
+\end{equation} 
+
+for $1<k<km$ (interior layer)
+\begin{equation} \label{Eq_dynhpg_zco}
+\left\{ \begin{aligned}
+					\left. \delta _{i+1/2} \left[  p^h 			 \right] \right|_{k} 
+&=					\left. \delta _{i+1/2} \left[  p^h 			 \right] \right|_{k-1} 
++    \frac{1}{2}\;g\;	\left. \delta _{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
+     					\left. \delta _{j+1/2} \left[  p^h  			 \right] \right|_{k} 
+&=     				\left. \delta _{j+1/2} \left[  p^h  			 \right] \right|_{k-1} 
++    \frac{1}{2}\;g\;	\left. \delta _{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
+\end{aligned} \right.
+\end{equation} 
+
+Note that the $1/2$ factor in (\ref{Eq_dynhpg_zco_surf}) is adequate because of 
+the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface 
+level ($z=0)$. 
+
+%--------------------------------------------------------------------------------------------------------------
+%           z-coordinate with partial step
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [$z$-coordinate with partial step (\np{ln\_dynhpg\_zps})]
+			{$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=.true.)}
+\label{DYN_hpg_zps}
+
+With partial bottom cells, tracers in horizontally adjacent cells generally live at 
+different depths. Before taking horizontal gradients between these tracer points, 
+a linear interpolation is used to approximate the deeper tracer as if it actually lived 
+at the depth of the shallower tracer point. 
+
+Apart from this modification, the horizontal hydrostatic pressure gradient evaluated 
+in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case. 
+As explained in detail in section \S\ref{TRA_zpshde}, the nonlinearity of pressure 
+effects in the equation of state is such that it is better to interpolate temperature and 
+salinity vertically before computing the density. Horizontal gradients of temperature 
+and salinity are needed for the TRA modules, which is the reason why the horizontal 
+gradients of density at the deepest model level are computed in module \mdl{zpsdhe} 
+located in the TRA directory and described in \S\ref{TRA_zpshde}.
+
+%--------------------------------------------------------------------------------------------------------------
+%           s- and s-z-coordinates
+%--------------------------------------------------------------------------------------------------------------
+\subsection{$s$- and $z$-$s$-coordinates}
+\label{DYN_hpg_sco}
+
+Pressure gradient formulations in $s$-coordinate have been the subject of a vast 
+literature ($e.g.$, \citet{Song1998, Sacha2003}). A number of different pressure 
+gradient options are coded, but they are not yet fully documented or tested. 
+
+$\bullet$ Traditional coding (see for example \citet{Madec1996}: (\np{ln\_dynhpg\_sco}=.true., 
+\np{ln\_dynhpg\_hel}=.true.)
+\begin{equation} \label{Eq_dynhpg_sco}
+\left\{ \begin{aligned}
+ - \frac{1}    					{\rho_o \, e_{1u}} \;	\delta _{i+1/2} \left[  p^h  \right] 
++ \frac{g\; \overline {\rho}^{i+1/2}}	{\rho_o \, e_{1u}} \;	\delta _{i+1/2} \left[  z_T  \right]    \\
+ - \frac{1}    					{\rho_o \, e_{2v}} \;	\delta _{j+1/2} \left[  p^h  \right]  
++ \frac{g\; \overline {\rho}^{j+1/2}}	{\rho_o \, e_{2v}} \;	\delta _{j+1/2} \left[  z_T  \right]    \\
+\end{aligned} \right.
+\end{equation} 
+
+Where the first term is the pressure gradient along coordinates, computed as in 
+\eqref{Eq_dynhpg_zco_surf} - \eqref{Eq_dynhpg_zco}, and $z_T$ is the depth of 
+the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point 
+($e_{3w}$). The version \np{ln\_dynhpg\_hel}=.true. has been added by Aike 
+Beckmann and involves a redefinition of the relative position of $T$-points relative 
+to $w$-points. 
+
+$\bullet$ Weighted density Jacobian (WDJ) \citep{Song1998} (\np{ln\_dynhpg\_wdj}=.true.)
+
+$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Sacha2003} 
+(\np{ln\_dynhpg\_djc}=.true.)
+
+$\bullet$ Rotated axes scheme (rot) \citep{Thiem2006} (\np{ln\_dynhpg\_rot}=.true.)
+
+Note that expression \eqref{Eq_dynhpg_sco} is used when the variable volume 
+formulation is activated (\key{vvl}) because in that case, even with a flat bottom, 
+the coordinate surfaces are not horizontal but follow the free surface 
+\citep{Levier2007}. The other pressure gradient options are not yet available.
+
+%--------------------------------------------------------------------------------------------------------------
+%           Time-scheme
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Time-scheme (\np{ln\_dynhpg\_imp}) ]
+			{Time-scheme (\np{ln\_dynhpg\_imp}=.true./.false.)}
+\label{DYN_hpg_imp}
+
+The default time differencing scheme used for the horizontal pressure gradient is 
+a leapfrog scheme and therefore the density used in all discrete expressions given 
+above is the  \textit{now} density, computed from the \textit{now} temperature and 
+salinity. In some specific cases (usually high resolution simulations over an ocean 
+domain which includes weakly stratified regions) the physical phenomenum that 
+controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for 
+doubling the stability limit associated with IGWs can be used \citep{Brown1978, 
+Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an 
+average over the three time levels $t-\Delta t$, $t$, and $t+\Delta t$ ($i.e.$  
+\textit{before},  \textit{now} and  \textit{after} time-steps), rather than at central 
+time level $t$ only, as in the standard leapfrog scheme. 
+
+$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=.true.):
+
+\begin{equation} \label{Eq_dynhpg_lf}
+\frac{u^{t+\Delta t}-u^{t-\Delta t}}{2\Delta t}
+=\;\cdots \;-\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right]
+\end{equation}
+
+$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=.true.):
+\begin{equation} \label{Eq_dynhpg_imp}
+\frac{u^{t+\Delta t}-u^{t-\Delta t}}{2\Delta t}
+=\;\cdots \;-\frac{1}{\rho _o \,e_{1u} } \delta _{i+1/2} \left[ \frac{ p_h^{t+\Delta t} +2p_h^t 
++p_h^{t-\Delta t} } { 4 }  \right]
+\end{equation}
+
+The semi-implicit time scheme \eqref{Eq_dynhpg_imp} is made possible without 
+significant additional computation since the density can be updated to time level 
+$t+\Delta t$ before computing the horizontal hydrostatic pressure gradient. It can 
+be easily shown that the stability limit associated with the hydrostatic pressure 
+gradient doubles using \eqref{Eq_dynhpg_imp} compared to that using the 
+standard leapfrog scheme \eqref{Eq_dynhpg_lf}. Note that \eqref{Eq_dynhpg_imp} 
+is equivalent to applying a time filter to the pressure gradient to eliminate high 
+frequency IGWs. Obviously, when using \eqref{Eq_dynhpg_imp}, the doubling of 
+the time-step is achievable only if no other factors control the time-step, such as 
+the stability limits associated with advection or diffusion.
+
+In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=.true.. 
+In this case, we choose to apply the time filter to temperature and salinity used in 
+the equation of state, instead of applying it to the hydrostatic pressure or to the 
+density, so that no additional storage array has to be defined. The density used to 
+compute the hydrostatic pressure gradient (whatever the formulation) is evaluated 
+as follows:
+\begin{equation} \label{Eq_rho_flt}
+	\rho^t = \rho( \widetilde{T},\widetilde {S},z_T)
+ \quad	  \text{with}	\quad 
+	\widetilde{\,\cdot\,} = \frac{  \,\cdot\,^{t+\Delta t} +2 \,\,\cdot\,^t + \,\cdot\,^{t-\Delta t}  } {4}
+\end{equation}
+\gmcomment{STEVEN: bullets look odd in this, could use X}
+
+Note that in the semi-implicit case, it is necessary to save the filtered density, an 
+extra three-dimensional field, in the restart file to restart the model with exact 
+reproducibility. This option is controlled by the namelist parameter 
+\np{nn\_dynhpg\_rst}=.true..
+
+% ================================================================
+% Surface Pressure Gradient
+% ================================================================
+\section  [Surface pressure gradient (\textit{dynspg}) ]
+		{Surface pressure gradient (\mdl{dynspg})}
+\label{DYN_hpg_spg}
+%-----------------------------------------nam_dynspg----------------------------------------------------
+\namdisplay{nam_dynspg} 
+%------------------------------------------------------------------------------------------------------------
+
+The form of the surface pressure gradient term is dependent on the representation 
+of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed 
+volume case (linear free surface or rigid lid) and the variable volume case 
+(nonlinear free surface, \key{vvl} is defined). In the linear free surface case 
+(\S\ref{PE_free_surface}) and the rigid lid case (\S\ref{PE_rigid_lid}), the vertical 
+scale factors $e_{3}$ are fixed in time, whilst in the nonlinear case 
+(\S\ref{PE_free_surface}) they are time-dependent. With both linear and nonlinear 
+free surface, external gravity waves are allowed in the equations, which imposes 
+a very small time step when an explicit time stepping is used. Two methods are 
+proposed to allow a longer time step for the three-dimensional equations: the 
+filtered free surface method, which involves a modification of the continuous 
+equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface method 
+described below. The extra term introduced in the filtered method is calculated 
+implicitly, so that the update of the $next$ velocities is done in module 
+\mdl{dynspg\_flt} and not in \mdl{dynnxt}.
+
+%--------------------------------------------------------------------------------------------------------------
+% Linear free surface formulation
+%--------------------------------------------------------------------------------------------------------------
+\subsection{Linear free surface formulation (\key{exp} or \textbf{\_ts} or \textbf{\_flt})}
+\label{DYN_spg_linear}
+
+In the linear free surface formulation, the sea surface height is assumed to be 
+small compared to the thickness of the ocean levels, so that $(a)$ the time 
+evolution of the sea surface height becomes a linear equation, and $(b)$ the 
+thickness of the ocean levels is assumed to be constant with time. 
+As mentioned in (\S\ref{PE_free_surface}) the linearization affects the 
+conservation of tracers. 
+
+%-------------------------------------------------------------
+% Explicit
+%-------------------------------------------------------------
+\subsubsection{Explicit (\key{dynspg\_exp})}
+\label{DYN_spg_exp}
+
+In the explicit free surface formulation, the model time step is chosen to be 
+small enough to describe the external gravity waves (typically a few tens of 
+seconds). The sea surface height is given by :
+\begin{equation} \label{Eq_dynspg_ssh}
+\frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} 
+e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u} 
+\right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)} 
+\end{equation}
+where EMP is the surface freshwater budget, expressed in Kg/m$^2$/s 
+(which is equal to mm/s), and $\rho _w$=1,000~Kg/m$^3$ is the volumic 
+mass of pure water. If river runoff is expressed as a surface freshwater flux 
+(see \S\ref{SBC}) then EMP can be written as the evaporation minus 
+precipitation, minus the river runoff. The sea-surface height is evaluated 
+using a leapfrog scheme in combination with an Asselin time filter, $i.e.$ 
+the velocity appearing in \eqref{Eq_dynspg_ssh} is centred in time 
+(\textit{now} velocity). 
+
+The surface pressure gradient, also evaluated using a leap-frog scheme, is 
+then simply given by :
+\begin{equation} \label{Eq_dynspg_exp}
+\left\{ \begin{aligned}
+ - \frac{1}{e_{1u}} \;	\delta _{i+1/2} \left[  \,\eta\,  \right] 	\\
+ - \frac{1}{e_{2v}} \;	\delta _{j+1/2} \left[  \,\eta\,  \right]  
+\end{aligned} \right.
+\end{equation} 
+
+Consistent with the linearization, a factor of $\left. \rho \right|_{k=1} / \rho _o$ 
+is omitted in \eqref{Eq_dynspg_exp}. 
+
+%-------------------------------------------------------------
+% Split-explicit time-stepping
+%-------------------------------------------------------------
+\subsubsection{Split-explicit time-stepping (\key{dynspg\_ts})}
+\label{DYN_spg_ts}
+%--------------------------------------------namdom----------------------------------------------------
+\namdisplay{namdom} 
+%--------------------------------------------------------------------------------------------------------------
+
+The split-explicit free surface formulation used in \NEMO follows the one 
+proposed by \citet{Griffies2004}. The general idea is to solve the free surface 
+equation with a small time step \np{rdtbt}, while the three dimensional 
+prognostic variables are solved with a longer time step that is a multiple of 
+\np{rdtbt} (Fig.\ref {Fig_DYN_dynspg_ts}). 
+
+%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
+\begin{figure}[!t] \label{Fig_DYN_dynspg_ts}
+\begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_DYN_dynspg_ts.pdf}
+\caption{Schematic of the split-explicit time stepping scheme for the external 
+and internal modes. Time increases to the right. 
+Internal mode time steps (which are also the model time steps) are denoted 
+by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
+The curved line represents a leap-frog time step, and the smaller time 
+steps $N \Delta t_e=\frac{3}{2}\Delta t$ are denoted by the zig-zag line. The vertically 
+integrated forcing \textbf{M}(t) computed at the model time step $t$ 
+represents the interaction between the external and internal motions. 
+While keeping \textbf{M} and freshwater forcing field fixed, a 
+leap-frog integration carries the external mode variables (surface height and vertically integrated velocity) from $t$ to $t+\frac{3}{2} \Delta t$ using N external time steps of length $\Delta t_e$. 
+Time averaging the external fields over the $\frac{2}{3}N+1$ time steps (endpoints 
+included) centers the vertically integrated velocity and the sea surface height at the model timestep $t+\Delta t$. These averaged values are used to update \textbf{M}(t) with both the surface pressure gradient and the Coriolis force. 
+A baroclinic leap-frog time step carries the surface height to The model time stepping scheme can then be achieved by 
+$t+\Delta t$ using the convergence of the time averaged vertically integrated 
+velocity taken from baroclinic time step t. }
+%%%
+\gmcomment{STEVEN: what does convergence mean in this context?}
+%%%
+\end{center}
+\end{figure}
+%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
+
+The split-explicit formulation has a damping effect on external gravity waves, 
+which is weaker damping than for the filtered free surface but still significant as 
+shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. 
+
+%-------------------------------------------------------------
+% Filtered formulation 
+%-------------------------------------------------------------
+\subsubsection{Filtered formulation (\key{dynspg\_flt})}
+\label{DYN_spg_flt}
+
+The filtered formulation follows the \citet{Roullet2000} implementation. The extra 
+term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic 
+solvers available in the code are documented in \S\ref{MISC}. The amplitude of 
+the extra term is given by the namelist variable \np{rnu}. The default value is 1, 
+as recommended by \citet{Roullet2000}
+
+\gmcomment{\np{rnu}=1 to be suppressed from namelist !}
+
+%-------------------------------------------------------------
+% Non-linear free surface formulation 
+%-------------------------------------------------------------
+\subsection{Non-linear free surface formulation (\key{vvl})}
+\label{DYN_spg_vvl}
+
+In the non-linear free surface formulation, the variations of volume are fully 
+taken into account. This option is presented in a report \citep{Levier2007} 
+available on the \NEMO web site. The three time-stepping methods (explicit, 
+split-explicit and filtered) are the same as in \S\ref{DYN_spg_linear} except 
+that the ocean depth is now time-dependent. In particular, this means that 
+in the filtered case, the matrix to be inverted has to be recomputed at each 
+time-step.
+
+%--------------------------------------------------------------------------------------------------------------
+%           Rigid-lid formulation
+%--------------------------------------------------------------------------------------------------------------
+\subsection{Rigid-lid formulation (\key{dynspg\_rl})}
+\label{DYN_spg_rl}
+
+With the rigid lid formulation, an elliptic equation has to be solved for 
+the barotropic streamfunction. For consistency this equation is obtained by 
+taking the discrete curl of the discrete vertical sum of the discrete 
+momentum equation: 
+\begin{equation}\label{Eq_dynspg_rl}
+\frac{1}{\rho _o }\nabla _h p_s \equiv \left( {{\begin{array}{*{20}c}
+ {\overline M_u +\frac{1}{H\;e_2 } \delta_ j \left[ \partial_t \psi \right]}		 \\
+ \\
+ {\overline M_v -\frac{1}{H\;e_1 }  \delta_i  \left[ \partial_t \psi \right]} 		 \\
+\end{array} }} \right)
+\end{equation}
+
+Here ${\rm {\bf M}}= \left( M_u,M_v \right)$ represents the collected 
+contributions of nonlinear, viscous and hydrostatic pressure gradient terms in 
+\eqref{Eq_PE_dyn} and the overbar indicates a vertical average over the 
+whole water column (i.e. from $z=-H$, the ocean bottom, to $z=0$, the rigid-lid). 
+The time derivative of $\psi$ is the solution of an elliptic equation:
+\begin{multline} \label{Eq_bsf}
+   \delta_{i+1/2} \left[ \frac{e_{2v}}{H_v\;e_{1v}} \delta_{i} \left[  \partial_t \psi \right] \right]
++ \delta_{j+1/2} \left[ \frac{e_{1u}}{H_u\;e_{2u}} \delta_{j} \left[  \partial_t \psi \right] \right]
+\\ =
+  \delta_{i+1/2} \left[ e_{2v} M_v  \right]
+- \delta_{j+1/2} \left[ e_{1u} M_u  \right]
+\end{multline}
+
+The elliptic solvers available in the code are documented in \S\ref{MISC}). 
+The boundary conditions must be given on all separate landmasses (islands), 
+which is done by integrating the vorticity equation around each island. This 
+requires identifying each island in the bathymetry file, a cumbersome task. 
+This explains why the rigid lid option is not recommended with complex 
+domains such as the global ocean. Parameters jpisl (number of islands) and 
+jpnisl (maximum number of points per island) of the \hf{par\_oce} file are 
+related to this option.
+
+
+% ================================================================
+% Lateral diffusion term
+% ================================================================
+\section  [Lateral diffusion term (\textit{dynldf})]
+		{Lateral diffusion term (\mdl{dynldf})}
+\label{DYN_ldf}
+%------------------------------------------nam_dynldf----------------------------------------------------
+\namdisplay{nam_dynldf} 
+%-------------------------------------------------------------------------------------------------------------
+
+The options available for lateral diffusion are for the choice of  laplacian 
+(rotated or not) or biharmonic operators. The coefficients may be constant 
+or spatially variable; the description of the coefficients is found in the chapter 
+on lateralphysics (Chap.\ref{LDF}). The lateral diffusion of momentum is 
+evaluated using a forward scheme, i.e. the velocity appearing in its expression 
+is the \textit{before} velocity in time, except for the pure vertical component 
+that appears when a tensor of rotation is used. This latter term is solved 
+implicitly together with the vertical diffusion term (see \S\ref{DOM_nxt}) 
+
+At the lateral boundaries either free slip, no slip or partial slip boundary 
+conditions are applied according to the user's choice (see Chap.\ref{LBC}).
+
+% ================================================================
+\subsection   [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ]
+			{Iso-level laplacian operator (\np{ln\_dynldf\_lap}=.true.)}
+\label{DYN_ldf_lap}
+
+For lateral iso-level diffusion, the discrete operator is: 
+\begin{equation} \label{Eq_dynldf_lap}
+\left\{ \begin{aligned}
+ D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} 
+\;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[ 
+{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
+\\
+ D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm} 
+\;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[ 
+{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
+\end{aligned} \right.
+\end{equation} 
+
+As explained in \S\ref{PE_ldf}, this formulation (as the gradient of a divergence 
+and curl of the vorticity) preserves symmetry and ensures a complete 
+separation between the vorticity and divergence parts. Note that in the full step 
+$z$-coordinate (\key{zco} is defined), $e_{3u} =e_{3v} =e_{3f}$ so that they 
+cancel in the rotational part of \eqref{Eq_dynldf_lap}.
+
+%--------------------------------------------------------------------------------------------------------------
+%           Rotated laplacian operator
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ]
+			{Rotated laplacian operator (\np{ln\_dynldf\_iso}=.true.)}
+\label{DYN_ldf_iso}
+
+A rotation of the lateral momentum diffusive operator is needed in several cases: 
+for iso-neutral diffusion in $z$-coordinate (\np{ln\_dynldf\_iso}=.true.) and for 
+either iso-neutral (\np{ln\_dynldf\_iso}=.true.) or geopotential 
+(\np{ln\_dynldf\_hor}=.true.) diffusion in $s$-coordinate. In the partial step 
+case, coordinates are horizontal excepted at the deepest level and no 
+rotation is performed when \np{ln\_dynldf\_hor}=.true.. The diffusive operator 
+is defined simply as the divergence of down gradient momentum fluxes on each 
+momentum component. It must be emphasized that this formulation ignores 
+constraints on the stress tensor such as symmetry. The resulting discrete 
+representation is:
+\begin{equation} \label{Eq_dyn_ldf_iso}
+\begin{split}
+ D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} }	\\
+&  \left\{\quad  {\delta _{i+1/2} \left[ {A_T^{lm}  \left( 
+	 {\frac{e_{2T} \; e_{3T} }{e_{1T} } \,\delta _{i}[u]
+	-e_{2T} \; r_{1T} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}}
+ \right)} \right]} 	\right.
+\\ 
+& \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} 
+}\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f} 
+\,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}} 
+\right)} \right] 
+\\ 
+&\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline 
+{\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } 
+\right.} \right. 
+\\ 
+&  \ \qquad \qquad \qquad \quad\ 
+- e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2}
+\\ 
+& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 
++\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} 
+\right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} 
+\\
+\\
+ D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} }    \\
+&  \left\{\quad  {\delta _{i+1/2} \left[ {A_f^{lm}  \left( 
+	 {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v]
+	-e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}}
+ \right)} \right]} 	\right.
+\\ 
+& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1T}\,e_{3T} }{e_{2T} 
+}\,\delta _{j} [v] - e_{1T}\, r_{2T} 
+\,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} 
+\right)} \right] 
+\\ 
+& \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
+{\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. 
+\\
+&  \ \qquad \qquad \qquad \quad\ 
+- e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2}
+\\ 
+& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 
++\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 
+\right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
+ \end{split}
+\end{equation}
+where $r_1$ and $r_2$ are the slopes between the surface along which the 
+diffusive operator acts and the surface of computation ($z$- or $s$-surfaces). 
+The way these slopes are evaluated is given in the lateral physics chapter 
+(Chap.\ref{LDF}).
+
+%--------------------------------------------------------------------------------------------------------------
+%           Iso-level bilaplacian operator
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})]
+			{Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=.true.)}
+\label{DYN_ldf_bilap}
+
+The lateral fourth order operator formulation on momentum is obtained by 
+applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on 
+boundary conditions: the first derivative term normal to the coast depends on 
+the free or no-slip lateral boundary conditions chosen, while the third 
+derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}).
+%%%
+\gmcomment{add a remark on the the change in the position of the coefficient}
+%%%
+
+% ================================================================
+%           Vertical diffusion term
+% ================================================================
+\section  [Vertical diffusion term (\mdl{dynzdf})]
+		{Vertical diffusion term (\mdl{dynzdf})}
+\label{DYN_zdf}
+%----------------------------------------------namzdf------------------------------------------------------
+\namdisplay{namzdf} 
+%-------------------------------------------------------------------------------------------------------------
+
+The large vertical diffusion coefficient found in the surface mixed layer together 
+with high vertical resolution implies that in the case of explicit time stepping there 
+would be too restrictive a constraint on the time step. Two time stepping schemes 
+can be used for the vertical diffusion term : $(a)$ a forward time differencing 
+scheme (\np{ln\_zdfexp}=.true.) using a time splitting technique 
+(\np{n\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme 
+(\np{ln\_zdfexp}=.false.) (see \S\ref{DOM_nxt}). Note that namelist variables 
+\np{ln\_zdfexp} and \np{n\_zdfexp} apply to both tracers and dynamics. 
+
+The formulation of the vertical subgrid scale physics is the same whatever 
+the vertical coordinate is. The vertical diffusion operators given by 
+\eqref{Eq_PE_zdf} take the following semi-discrete space form:
+\begin{equation} \label{Eq_dynzdf}
+\left\{   \begin{aligned}
+D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
+                              \ \delta _{k+1/2} [\,u\,]         \right]     \\
+\\
+D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
+                              \ \delta _{k+1/2} [\,v\,]         \right]
+\end{aligned}   \right.
+\end{equation} 
+where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and 
+diffusivity coefficients. The way these coefficients are evaluated 
+depends on the vertical physics used (see \S\ref{ZDF}).
+
+The surface boundary condition on momentum is given by the stress exerted by 
+the wind. At the surface, the momentum fluxes are prescribed as the boundary 
+condition on the vertical turbulent momentum fluxes,
+\begin{equation} \label{Eq_dynzdf_sbc}
+\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
+	 = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
+\end{equation}
+where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress 
+vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients 
+in the surface mixed layer ensure that the surface wind stress is distributed in 
+the vertical over the mixed layer depth. If the vertical mixing coefficient 
+is small (when no mixed layer scheme is used) the surface stress enters only 
+the top model level, as a body force. The surface wind stress is calculated 
+in the surface module routines (SBC, see Chap.\ref{SBC})
+
+The turbulent flux of momentum at the bottom of the ocean is specified through 
+a bottom friction parameterization (see \S\ref{ZDF_bfr})
+
+% ================================================================
+% External Forcing
+% ================================================================
+\section{External Forcings}
+\label{DYN_forcing}
+
+Besides the surface and bottom stresses (see the above section) which are 
+introduced as boundary conditions on the vertical mixing, two other forcings 
+enter the dynamical equations. 
+
+One is the effect of atmospheric pressure on the ocean dynamics (to be 
+introduced later). 
+
+Another forcing term is the tidal potential, which will be introduced in the 
+reference version soon. 
+
+% ================================================================
+% Time evolution term 
+% ================================================================
+\section  [Time evolution term (\textit{dynnxt})]
+		{Time evolution term (\mdl{dynnxt})}
+\label{DYN_nxt}
+
+%----------------------------------------------namdom----------------------------------------------------
+\namdisplay{namdom} 
+%-------------------------------------------------------------------------------------------------------------
+
+The general framework for dynamics time stepping is a leap-frog scheme, 
+$i.e.$ a three level centred time scheme associated with an Asselin time filter 
+(cf. \S\ref{DOM_nxt})
+\begin{equation} \label{Eq_dynnxt}
+\begin{split}
+&u^{t+\Delta t} = u^{t-\Delta t} + 2 \, \Delta t  \ \text{RHS}_u^t  	\\
+\\
+&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\Delta t} -2u^t+u^{t+\Delta t}} \right]
+\end{split}
+\end{equation} 
+where RHS is the right hand side of the momentum equation, the subscript $f$ 
+denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is 
+initialized as \np{atfp} (namelist parameter). Its default value is \np{atfp} = 0.1.
+
+Note that whith the filtered free surface, the update of the \textit{next} velocities 
+is done in the \mdl{dynsp\_flt} module, and only the swap of arrays 
+and Asselin filtering is done in \mdl{dynnxt.}
+
+% ================================================================
+% Diagnostic variables
+% ================================================================
+\section{Diagnostic variables ($\zeta$, $\chi$, $w$)}
+\label{DYN_divcur_wzv}
+
+%--------------------------------------------------------------------------------------------------------------
+%           Horizontal divergence and relative vorticity
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Horizontal divergence and relative vorticity (\textit{divcur})]
+			{Horizontal divergence and relative vorticity (\mdl{divcur})}
+\label{DYN_divcur}
+
+The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows:
+\begin{equation} \label{Eq_divcur_cur}
+\zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right]
+						        -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
+\end{equation} 
+
+The horizontal divergence is defined at a $T$-point. It is given by:
+\begin{equation} \label{Eq_divcur_div}
+\chi =\frac{1}{e_{1T}\,e_{2T}\,e_{3T} }
+		\left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right]
+			  +\delta _j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
+\end{equation} 
+
+Note that in the $z$-coordinate with full step (\key{zco} is defined), $e_{3u} =e_{3v} =e_{3f}$ so that they cancel in \eqref{Eq_divcur_div}.
+
+Note also that whereas the vorticity have the same discrete expression in $z$- 
+and $s$-coordinate, its physical meaning is not identical. $\zeta$ is a pseudo 
+vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along 
+geopotential surfaces, but are no more necessary defined at the same depth).
+
+The vorticity and divergence at the \textit{before} step are used in the computation 
+of the horizontal diffusion of momentum. Note that because they have been 
+calculated prior to the Asselin filtering of the \textit{before} velocities, the 
+\textit{before} vorticity and divergence arrays must be included in the restart file 
+to ensure perfect restartability. The vorticity and divergence at the \textit{now} 
+time step are used for the computation of the nonlinear advection and of the 
+vertical velocity respectively. 
+
+%--------------------------------------------------------------------------------------------------------------
+%           Vertical Velocity
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Vertical velocity (\textit{wzvmod})]
+			{Vertical velocity (\mdl{wzvmod})}
+\label{DYN_wzv}
+
+The vertical velocity is computed by an upward integration of the horizontal 
+divergence from the bottom :
+
+\begin{equation} \label{Eq_wzv}
+\left\{   \begin{aligned}
+&\left. w \right|_{3/2} \quad= 0   	\\
+\\
+&\left. w \right|_{k+1/2}     = \left. w \right|_{k+1/2}  + e_{3t}\;  \left. \chi \right|_k  
+\end{aligned}   \right.
+\end{equation} 
+
+With a free surface, the top vertical velocity is non-zero, due to the 
+freshwater forcing and the variations of the free surface elevation. With a 
+linear free surface or with a rigid lid, the upper boundary condition 
+applies at a fixed level $z=0$. Note that in the rigid-lid case (\key{dynspg\_rl} 
+is defined), the surface boundary condition ($\left. w \right|_\text{surface}=0)$ is 
+automatically achieved at least at computer accuracy, due to the the way the 
+surface pressure gradient is expressed in discrete form (Appendix~\ref{Apdx_C}).
+
+Note also that whereas the vertical velocity has the same discrete 
+expression in $z$- and $s$-coordinate, its physical meaning is not the same: 
+in the second case, $w$ is the velocity normal to the $s$-surfaces. 
+
+With the variable volume option, the calculation of the vertical velocity is 
+modified (see \citet{Levier2007}, report available on the \NEMO web site). 
+
+% ================================================================
Index: trunk/DOC/TexFiles/Chapters/Chap_LBC.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_LBC.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_LBC.tex	(revision 994)
@@ -0,0 +1,715 @@
+% ================================================================
+% Chapter Ñ Lateral Boundary Condition (LBC) 
+% ================================================================
+\chapter{Lateral Boundary Condition (LBC) }
+\label{LBC}
+\minitoc
+
+%gm% add here introduction to this chapter
+
+% ================================================================
+% Boundary Condition at the Coast
+% ================================================================
+\section{Boundary Condition at the Coast (\np{shlat})}
+\label{LBC_coast}
+%--------------------------------------------nam_lbc-------------------------------------------------------
+\namdisplay{nam_lbc} 
+%--------------------------------------------------------------------------------------------------------------
+
+%The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt (no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip). They are handled automatically by the mask system (see \S\ref{DOM_msk}). 
+
+%OPA allows land and topography grid points in the computational domain due to the presence of continents or islands, and includes the use of a full or partial step representation of bottom topography. The computation is performed over the whole domain, i.e. we do not try to restrict the computation to ocean-only points. This choice has two motivations. Firstly, working on ocean only grid points overloads the code and harms the code readability. Secondly, and more importantly, it drastically reduces the vector portion of the computation, leading to a dramatic increase of CPU time requirement on vector computers.  The current section describes how the masking affects the computation of the various terms of the equations with respect to the boundary condition at solid walls. The process of defining which areas are to be masked is described in \S\ref{DOM_msk}.
+
+The discrete representation of a domain with complex boundaries (coastlines and 
+bottom topography) leads to arrays that include large portions where a computation 
+is not required as the model variables remain at zero. Nevertheless, vectorial 
+supercomputers are far more efficient when computing over a whole array, and the 
+readability of a code is greatly improved when boundary conditions are applied in 
+an automatic way rather than by a specific computation before or after each 
+computational loop. An efficient way to work over the whole domain while specifying 
+the boundary conditions, is to use multiplication by mask arrays in the computation. 
+A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ 
+elsewhere. A simple multiplication of a variable by its own mask ensures that it will 
+remain zero over land areas. Since most of the boundary conditions consist of a 
+zero flux across the solid boundaries, they can be simply applied by multiplying 
+variables by the correct mask arrays, $i.e.$ the mask array of the grid point where 
+the flux is evaluated. For example, the heat flux in the \textbf{i}-direction is evaluated 
+at $u$-points. Evaluating this quantity as,
+
+\begin{equation} \label{Eq_lbc_aaaa}
+\frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 
+}{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u 
+\end{equation}
+(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is 
+zero inside land and at the boundaries, since mask$_{u}$ is zero at solid boundaries 
+which in this case are defined at $u$-points (normal velocity $u$ remains zero at 
+the coast) (Fig.~\ref{Fig_LBC_uv}). 
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_LBC_uv}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_LBC_uv.pdf}
+\caption {Lateral boundary (thick line) at T-level. The velocity normal to the 
+		 boundary is set to zero.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+For momentum the situation is a bit more complex as two boundary conditions 
+must be provided along the coast (one each for the normal and tangential velocities). 
+The boundary of the ocean in the C-grid is defined by the velocity-faces. 
+For example, at a given $T$-level, the lateral boundary (a coastline or an intersection 
+with the bottom topography) is made of segments joining $f$-points, and normal 
+velocity points are located between two $f-$points (Fig.~\ref{Fig_LBC_uv}). 
+The boundary condition on the normal velocity (no flux through solid boundaries) 
+can thus be easily implemented using the mask system. The boundary condition 
+on the tangential velocity requires a more specific treatment. This boundary 
+condition influences the relative vorticity and momentum diffusive trends, and is 
+required in order to compute the vorticity at the coast. Four different types of 
+lateral boundary condition are available, controlled by the value of the \np{shlat} 
+namelist parameter. (The value of the mask$_{f}$ array along the coastline is set 
+equal to this parameter.) These are:
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!p] \label{Fig_LBC_shlat}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_LBC_shlat.pdf}
+\caption {lateral boundary condition (a) free-slip ($shlat=0$) ; (b) no-slip ($shlat=2$) ; (c) "partial" free-slip ($0<shlat<2$) and (d) "strong" no-slip ($2<shlat$). Implied "ghost" velocity inside land area is display in grey. }
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+\begin{description}
+
+\item[free-slip boundary condition (\np{shlat}=0): ]  the tangential velocity at the 
+coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the 
+tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set 
+to zero inside the land and just at the coast (Fig.~\ref{Fig_LBC_shlat}-a).
+
+\item[no-slip boundary condition (\np{shlat}=2): ] the tangential velocity vanishes 
+at the coastline. Assuming that the tangential velocity decreases linearly from 
+the closest ocean velocity grid point to the coastline, the normal derivative is 
+evaluated as if the velocities at the closest land velocity gridpoint and the closest 
+ocean velocity gridpoint were of the same magnitude but in the opposite direction 
+(Fig.~\ref{Fig_LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by: 
+
+\begin{equation*}
+\zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ ,
+\end{equation*}
+where $u$ and $v$ are masked fields. Setting the mask$_{f}$ array to $2$ along 
+the coastline provides a vorticity field computed with the no-slip boundary condition, 
+simply by multiplying it by the mask$_{f}$ :
+\begin{equation} \label{Eq_lbc_bbbb}
+\zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2} 
+\left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]} 
+\right)\;\mbox{mask}_f 
+\end{equation}
+
+\item["partial" free-slip boundary condition (0$<$\np{shlat}$<$2): ] the tangential 
+velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral 
+friction but not strong enough to make the tangential velocity at the coast vanish 
+(Fig.~\ref{Fig_LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$ 
+strictly inbetween $0$ and $2$.
+
+\item["strong" no-slip boundary condition (2$<$\np{shlat}): ] the viscous boundary 
+layer is assumed to be smaller than half the grid size (Fig.~\ref{Fig_LBC_shlat}-d). 
+The friction is thus larger than in the no-slip case.
+
+\end{description}
+
+Note that when the bottom topography is entirely represented by the $s$-coor-dinates 
+(pure $s$-coordinate), the lateral boundary condition on tangential velocity is of much 
+less importance as it is only applied next to the coast where the minimum water depth 
+can be quite shallow.
+
+The alternative numerical implementation of the no-slip boundary conditions for an 
+arbitrary coast line of \citet{Shchepetkin1996} is also available through the 
+\key{noslip\_accurate} CPP key. It is based on a fourth order evaluation of the shear at the 
+coast which, in turn, allows a true second order scheme in the interior of the domain 
+($i.e.$ the numerical boundary scheme simulates the truncation error of the numerical 
+scheme used in the interior of the domain). \citet{Shchepetkin1996} found that such a 
+technique considerably improves the quality of the numerical solution. In \NEMO, such 
+spectacular improvements have not been found in the half-degree global ocean 
+(ORCA05), but significant reductions of numerically induced coastal upwellings were 
+found in an eddy resolving simulation of the Alboran Sea \citep{OlivierPh2001}. 
+Nevertheless, since a no-slip boundary condition is not recommended in an eddy 
+permitting or resolving simulation \citep{Penduff2007}, the use of this option is also 
+not recommended.
+
+In practice, the no-slip accurate option changes the way the curl is evaluated at the 
+coast (see \mdl{divcur} module), and requires the nature of each coastline grid point 
+(convex or concave corners, straight north-south or east-west coast) to be specified.  
+This is performed in routine \rou{dom\_msk\_nsa} in the \mdl{domask} module.
+
+% ================================================================
+% Boundary Condition around the Model Domain
+% ================================================================
+\section{Model Domain Boundary Condition (\jp{jperio})}
+\label{LBC_jperio}
+
+At the model domain boundaries several choices are offered: closed, cyclic east-west, south symmetric across the equator, a north-fold, and combination closed-north fold or cyclic-north-fold. The north-fold boundary condition is associated with the 3-pole ORCA mesh. 
+
+% -------------------------------------------------------------------------------------------------------------
+%        Closed, cyclic, south symmetric (\jp{jperio} = 0, 1 or 2) 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Closed, cyclic, south symmetric (\jp{jperio} = 0, 1 or 2)}
+\label{LBC_jperio012}
+
+The choice of closed, cyclic or symmetric model domain boundary condition is made 
+by setting \jp{jperio} to 0, 1 or 2 in file \mdl{par\_oce}. Each time such a boundary 
+condition is needed, it is set by a call to routine \mdl{lbclnk}. The computation of 
+momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to 
+$j=jpj-1$, $i.e.$ in the model interior. To choose a lateral model boundary condition 
+is to specify the first and last rows and columns of the model variables. 
+
+\begin{description}
+
+\item[For closed boundary (\textit{jperio=0})], solid walls are imposed at all model 
+boundaries: first and last rows and columns are set to zero.
+
+\item[For cyclic east-west boundary (\textit{jperio=1})], first and last rows are set 
+to zero (closed) whilst the first column is set to the value of the last-but-one column 
+and the last column to the value of the second one (Fig.~\ref{Fig_LBC_jperio}-a). 
+Whatever flows out of the eastern (western) end of the basin enters the western 
+(eastern) end. Note that there is no option for north-south cyclic or for doubly 
+cyclic cases.
+
+\item[For symmetric boundary condition across the equator (\textit{jperio=2})], 
+last rows, and first and last columns are set to zero (closed). The row of symmetry 
+is chosen to be the $u$- and $T-$points equator line ($j=2$, i.e. at the southern 
+end of the domain). For arrays defined at $u-$ or $T-$points, the first row is set 
+to the value of the third row while for most of $v$- and $f$-point arrays ($v$, $\zeta$, 
+$j\psi$, but \gmcomment{not sure why this is "but"} scalar arrays such as eddy coefficients) 
+the first row is set to minus the value of the second row (Fig.~\ref{Fig_LBC_jperio}-b). 
+Note that this boundary condition is not yet available for the case of a massively 
+parallel computer (\textbf{key{\_}mpp} defined).
+
+\end{description}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_LBC_jperio}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./Figures/Fig_LBC_jperio.pdf}
+\caption {setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+%        North fold (\textit{jperio = 3 }to $6)$ 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{North-fold (\textit{jperio = 3 }to $6)$ }
+\label{LBC_north_fold}
+
+The north fold boundary condition has been introduced in order to handle the north boundary of a three-polar ORCA grid. Such a grid has two poles in the northern hemisphere. \colorbox{yellow}{to be completed...}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_North_Fold_T}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_North_Fold_T.pdf}
+\caption {North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$), as used in ORCA 2, 1/4, and 1/12. Pink shaded area corresponds to the inner domain mask (see text). }
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% ====================================================================
+% Exchange with neighbouring processors 
+% ====================================================================
+\section  [Exchange with neighbouring processors (\textit{lbclnk}, \textit{lib\_mpp})]
+		{Exchange with neighbouring processors (\mdl{lbclnk}, \mdl{lib\_mpp})}
+\label{LBC_mpp}
+
+For massively parallel processing (mpp), a domain decomposition method is used. 
+The basic idea of the method is to split the large computation domain of a numerical 
+experiment into several smaller domains and solve the set of equations by addressing 
+independent local problems. Each processor has its own local memory and computes 
+the model equation over a subdomain of the whole model domain. The subdomain 
+boundary conditions are specified through communications between processors 
+which are organized by explicit statements (message passing method). 
+
+A big advantage is that the method does not need many modifications of the initial 
+FORTRAN code. From the modeller's point of view, each sub domain running on 
+a processor is identical to the "mono-domain" code. In addition, the programmer 
+manages the communications between subdomains, and the code is faster when 
+the number of processors is increased. The porting of OPA code on an iPSC860 
+was achieved during Guyon's PhD [Guyon et al. 1994, 1995] in collaboration with 
+CETIIS and ONERA. The implementation in the operational context and the studies 
+of performance on a T3D and T3E Cray computers have been made in collaboration 
+with IDRIS and CNRS. The present implementation is largely inspired by Guyon's 
+work  [Guyon 1995].
+
+The parallelization strategy is defined by the physical characteristics of the 
+ocean model. Second order finite difference schemes lead to local discrete 
+operators that depend at the very most on one neighbouring point. The only 
+non-local computations concern the vertical physics (implicit diffusion, 1.5 
+turbulent closure scheme, ...) (delocalization over the whole water column), 
+and the solving of the elliptic equation associated with the surface pressure 
+gradient computation (delocalization over the whole horizontal domain). 
+Therefore, a pencil strategy is used for the data sub-structuration \gmcomment{no 
+idea what this means!}: the 3D initial domain is laid out on local processor 
+memories following a 2D horizontal topological splitting. Each sub-domain 
+computes its own surface and bottom boundary conditions and has a side 
+wall overlapping interface which defines the lateral boundary conditions for 
+computations in the inner sub-domain. The overlapping area consists of the 
+two rows at each edge of the sub-domain. After a computation, a communication 
+phase starts: each processor sends to its neighbouring processors the update 
+values of the points corresponding to the interior overlapping area to its 
+neighbouring sub-domain (i.e. the innermost of the two overlapping rows). The communication is done through message passing. Usually the parallel virtual 
+language, PVM, is used as it is a standard language available on  nearly  all 
+MPP computers. More specific languages (i.e. computer dependant languages) 
+can be easily used to speed up the communication, such as SHEM on a T3E 
+computer. The data exchanges between processors are required at the very 
+place where lateral domain boundary conditions are set in the mono-domain 
+computation (\S III.10-c): the lbc\_lnk routine which manages such conditions 
+is substituted by mpplnk.F or mpplnk2.F routine when running on an MPP 
+computer (\key{mpp\_mpi} defined). It has to be pointed out that when using 
+the MPP version of the model, the east-west cyclic boundary condition is done 
+implicitly, whilst the south-symmetric boundary condition option is not available.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_mpp}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_mpp.pdf}
+\caption {Positioning of a sub-domain when massively parallel processing is used. }
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+In the standard version of the OPA model, the splitting is regular and arithmetic.
+ the i-axis is divided by \jp{jpni} and the j-axis by \jp{jpnj} for a number of processors 
+ \jp{jpnij} most often equal to $jpni \times jpnj$ (model parameters set in 
+ \mdl{par\_oce}). Each processor is independent and without message passing 
+ or synchronous process \gmcomment{how does a synchronous process relate to this?}, 
+ programs run alone and access just its own local memory. For this reason, the 
+ main model dimensions are now the local dimensions of the subdomain (pencil) 
+ that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. These dimensions include the internal 
+ domain and the overlapping rows. The number of rows to exchange (known as 
+ the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}). The whole domain 
+ dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}. The relationship between 
+ the whole domain and a sub-domain is:
+\begin{eqnarray} 
+      jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\
+      jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{Eq_lbc_jpi}
+\end{eqnarray}
+where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis.
+
+\colorbox{yellow}{Figure IV.3: example of a domain splitting with 9 processors and no east-west cyclic boundary conditions.}
+
+One also defines variables nldi and nlei which correspond to the internal 
+domain bounds, and the variables nimpp and njmpp which are the position 
+of the (1,1) grid-point in the global domain. An element of $T_{l}$, a local array 
+(subdomain) corresponds to an element of $T_{g}$, a global array 
+(whole domain) by the relationship: 
+\begin{equation} \label{Eq_lbc_nimpp}
+T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
+\end{equation}
+with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
+
+Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable 
+nproc. In the standard version, a processor has no more than four neighbouring 
+processors named nono (for north), noea (east), noso (south) and nowe (west) 
+and two variables, nbondi and nbondj, indicate the relative position of the processor 
+\colorbox{yellow}{(see Fig.IV.3)}:
+\begin{itemize}
+\item 		nbondi = -1 	an east neighbour, no west processor,
+\item 		nbondi =  0	an east neighbour, a west neighbour,
+\item 		nbondi =  1 	no east processor, a west neighbour,
+\item 		nbondi =  2 	no splitting following the i-axis.
+\end{itemize}
+	During the simulation, processors exchange data with their neighbours. If there is effectively a neighbour, the processor receives variables from this processor on its overlapping row, and sends the data issued from internal domain corresponding to the overlapping row of the other processor.
+		 
+
+\colorbox{yellow}{Figure IV.4: pencil splitting with the additional outer halos }
+
+
+
+The OPA model computes equation terms with the help of mask arrays (0 on land 
+points and 1 on sea points). It is easily readable and very efficient in the context of 
+a computer with vectorial architecture. However, in the case of a scalar processor, 
+computations over the land regions become more expensive in terms of CPU time. 
+It is worse when we use a complex configuration with a realistic bathymetry like the 
+global ocean where more than 50 \% of points are land points. For this reason, a 
+pre-processing tool can be used to choose the mpp domain decomposition with a 
+maximum number of only land points processors, which can then be eliminated. 
+(For example, the mpp\_optimiz tools, available from the DRAKKAR web site.) 
+This optimisation is dependent on the specific bathymetry employed. The user 
+then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with 
+$jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$ 
+land processors. When those parameters are specified in module \mdl{par\_oce}, 
+the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound, 
+nono, noea,...) so that the land-only processors are not taken into account. 
+
+\colorbox{yellow}{Note that the inimpp2 routine is general so that the original inimpp routine should be suppressed from the code.}
+
+When land processors are eliminated, the value corresponding to these locations in the model output files is zero. Note that this is a problem for a mesh output file written by such a model configuration, because model users often divide by the scale factors ($e1t$, $e2t$, etc) and do not expect the grid size to be zero, even on land. It may be best not to eliminate land processors when running the model especially to write the mesh files as outputs (when \np{nmsh} namelist parameter differs from 0).
+\gmcomment{Steven : dont understand this, no land processor means no output file covering this part of globe; its only when files are stitched together into one that you can leave a hole}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \label{Fig_mppini2}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_mppini2.pdf}
+\caption {Example of Atlantic domain defined for the CLIPPER projet. Initial grid is 
+composed of 773 x 1236 horizontal points. (a) the domain is split onto 9 \time 20 
+subdomains (jpni=9, jpnj=20). 52 subdomains are land areas. (b) 52 subdomains 
+are eliminated (white rectangles) and the resulting number of processors really 
+used during the computation is jpnij=128.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+% ================================================================
+% Open Boundary Conditions 
+% ================================================================
+\section{Open Boundary Conditions (\key{obc})}
+\label{LBC_obc}
+%-----------------------------------------nam_obc  -------------------------------------------
+%-    nobc_dta    =    0     !  = 0 the obc data are equal to the initial state
+%-                           !  = 1 the obc data are read in 'obc   .dta' files
+%-    rdpein      =    1.    !  ???
+%-    rdpwin      =    1.    !  ???
+%-    rdpnin      =   30.    !  ???
+%-    rdpsin      =    1.    !  ???
+%-    rdpeob      = 1500.    !  time relaxation (days) for the east  open boundary
+%-    rdpwob      =   15.    !    "        "           for the west  open boundary
+%-    rdpnob      =  150.    !    "        "           for the north open boundary
+%-    rdpsob      =   15.    !    "        "           for the south open boundary
+%-    zbsic1      =  140.e+6 !  barotropic stream function on first  isolated coastline
+%-    zbsic2      =    1.e+6 !    "                   "    on second isolated coastline
+%-    zbsic3      =    0.    !    "                   "    on thrid  isolated coastline
+%-    ln_obc_clim = .true.   !  climatological obc data files (default T)
+%-    ln_vol_cst  = .true.   !  total volume conserved
+\namdisplay{namobc} 
+
+It is often necessary to implement a model configuration limited to an oceanic 
+region or a basin, which communicates with the rest of the global ocean through 
+''open boundaries''. As stated by \citet{Roed1986}, an open boundary is a 
+computational border where the aim of the calculations is to allow the perturbations 
+generated inside the computational domain to leave it without deterioration of the 
+inner model solution. However, an open boundary also has to let information from 
+the outer ocean enter the model and should support inflow and outflow conditions. 
+
+The open boundary package OBC is the first open boundary option developed in 
+NEMO (originally in OPA8.2). It allows the user to 
+\begin{itemize}
+\item tell the model that a boundary is ''open'' and not closed by a wall, for example by modifying the calculation of the divergence of velocity there;
+\item impose values of tracers and velocities at that boundary (values which may be taken from a climatology): this is the``fixed OBC'' option. 
+\item calculate boundary values by a sophisticated algorithm combining radiation and relaxation (``radiative OBC'' option)
+\end{itemize}
+
+The package resides in the OBC directory. It is described here in four parts: the 
+boundary geometry (parameters to be set in \mdl{obc\_par}), the forcing data at 
+the boundaries (module \mdl{obcdta}),  the radiation algorithm involving the 
+namelist and module \mdl{obcrad}, and a brief presentation of boundary update 
+and restart files.
+
+%----------------------------------------------
+\subsection{Boundary geometry}
+\label{OBC_geom}
+%
+First one has to realize that open boundaries may not necessarily be located 
+at the extremities of the computational domain. They may exist in the middle 
+of the domain, for example at Gibraltar Straits if one wants to avoid including 
+the Mediterranean in an Atlantic domain. This flexibility has been found necessary 
+for the CLIPPER project \citep{Treguier2001}. Because of the complexity of the 
+geometry of ocean basins, it may even be necessary to have more than one 
+''west'' open boundary, more than one ''north'', etc. This is not possible with 
+the OBC option: only one open boundary of each kind, west, east, south and 
+north is allowed; these names refer to the grid geometry (not to the direction 
+of the geographical ''west'', ''east'', etc).
+
+The open boundary geometry is set by a series of parameters in the module 
+\mdl{obc\_par}. For an eastern open boundary, parameters are \jp{lp\_obc\_east} 
+(true if an east open boundary exists), \jp{jpieob} the $i$-index along which 
+the eastern open boundary (eob) is located, \jp{jpjed} the $j$-index at which 
+it starts, and \jp{jpjef} the $j$-index where it ends (note $d$ is for ''d\'{e}but'' 
+and $f$ for ''fin'' in French). Similar parameters exist for the west, south and 
+north cases (Table~\ref{Tab_obc_param}).
+
+
+%--------------------------------------------------TABLE--------------------------------------------------
+
+\begin{table}[htbp]  \label{Tab_obc_param}
+\begin{center}
+\begin{tabular}{|l|c|c|c|}
+\hline
+Boundary and  & Constant index  & Starting index (d\'{e}but) & Ending index (fin) \\
+Logical flag  &                 &                            &                     \\
+\hline
+West          & \jp{jpiwob} $>= 2$         &  \jp{jpjwd}$>= 2$          &  \jp{jpjwf}<= \jp{jpjglo}-1 \\
+lp\_obc\_west & $i$-index of a $u$ point   & $j$ of a $T$ point   &$j$ of a $T$ point \\
+\hline
+East            & \jp{jpieob}$<=$\jp{jpiglo}-2&\jp{jpjed} $>= 2$         & \jp{jpjef}$<=$ \jp{jpjglo}-1 \\
+ lp\_obc\_east  & $i$-index of a $u$ point    & $j$ of a $T$ point & $j$ of a $T$ point \\
+\hline
+South           & \jp{jpjsob} $>= 2$         & \jp{jpisd} $>= 2$          & \jp{jpisf}$<=$\jp{jpiglo}-1 \\
+lp\_obc\_south  & $j$-index of a $v$ point   & $i$ of a $T$ point   & $i$ of a $T$ point \\
+\hline
+North           & \jp{jpjnob} $<=$ \jp{jpjglo}-2& \jp{jpind} $>= 2$        & \jp{jpinf}$<=$\jp{jpiglo}-1 \\
+lp\_obc\_north  & $j$-index of a $v$ point      & $i$  of a $T$ point & $i$ of a $T$ point \\
+\hline
+\end{tabular}
+\end{center}
+\caption{Names of different indices relating to the open boundaries. In the case 
+of a completely open ocean domain with four ocean boundaries, the parameters 
+take exactly the values indicated.}
+\end{table}
+
+The open boundaries must be along coordinate lines. On the C-grid, the boundary 
+itself is along a line of normal velocity points: $v$ points for a zonal open boundary 
+(the south or north one), and $u$ points for a meridional open boundary (the west 
+or east one). Another constraint is that there still must be a row of masked points 
+all around the domain, as if the domain were a closed basin (unless periodic conditions 
+are used together with open boundary conditions). Therefore, an open boundary 
+cannot be located at the first/last index, namely, 1, \jp{jpiglo} or \jp{jpjglo}. Also, 
+the open boundary algorithm involves calculating the normal velocity points situated 
+just on the boundary, as well as the tangential velocity and temperature and salinity 
+just outside the boundary. This means that for a west/south boundary, normal 
+velocities and temperature are calculated at the same index \jp{jpiwob} and 
+\jp{jpjsob}, respectively. For an east/north boundary, the normal velocity is 
+calculated at index \jp{jpieob} and \jp{jpjnob}, but the ``outside'' temperature is 
+at index \jp{jpieob}+1 and \jp{jpjnob}+1. This means that \jp{jpieob}, \jp{jpjnob} 
+cannot be bigger than \jp{jpiglo}-2, \jp{jpjglo}-2. 
+
+
+The starting and ending indices are to be thought of as $T$ point indices: in many 
+cases they indicate the first land $T$-point, at the extremity of an open boundary 
+(the coast line follows the $f$ grid points, see Fig.~\ref{Fig_obc_north} for an example 
+of a northern open boundary). All indices are relative to the global domain. In the 
+free surface case it is possible to have ``ocean corners'', that is, an open boundary 
+starting and ending in the ocean.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_obc_north}  \begin{center}
+\includegraphics[width=0.70\textwidth]{./Figures/Fig_obc_north.pdf}
+\caption {Localization of the North open boundary points.}
+\end{center}  
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Although not compulsory, it is highly recommended that the bathymetry in the 
+vicinity of an open boundary follows the following rule: in the direction perpendicular 
+to the open line, the water depth should be constant for 4 grid points. This is in 
+order to ensure that the radiation condition, which involves model variables next 
+to the boundary, is calculated in a consistent way. On Fig.\ref{Fig_obc_north} we 
+indicate by an $=$ symbol, the points which should have the same depth. It means 
+that at the 4 points near the boundary, the bathymetry is cylindrical \gmcomment{not sure 
+why cylindrical}. The line behind the open $T$-line must be 0 in the bathymetry file 
+(as shown on Fig.\ref{Fig_obc_north} for example).
+
+%----------------------------------------------
+\subsection{Boundary data}
+\label{OBC_data}
+
+It is necessary to provide information at the boundaries. The simplest case is 
+when this information does not change in time and is equal to the initial conditions 
+(namelist variable \np{nobc\_dta}=0). This is the case for the standard configuration 
+EEL5 with open boundaries. When (\np{nobc\_dta}=1), open boundary information 
+is read from netcdf files. For convenience the input files are supposed to be similar 
+to the ''history'' NEMO output files, for dimension names and variable names. 
+Open boundary arrays must be dimensioned according to the parameters of table~
+\ref{Tab_obc_param}: for example, at the western boundary, arrays have a 
+dimension of \jp{jpwf}-\jp{jpwd}+1 in the horizontal and \jp{jpk} in the vertical. 
+
+When ocean observations are used to generate the boundary data (a hydrographic 
+section for example, as in \citet{Treguier2001}) it happens often that only the velocity 
+normal to the boundary is known, which is the reason why the initial OBC code 
+assumes that only $T$, $S$, and the normal velocity ($u$ or $v$) needs to be 
+specified. As more and more global model solutions and ocean analysis products 
+become available, it will be possible to provide information about all the variables 
+(including the tangential velocity) so that the specification of four variables at each 
+boundaries will become standard. For the sea surface height, one must distinguish 
+between the filtered free surface case and the time-splitting or explicit treatment of 
+the free surface.
+ In the first case, it is assumed that the user does not wish to represent high 
+ frequency motions such as tides. The boundary condition is thus one of zero 
+ normal gradient of sea surface height at the open boundaries, following \citet{Marchesiello2001}. 
+No information other than the total velocity needs to be provided at the open 
+boundaries in that case. In the other two cases (time splitting or explicit free surface), 
+the user must provide barotropic information (sea surface height and barotropic 
+velocities) and the use of the Flather algorithm for barotropic variables is 
+recommanded. However, this algorithm has not yet been fully tested and bugs 
+remain in NEMO v2.3. Users should read the code carefully before using it. Finally, 
+in the case of the rigid lid approximation the barotropic streamfunction must be 
+provided, as documented in \citet{Treguier2001}). This option is no longer 
+recommended but remains in NEMO V2.3.
+
+One frequently encountered case is when an open boundary domain is constructed 
+from a global or larger scale NEMO configuration. Assuming the domain corresponds 
+to indices $ib:ie$, $jb:je$ of the global domain, the bathymetry and forcing of the 
+small domain can be created by using the following netcdf utility on the global files: 
+ncks -F $-d\;x,ib,ie$ $-d\;y,jb,je$ (part of the nco series of utilities, see http://nco.sourceforge.net). The open boundary files can be constructed using ncks 
+commands, following table~\ref{Tab_obc_ind}. 
+
+%--------------------------------------------------TABLE--------------------------------------------------
+
+\begin{table}[htbp]  \label{Tab_obc_ind}
+\begin{center}
+\begin{tabular}{|l|c|c|c|c|c|}
+\hline
+OBC  & Variable   & file name      & Index  & Start  & end  \\
+West &  T,S       &   obcwest\_TS.nc &  $ib$+1     &   $jb$+1 &  $je-1$  \\
+     &    U       &   obcwest\_U.nc  &  $ib$+1     &   $jb$+1 &  $je-1$  \\ 
+     &    V       &   obcwest\_V.nc  &  $ib$+1     &   $jb$+1 &  $je-1$  \\       
+\hline
+East &  T,S       &   obceast\_TS.nc &  $ie$-1     &   $jb$+1 &  $je-1$  \\
+     &    U       &   obceast\_U.nc  &  $ie$-2     &   $jb$+1 &  $je-1$  \\ 
+     &    V       &   obceast\_V.nc  &  $ie$-1     &   $jb$+1 &  $je-1$  \\       
+\hline         
+South &  T,S      &   obcsouth\_TS.nc &  $jb$+1     &  $ib$+1 &  $ie-1$  \\
+      &    U      &   obcsouth\_U.nc  &  $jb$+1     &  $ib$+1 &  $ie-1$  \\ 
+      &    V      &   obcsouth\_V.nc  &  $jb$+1     &  $ib$+1 &  $ie-1$  \\    
+\hline
+North &  T,S      &   obcnorth\_TS.nc &  $je$-1     &  $ib$+1 &  $ie-1$  \\
+      &    U      &   obcnorth\_U.nc  &  $je$-1     &  $ib$+1 &  $ie-1$  \\ 
+      &    V      &   obcnorth\_V.nc  &  $je$-2     &  $ib$+1 &  $ie-1$  \\  
+\hline
+\end{tabular}
+\end{center}
+\caption{Requirements for creating open boundary files from a global configuration, 
+appropriate for the subdomain of indices $ib:ie$, $jb:je$. ``Index'' designates the 
+$i$ or $j$ index along which the $u$ of $v$ boundary point is situated in the global 
+configuration, starting and ending with the $j$ or $i$ indices indicated. 
+For example, to generate file obcnorth\_V.nc, use the command ncks 
+$-F$ $-d\;y,je-2$  $-d\;x,ib+1,ie-1$ } 
+\end{table}
+
+It is assumed that the open boundary files contain the variables for the period of 
+the model integration. If the boundary files contain one time frame, the boundary 
+data is held fixed in time. If the files contain 12 values, it is assumed that the input 
+is a climatology for a repeated annual cycle (corresponding to the case \np{ln\_obc\_clim} 
+= .True.). The case of an arbitrary number of time frames is not yet implemented 
+correctly; the user is required to write his own code in the module \mdl{obc\_dta} 
+to deal with this situation. 
+
+\subsection{Radiation algorithm}
+\label{OBC_rad}
+
+The art of open boundary management consists in applying a constraint strong 
+enough that the inner domain "feels" the rest of the ocean, but weak enough
+that perturbations are allowed to leave the domain with minimum false reflections 
+of energy. The constraints are specified separately at each boundary as time 
+scales for ''inflow'' and ''outflow'' as defined below. The time scales are set (in days) 
+by namelist parameters such as \np{rdpein}, \np{rdpeob} for the eastern open 
+boundary for example. When both time scales are zero for a given boundary 
+($e.g.$ for the western boundary, \jp{lp\_obc\_west}=.True., \np{rdpwob}=0 and 
+\np{rdpwin}=0) this means that the boundary in question is a ''fixed '' boundary 
+where the solution is set exactly by the boundary data. This is not recommended, 
+except in combination with increased viscosity in a ''sponge'' layer next to the 
+boundary in order to avoid spurious reflections.  
+
+
+The radiation\/relaxation \gmcomment{the / doesnt seem to appear in the output} 
+algorithm is applied when either relaxation time (for ''inflow'' or ''outflow'') is 
+non-zero. It has been developed and tested in the SPEM model and its 
+successor ROMS \citep{Barnier1996, Marchesiello2001}, which is an 
+$s$-coordinate model on an Arakawa C-grid. Although the algorithm has 
+been numerically successful in the CLIPPER Atlantic models, the physics 
+do not work as expected \citep{Treguier2001}. Users are invited to consider 
+open boundary conditions (OBC hereafter) with some scepticism 
+\citep{Durran2001, Blayo2005}.
+
+The first part of the algorithm calculates a phase velocity to determine 
+whether perturbations tend to propagate toward, or away from, the 
+boundary. Let us consider a model variable $\phi$. 
+The phase velocities ($C_{\phi x}$,$C_{\phi y}$) for the variable $\phi$, 
+in the directions normal and tangential to the boundary are
+\begin{equation} \label{Eq_obc_cphi}
+C_{\phi x} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{x} 
+\;\;\;\;\; \;\;\; 
+C_{\phi y} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{y}. 
+\end{equation}
+Following \citet{Treguier2001} and \citet{Marchesiello2001} we retain only 
+the normal component of the velocity, $C_{\phi x}$, setting $C_{\phi y} =0$ 
+(but unlike the original Orlanski radiation algorithm we retain $\phi_{y}$ in 
+the expression for $C_{\phi x}$).  
+
+The discrete form of (\ref{Eq_obc_cphi}), described by \citet{Barnier1998},
+takes into account the two rows of grid points situated inside the domain 
+next to the boundary, and the three previous time steps ($n$, $n-1$,
+and $n-2$). The same equation can then be discretized at the boundary at
+time steps $n-1$, $n$ and $n+1$ \gmcomment{since the original was three time-level} 
+in order to extrapolate for the new boundary value $\phi^{n+1}$. 
+
+In the open boundary algorithm as implemented in NEMO v2.3, the new boundary 
+values are updated differently depending on the sign of $C_{\phi x}$. Let us take 
+an eastern boundary as an example. The solution for variable $\phi$ at the 
+boundary is given by a generalized wave equation with phase velocity $C_{\phi}$, 
+with the addition of a relaxation term, as:
+\begin{eqnarray}
+\phi_{t} &  =  & -C_{\phi x} \phi_{x} + \frac{1}{\tau_{o}} (\phi_{c}-\phi) 
+                        \;\;\; \;\;\; \;\;\; (C_{\phi x} > 0), \label{Eq_obc_rado} \\
+\phi_{t} &  =  & \frac{1}{\tau_{i}} (\phi_{c}-\phi) 
+\;\;\; \;\;\; \;\;\;\;\;\; (C_{\phi x} < 0), \label{Eq_obc_radi}
+\end{eqnarray}
+where $\phi_{c}$ is the estimate of $\phi$ at the boundary, provided as boundary 
+data. Note that in (\ref{Eq_obc_rado}), $C_{\phi x}$ is bounded by the ratio 
+$\delta x/\delta t$ for stability reasons. When $C_{\phi x}$ is eastward (outward 
+propagation), the radiation condition (\ref{Eq_obc_rado}) is used. 
+When  $C_{\phi x}$ is westward (inward propagation), (\ref{Eq_obc_radi}) is 
+used with a strong relaxation to climatology (usually $\tau_{i}=\np{rdpein}=$1~day).
+Equation (\ref{Eq_obc_radi}) is solved with a Euler time-stepping scheme. As a 
+consequence, setting $\tau_{i}$ smaller than, or equal to the time step is equivalent 
+to a fixed boundary condition. A time scale of one day is usually a good compromise 
+which guarantees that the inflow conditions remain close to climatology while ensuring 
+numerical stability. 
+
+In  the case of a western boundary located in the Eastern Atlantic, \citet{Penduff2000} 
+have been able to implement the radiation algorithm without any boundary data, 
+using persistence from the previous time step instead. This solution has not worked 
+in other cases \citep{Treguier2001}, so that the use of boundary data is recommended. 
+Even in the outflow condition (\ref{Eq_obc_rado}), we have found it desirable to 
+maintain a weak relaxation to climatology. The time step is usually chosen so as to 
+be larger than typical turbulent scales (of order 1000~days \gmcomment{or maybe seconds?}).
+
+The radiation condition is applied to the model variables: temperature, salinity, 
+tangential and normal velocities. For normal and tangential velocities, $u$ and $v$, 
+radiation is applied with phase velocities calculated from $u$ and $v$ respectively.  
+For the radiation of tracers, we use the phase velocity calculated from the tangential 
+velocity in order to avoid calculating too many independent radiation velocities and 
+because tangential velocities and tracers have the same position along the boundary 
+on a C-grid.  
+
+\subsection{Domain decomposition (\key{mpp\_mpi})}
+\label{OBC_mpp}
+When \key{mpp\_mpi} is active in the code, the computational domain is divided 
+into rectangles that are attributed each to a different processor. The open boundary 
+code is ``mpp-compatible'' up to a certain point. The radiation algorithm will not 
+work if there is an mpp subdomain boundary parallel to the open boundary at the 
+index of the boundary, or the grid point after (outside), or three grid points before 
+(inside). On the other hand, there is no problem if an mpp subdomain boundary 
+cuts the open boundary perpendicularly. These geometrical limitations must be 
+checked for by the user (there is no safeguard in the code).  
+The general principle for the open boundary mpp code is that loops over the open 
+boundaries {not sure what this means} are performed on local indices (nie0, 
+nie1, nje0, nje1 for an eastern boundary for instance) that are initialized in module 
+\mdl{obc\_ini}. Those indices have relevant values on the processors that contain 
+a segment of an open boundary. For processors that do not include an open 
+boundary segment, the indices are such that the calculations within the loops are 
+not performed.
+\gmcomment{I dont understand most of the last few sentences}
+ 
+Arrays of climatological data that are read from files are seen by all processors 
+and have the same dimensions for all (for instance, for the eastern boundary, 
+uedta(jpjglo,jpk,2)). On the other hand, the arrays for the calculation of radiation 
+are local to each processor (uebnd(jpj,jpk,3,3) for instance).  This allowed the 
+CLIPPER model for example, to save on memory where the eastern boundary 
+crossed 8 processors so that \jp{jpj} was much smaller than (\jp{jpjef}-\jp{jpjed}+1). 
+
+\subsection{Volume conservation}
+\label{OBC_vol}
+
+It is necessary to control the volume inside a domain when using open boundaries. 
+With fixed boundaries, it is enough to ensure that the total inflow/outflow has 
+reasonable values (either zero or a value compatible with an observed volume 
+balance). When using radiative boundary conditions it is necessary to have a 
+volume constraint because each open boundary works independently from the 
+others. The methodology used to control this volume is identical to the one 
+coded in the ROMS model \citep{Marchesiello2001}.
+
+
+%---------------------------------------- EXTRAS
+\colorbox{yellow}{Explain obc\_vol{\ldots}}
+
+\colorbox{yellow}{OBC algorithm for update, OBC restart, list of routines where obc key appears{\ldots}}
+
+\colorbox{yellow}{OBC rigid lid? {\ldots}}
+
+
+
+
+% ====================================================================
+% Flow Relaxation Scheme 
+% ====================================================================
+\section{Flow Relaxation Scheme (???)}
+\label{LBC_bdy}
+
+%gm% to be updated by Met Office
Index: trunk/DOC/TexFiles/Chapters/Chap_LDF.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_LDF.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_LDF.tex	(revision 994)
@@ -0,0 +1,297 @@
+
+% ================================================================
+% Chapter Ñ Lateral Ocean Physics (LDF)
+% ================================================================
+\chapter{Lateral Ocean Physics (LDF)}
+\label{LDF}
+\minitoc
+
+$\ $\newline    % force a new ligne
+
+The lateral physics on momentum and tracer equations have been given in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf} and \S\ref{DYN_ldf}). In this section we 
+further discuss the choices that underlie each lateral physics option. 
+Choosing one lateral physics means for the user defining, (1) the space and time variations of the eddy coefficients ; (2) the direction along which the lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal surfaces); and (3) the type of operator used (harmonic, or biharmonic operators, and for tracers only, eddy induced advection on tracers). These 
+three aspects of the lateral diffusion are set through namelist parameters and CPP keys (see the nam\_traldf and nam\_dynldf below).
+%-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
+\namdisplay{nam_traldf} 
+\namdisplay{nam_dynldf} 
+%--------------------------------------------------------------------------------------------------------------
+
+
+% ================================================================
+% Lateral Mixing Coefficients
+% ================================================================
+\section{Lateral Mixing Coefficient (\textbf{key\_ldftra\_c.d} - \textbf{key\_ldfdyn\_c.d)} }
+\label{LDF_coef}
+
+
+Introducing a space variation in the lateral eddy mixing coefficients changes 
+the model core memory requirement, adding up to four three-dimensional 
+arrays for geopotential or isopycnal second order operator applied to 
+momentum. Six cpp keys control the space variation of eddy 
+coefficients: three for momentum and three for tracer. They 
+allow to specify a space variation in the three space directions, in the 
+horizontal plane, or in the vertical only. The default option is a constant 
+value over the whole ocean on momentum and tracers. 
+
+The number of additional arrays that have to be defined and the gridpoint 
+position at which they are defined depend on both the space variation chosen 
+and the type of operator used. The resulting eddy viscosity and 
+diffusivity coefficients can be either single or multiple valued functions. 
+Changes in the computer code when switching from one option to another have 
+been minimized by introducing the eddy coefficients as statement function 
+(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions are replaced by their actual meaning during the preprocessing step (cpp capability). The specification of the space variation of the coefficient is settled in \mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files \textit{ldftra\_cNd.h90} and \textit{ldfdyn\_cNd.h90}, with N=1, 2 or 3. The user can change these include files following his desiderata. The way the mixing coefficient are set in the reference version can be briefly described as follows:
+
+\subsubsection{Constant Mixing Coefficients (default option)}
+When none of the \textbf{key\_ldfdyn\_...} and \textbf{key\_ldftra\_...} keys are defined, a constant value over the whole ocean on momentum and tracers that is specified through 
+\np{ahm0} and \np{aht0} namelist parameters.
+
+\subsubsection{Vertically varying Mixing Coefficients (\key{ldftra\_c1d} and \key{ldfdyn\_c1d})} 
+The 1D option is only available in $z$-coordinate with full step. Indeed in all the other type of vertical coordinate, the depth is a 3D function of (\textbf{i},\textbf{j},\textbf{j}) and therefore, introducing depth-dependant mixing coefficients will requires 3D arrays, $i.e.$ \key{ldftra\_c3d} and \key{ldftra\_c3d}.  In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced in which the surface value is \np{aht0} (\np{ahm0}), the bottom value is 1/4 of the surface value, and the transition is round z=300~m with a width of 300~m ($i.e.$ both the depth and the width of the inflection point are set to 300~m). This profile is hard coded in \hf{ldftra\_c1d} file, but can be easily modified by users.
+
+\subsubsection{Horizontally Varying Mixing Coefficients (\key{ldftra\_c2d} and \key{ldfdyn\_c2d})}
+
+By default the horizontal variation of the eddy coefficient depend on the local mesh size and the type of operator used:
+\begin{equation} \label{Eq_title}
+  A_l = \left\{     
+   \begin{aligned}
+         & \frac{\max(e_1,e_2)}{e_{max}} A_o^l  			& \text{for laplacian operator } \\
+         & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l  & \text{for bilaplacian operator } 
+   \end{aligned}    \right.
+\quad \text{comments}
+\end{equation}
+where $e_{max}$ is the max of $e_1$ and $e_2$ taken over the whole masked ocean domain, and $A_o^l$ is \np{ahm0} (momentum) or \np{aht0} (tracer) namelist parameters. This variation is intended to reflect the lesser need for subgrid scale eddy mixing where the grid size is smaller in the domain. It was introduced in the context of the DYNAMO modelling project \citep{Willebrand2001}. 
+%%%
+\gmcomment { not only that! stability reasons: with non uniform grid size, it is common to face a blow up of the model due to to large diffusive coefficient compare to the smallest grid size... this is especially true for bilaplacian (to be added in the text!)  }
+
+Other formulations can be introduced by the user for a given configuration. For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian viscous operator uses \np{ahm0}~=~$4.10^4 m^2.s^{-1}$ poleward of 20$^{\circ}$ north and south and decreases to \np{aht0}~=~$2.10^3 m^2.s^{-1}$ at the equator \citep{Madec1996, Delecluse_Madec_Bk00}. This specification can be found in \rou{ldf\_dyn\_c2d\_orca} routine defined in \mdl{ldfdyn\_c2d}. Similar specific horizontal variation can be found for Antarctic or Arctic sub-domain of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}).
+
+\subsubsection{Space Varying Mixing Coefficients (\key{ldftra\_c3d} and \key{ldfdyn\_c3d})}
+
+The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with a grid size dependence of the magnitude of the coefficient. 
+
+\subsubsection{Space and time Varying Mixing Coefficients}
+
+There is no default specification of space and time varying mixing coefficient. The only case available is specific to ORCA2 and ORCA05 global ocean configurations (\key{orca\_r2} or \key{orca\_r05}). It provides only a tracer mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and eddy induced velocity (ORCA05) that depends on the local growth rate of baroclinic instability. This specification is actually used when a ORCA key plus \key{traldf\_eiv} plus \key{traldf\_c2d} are defined.
+
+A space variation in the eddy coefficient appeals several remarks:
+
+(1) the momentum diffusive operator acting along model level surfaces is 
+written in terms of curl and divergent components of the horizontal current 
+(see \S\ref{PE_ldf}). Although the eddy coefficient can be set to different values 
+in these two terms, this option is not available. 
+
+(2) with a horizontal varying viscosity, the quadratic integral constraints 
+on enstrophy and on the square of the horizontal divergence for operators 
+acting along model-surfaces are no more satisfied (\colorbox{yellow}{Appendix C}).
+
+(3) for isopycnal diffusion on momentum or tracers, an additional purely 
+horizontal background diffusion with uniform coefficient can be added by 
+setting a non zero value of \np{ahmb0} or \np{ahtb0}, a background horizontal eddy 
+viscosity or diffusivity coefficient (\textbf{namelist parameters} which default value are $0$). Nevertheless, the technique used to compute the isopycnal slopes allows to get rid of such a background diffusion which introduces spurious diapycnal diffusion (see {\S\ref{LDF_slp}).
+
+(4) when an eddy induced advection is used (\key{trahdf\_eiv}), $A^{eiv}$ , the eddy induced coefficient has to be defined. Its space variations are controlled by the same CPP variable as for the eddy diffusivity coefficient (i.e. \textbf{key\_traldf\_cNd}). 
+
+(5) the eddy coefficient associated to a biharmonic operator must be set to a \emph{negative} value.
+
+
+% ================================================================
+% Direction of lateral Mixing
+% ================================================================
+\section  [Direction of Lateral Mixing (\textit{ldfslp})]
+		{Direction of Lateral Mixing (\mdl{ldfslp})}
+\label{LDF_slp}
+
+%%%
+\gmcomment{  we should emphasize here that the implementation is a rather old one. Better work can be achieved by using \citet{Griffies1998, Griffies2004} iso-neutral scheme. }
+
+
+A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. This occurs when $(a)$ horizontal mixing is required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$ or mixed $s$-$z$-coordinate, and $(b)$ isoneutral mixing is required whatever the vertical coordinate is. This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the quantity to be diffused. For tracer, this leads to the following four slopes : $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while for momentum the slopes are  $r_{1T}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and  $r_{1f}$, $r_{1vw}$, $r_{2T}$, $r_{2vw}$ for $v$. 
+%gm% add here afigure of the slope in i-direction
+
+\subsection{slopes for tracer geopotential mixing in $s$-coordinate}
+
+In $s$-coordinates, geopotential mixing ($i.e.$ horizontal one) $r_1$ and $r_2$ are the slopes between the geopotential and computational surfaces. Their discrete formulation is found by locally 
+vanishing the diffusive fluxes when $T$ is horizontally uniform, i.e. by replacing in \eqref{Eq_tra_ldf_iso} $T$ by $z_T$, the depth of $T$-point, and setting to zero the diffusive fluxes in the three directions. This leads to the following expression for the slopes:
+
+\begin{equation} \label{Eq_ldfslp_geo}
+\begin{aligned}
+ r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
+ 			  \;\delta_{i+1/2}[z_T] 
+ 		&\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_T] 
+\\
+ r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 
+ 			  \;\delta_{j+1/2} [z_T] 
+		&\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_T] 
+\\
+ r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_T]}}^{\,i,\,k+1/2}
+ 		&\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}] 
+ \\
+ r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_T]}}^{\,j,\,k+1/2}
+		&\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] 
+ \\
+\end{aligned}
+\end{equation}
+
+%gm%  caution I'm not sure the simplification was a good idea! 
+These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=T and \np{ln\_traldf\_hor}=T or \np{ln\_dynldf\_hor}=T. 
+
+\subsection{slopes for tracer iso-neutral mixing}
+In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral and computational 
+surfaces. Their formulation does not depend on the vertical coordinate used. Their discrete formulation is found using the fact that the diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density) vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting to zero diffusive fluxes in the three directions leads to the following definition for the neutral slopes:
+
+\begin{equation} \label{Eq_ldfslp_iso}
+\begin{split}
+ r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
+ 								{\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}
+\\
+ r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
+ 								{\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}
+\\
+ r_{1w} &= \frac{e_{3w}}{e_{1w}}\; 
+ 			\frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
+				 {\delta_{k+1/2}[\rho]}
+\\
+ r_{2w} &= \frac{e_{3w}}{e_{2w}}\; 
+ 			\frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
+				 {\delta_{k+1/2}[\rho]}
+\\
+\end{split}
+\end{equation}
+
+%gm% rewrite this as the explanation in not very clear !!!
+%In practice, \eqref{Eq_ldfslp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \eqref{Eq_ldfslp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth. 
+
+%By definition, neutral surfaces are tangent to the local $in situ$ density \citep{McDougall1987}, therefore in \eqref{Eq_ldfslp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters).
+
+%In $z$-coordinate, the derivative of the  \eqref{Eq_ldfslp_iso} numerator is evaluated at a same depth ($T$-level which is also $u$- and $v$-levels), so  the $in situ$ density can be used for its evaluation. 
+
+As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \eqref{Eq_ldfslp_iso} have to be evaluated at the same local pressure (which, in decibars, is approximated by the depth in meters in the model). Therefore \eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is needed depending on the vertical coordinate used:
+
+\begin{description}
+
+\item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth, thus the $in situ$ density can be used. it is not the case for the vertical derivatives. $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{McDougall1987} (see \S\ref{TRA_bn2}). 
+
+\item[$z$-coordinate with partial step : ] the technique is identical to the full step case except that at partial step level, the \emph{horizontal} density gradient is evaluated as described in \S\ref{TRA_zpshde}.
+
+\item[$s$- or hybrid $s$-$z$ coordinate : ] in the current release of \NEMO, there is no specific treatment for iso-neutral mixing in $s$-coordinate. In other word, iso-neutral mixing will only be accurately represented with a linear equation of state (\np{neos}=1 or 2). In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} will include a pressure dependent part, leading to a wrong evaluation of the neutral slopes.
+
+%gm% 
+Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for the constraint on iso-neutral fluxes. Following \citet{Griffies2004}, instead of specifying directly that there is a zero neutral diffusive flux of locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between the neutral direction diffusive fluxes of potential temperature and salinity:
+\begin{equation}
+\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
+\end{equation}
+
+This constraint leads to the following definition for the slopes:
+
+\begin{equation} \label{Eq_ldfslp_iso2}
+\begin{split}
+ r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
+ 		{\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
+		{\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
+		 -\beta_u  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }
+\\
+ r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
+ 		{\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
+		{\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
+		 -\beta_v  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }
+\\
+ r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
+		{\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
+		 -\beta_w  \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
+ 		{\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
+\\
+ r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
+		{\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
+		 -\beta_w  \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
+ 		{\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
+\\
+\end{split}
+\end{equation}
+where $\alpha$ and $\beta$, the thermal expansion and saline contracion coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three velocity point. Inorder to save computation time, they should be approximated by the mean of their values at $T$-points (for example in the case of $\alpha$:  $\alpha_u=\overline{\alpha_T}^{i+1/2}$,  $\alpha_v=\overline{\alpha_T}^{j+1/2}$ and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).
+
+Note that such a formulation could be also used in $z$ and $zps$ cases.
+
+
+\end{description}
+
+This implementation is a rather old one. It is similar to the one proposed by Cox [1987], except for 
+the background horizontal diffusion. Indeed, the Cox implementation of isopycnal diffusion in GFDL-type models requires a minimum background horizontal diffusion for numerical stability reasons. To overcome this problem, several techniques have been proposed in which the numerical 
+schemes of the OGCM are modified \citep{Weaver1997, Griffies1998}. 
+Here, another strategy has been chosen \citep{Lazar1997}: a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of grid point noise generated by the iso-neutral diffusive operator (Fig.~\ref{Fig_LDF_ZDF1}). 
+This allows an iso-neutral diffusion scheme without additional background horizontal mixing. This technique can be viewed as a diffusive operator that acts along large-scale (2~$\Delta$x) iso-neutral surfaces. The diapycnal diffusion required for numerical stability is thus minimized and its net effect on the flow is quite small when compared to the effect of a horizontal background mixing. 
+
+Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, contrary to the \citet{Griffies1998} operator which has that property. 
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \label{Fig_LDF_ZDF1}  \begin{center}
+\includegraphics[width=0.70\textwidth]{./Figures/Fig_LDF_ZDF1.pdf}
+\caption {averaging procedure for isopycnal slope computation.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+%There is three additional questions about the slope calculation. First the expression of the rotation tensor used have been obtain assuming the "small slope" approximation, so a bound has to be specified on slopes. Second, numerical stability issues also require a bound on slopes. Third, the question of boundary condition spefified on slopes...
+
+%from griffies: chapter 13.1....
+
+
+
+In addition and also for numerical stability reasons \citep{Cox1987, Griffies2004}, the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly to zero fom $70$ meters depth 
+and the surface (the fact that the eddies "feel" the surface motivates this 
+flattening of isopycnals near the surface).
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \label{Fig_eiv_slp}  \begin{center}
+\includegraphics[width=0.70\textwidth]{./Figures/Fig_eiv_slp.pdf}
+\caption {Vertical profile of the slope used for lateral mixing in the mixed layer : \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior and their have to adjust to the surface boundary (i.e. tend to zero at the surface as there is no mixing across the air-sea interface: wall boundary condition). Nevertheless, the profile between surface zero value and interior iso-neutral one is unknown, and especially the value at the based of the mixed layer ; \textit{(b)} profile of slope using a linear tapering of the slope near the surface and imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actuelly used in \NEMO: linear decrease of the slope from zero at the surface to its ocean interior value computed just below the mixed layer. Note the huge change in the slope at the based of the mixed layer between  \textit{(b)}  and \textit{(c)}. 
+.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs  tapering the coefficient.}
+
+\subsection{slopes for momentum iso-neutral mixing}
+
+The diffusive iso-neutral operator on momentum is the same as the on used on tracer but applied to each component of the velocity (see \eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the surface along which the diffusive operator acts and the surface of computation ($z$- or $s$-surfaces) are defined at $T$-, $f-$, and \textit{uw-}points for the $u$-component, and $f-T$-, \textit{vw}-points for the $v$-component. They are computed as follows from the slopes used for tracer diffusion, i.e. \eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :
+
+\begin{equation} \label{Eq_ldfslp_dyn}
+\begin{aligned}
+&r_{1T}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
+&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& 	r_{2T}\ &= \overline{r_{2v}}^{\,j} \\
+&r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
+&r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
+\end{aligned}
+\end{equation}
+
+The major issue remains in the specification of the boundary conditions. The 
+choice made consists in keeping the same boundary conditions as for lateral 
+diffusion along model level surfaces, i.e. using the shear computed along 
+the model levels and with no additional friction at the ocean bottom (see 
+{\S\ref{LBC_coast}).
+
+
+% ================================================================
+% Eddy Induced Mixing
+% ================================================================
+\section  [Eddy Induced Velocity (\textit{traadv\_eiv}, \textit{ldfeiv})]
+		{Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})}
+\label{LDF_eiv}
+
+When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), an eddy induced tracer advection term is added, the formulation of which depends on the slopes of iso-neutral surfaces. Contrary to iso-neutral mixing, the slopes use here are referenced to the geopotential surfaces, i.e. \eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo}  + \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
+\begin{equation} \label{Eq_ldfeiv}
+\begin{split}
+ u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
+v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
+w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + \delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\
+\end{split}
+\end{equation}
+where $A^{eiv}$ is the eddy induced velocity coefficient set through \np{aeiv}, a \textit{nam\_traldf} namelist parameter. 
+The three components of the eddy induced velocity are computed and add to the eulerian velocity in the mdl{traadv\_eiv}. This has been preferred to a separate computation of the advective trends associated to the eiv velocity as it allows to take advantage of all the advection schemes offered for the tracers (see \S\ref{TRA_adv}) and not only the $2^{nd}$ order advection scheme as in previous release of OPA \citep{Madec1998}. This is particularly useful for passive tracers where \emph{positivity}of the advection scheme is of paramount importance. 
+
+At surface, lateral and bottom boundaries, the eddy induced velocity and thus the 
+advective eddy fluxes of heat and salt are set to zero. 
+
+
+
+
+
Index: trunk/DOC/TexFiles/Chapters/Chap_MISC.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_MISC.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_MISC.tex	(revision 994)
@@ -0,0 +1,664 @@
+% ================================================================
+% Chapter Ñ Miscellaneous Topics
+% ================================================================
+\chapter{Miscellaneous Topics (xxx)}
+\label{MISC}
+\minitoc
+
+% ================================================================
+% Representation of Unresolved Straits
+% ================================================================
+\section{Representation of Unresolved Straits}
+\label{MISC_strait}
+
+% In climate modeling, it is often necessary to allow water masses that are separated by land to exchange properties. This situation arises in models when the grid mesh is too coarse to resolve narrow passageways that in reality provide crucial connections between water masses. For example, coarse grid spacing typically closes off the Mediterranean from the Atlantic at the Straits of Gibraltar. In this case, it is important for climate models to include the effects of salty water entering the Atlantic from the Mediterranean. Likewise, it is important for the Mediterranean to replenish its supply of water from the Atlantic to balance the net evaporation occurring over the Mediterranean region. 
+% Same problem occurs as soon as important straits are not properly resolved by the mesh. For example it arises in ORCA 1/4\r{} in several straits of the Indonesian archipelago (Ombai, Lombok...).
+% We describe here the three methods that can be used in \NEMO to handle such improperly resolved straits. 
+%The problem is resolved here by allowing the mixing of tracers and mass/volume between non-adjacent water columns at nominated regions within the model. Momentum is not mixed. The scheme conserves total tracer content, and total volume (the latter in $z*$- or $s*$-coordinate), and maintains compatibility between the tracer and mass/volume budgets.  
+
+% Note: such changes are so specific to a given configuration that no attempt has been made to set them in a generic way. Nevertheless, an example of how they can be set up is given in the ORCA2 configuration (\key{ORCA_R2}).
+
+% -------------------------------------------------------------------------------------------------------------
+%       Hand made geometry changes
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Hand made geometry changes}
+\label{MISC_strait_hand}
+
+$\bullet$ reduced scale factor, also called partially open face (Hallberg, personnal communication 2006)
+%also called partially open cell  or partial closed cells
+
+$\bullet$ increase of the viscous boundary layer thickness by local increase of the fmask value at the coast
+
+\colorbox{yellow}{Add a short description of scale factor changes staff and fmask increase}
+
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tbp] \label{Fig_MISC_strait_hand}  \begin{center}
+\includegraphics[width=0.80\textwidth]{./Figures/Fig_Gibraltar.pdf}
+\includegraphics[width=0.80\textwidth]{./Figures/Fig_Gibraltar2.pdf}
+\caption {Example of the Gibraltar strait defined in a 1\r{} x 1\r{} mesh. 
+\textit{Top}: using partially open cells. The meridional scale factor at $v$-point 
+is reduced on both sides of the strait to account for the real width of the strait 
+(about 20 km). Note that the scale factors of the strait $T$-point remains unchanged. \textit{Bottom}: using viscous boundary layers. The four fmask parameters 
+along the strait coastlines are set to a value larger than 4, $i.e.$ "strong" no-slip 
+case (see Fig.\ref{Fig_LBC_shlat}) creating a large viscous boundary layer 
+that allows a reduced transport through the strait.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+% Cross Land Advection 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Cross Land Advection (\textit{tracla} module)}
+\label{MISC_strait_cla}
+
+%--------------------------------------------namcro--------------------------------------------------------
+\namdisplay{namcro} 
+%--------------------------------------------------------------------------------------------------------------
+
+\colorbox{yellow}{Add a short description of CLA staff here or in lateral boundary condition chapter?}
+
+% ================================================================
+% Closed seas
+% ================================================================
+\section{Closed seas}
+\label{MISC_closea}
+
+
+% ================================================================
+% Sub-Domain Functionality (\textit{nizoom, njzoom}, namelist parameters)
+% ================================================================
+\section{Sub-Domain Functionality (\jp{jpizoom}, \jp{jpjzoom})}
+\label{MISC_zoom}
+
+The sub-domain functionality, also improperly called the zoom option 
+(improperly because it is not associated with a change in model resolution) 
+is a quite simple function that allows a simulation over a sub-domain of an 
+already defined configuration ($i.e.$ without defining a new mesh, initial 
+state and forcings). This option can be useful for testing the user settings 
+of surface boundary conditions, or the initial ocean state of a huge ocean 
+model configuration while having a small computer memory requirement. 
+It can also be used to easily test specific physics in a sub-domain (for example, 
+see \citep{Madec1996} for a test of the coupling used in the global ocean 
+version of OPA between sea-ice and ocean model over the Arctic or Antarctic 
+ocean, using a sub-domain). In the standard model, this option does not 
+include any specific treatment for the ocean boundaries of the sub-domain: 
+they are considered as artificial vertical walls. Nevertheless, it is quite easy 
+to add a restoring term toward a climatology in the vicinity of such boundaries 
+(see \S\ref{TRA_dmp}).
+
+In order to easily define a sub-domain over which the computation can be 
+performed, the dimension of all input arrays (ocean mesh, bathymetry, 
+forcing, initial state, ...) are defined as \jp{jpidta}, \jp{jpjdta} and \jp{jpkdta} 
+(\mdl{par\_oce} module), while the computational domain is defined through 
+\jp{jpiglo}, \jp{jpjglo} and \jp{jpk} (\mdl{par\_oce} module). When running the 
+model over the whole domain, the user sets \jp{jpiglo}=\jp{jpidta} \jp{jpjglo}=\jp{jpjdta} 
+and \jp{jpk}=\jp{jpkdta}. When running the model over a sub-domain, the user 
+has to provide the size of the sub-domain, (\jp{jpiglo}, \jp{jpjglo}, \jp{jpkglo}), 
+and the indices of the south western corner as \jp{jpizoom} and \jp{jpjzoom} in 
+the \mdl{par\_oce} module (Fig.~\ref{Fig_LBC_zoom}). 
+
+Note that a third set of dimensions exist, \jp{jpi}, \jp{jpj} and \jp{jpk} which is 
+actually used to perform the computation. It is set by default to \jp{jpi}=\jp{jpjglo} 
+and \jp{jpj}=\jp{jpjglo}, except for massively parallel computing where the 
+computational domain is laid out on local processor memories following a 2D 
+horizontal splitting. % (see {\S}IV.2-c) ref to the section to be updated
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \label{Fig_LBC_zoom}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_LBC_zoom.pdf}
+\caption {Position of a model domain compared to the data input domain when the zoom functionality is used.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+% ================================================================
+% 1D model functionality
+% ================================================================
+\section{Water column model: 1D model (\key{cfg\_1d})}
+\label{MISC_1D}
+
+The 1D model option simulates a stand alone water column within the 3D \NEMO 
+system. It can be applied to the ocean alone or to the ocean-ice system and can 
+include passive tracers or a biogeochemical model. It is set up by defining the 
+\key{cfg\_1d} CPP key. This 1D model is a very useful tool  \textit{(a)} to learn 
+about the physics and numerical treatment of vertical mixing processes ; \textit{(b)} 
+to investigate suitable parameterizations of unresolved turbulence (wind steering, 
+langmuir circulation, skin layers) ; \textit{(c)} to compare the behaviour of different 
+vertical mixing schemes  ; \textit{(d)} to perform sensitivity studies on the vertical 
+diffusion at a particular point of the ocean global domain ; \textit{(d)} to produce 
+extra diagnostics, without the large memory requirement of the full 3D model.
+
+The methodology is based on the use of the zoom functionality (see 
+\S\ref{MISC_zoom}), with some extra routines. There is no need to define a new 
+mesh, bathymetry, initial state or forcing, since the 1D model will use those of the 
+configuration it is a zoom of. 
+
+% ================================================================
+% Accelerating the Convergence 
+% ================================================================
+\section{Accelerating the Convergence (\np{nn\_acc} = 1)}
+\label{MISC_acc}
+%--------------------------------------------namdom-------------------------------------------------------
+\namdisplay{namdom} 
+%--------------------------------------------------------------------------------------------------------------
+
+% Steven update just bellow...   not sure it  is what I vant to say
+%Searching an equilibrium state with an ocean model requires repeated very long time 
+%integrations (each of a few thousand years for a global model). Due to the size of 
+Searching an equilibrium state with an ocean model requires very long time 
+integration (a few thousand years for a global model). Due to the size of 
+the time step required for numerical stability (less than a 
+few hours), this usually requires a large elapsed time. In order to overcome 
+this problem, \citet{Bryan1984} introduces a technique that is intended to accelerate 
+the spin up to equilibrium. It uses a larger time step in 
+the thermodynamic evolution equations than in the dynamic evolution 
+equations. It does not affect the equilibrium solution but modifies the 
+trajectory to reach it.
+
+The acceleration of convergence option is used when \np{nn\_acc}=1. In that case, 
+$\Delta t=rdt$ is the time step of dynamics while $\widetilde{\Delta t}=rdttra$ is the 
+tracer time-step. Their values are set from the \np{rdt} and \np{rdttra}  namelist 
+parameters. The set of prognostic equations to solve becomes:
+\begin{equation} \label{Eq_acc}
+\begin{split}
+\frac{\partial \textbf{U}_h }{\partial t} 
+	&\equiv \frac{\textbf{U}_h ^{t+1}-\textbf{U}_h^{t-1} }{2\Delta t} = \ldots \\ 
+\frac{\partial T}{\partial t} &\equiv \frac{T^{t+1}-T^{t-1}}{2 \widetilde{\Delta t}} = \ldots \\ 
+\frac{\partial S}{\partial t} &\equiv \frac{S^{t+1} -S^{t-1}}{2 \widetilde{\Delta t}} = \ldots \\ 
+\end{split}
+\end{equation}
+
+\citet{Bryan1984} has examined the consequences of this distorted physics. 
+Free waves have a slower phase speed, their meridional structure is slightly 
+modified, and the growth rate of baroclinically unstable waves is reduced 
+but with a wider range of instability. This technique is efficient for 
+searching for an equilibrium state in coarse resolution models. However its 
+application is not suitable for many oceanic problems: it cannot be used for 
+transient or time evolving problems (in particular, it is very questionable 
+to use this technique when there is a seasonal cycle in the forcing fields), 
+and it cannot be used in high-resolution models where baroclinically 
+unstable processes are important. Moreover, the vertical variation of 
+$\widetilde{ \Delta t}$ implies that the heat and salt contents are no longer 
+conserved due to the vertical coupling of the ocean level through both 
+advection and diffusion.
+% first mention of depth varying tilda-delta-t!   and namelist parameter associated to that are to be described
+
+
+% ================================================================
+% Model optimisation, Control Print and Benchmark
+% ================================================================
+\section{Model Optimisation, Control Print and Benchmark}
+\label{MISC_opt}
+%--------------------------------------------namdom-------------------------------------------------------
+\namdisplay{nam_ctl} 
+%--------------------------------------------------------------------------------------------------------------
+
+% \gmcomment{why not make these bullets into subsections?}
+
+Three issues to be described here:
+
+$\bullet$ Vector and memory optimisation:
+
+\key{vectopt\_loop} enables internal loop collapse, a very efficient way to increase 
+the length of vector calculations and thus speed up the model on vector computers.
+ 
+% Add here also one word on NPROMA technique that has been found useless, since compiler have made significant progress during the last decade.
+ 
+% Add also one word on NEC specific optimisation (Novercheck option for example)
+
+\key{vectopt\_memory} has been introduced in order to reduce the memory 
+requirement of the model. This is obviously done at the cost of increasing the 
+CPU time requirement, since it suppress intermediate computations which would 
+have been saved in in-core memory. This feature has not been intensively used. 
+In fact, currently it is only implemented for the TKE physics, in which,  when 
+\key{vectopt\_memory} is defined, the coefficients used for horizontal smoothing 
+of $A_v^T$ and $A_v^m$ are no longer computed once and for all. This reduces 
+the memory requirement by three 2D arrays.
+
+ 
+$\bullet$ Control print %: describe here 4 things:
+
+1- \np{ln\_ctl} : compute and print the trends averaged over the interior domain 
+in all TRA, DYN, LDF and ZDF modules. This option is very helpful when 
+diagnosing the origin of an undesired change in model results. 
+
+2- also \np{ln\_ctl} but using the nictl and njctl namelist parameters to check 
+the source of differences between mono and multi processor runs.
+
+3- \key{esopa} (to be rename key\_nemo) : which is another option for model 
+management. When defined, this key forces the activation of all options and 
+CPP keys. For example, all tracer and momentum advection schemes are called! 
+There is therefore no physical meaning associated with the model results. 
+However, this option forces both the compiler and the model to run through 
+all the \textsc{Fortran} lines of the model. This allows the user to check for obvious 
+compilation or execution errors with all CPP options, and errors in namelist options.
+
+3- \key{esopa} (to be rename key\_nemo) : which is another option for model 
+management. When defined, this key forces the activation of all options and CPP keys. 
+For example, all tracer and momentum advection schemes are called! There is 
+therefore no physical meaning associated with the model results. However, this option 
+forces both the compiler and the model to run through all the Fortran lines of the model. 
+This allows the user to check for obvious compilation or execution errors with all CPP 
+options, and errors in namelist options.
+
+4- last digit comparison (\np{nbit\_cmp}). In an MPP simulation, the computation of 
+a sum over the whole domain is performed as the summation over all processors of 
+each of their sums over their interior domains. This double sum never gives exactly 
+the same result as a single sum over the whole domain, due to truncation differences. 
+The "bit comparison" option has been introduced in order to be able to check that 
+mono-processor and multi-processor runs give exactly the same results. 
+
+$\bullet$  Benchmark (\np{nbench}). This option defines a benchmark run based on 
+a GYRE configuration in which the resolution remains the same whatever the domain 
+size. This allows a very large model domain to be used, just by changing the domain 
+size (\jp{jpiglo}, \jp{jpjglo}) and without adjusting either the time-step or the physical 
+parameterizations. 
+
+
+% ================================================================
+% Elliptic solvers (SOL)
+% ================================================================
+\section{Elliptic solvers (SOL directory)}
+\label{MISC_sol}
+%--------------------------------------------namdom-------------------------------------------------------
+\namdisplay{namsol} 
+%--------------------------------------------------------------------------------------------------------------
+
+The computation of the surface pressure gradient with a rigid lid assumption 
+requires the computation of $\partial_t \psi$, the time evolution of the 
+barotropic streamfunction. $\partial_t \psi$ is the solution of an elliptic 
+equation \eqref{Eq_PE_psi} for which three solvers are available: a 
+Successive-Over-Relaxation scheme (SOR), a preconditioned conjugate gradient 
+scheme(PCG) \citep{Madec1988, Madec1990} and a Finite Elements Tearing and 
+Interconnecting scheme (FETI) \citep{Guyon_al_EP99, Guyon_al_CalPar99}. 
+The PCG is a very efficient method for solving elliptic equations on vector computers. 
+It is a fast and rather easy method to use; which are attractive features for a large 
+number of ocean situations (variable bottom topography, complex coastal geometry, 
+variable grid spacing, islands, open or cyclic boundaries, etc ...). It does not require 
+a search for an optimal parameter as in the SOR method. However, the SOR has 
+been retained because it is a linear solver, which is a very useful property when 
+using the adjoint model of \NEMO. The FETI solver is a very efficient method on 
+massively parallel computers. However, it has never been used since OPA 8.0. 
+The current version in \NEMO should not even successfully go through the 
+compilation phase. 
+
+The surface pressure gradient is computed in \mdl{dynspg}. The solver used 
+(PCG or SOR) depending on the value of \np{nsolv} (namelist parameter). 
+At each time step the time derivative of the barotropic streamfunction is 
+the solution of \eqref{Eq_psi_total}. Introducing the following coefficients:
+
+\begin{equation} 
+\begin{aligned}
+&C_{i,j}^{NS} &&= \frac{e_{2v}(i,j)}{\left( H_v (i,j) e_{1v} (i,j)\right)}\\
+&C_{i,j}^{EW} &&= \frac{e_{1u}(i,j)}{\left( H_u (i,j) e_{2u} (i,j)\right)}\\
+&B_{i,j} &&= \delta_i \left( e_{2v}M_v \right) - \delta_j \left( e_{1u}M_u \right)\\
+\end{aligned}
+\end{equation}
+
+the five-point finite difference equation \eqref{Eq_psi_total} can be rewritten as:
+
+\begin{multline} \label{Eq_solmat}
+C_{i+1,j}^{NS} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i+1,j} + \;
+C_{i,j+1}^{EW}{   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j+1} + \;
+C_{i,j}    ^{NS} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i-1,j} + \;
+C_{i,j}    ^{EW}{   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j-1}\\
+-\left(C_{i+1,j}^{NS} + C_{i,j+1}^{EW} + C_{i,j}^{NS} + C_{i,j}^{EW} \right)   {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j}  =  B_{i,j}
+\end{multline}
+\eqref{Eq_solmat} is a linear symmetric system of equations. All the elements of the 
+corresponding matrix \textbf{A} vanish except those of five diagonals. With 
+the natural ordering of the grid points (i.e. from west to east and from 
+south to north), the structure of \textbf{A} is block-tridiagonal with 
+tridiagonal or diagonal blocks. \textbf{A} is a positive-definite symmetric 
+matrix of size $(jpi \cdot jpj)^2$, and \textbf{B}, the right hand side of 
+\eqref{Eq_solmat}, is a vector.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Successive Over Relaxation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Successive Over Relaxation \np{nsolv}=2}
+\label{MISC_solsor}
+
+Let us introduce the four cardinal coefficients: $A_{i,j}^S = C_{i,j}^{NS}/D_{i,j}\,$, 
+$A_{i,j}^W = C_{i,j}^{EW}/D_{i,j}\,$, $A_{i,j}^E = C_{i,j+1}^{EW}/D_{i,j}\,$ and $A_{i,j}^N = C_{i+1,j}^{NS}/D_{i,j}\,$, and define 
+$\tilde B_{i,j} = B_{i,j}/D_{i,j}\,$, where $D_{i,j} = C_{i,j}^{NS}+ C_{i+1,j}^{NS} + C_{i,j}^{EW} + C_{i,j+1}^{EW} $ (i.e. the diagonal of 
+\textbf{A}). (VII.5.1) can be rewritten as:
+
+\begin{multline} \label{Eq_solmat_p}
+    A_{i,j}^{N}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i+1,j   } 
++\,A_{i,j}^{E}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i    ,j+1} 		\\
++\,A_{i,j}^{S}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i-1 ,j    } 
++\,A_{i,j}^{W} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i    ,j-1} 	
+               -  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j} = \tilde B_{i,j}
+\end{multline}
+
+The SOR method used is an iterative method. Its algorithm can be summarised 
+as follows [see \citet{Haltiner1980} for a further discussion]:
+
+initialisation (evaluate a first guess from previous time step computations)
+\begin{equation}
+\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^0 = 2{\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^t} - {\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^{t-1}}
+\end{equation}
+iteration $n$, from $n=0$ until convergence, do :
+\begin{multline} 
+R_{i,j}^n
+= A_{i,j}^{N}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i+1,j}^n 
++\,A_{i,j}^{E}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j+1} ^n		\\
++\,A_{i,j}^{S}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i-1,j} ^{n+1}
++\,A_{i,j}^{W} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j-1} ^{n+1}	
+                   -  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j}^n - \tilde B_{i,j}
+\end{multline}
+\begin{equation} 
+     {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j} ^{n+1}
+  = {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j} ^{n}
+  + \omega \;R_{i,j}^n
+\end{equation}
+where \textit{$\omega $ }satisfies $1\leq \omega \leq 2$. An optimal value exists for 
+\textit{$\omega$} which significantly accelerates the convergence, but it has to be 
+adjusted empirically for each model domain (except for a uniform grid where an 
+analytical expression for \textit{$\omega$} can be found \citep{Richtmyer1967}). 
+The value of $\omega$ is set using \textit{sor}, a \textbf{namelist} parameter. 
+The convergence test is of the form:
+\begin{equation}
+\delta = \frac{\sum\limits_{i,j}{R_{i,j}^n}{R_{i,j}^n}}{\sum\limits_{i,j}{ \tilde B_{i,j}^n}{\tilde B_{i,j}^n}} \leq \epsilon
+\end{equation}
+where $\epsilon$ is the absolute precision that is required. It is recommended 
+that a value smaller or equal to $10^{-3}$ is used for $\epsilon$ since larger 
+values may lead to numerically induced basin scale barotropic oscillations. In fact, 
+for an eddy resolving configuration or in a filtered free surface case, a value three 
+orders of magnitude smaller than this should be used. The precision is specified by 
+setting \textit{eps} (\textbf{namelist parameter}). In addition, two other tests are 
+used to halt the iterative algorithm. They involve the number of iterations and the 
+modulus of the right hand side. If the former exceeds a specified value, \textit{nmax} 
+(\textbf{namelist parameter}), or the latter is greater than $10^{15}$, the whole 
+model computation is stopped and the last computed time step fields are saved 
+in the standard output file. In both cases, this usually indicates that there is something 
+wrong in the model configuration (an error in the mesh, the initial state, the input forcing, 
+or the magnitude of the time step or of the mixing coefficients). A typical value of 
+$nmax$ is a few hundred when $\epsilon = 10^{-6}$, increasing to a few 
+thousand when $\epsilon = 10^{-12}$.
+The vectorization of the SOR algorithm is not straightforward. The scheme
+contains two linear recurrences on $i$ and $j$. This inhibits the vectorisation. 
+Therefore it has been rewritten as:
+
+\begin{multline} 
+R_{i,j}^n
+= A_{i,j}^{N}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i+1,j}^n 
++\,A_{i,j}^{E}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j+1} ^n		\\
++\,A_{i,j}^{S}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i-1,j} ^{n}
++\,A_{i,j}^{W} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j-1} ^{n}	
+                   -  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j}^n - \tilde B_{i,j}
+\end{multline}
+
+\begin{equation}
+R_{i,j}^n = R_{i,j}^n - \omega \;A_{i,j}^{S}\; R_{i,j-1}^n
+\end{equation}
+
+\begin{equation}
+R_{i,j}^n = R_{i,j}^n - \omega \;A_{i,j}^{W}\; R_{i-1,j}^n
+\end{equation}
+
+The SOR method is very flexible and can be used under a wide 
+range of conditions, including irregular boundaries, interior boundary 
+points, etc. Proofs of convergence, etc. may be found in the standard 
+numerical methods texts for partial differential equations.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Preconditioned Conjugate Gradient
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Preconditioned Conjugate Gradient}
+\label{MISC_solpcg}
+
+\begin{flushright}
+(\textit{nbsfs=1}, \textbf{namelist parameter})
+\end{flushright}
+
+\textbf{A} is a definite positive symmetric matrix, thus solving the linear 
+system \eqref{Eq_solmat} is equivalent to the minimisation of a quadratic 
+functional:
+\begin{equation*}
+\textbf{Ax} = \textbf{b} \leftrightarrow \textbf{x} =\text{inf}_{y} \,\phi (\textbf{y})
+\quad , \qquad
+\phi (\textbf{y}) = 1/2 \langle \textbf{Ay},\textbf{y}\rangle - \langle \textbf{b},\textbf{y} \rangle 
+\end{equation*}
+where $\langle , \rangle$ is the canonical dot product. The idea of the 
+conjugate gradient method is to search for the solution in the following 
+iterative way: assuming that $\textbf{x}^n$ has been obtained, $\textbf{x}^{n+1}$ 
+is found from $\textbf {x}^{n+1}={\textbf {x}}^n+\alpha^n{\textbf {d}}^n$ which satisfies:
+\begin{equation*}
+{\textbf{ x}}^{n+1}=\text{inf} _{{\textbf{ y}}={\textbf{ x}}^n+\alpha^n \,{\textbf{ d}}^n} \,\phi ({\textbf{ y}})\;\;\Leftrightarrow \;\;\frac{d\phi }{d\alpha}=0
+\end{equation*}
+and expressing $\phi (\textbf{y})$ as a function of \textit{$\alpha $}, we obtain the value that minimises the functional: 
+\begin{equation*}
+\alpha ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle  / \langle {\textbf{ A d}^n, \textbf{d}^n} \rangle
+\end{equation*}
+where $\textbf{r}^n = \textbf{b}-\textbf{A x}^n = \textbf{A} (\textbf{x}-\textbf{x}^n)$ 
+is the error at rank $n$. The descent vector $\textbf{d}^n$ s chosen to be dependent 
+on the error: $\textbf{d}^n = \textbf{r}^n + \beta^n \,\textbf{d}^{n-1}$. $\beta ^n$ 
+is searched such that the descent vectors form an orthogonal basis for the dot 
+product linked to \textbf{A}. Expressing the condition 
+$\langle \textbf{A d}^n, \textbf{d}^{n-1} \rangle = 0$ the value of $\beta ^n$ is found:
+ $\beta ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle  / \langle {\textbf{r}^{n-1}, \textbf{r}^{n-1}} \rangle$. As a result, the errors $ \textbf{r}^n$ form an orthogonal 
+base for the canonic dot product while the descent vectors $\textbf{d}^n$ form 
+an orthogonal base for the dot product linked to \textbf{A}. The resulting 
+algorithm is thus the following one:
+
+initialisation :
+
+\begin{equation*} 
+\textbf{x}^0 = \left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^0 = 2{\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^t} - {\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^{t-1}}
+, \text{the initial guess }
+\end{equation*}
+
+\begin{equation*}
+\textbf{r}^0 = \textbf{d}^0 = \textbf{b} - \textbf{A x}^0
+\end{equation*}
+\begin{equation*}
+\gamma_0 = \langle{ \textbf{r}^0 , \textbf{r}^0} \rangle
+\end{equation*}
+iteration $n,$ from $n=0$ until convergence, do :
+
+\begin{equation} 
+\begin{split}
+\text{z}^n& = \textbf{A d}^n \\
+\alpha_n &= \gamma_n  \langle{ \textbf{z}^n , \textbf{d}^n} \rangle \\
+\textbf{x}^{n+1} &= \textbf{x}^n + \alpha_n \,\textbf{d}^n \\
+\textbf{r}^{n+1} &= \textbf{r}^n - \alpha_n \,\textbf{z}^n \\
+\gamma_{n+1} &= \langle{ \textbf{r}^{n+1} , \textbf{r}^{n+1}} \rangle \\
+\beta_{n+1} &= \gamma_{n+1}/\gamma_{n}  \\
+\textbf{d}^{n+1} &= \textbf{r}^{n+1} + \beta_{n+1}\; \textbf{d}^{n}\\
+\end{split}
+\end{equation}
+
+
+The convergence test is:
+
+\begin{equation}
+\delta = \gamma_{n}\; / \langle{ \textbf{b} , \textbf{b}} \rangle \leq \epsilon
+\end{equation}
+where $\epsilon $ is the absolute precision that is required. As for the SOR algorithm, 
+the whole model computation is stopped when the number of iterations, $nmax$, or 
+the modulus of the right hand side of the convergence equation exceeds a 
+specified value (see \S\ref{MISC_solsor} for a further discussion). The required 
+precision and the maximum number of iterations allowed are specified by setting 
+$eps$ and $nmax$ (\textbf{namelist parameters}).
+
+It can be demonstrated that the above algorithm is optimal, provides the exact 
+solution in a number of iterations equal to the size of the matrix, and that 
+the convergence rate is faster as the matrix is closer to the identity matrix,
+$i.e.$ its eigenvalues are closer to 1. Therefore, it is more efficient to solve a 
+better conditioned system which has the same solution. For that purpose, we 
+introduce a preconditioning matrix \textbf{Q} which is an approximation of 
+\textbf{A} but much easier to invert than \textbf{A}, and solve the system:
+\begin{equation} \label{Eq_pmat}
+\textbf{Q}^{-1} \textbf{A x} = \textbf{Q}^{-1} \textbf{b}
+\end{equation}
+
+The same algorithm can be used to solve \eqref{Eq_pmat} if instead of the 
+canonical dot product the following one is used: 
+${\langle{ \textbf{a} , \textbf{b}} \rangle}_Q = \langle{ \textbf{a} , \textbf{Q b}} \rangle$, 
+and if $\textbf{\~{b}} = \textbf{Q}^{-1}\;\textbf{b}$ and $\textbf{\~{A}} = \textbf{Q}^{-1}\;\textbf{A}$ are substituted to \textbf{b} and \textbf{A} \citep{Madec1988}. 
+In \NEMO, \textbf{Q} is chosen as the diagonal of \textbf{ A}, i.e. the simplest form for \textbf{Q} so that it can be easily inverted. In this case, the discrete formulation of 
+\eqref{Eq_pmat} is in fact given by \eqref{Eq_solmat_p} and thus the matrix and 
+right hand side are computed independently from the solver used.
+
+% -------------------------------------------------------------------------------------------------------------
+%       FETI
+% -------------------------------------------------------------------------------------------------------------
+\subsection{FETI}
+\label{MISC_solfeti}
+
+FETI is a powerful solver that was developed by Marc Guyon  \citep{Guyon_al_EP99, 
+Guyon_al_CalPar99}. It has been converted from Fortan 77 to 90, but never 
+successfully tested after that. 
+
+Its main advantage is to save a lot of CPU time when compared to the SOR or PCG 
+algorithms. However, its main drawback is that the solution is dependent on the 
+domain decomposition chosen. Using a different number of processors, the solution 
+is the same at the precision required, but not the same at the computer precision. 
+This make it hard to debug.  
+
+% -------------------------------------------------------------------------------------------------------------
+%       Boundary Conditions --- Islands
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Boundary Conditions --- Islands (\key{islands})}
+\label{MISC_solisl}
+
+The boundary condition used for both recommended solvers is that the time 
+derivative of the barotropic streamfunction is zero along all the coastlines. 
+When islands are present in the model domain, additional computations must 
+be performed to determine the barotropic streamfunction with the correct boundary 
+conditions. This is detailed below.
+
+The model does not have specialised code for islands. They must instead be 
+identified to the solvers by the user via bathymetry information, i.e. the $mbathy$ 
+array should equal $-1$ over the first island, $-2$ over the second, ... , $-N$ over 
+the $N^{th}$ island. The model determines the position of island grid-points and 
+defines a closed contour around each island which is used to compute the circulation 
+around each one. The closed contour is formed from the ocean grid-points which 
+are the closest to the island.
+
+First, the island barotropic streamfunctions $\psi_n$ are computed using the SOR 
+or PCG scheme (they are solutions of \eqref{Eq_solmat} with the right-hand side 
+equal to zero and with $\psi_n = 1$ along the island $n$ and $\psi_n = 0$ along 
+the other coastlines) (Note that specifying $1$ as boundary condition on an island 
+for $\psi$ is equivalent to defining a specific right hand side for \eqref{Eq_solmat}). 
+The requested precision of this computation can be very high since it is only 
+performed once. The absolute precision, $epsisl$, and the maximum number of 
+iterations allowed, $nmisl$, are the \textbf{namelist parameters} used for this 
+computation. Their typical values are $epsisl = 10^{-10}$ and $nmisl = 4000$. 
+Then the island matrix A is computed from \eqref{Eq_PE_psi_isl_matrix} and 
+inverted. At each time step, $\psi_0$, the solution of \eqref{Eq_solmat} with $\psi = 0$ 
+along all coastlines, is computed using either SOR or PCG. (It should
+be noted that the first guess of this computation remains as usual except that 
+$\partial_t \psi_0$ is used, instead of $\partial_t \psi$. Indeed we are 
+computing $\partial_t \psi_0$ which is usually very different from $\partial_t \psi$.)
+Then, it is easy to find the time evolution of the barotropic streamfunction on each 
+island and to deduce $\partial_t \psi$, and to use it to compute the surface pressure 
+gradient. Note that the value of the barotropic streamfunction itself is also computed 
+as the time integration of $\partial_t \psi$ for further diagnostics.
+
+% ================================================================
+% Diagnostics
+% ================================================================
+\section{Diagnostics }
+\label{MISC_diag}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Standard Model Output 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Standard Model Output (default option or \key{dimg})}
+\label{MISC_iom}
+
+%to be updated with Seb documentation on the IO
+\amtcomment{ 
+The model outputs are of three types: the restart file, the output listing, 
+and the output file(s). The restart file is used internally by the code when 
+the user wants to start the model with initial conditions defined by a 
+previous simulation. It contains all the information that is necessary in 
+order for there to be no changes in the model results (even at the computer 
+precision) between a run performed with several restarts and the same run 
+performed in one step. It should be noted that this requires that the restart file 
+contain two consecutive time steps for all the prognostic variables, and 
+that it is saved in the same binary format as the one used by the computer 
+that is to read it (in particular, 32 bits binary IEEE format must not be used for 
+this file). The output listing and file(s) are predefined but should be checked 
+and eventually adapted to the user's needs. The output listing is stored in 
+the $ocean.output$ file. The information is printed from within the code on the 
+logical unit $numout$. To locate these prints, use the UNIX command 
+"$grep -i numout^*$" in the source code directory.
+
+In the standard configuration, the user will find the model results in two 
+output files containing values for every time-step where output is demanded: 
+a \textbf{VO} file containing all the three dimensional fields in logical unit 
+$numwvo$, and a \textbf{SO} file containing all the two dimensional (horizontal) 
+fields in logical unit $numwso$. These outputs are defined in the $diawri.F$ 
+routine. The default and user-selectable diagnostics are described in {\S}III-12-c.
+
+The default output (for all output files apart from the restart file) is 32 bits binary 
+IEEE format, compatible with the Vairmer software (see the Climate Modelling 
+and global Change team WEB server at CERFACS: http://www.cerfacs.fr). 
+The model's reference directory also contains a visualisation tool based on 
+\textbf{NCAR Graphics} (http://ngwww.ucar.edu). If a user has access to the 
+NCAR software, he or she can copy the \textbf{LODMODEL/UTILS/OPADRA} 
+directory from the reference and, following the \textbf{README}, create 
+graphical outputs from the model's results.
+}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Tracer/Dynamics Trends
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Tracer/Dynamics Trends (\key{trdlmd}, \key{diatrdtra}, \key{diatrddyn})}
+\label{MISC_tratrd}
+
+%to be udated this corresponds to OPA8
+When \key{diatrddyn} and/or \key{diatrddyn} cpp variables are defined, each 
+trend of the dynamics and/or temperature and salinity time evolution equations 
+is stored in three-dimensional arrays just after their computation ($i.e.$ at the end 
+of each $dyn\cdots .F90$ and/or $tra\cdots .F90$ routine). These trends are then 
+used in diagnostic routines $diadyn.F90$ and $diatra.F90$ respectively. 
+In the standard model, these routines check the basin averaged properties of 
+the momentum and tracer equations every \textit{ntrd } time-steps (\textbf{namelist 
+parameter}). These routines are supplied as an example; they must be adapted by 
+the user to his/her requirements.
+
+These two options imply the creation of several extra arrays in the in-core 
+memory, increasing quite seriously the code memory requirements.
+
+% -------------------------------------------------------------------------------------------------------------
+%       On-line Floats trajectories
+% -------------------------------------------------------------------------------------------------------------
+\subsection{On-line Floats trajectories}
+\label{FLO}
+%--------------------------------------------namflo-------------------------------------------------------
+\namdisplay{namflo} 
+%--------------------------------------------------------------------------------------------------------------
+
+The on-line computation of floats adevected either by the three dimensional velocity 
+field or constraint to remain at a given depth ($w = 0$ in the computation) have been introduced in the system during the CLIPPER project. The algorithm used is based on 
+the work of \cite{Blanke_Raynaud_JPO97}. (see also the web site describing the off-line 
+use of this marvellous diagnostic tool (http://stockage.univ-brest.fr/~grima/Ariane/).
+
+% -------------------------------------------------------------------------------------------------------------
+%       Other Diagnostics
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Other Diagnostics}
+\label{MISC_diag_others}
+
+%To be updated  this mainly corresponds to OPA 8
+
+Aside from the standard model variables, other diagnostics can be computed 
+on-line or can be added to the model. The available ready-to-add diagnostics 
+routines can be found in directory DIA. Among the available diagnostics are: 
+
+- the mixed layer depth (based on a density criterion) (\mdl{diamxl})
+
+- the turbocline depth (based on a turbulent mixing coefficient criterion) (\mdl{diamxl})
+
+- the depth of the 20\r{}C isotherm (\mdl{diahth})
+
+- the depth of the thermocline (maximum of the vertical temperature gradient) (\mdl{diahth})
+
+- the meridional heat and salt transports and their decomposition (\mdl{diamfl})
+
+- the surface pressure (\mdl{diaspr})
Index: trunk/DOC/TexFiles/Chapters/Chap_Model_Basics.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_Model_Basics.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_Model_Basics.tex	(revision 994)
@@ -0,0 +1,987 @@
+% ================================================================
+% Chapter 1 Ñ Model Basics
+% ================================================================
+
+\chapter{Model basics}
+\label{PE}
+\minitoc
+
+
+% ================================================================
+% Primitive Equations
+% ================================================================
+\section{Primitive Equations}
+\label{PE_PE}
+
+% -------------------------------------------------------------------------------------------------------------
+%        Vector Invariant Formulation 
+% -------------------------------------------------------------------------------------------------------------
+
+\subsection{Vector Invariant Formulation}
+\label{PE_Vector}
+
+
+The ocean is a fluid that can be described to a good approximation by the primitive equations, $i.e.$ the Navier-Stokes equations along with a nonlinear equation of state which couples the two active tracers (temperature and salinity) to the fluid velocity, plus the following additional assumptions made from scale considerations:
+
+\textit{(1) spherical earth approximation: }the geopotential surfaces are assumed to be spheres so that gravity (local vertical) is parallel to the earth's radius
+
+\textit{(2) thin-shell approximation: }the ocean depth is neglected compared to the earth's radius
+
+\textit{(3) turbulent closure hypothesis: }the turbulent fluxes (which represent the effect of small scale processes on the large-scale) are expressed in terms of large-scale features
+
+\textit{(4) Boussinesq hypothesis:} density variations are neglected except in their contribution to the buoyancy force
+
+\textit{(5) Hydrostatic hypothesis: }the vertical momentum equation is reduced to a balance between the vertical pressure gradient and the buoyancy force (this removes convective processes from 
+the initial Navier-Stokes equations and so convective processes must be parameterized instead)
+
+\textit{(6) Incompressibility hypothesis: }the three dimensional divergence of the velocity vector is assumed to be zero.
+
+Because the gravitational force is so dominant in the equations of large-scale motions, it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k}, $i.e.$ tangent to the geopotential surfaces. Let us define the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$ (the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane), $T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density. The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k}) vector system provides the following six equations (namely the momentum balance, the hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation equations and an equation of state):
+\begin{subequations} \label{Eq_PE}
+  \begin{equation}     \label{Eq_PE_dyn}
+\frac{\partial {\rm {\bf U}}_h }{\partial t}=
+-\left[    {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}}
+            +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)}    \right]_h
+ -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h 
+-\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}}
+  \end{equation}
+  \begin{equation}     \label{Eq_PE_hydrostatic}
+\frac{\partial p }{\partial z} = - \rho \ g
+  \end{equation}
+  \begin{equation}     \label{Eq_PE_continuity}
+\nabla \cdot {\bf U}=  0
+  \end{equation}
+\begin{equation} \label{Eq_PE_tra_T}
+\frac{\partial T}{\partial t} = - \nabla \cdot  \left( T \ \rm{\bf U} \right) + D^T + F^T
+  \end{equation}
+  \begin{equation}     \label{Eq_PE_tra_S}
+\frac{\partial S}{\partial t} = - \nabla \cdot  \left( S \ \rm{\bf U} \right) + D^S + F^S
+  \end{equation}
+  \begin{equation}     \label{Eq_PE_eos}
+\rho = \rho \left( T,S,p \right)
+  \end{equation}
+\end{subequations}
+where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, $t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by the equation of state (\ref{Eq_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, $f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's angular velocity vector), and $g$ is the gravitational acceleration. 
+${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterizations of small-scale physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$ surface forcing terms. Their nature and formulation are discussed in \S\ref{PE_zdf_ldf} and page \S\ref{PE_boundary_condition}.
+
+.
+
+% -------------------------------------------------------------------------------------------------------------
+% Boundary condition
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Boundary Conditions}
+\label{PE_boundary_condition}
+
+An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea or ice-sea interface at its top. These boundaries can be defined by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, chosen as a mean sea surface (Fig.~\ref{Fig_ocean_bc}). Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth, the continental margins, the sea ice and the atmosphere. However, some of these fluxes are so weak that even on climatic time scales of thousands of years they can be neglected. In the following, we briefly review the fluxes exchanged at the interfaces between the ocean and the other components of the earth system.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \label{Fig_ocean_bc}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_I_ocean_bc.pdf}
+\caption{The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, where $H$ is the depth of the sea floor and $\eta$ the height of the sea surface. Both $H$ and $\eta $ are referenced to $z=0$.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+\begin{description}
+\item[Land - ocean interface:] the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff. Such an exchange modifies the sea surface salinity especially in the vicinity of major river mouths. It can be neglected for short range integrations but has to be taken into account for long term integrations as it influences the characteristics of water masses formed (especially at high latitudes). It is required in order to close the water cycle of the climate system. It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths.
+\item[Solid earth - ocean interface:] heat and salt fluxes through the sea floor are small, except in special areas of little extent. They are usually neglected in the model \footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world ocean (see \ref{TRA_bbc}).}. The boundary condition is thus set to no flux of heat and salt across solid boundaries. For momentum, the situation is different. There is no flow across solid boundaries, $i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words, the bottom velocity is parallel to solid boundaries). This kinematic boundary condition can be expressed as:
+\begin{equation} \label{Eq_PE_w_bbc}
+w = -{\rm {\bf U}}_h \cdot  \nabla _h \left( H \right)
+\end{equation}
+In addition, the ocean exchanges momentum with the earth through frictional processes. Such momentum transfer occurs at small scales in a boundary layer. It must be parameterized in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification depends on the nature of the physical parameterization used for ${\rm {\bf D}}^{\rm {\bf U}}$ in \eqref{Eq_PE_dyn}. It is discussed in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9.
+\item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux of fresh water PE  (the precipitation minus evaporation budget) leads to: 
+\begin{equation} \label{Eq_PE_w_sbc}
+w = \frac{\partial \eta }{\partial t} 
+    + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot  \nabla _h \left( \eta \right) 
+    + P-E
+\end{equation}
+The dynamic boundary condition, neglecting the surface tension (which removes capillary waves from the system) leads to the continuity of pressure across the interface $z=\eta$. The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat.
+\item[Sea ice - ocean interface:] the ocean and sea ice exchange heat, salt, fresh water and momentum. The sea surface temperature is constrained to be at the freezing point at the interface. Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the ocean ($\sim34 \,psu$). The cycle of freezing/melting is associated with fresh water and salt fluxes that cannot be neglected.
+\end{description}
+
+
+% ================================================================
+% The Horizontal Pressure Gradient
+% ================================================================
+\section{The Horizontal Pressure Gradient }
+\label{PE_hor_pg}
+
+% -------------------------------------------------------------------------------------------------------------
+% Pressure Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Pressure Formulation}
+\label{PE_p_formulation}
+
+The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that: $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ref{Eq_PE_hydrostatic}), assuming that pressure in decibars can be approximated by depth in meters in (\ref{Eq_PE_eos}). The hydrostatic pressure is then given by:
+\begin{equation} \label{Eq_PE_pressure}
+p_h \left( {i,j,z,t} \right)
+ = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,z} \right)\;d\varsigma } 
+\end{equation}
+ Two strategies can be considered for the surface pressure term: $(a)$ introduce of a new variable $\eta$, the free-surface elevation, for which a prognostic equation can be established and solved; $(b)$ assume that the ocean surface is a rigid lid, on which the pressure (or its horizontal gradient) can be diagnosed. When the former strategy is used, one solution of the free-surface elevation consists of the excitation of external gravity waves. The flow is barotropic and the surface moves up and down with gravity as the restoring force. The phase speed of such waves is high (some hundreds of metres per second) so that the time step would have to be very short if they were present in the model. The latter strategy filters out these waves since the rigid lid approximation implies $\eta=0$, $i.e.$ the sea surface is the surface $z=0$. This well known approximation increases the surface wave speed to infinity and modifies certain other longwave dynamics ($e.g.$ barotropic Rossby or planetary waves). In the present release of \NEMO, both strategies are still available. They are further described in the next two sub-sections.
+
+% -------------------------------------------------------------------------------------------------------------
+% Free Surface Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Free Surface Formulation}
+\label{PE_free_surface}
+
+In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced which describes the shape of the air-sea interface. This variable is solution of a prognostic equation which is established by forming the vertical average of the kinematic surface condition (\ref{Eq_PE_w_bbc}):
+\begin{equation} \label{Eq_PE_ssh}
+\frac{\partial \eta }{\partial t}=-D+P-E
+ 	\quad \text{where} \ 
+D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]
+\end{equation}
+and using (\ref{Eq_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.
+
+Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as a class of solution of the primitive equations. These waves are barotropic because of hydrostatic assumption, and their phase speed is quite high. Their time scale is short with respect to the other processes described by the primitive equations.
+
+Three choices can be made regarding the implementation of the free surface in the model, depending on the physical processes of interest. 
+
+$\bullet$ If one is interested in EGWs, in particular the tides and their interaction 
+with the baroclinic structure of the ocean (internal waves) possibly in 
+shallow seas, then a non linear free surface is the most appropriate. This 
+means that no approximation is made in (\ref{Eq_PE_ssh}) and that the variation of the 
+ocean volume is fully taken into account. Note that in order to study the 
+fast time scales associated with EGWs it is necessary to minimize time 
+filtering effects (use an explicit time scheme with very small time step, or 
+a split-explicit scheme with reasonably small time step, see \S\ref{DYN_spg_exp} or
+\S\ref{DYN_spg_ts}.
+
+$\bullet$ If one is not interested in EGW but rather sees them as high frequency 
+noise, it is possible to apply an explicit filter to slow down the fastest waves while 
+not altering the slow barotropic Rossby waves. If further, an approximative conservation 
+of heat and salt contents is sufficient for the problem solved, then it is 
+sufficient to solve a linearized version of (\ref{Eq_PE_ssh}), which still allows 
+to take into account freshwater fluxes applied at the ocean surface \citep{Roullet2000}.
+
+$\bullet$ For process studies not involving external waves nor surface freshwater 
+fluxes, it is possible to use the rigid lid approximation see (next 
+section). The ocean surface is then considered as a fixed surface, so that all 
+external waves are removed from the system. 
+
+The filtering of EGWs in models with a free surface is usually a matter of discretisation of the temporal derivatives, using the time splitting method \citep{Killworth1991, Zhang1992} or the implicit scheme \citep{Dukowicz1994}. In \NEMO, we use a slightly different approach developed by \citet{Roullet2000}: the damping of EGWs is ensured by introducing an additional force in the momentum equation. \eqref{Eq_PE_dyn} becomes: 
+\begin{equation} \label{Eq_PE_flt}
+\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
+- g \nabla \left( \tilde{\rho} \ \eta \right) 
+- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) 
+\end{equation}
+where $T_c$, is a parameter with dimensions of time which characterizes the force, $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, non-linear and viscous terms in \eqref{Eq_PE_dyn}.
+
+The new force can be interpreted as a diffusion of vertically integrated volume flux divergence. The time evolution of $D$ is thus governed by a balance of two terms, $-g$ \textbf{A} $\eta$ and $g \, T_c \,$ \textbf{A} $D$, associated with a propagative regime and a diffusive regime in the temporal spectrum, respectively. In the diffusive regime, the EGWs no longer propagate, $i.e.$ they are stationary and damped. The diffusion regime applies to the modes shorter than $T_c$. For longer ones, the diffusion term vanishes. Hence, the temporally unresolved EGWs can be damped by choosing $T_c > \Delta t$. \citet{Roullet2000} demonstrate that (\ref{Eq_PE_flt}) can be integrated with a leap frog scheme except the additional term which has to be computed implicitly. This is not surprising since the use of a large time step has a necessarily numerical cost. Two gains arise in comparison with the previous formulations. Firstly, the damping of EGWs can be quantified through the magnitude of the additional term. Secondly, the numerical scheme does not need any tuning. Numerical stability is ensured as soon as $T_c > \Delta t$.
+
+When the variations of free surface elevation are small compared to the thickness of the first model layer, the free surface equation (\ref{Eq_PE_ssh}) can be linearized. As emphasized by \citet{Roullet2000} the linearization of (\ref{Eq_PE_ssh}) has consequences on the conservation of salt in the model. With the nonlinear free surface equation, the time evolution of the total salt content is 
+\begin{equation} \label{Eq_PE_salt_content}
+\frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} =\int\limits_S 
+{S\;(-\frac{\partial \eta }{\partial t}-D+P-E)\;ds}
+\end{equation}
+where $S$ is the salinity, and the total salt is integrated over the whole ocean volume $D_\eta$ bounded by the time-dependent free surface. The right hand side (which is an integral over the free surface) vanishes when the nonlinear equation (\ref{Eq_PE_ssh}) is satisfied, so that the salt is perfectly conserved. When the free surface equation is linearized, \citet{Roullet2000} show that the total salt content integrated in the fixed volume $D$ (bounded by the surface $z=0$) is no longer conserved:
+\begin{equation} \label{Eq_PE_salt_content_linear}
+\frac{\partial }{\partial t}\int\limits_D {S\;dv} =-\int\limits_S 
+{S\;\frac{\partial \eta }{\partial t}ds} 
+\end{equation}
+
+The right hand side of (\ref{Eq_PE_salt_content_linear}) is small in equilibrium solutions \citep{Roullet2000}. It can be significant when the freshwater forcing is not balanced and the globally averaged free surface is drifting. An increase in sea surface height \textit{$\eta $} results in a decrease of the salinity in the fixed volume $D$. Even in that case though, the total salt integrated in the variable volume $D_{\eta}$ varies much less, since (\ref{Eq_PE_salt_content_linear}) can be rewritten as 
+\begin{equation} \label{Eq_PE_salt_content_corrected}
+\frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} 
+=\frac{\partial}{\partial t} \left[ \;{\int\limits_D {S\;dv} +\int\limits_S {S\eta \;ds} } \right]
+=\int\limits_S {\eta \;\frac{\partial S}{\partial t}ds}
+\end{equation}
+
+Although the total salt content is not exactly conserved with the linearized free surface, its variations are driven by correlations of the time variation of surface salinity with the sea surface height, which is a negligible term. This situation contrasts with 
+the case of the rigid lid approximation (following section) in which case freshwater forcing is represented by a virtual salt flux, leading to a spurious source of salt at the ocean surface \citep{Roullet2000}.
+
+% -------------------------------------------------------------------------------------------------------------
+% Rigid-Lid Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Rigid-Lid formulation}
+\label{PE_rigid_lid}
+
+With the rigid lid approximation, we assume that the ocean surface ($z=0$) is a rigid lid on which a pressure $p_s$ is exerted. This implies that the vertical velocity at the surface is equal to zero. From the continuity equation \eqref{Eq_PE_continuity} and the kinematic condition at the bottom \eqref{Eq_PE_w_bbc} (no flux across the bottom), it can be shown that the vertically integrated flow $H{\rm {\bf \bar {U}}}_h$ is non-divergent (where the overbar indicates a vertical average over the whole water column, i.e. from $z=-H$, the ocean bottom, to $z=0$ , the rigid-lid). Thus, $\rm {\bf \bar {U}}_h$ can be derived from a volume transport streamfunction $\psi$:
+\begin{equation} \label{Eq_PE_u_psi}
+\overline{\vect{U}}_h =\frac{1}{H}\left(   \vect{k} \times \nabla \psi   \right)
+\end{equation}
+
+As $p_s$ does not depend on depth, its horizontal gradient is obtained by forming the vertical average of \eqref{Eq_PE_dyn} and using \eqref{Eq_PE_u_psi}:
+
+\begin{equation} \label{Eq_PE_u_barotrope}
+\frac{1}{\rho _o }\nabla _h p_s 
+=\overline{\vect{M}} -\frac{\partial \overline{\vect{U}} _h }{\partial t}
+=\overline{\vect{M}} 
+-\frac{1}{H} \left[   \vect{k} \times \nabla \left( \frac{\partial \psi}{\partial t} \right)   \right]
+\end{equation}
+
+Here ${\rm {\bf M}} = \left( M_u,M_v \right)$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, nonlinear and viscous terms in \eqref{Eq_PE_dyn}. The time derivative of $\psi $ is the solution of an elliptic equation which is obtained from the vertical component of the curl of (\ref{Eq_PE_u_barotrope}):
+\begin{equation} \label{Eq_PE_psi}
+\left[   {\nabla \times \left[ {\frac{1}{H} \vect{\bf k} 
+  \times \nabla \left(   {\frac{\partial \psi }{\partial t}} \right)}   \right]} \; \right]_z 
+=\left[   {\nabla \times \overline{\vect{M}} }   \right]_z 
+\end{equation}
+
+Using the proper boundary conditions, \eqref{Eq_PE_psi} can be solved to find $\partial_t \psi$ and thus using \eqref{Eq_PE_u_barotrope} the horizontal surface pressure gradient. It should be noted that $p_s$ can be computed by taking the divergence of \eqref{Eq_PE_u_barotrope} and solving the resulting elliptic equation. Thus the surface pressure is a diagnostic quantity that can be recovered for analysis purposes.
+
+A difficulty lies in the determination of the boundary condition on $\partial_t \psi$. The boundary condition on velocity is that there is no flow normal to a solid wall, $i.e.$ the coastlines are streamlines. Therefore \eqref{Eq_PE_psi} is solved with the following Dirichlet boundary condition: $\partial_t \psi$ is constant along each coastline of the same continent or of the same island. When all the coastlines are connected (there are no islands), the constant value of $\partial_t \psi$ along the coast can be arbitrarily chosen to be zero. When islands are present in the domain, the value of the barotropic streamfunction will generally be different for each island and for the continent, and will vary with respect to time. So the boundary condition is: $\psi=0$ along the continent and $\psi=\mu_n$ along island $n$ ($1 \leq n \leq Q$), where $Q$ is the number of islands present in the domain and $\mu_n$ is a time dependent variable. A time evolution equation of the unknown $\mu_n$ can be found by evaluating the circulation of the time derivative of the vertical average (barotropic) velocity field along a closed contour around each island. Since the circulation of a gradient field along a closed contour is zero, from \eqref{Eq_PE_u_barotrope} we have:
+\begin{equation} \label{Eq_PE_isl_circulation}
+\oint_n {\frac{1}{H}\left[ {{\rm {\bf k}}\times \nabla \left( 
+{\frac{\partial \psi }{\partial t}} \right)} \right] \cdot {\rm {\bf d}}\ell } 
+= \oint_n {\overline {\rm {\bf M}} \cdot {\rm {\bf d}}\ell } 
+\qquad  1 \leq n \leq Q
+\end{equation}
+
+Since (\ref{Eq_PE_psi}) is linear, its solution \textit{$\psi $} can be decomposed 
+as follows:
+\begin{equation} \label{Eq_PE_psi_isl}
+\psi =\psi _o +\sum\limits_{n=1}^{n=Q} {\mu _n \psi _n } 
+\end{equation}
+where $\psi _o$ is the solution of \eqref{Eq_PE_psi} with $\psi _o=0$ long all 
+the coastlines, and where $\psi _n$ is the solution of \eqref{Eq_PE_psi} with 
+the right-hand side equal to $0$, and with $\psi _n =1$ long the island $n$, 
+$\psi _n =0$ along the other boundaries. The function $\psi _n$ is thus 
+independent of time. Introducing \eqref{Eq_PE_psi_isl} into 
+\eqref{Eq_PE_isl_circulation} yields:
+\begin{multline} \label{Eq_PE_psi_isl_circulation}
+\left[ {\oint_n {\frac{1}{H}  \left[ {{\rm {\bf k}}\times \nabla \psi _m } \right]\cdot {\rm {\bf d}}\ell } } \right]_{1\leq m\leqslant Q \atop 1\leq n\leqslant Q  }
+ \left( {\frac{\partial \mu _n }{\partial t}} 
+\right)_{1\leqslant n\leqslant Q}        \\
+ =\left( {\oint_n {\left[ {\overline {\rm 
+{\bf M}} -\frac{1}{H}\left[ {{\rm {\bf k}}\times \nabla \left( 
+{\frac{\partial \psi _o }{\partial t}} \right)} \right]} \right]\cdot {\rm 
+{\bf d}}\ell } } \right)_{1\leqslant n\leqslant Q} 
+\end{multline}
+which can be rewritten as:
+\begin{equation} \label{Eq_PE_psi_isl_matrix}
+{\rm {\bf A}}\;\left( {\frac{\partial \mu _n }{\partial t}} 
+\right)_{1\leqslant n\leqslant Q} ={\rm {\bf B}}
+\end{equation}
+where \textbf{A} is a $Q  \times Q$ matrix and \textbf{B} is a time dependent vector. As \textbf{A} is independent of time, it can be calculated and inverted once. The time derivative of the streamfunction when islands are present is thus given by:
+\begin{equation} \label{Eq_PE_psi_isl_dt}
+\frac{\partial \psi }{\partial t}=\frac{\partial \psi _o }{\partial 
+t}+\sum\limits_{n=1}^{n=Q} {{\rm {\bf A}}^{-1}{\rm {\bf B}}\;\psi _n } 
+\end{equation}
+
+
+
+% ================================================================
+% Curvilinear z-coordinate System
+% ================================================================
+\section{Curvilinear \textit{z-}coordinate System}
+\label{PE_zco}
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Tensorial Formalism
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Tensorial Formalism}
+\label{PE_tensorial}
+
+In many ocean circulation problems, the flow field has regions of enhanced dynamics ($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts). The representation of such dynamical processes can be improved by specifically increasing the model resolution in these regions. As well, it may be convenient to use a lateral boundary-following coordinate system to better represent coastal dynamics. Moreover, the common geographical coordinate system has a singular point at the North Pole that cannot be easily treated in a global model without filtering. A solution consists of introducing an appropriate coordinate transformation that shifts the singular point onto land \citep{MadecImb1996, Murray1996}. As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems. An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism. This formalism is suited to any multidimensional curvilinear coordinate system. Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation), with preservation of the local vertical. Here we give the simplified equations for this particular case. The general case is detailed by \citet{Eiseman1980} in their survey of the conservation laws of fluid dynamics.
+
+Let (\textbf{i},\textbf{j},\textbf{k}) be a set of orthogonal curvilinear coordinates on the sphere associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (Fig.\ref{Fig_referential}). Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea level (Fig.\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$, the three scale factors:
+\begin{equation} \label{Eq_scale_factors}
+\begin{aligned}
+ e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda 
+}{\partial i}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi 
+}{\partial i}} \right)^2} \right]^{1/2} \\ 
+ e_2 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda 
+}{\partial j}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi 
+}{\partial j}} \right)^2} \right]^{1/2} \\ 
+ e_3 &=\left( {\frac{\partial z}{\partial k}} \right) \\ 
+ \end{aligned}
+ \end{equation}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \label{Fig_referential}  \begin{center}
+\includegraphics[width=0.60\textwidth]{./Figures/Fig_I_earth_referential.pdf}
+\caption{the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear coordinate system (\textbf{i},\textbf{j},\textbf{k}). }
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by $a$ in (\ref{Eq_scale_factors}) (thin-shell approximation). The resulting horizontal scale factors $e_1$, $e_2$  are independent of $k$ while the vertical scale factor is a single function of $k$ as \textbf{k} is parallel to \textbf{z}. The scalar and vector operators that appear in the primitive equations (Eqs. \eqref{Eq_PE_dyn} to \eqref{Eq_PE_eos}) can be written in the tensorial form, invariant in any orthogonal horizontal curvilinear coordinate system transformation:
+\begin{subequations} \label{Eq_PE_discrete_operators}
+\begin{equation} \label{Eq_PE_grad}
+\nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf 
+i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3 
+}\frac{\partial q}{\partial k}\;{\rm {\bf k}}    \\
+\end{equation}
+\begin{equation} \label{Eq_PE_div}
+\nabla \cdot {\rm {\bf A}} 
+= \frac{1}{e_1 \; e_2} \left[ 
+  \frac{\partial \left(e_2 \; a_1\right)}{\partial i }
++\frac{\partial \left(e_1 \; a_2\right)}{\partial j }    	\right]
++ \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k }   \right]
+\end{equation}
+\begin{equation} \label{Eq_PE_curl}
+   \begin{split}
+\nabla \times \vect{A} = 
+    \left[ {\frac{1}{e_2 }\frac{\partial a_3}{\partial j}
+            -\frac{1}{e_3 }\frac{\partial a_2 }{\partial k}} \right] \; \vect{i}
+&+\left[ {\frac{1}{e_3 }\frac{\partial a_1 }{\partial k}
+           -\frac{1}{e_1 }\frac{\partial a_3 }{\partial i}} \right] \; \vect{j} 		\\
+&+\frac{1}{e_1 e_2 } \left[ {\frac{\partial \left( {e_2 a_2 } \right)}{\partial i}
+                                       -\frac{\partial \left( {e_1 a_1 } \right)}{\partial j}} \right] \; \vect{k} 
+   \end{split}
+\end{equation}
+\begin{equation} \label{Eq_PE_lap}
+\Delta q = \nabla \cdot \left(  \nabla q \right)
+\end{equation}
+\begin{equation} \label{Eq_PE_lap_vector}
+\Delta {\rm {\bf A}} =
+  \nabla \left( \nabla \cdot {\rm {\bf A}} \right)
+- \nabla \times \left(  \nabla \times {\rm {\bf A}} \right)
+\end{equation}
+\end{subequations}
+where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinate system.
+
+% -------------------------------------------------------------------------------------------------------------
+% Continuous Model Equations
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Continuous Model Equations}
+\label{PE_zco_Eq}
+
+In order to express the Primitive Equations in tensorial formalism, it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using \eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}. Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by:
+\begin{equation} \label{Eq_PE_curl_Uh}
+\zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} 
+\right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}} 
+\right]
+\end{equation}
+\begin{equation} \label{Eq_PE_div_Uh}
+\chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u} 
+\right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}} 
+\right]
+\end{equation}
+
+Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that $e_3$  is a function of the single variable $k$, the nonlinear term of \eqref{Eq_PE_dyn} can be transformed as follows:
+\begin{flalign*}
+&\left[ {\left( {	\nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
++\frac{1}{2}	\nabla \left( {{\rm {\bf U}}^2} \right)}	 \right]_h 			&
+\end{flalign*}
+\begin{flalign*}
+&\qquad=\left( {{\begin{array}{*{20}c}
+ {\left[ 	{	 \frac{1}{e_3}	\frac{\partial u	}{\partial k}
+			-\frac{1}{e_1}	\frac{\partial w	}{\partial i} } \right] w - \zeta \; v } 	 	\\
+ 		{\zeta \; u - \left[ {	 \frac{1}{e_2} \frac{\partial w}{\partial j}
+							-\frac{1}{e_3} \frac{\partial v}{\partial k} } \right] \ w}	 \\
+		 \end{array} }} \right)			
++\frac{1}{2}	\left( {{\begin{array}{*{20}c}
+		 {	\frac{1}{e_1}	\frac{\partial \left( u^2+v^2+w^2 \right)}{\partial i}} 	\hfill	 \\
+		 {	\frac{1}{e_2}	\frac{\partial \left( u^2+v^2+w^2 \right)}{\partial j}} 	\hfill	 \\
+		 \end{array} }} \right)			&
+\end{flalign*}
+\begin{flalign*}
+& \qquad =\left( {{	\begin{array}{*{20}c}
+ {-\zeta \; v} \hfill \\
+ { \zeta \; u} \hfill \\
+			\end{array} }} \right)
++\frac{1}{2}\left( {{	\begin{array}{*{20}c}
+ {\frac{1}{e_1 }\frac{\partial \left( {u^2+v^2} \right)}{\partial i}} \hfill 	\\
+ {\frac{1}{e_2 }\frac{\partial \left( {u^2+v^2} \right)}{\partial j}} \hfill 	\\
+						\end{array} }} \right)			
++\frac{1}{e_3 }\left( {{		\begin{array}{*{20}c}
+ { w \; \frac{\partial u}{\partial k}} 	\\
+ { w \; \frac{\partial v}{\partial k}} 	\\
+							\end{array} }} \right)	
+-\left( {{	\begin{array}{*{20}c}
+ {\frac{w}{e_1}\frac{\partial w}{\partial i}
+ -\frac{1}{2e_1}\frac{\partial w^2}{\partial i}} \hfill \\
+ {\frac{w}{e_2}\frac{\partial w}{\partial j}
+  -\frac{1}{2e_2}\frac{\partial w^2}{\partial j}} \hfill \\
+			\end{array} }} \right) 			&
+\end{flalign*}
+
+The last term of the right hand side is obviously zero, and thus the nonlinear term of \eqref{Eq_PE_dyn} is written in the $(i,j,k)$ coordinate system:
+\begin{equation} \label{Eq_PE_vector_form}
+\left[ {\left( {	\nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
++\frac{1}{2}	\nabla \left( {{\rm {\bf U}}^2} \right)}	 \right]_h 
+=\zeta 
+\;{\rm {\bf k}}\times {\rm {\bf U}}_h +\frac{1}{2}\nabla _h \left( {{\rm 
+{\bf U}}_h^2 } \right)+\frac{1}{e_3 }w\frac{\partial {\rm {\bf U}}_h 
+}{\partial k}   	
+\end{equation}
+
+This is the so-called \textit{vector invariant form} of the momentum advection term. For some purposes, it can be advantageous to write this term in the so-called flux form, $i.e.$ to write it as the divergence of fluxes. For example, the first component of \eqref{Eq_PE_vector_form} (the $i$-component) is transformed as follows:
+\begin{flalign*}
+&{	\begin{array}{*{20}l}
+\left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U}
+          +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i   % \\
+%\\
+     = - \zeta \;v 
+     + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
+     + \frac{1}{e_3}w \ \frac{\partial u}{\partial k} 			\\
+\\
+\qquad =\frac{1}{e_1 \; e_2} \left( 	-v\frac{\partial \left( {e_2 \,v} \right)}{\partial i}
+							+v\frac{\partial \left( {e_1 \,u} \right)}{\partial j}	 \right)
++\frac{1}{e_1 e_2 }\left( 	+e_2 \; u\frac{\partial u}{\partial i}
+ 							+e_2 \; v\frac{\partial v}{\partial i}	  				 \right) 
++\frac{1}{e_3}       \left( 	w\;\frac{\partial u}{\partial k} 		\right)   \\
+\end{array} } 			&
+\end{flalign*}
+\begin{flalign*}
+&{	\begin{array}{*{20}l}
+\qquad =\frac{1}{e_1 \; e_2} 	\left\{ 
+ -\left(   	    v^2 	\frac{\partial e_2                                }{\partial i} 
+ 		+e_2 \,v		\frac{\partial v                                   }{\partial i}  	\right)
++\left( 				\frac{\partial \left( {e_1 \,u\,v} 	\right)}{\partial j}
+		-e_1 \,u		\frac{\partial v                                   }{\partial j} 	\right) 	\right. 
+\\  \left.	\qquad \qquad \quad
++\left( 				\frac{\partial \left( {e_2 u\,u}     \right)}{\partial i}
+		-u			\frac{\partial \left( {e_2 u}         \right)}{\partial i} 	\right)
++e_2 v		 		\frac{\partial v                                    }{\partial i}
+ 						\right\} 
++\frac{1}{e_3} \left(
+ 					\frac{\partial \left( {w\,u} \right)         }{\partial k}
+		 -u			\frac{\partial w 						   }{\partial k} 	\right) \\
+\end{array} } 		&
+\end{flalign*}
+\begin{flalign*}
+&{	\begin{array}{*{20}l}
+\qquad =\frac{1}{e_1 \; e_2} 	\left( 
+					\frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
+		+			\frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j}	\right)
++\frac{1}{e_3 }		\frac{\partial \left( {w\,u		 } \right)}{\partial k}
+\\  \qquad \qquad \quad
++\frac{1}{e_1 e_2 }		\left( 
+ 		-u	\left( 	\frac{\partial \left( {e_1 v	 } \right)}{\partial j}
+	 				-v\,\frac{\partial e_1 }{\partial j}				 \right)
+		-u			\frac{\partial \left( {e_2 u	 } \right)}{\partial i}
+						\right)
+ -\frac{1}{e_3 }	 	\frac{\partial w}{\partial k}	u
+ +\frac{1}{e_1 e_2 }\left( 	-v^2\frac{\partial e_2	 }{\partial i}		 \right)	
+\end{array} } 		&
+\end{flalign*}
+\begin{flalign*}
+&{	\begin{array}{*{20}l}
+\qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u}	\right)
+-   \left( \nabla \cdot {\rm {\bf U}} \right) \ u
++\frac{1}{e_1 e_2 }\left( 
+		-v^2		\frac{\partial e_2 }{\partial i}
+		+uv	\,		\frac{\partial e_1 }{\partial j} 	\right) \\
+\end{array} } 		&
+\end{flalign*}
+as $\nabla \cdot {\rm {\bf U}}\;=0$ (incompressibility) it comes:
+\begin{flalign*}
+&{	\begin{array}{*{20}l}
+\qquad = \nabla \cdot \left( 	{{\rm {\bf U}}\,u}		\right)
++	\frac{1}{e_1 e_2 }   \left( v \; \frac{\partial e_2}{\partial i}
+						       -u \; \frac{\partial e_1}{\partial j} 	\right)  \left( -v \right)	
+\end{array} } 		&
+\end{flalign*}
+
+The flux form of the momentum advection term is therefore given by:
+\begin{multline} \label{Eq_PE_flux_form}
+		\left[ 
+  \left( 	{\nabla \times {\rm {\bf U}}} 	\right) \times {\rm {\bf U}}
++\frac{1}{2}	\nabla \left( 	{{\rm {\bf U}}^2} 	\right)
+		\right]_h 
+\\
+= \nabla \cdot 	\left( {{\begin{array}{*{20}c}	{\rm {\bf U}} \, u 	\hfill \\
+												{\rm {\bf U}} \, v	\hfill \\
+						\end{array} }} 	
+				\right)
++\frac{1}{e_1 e_2 }		\left( 
+		 v\frac{\partial e_2}{\partial i}
+		-u\frac{\partial e_1}{\partial j} 
+						\right) {\rm {\bf k}} \times {\rm {\bf U}}_h
+\end{multline}
+
+The flux form has two terms, the first one is expressed as the divergence of momentum fluxes (hence the flux form name given to this formulation) and the second one is due to the curvilinear nature of the coordinate system used. The latter is called the \emph{metric} term and can be viewed as a modification of the Coriolis parameter: 
+\begin{equation} \label{Eq_PE_cor+metric}
+f \to f + \frac{1}{e_1 \; e_2}	\left( 	 v \frac{\partial e_2}{\partial i}
+										-u \frac{\partial e_1}{\partial j} 	\right)
+\end{equation}
+
+Note that in the case of geographical coordinate, $i.e.$ when $(i,j) \to (\lambda ,\varphi )$ and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, we recover the commonly used modification of the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
+
+To sum up, the equations solved by the ocean model can be written in the following tensorial formalism:
+
+\vspace{+10pt}
+$\bullet$ \textit{momentum equations} :
+
+vector invariant form :
+\begin{subequations} \label{Eq_PE_dyn_vect}
+\begin{multline} \label{Eq_PE_dyn_vect_u}
+\frac{\partial u}{\partial t}=
+	+   \left( {\zeta +f} \right)\,v                                    
+	-   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
+	-   \frac{1}{e_3} w \frac{\partial u}{\partial k}       \\
+	-   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho _o}    \right)    
++   D_u^{\vect{U}}  +   F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Eq_PE_dyn_vect_v}
+\frac{\partial v}{\partial t}=
+	-   \left( {\zeta +f} \right)\,u   
+	-   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)     	-   \frac{1}{e_3 }w\frac{\partial v}{\partial k}         \\
+	-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)    
++  D_v^{\vect{U}}  +   F_v^{\vect{U}}
+\end{multline}
+\end{subequations}
+
+flux form:
+\begin{subequations} \label{Eq_PE_dyn_flux}
+\begin{multline} \label{Eq_PE_dyn_flux_u}
+\frac{\partial u}{\partial t}=
++   \left( { f + \frac{1}{e_1 \; e_2}
+					\left( 	 v \frac{\partial e_2}{\partial i}
+						-u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, v    \\
+- \frac{1}{e_1 \; e_2} 	\left( 
+					\frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
+		+			\frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j}	\right)
+                 - \frac{1}{e_3 }\frac{\partial \left( {         w\,u} \right)}{\partial k}    \\
+-   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho _o}   \right)
++   D_u^{\vect{U}} +   F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Eq_PE_dyn_flux_v}
+\frac{\partial v}{\partial t}=
+-   \left( { f + \frac{1}{e_1 \; e_2}
+					\left( 	 v \frac{\partial e_2}{\partial i}
+						-u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, u   \\
+ \frac{1}{e_1 \; e_2} 	\left( 
+					\frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i}
+		+			\frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j}	\right)
+                 - \frac{1}{e_3 } \frac{\partial \left( {        w\,v} \right)}{\partial k}    \\
+-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}    \right)
++  D_v^{\vect{U}} +  F_v^{\vect{U}} 
+\end{multline}
+\end{subequations}
+where $\zeta$ is given by \eqref{Eq_PE_curl_Uh} and the surface pressure gradient formulation depends on the one of the free surface:
+
+$*$ free surface formulation
+\begin{equation}\label{Eq_PE_dyn_sco}
+\frac{1}{\rho _o }\nabla _h p_s =\left( {{\begin{array}{*{20}c}
+ {\frac{g}{\;e_1 }\frac{\partial \eta }{\partial i}} \hfill \\
+ {\frac{g}{\;e_2 }\frac{\partial \eta }{\partial j}} \hfill \\
+\end{array} }} \right)
+\qquad \text{where $\eta$ is solution of \eqref{Eq_PE_ssh} }
+\end{equation}
+
+$*$ rigid-lid approximation
+\begin{equation}\label{Eq_PE_dyn_zco}
+\frac{1}{\rho _o }\nabla _h p_s =\left( {{\begin{array}{*{20}c}
+ {\overline M _u +\frac{1}{H\;e_2 }\frac{\partial }{\partial j}\left( 
+{\frac{\partial \psi }{\partial t}} \right)}		 \\
+ {\overline M _v -\frac{1}{H\;e_1 }\frac{\partial }{\partial i}\left( 
+{\frac{\partial \psi }{\partial t}} \right)} 		 \\
+\end{array} }} \right)
+\end{equation}
+where ${\vect{M}}= \left( M_u,M_v \right)$ represents the collected contributions of nonlinear, 
+viscosity and hydrostatic pressure gradient terms in \eqref{Eq_PE_dyn_vect} and the overbar indicates a vertical average over the whole water column ($i.e.$ from $z=-H$, the ocean bottom, to $z=0$, the rigid-lid), and where the time derivative of $\psi$ is the solution of an elliptic equation:
+\begin{multline} \label{Eq_psi_total}
+  \frac{\partial }{\partial i}\left[ {\frac{e_2 }{H\,e_1}\frac{\partial}{\partial i}
+								 \left( {\frac{\partial \psi }{\partial t}} \right)}   \right]
++\frac{\partial }{\partial j}\left[ {\frac{e_1 }{H\,e_2}\frac{\partial }{\partial j}
+								 \left( {\frac{\partial \psi }{\partial t}} \right)} \right]
+= \\
+  \frac{\partial }{\partial i}\left( {e_2 \overline M _v } \right)
+- \frac{\partial }{\partial j}\left( {e_1 \overline M _u } \right)
+\end{multline}
+
+The vertical velocity and the hydrostatic pressure are diagnosed from the following equations:
+\begin{equation} \label{Eq_w_diag}
+\frac{\partial w}{\partial k}=-\chi \;e_3 
+\end{equation}
+\begin{equation} \label{Eq_hp_diag}
+\frac{\partial p_h }{\partial k}=-\rho \;g\;e_3 
+\end{equation}
+
+where the divergence of the horizontal velocity, $\chi$ is given by \eqref{Eq_PE_div_Uh}.
+
+\vspace{+10pt}
+$\bullet$ \textit{tracer equations} :
+\begin{equation} \label{Eq_S}
+\frac{\partial T}{\partial t} = 
+-\frac{1}{e_1 e_2 }\left[ {	   \frac{\partial \left( {e_2 T\,u} \right)}{\partial i}
+						+\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right]
+-\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T
+\end{equation}
+\begin{equation} \label{Eq_T}
+\frac{\partial S}{\partial t} = 
+-\frac{1}{e_1 e_2 }\left[ 	  {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i}
+						+\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right]
+-\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S
+\end{equation}
+\begin{equation} \label{Eq_rho}
+\rho =\rho \left( {T,S,z(k)} \right)
+\end{equation}
+
+The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid 
+scale parameterization used. It will be defined in \S\ref{PE_zdf}. The nature and formulation of ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed in Chapter~\ref{SBC}.
+
+\newpage 
+% ================================================================
+% Curvilinear z*-coordinate System
+% ================================================================
+\section{Curvilinear \textit{z*}-coordinate System}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!b] \label{Fig_z_zstar}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./Figures/Fig_z_zstar.pdf}
+\caption{(a) $z$-coordinate in linear free-surface case ; (b) $z-$coordinate in non-linear free surface case (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate \citep{Adcroft_Campin_OM04} ).}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. These coordinates systems is presented in a report 
+\citep{Levier2007} available on the \NEMO web site. 
+
+\gmcomment{
+The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation which allows one to deal with large amplitude free-surface
+variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. In
+the  \textit{z*} formulation, the variation of the column thickness due to sea-surface
+undulations is not concentrated in the surface level, as in the $z$-coordinate formulation,
+but is equally distributed over the full water column. Thus vertical
+levels naturally follow sea-surface variations, with a linear attenuation with
+depth, as illustrated by figure fig.1c . Note that with a flat bottom, such as in
+fig.1c, the bottom-following  $z$ coordinate and  \textit{z*} are equivalent.
+The definition and modified oceanic equations for the rescaled vertical coordinate
+ \textit{z*}, including the treatment of fresh-water flux at the surface, are
+detailed in Adcroft and Campin (2004). The major points are summarized
+here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as:
+
+$H +  \textit{z*} = (H + z) / r$ and  $\delta \textit{z*} = \delta z / r$ with $r = \frac{H+\eta} {H}$
+
+Since the vertical displacement of the free surface is incorporated in the vertical
+coordinate  \textit{z*}, the upper and lower boundaries are at fixed  \textit{z*} position,  $\textit{z*} = 0$ and  $\textit{z*} = ?H$ respectively. Also the divergence of the flow field is no longer zero as shown by the continuity equation:
+
+$\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right)
+		\left( r \; w\textit{*} \right) = 0 $
+
+}
+
+
+\newpage 
+% ================================================================
+% Curvilinear s-coordinate System
+% ================================================================
+\section{Curvilinear \textit{s}-coordinate System}
+\label{PE_sco}
+
+% -------------------------------------------------------------------------------------------------------------
+% Introduction
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Introduction}
+
+Several important aspects of the ocean circulation are influenced by bottom topography. Of course, the most important is that bottom topography determines deep ocean sub-basins, barriers, sills and channels that strongly constrain the path of water masses, but more subtle effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes. Topographic Rossby waves can be excited and can interact with the mean current. In the $z-$coordinate system presented in the previous section (\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom and to large localized depth gradients associated with large localized vertical velocities. The response to such a velocity field often leads to numerical dispersion effects. One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}. Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate) 
+
+The $s$-coordinate avoids the discretisation error in the depth field since the layers of computation are gradually adjusted with depth to the ocean bottom. Relatively small topographic features as well as  gentle, large-scale slopes of the sea floor in the deep ocean, which would be ignored in typical $z$-model applications with the largest grid spacing at greatest depths, can easily be represented (with relatively low vertical resolution). A terrain-following model (hereafter $s-$model) also facilitates the modelling of the boundary layer flows over a large depth range, which in the framework of the $z$-model would require high vertical resolution over the whole depth range. Moreover, with a $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface as the only boundaries of the domain (nomore lateral boundary condition to specify). Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a homogeneous ocean, it has strong limitations as soon as stratification is introduced. The main two problems come from the truncation error in the horizontal pressure gradient and a possibly increased diapycnal diffusion. The horizontal pressure force in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}),
+
+\begin{equation} \label{Eq_PE_p_sco}
+\left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial 
+p}{\partial s}\left. {\nabla z} \right|_s 
+\end{equation}
+
+The second term in \eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface and introduces a truncation error that is not present in a $z$-model. In the special case of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), \citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude of this truncation error. It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state, and the finite difference scheme. This error limits the possible topographic slopes that a model can handle at a given horizontal and vertical resolution. This is a severe restriction for large-scale applications using realistic bottom topography. The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive. This problem can be at least partially overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec1996}. However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for a realistic bottom topography: a envelope topography is defined in $s$-coordinate on which a full or partial step bottom topography is then applied in order to adjust the model depth to the observed one (see \S\ref{DOM_zgr}.
+
+For numerical reasons a minimum of diffusion is required along the coordinate surfaces of any finite difference model. It causes spurious diapycnal mixing when coordinate surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as well as for a $s$-model. 
+However, density varies more strongly on $s-$surfaces than on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a $z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation. For example, imagine an isolated bump of topography in an ocean at rest with a horizontally uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast, the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column (i.e. the main thermocline) \citep{Madec1996}. An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \S\ref{PE_ldf}. Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}). 
+
+The $s-$coordinates introduced here \citep{Lott1990,Madec1996} differ mainly in two aspects from similar models:  it allows  a representation of bottom topography with mixed full or partial step-like/terrain following topography ; It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate.
+
+% -------------------------------------------------------------------------------------------------------------
+% The s-coordinate Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{The \textit{s-}coordinate Formulation}
+
+Starting from the set of equations established in \S\ref{PE_zco} for the special case $k=z$ and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes $z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and $s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed equations is given in Appendix~\ref{Apdx_A}. Let us define the vertical scale factor by $e_3=\partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by :
+\begin{equation} \label{Eq_PE_sco_slope}
+\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 
+\quad \text{, and } \quad 
+\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
+\end{equation}
+We also introduce  $\omega $, a dia-surface velocity component, defined as the velocity relative to the moving $s$-surfaces and normal to them:
+\begin{equation} \label{Eq_PE_sco_w}
+\omega  = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v    \\
+\end{equation}
+
+The equations solved by the ocean model \eqref{Eq_PE} in $s-$coordinate can be written as follows:
+
+ \vspace{0.5cm}
+* momentum equation:
+\begin{multline} \label{Eq_PE_sco_u}
+\frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}=
+	+   \left( {\zeta +f} \right)\,v                                    
+	-   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
+	-   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
+	-   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)    
+	+  g\frac{\rho }{\rho _o}\sigma _1 
+	+   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
+\end{multline}
+\begin{multline} \label{Eq_PE_sco_v}
+\frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}=
+	-   \left( {\zeta +f} \right)\,u   
+	-   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)     	
+	-   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
+	-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right) 
+	 +  g\frac{\rho }{\rho _o }\sigma _2   
+	+  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
+\end{multline}
+where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic pressure have the same expressions as in $z$-coordinates although they do not represent exactly the same quantities. $\omega$ is provided by the continuity equation (see Appendix~\ref{Apdx_A}):
+
+\begin{equation} \label{Eq_PE_sco_continuity}
+\frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0   
+\qquad \text{with }\;\;  
+\chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u} 
+\right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial 
+j}} \right]
+\end{equation}
+
+ \vspace{0.5cm}
+* tracer equations:
+\begin{multline} \label{Eq_PE_sco_t}
+\frac{1}{e_3} \frac{\partial \left(  e_3\,T  \right) }{\partial t}=
+-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i}
+                                           +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right]   \\
+-\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k}   + D^T + F^S   \qquad
+\end{multline}
+
+\begin{multline} \label{Eq_PE_sco_s}
+\frac{1}{e_3} \frac{\partial \left(  e_3\,S  \right) }{\partial t}=
+-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i}
+                                           +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right]    \\
+-\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k}     + D^S + F^S   \qquad
+\end{multline}
+
+The equation of state has the same expression as in $z$-coordinate, and similar expressions are used for mixing and forcing terms.
+
+\gmcomment{
+\colorbox{yellow}{ to be updated $= = >$}
+Add a few works on z and zps and s and underlies the differences between all of them
+\colorbox{yellow}{ $< = =$ end update}  }
+
+\newpage 
+% ================================================================
+% Subgrid Scale Physics
+% ================================================================
+\section{Subgrid Scale Physics}
+\label{PE_zdf_ldf}
+
+The primitive equations describe the behaviour of a geophysical fluid at 
+space and time scales larger than a few kilometres in the horizontal, a few 
+meters in the vertical and a few minutes. They are usually solved at larger 
+scales: the specified grid spacing and time step of the numerical model. The 
+effects of smaller scale motions (coming from the advective terms in the 
+Navier-Stokes equations) must be represented entirely in terms of 
+large-scale patterns to close the equations. These effects appear in the 
+equations as the divergence of turbulent fluxes ($i.e.$ fluxes associated with 
+the mean correlation of small scale perturbations). Assuming a turbulent 
+closure hypothesis is equivalent to choose a formulation for these fluxes. 
+It is usually called the subgrid scale physics. It must be emphasized that 
+this is the weakest part of the primitive equations, but also one of the 
+most important for long-term simulations as small scale processes \textit{in fine} balance 
+the surface input of kinetic energy and heat.
+
+The control exerted by gravity on the flow induces a strong anisotropy 
+between the lateral and vertical motions. Therefore subgrid-scale physics  \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$  in \eqref{Eq_PE_dyn}, \eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part  \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part  \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms and their underlying physics are briefly discussed in the next two subsections.
+
+% -------------------------------------------------------------------------------------------------------------
+% Vertical Subgrid Scale Physics
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Vertical Subgrid Scale Physics}
+\label{PE_zdf}
+
+The model resolution is always larger than the scale at which the major 
+sources of vertical turbulence occur (shear instability, internal wave 
+breaking...). Turbulent motions are thus never explicitly solved, even 
+partially, but always parameterized. The vertical turbulent fluxes are 
+assumed to depend linearly on the gradients of large-scale quantities (for 
+example, the turbulent heat flux is given by $\overline{T'w'}=-A^{vT} \partial_z \overline T$, where $A^{vT}$ is an eddy coefficient). This formulation is 
+analogous to that of molecular diffusion and dissipation. This is quite 
+clearly a necessary compromise: considering only the molecular viscosity 
+acting on large scale severely underestimates the role of turbulent 
+diffusion and dissipation, while an accurate consideration of the details of 
+turbulent motions is simply impractical. The resulting vertical momentum and 
+tracer diffusive operators are of second order:
+\begin{equation} \label{Eq_PE_zdf}
+   \begin{split}
+{\vect{D}}^{v \vect{U}}
+&=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\         D^{vT} &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ ,
+\quad
+D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right)
+   \end{split}
+\end{equation}
+where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat and salt must be specified (see Chap.~\ref{SBC} and \ref{ZDF} and \S\ref{TRA_bbl}). All the vertical physics is embedded in the specification of the eddy coefficients. They can be assumed to be either constant, or function of the local fluid properties ($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a turbulent closure model. The choices available in \NEMO are discussed in \S\ref{ZDF}).
+
+% -------------------------------------------------------------------------------------------------------------
+% Lateral Diffusive and Viscous Operators Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Lateral Diffusive and Viscous Operators Formulation}
+\label{PE_ldf}
+
+Lateral turbulence can be roughly divided into a mesoscale turbulence 
+associated with eddies (which can be solved explicitly if the resolution is 
+sufficient since their underlying physics are included in the primitive 
+equations), and a sub mesoscale turbulence which is never explicitly solved 
+even partially, but always parameterized. The formulation of lateral eddy 
+fluxes depends on whether the mesoscale is below or above the grid-spacing 
+($i.e.$ the model is eddy-resolving or not).
+
+In non-eddy-resolving configurations, the closure is similar to that used 
+for the vertical physics. The lateral turbulent fluxes are assumed to depend 
+linearly on the lateral gradients of large-scale quantities. The resulting 
+lateral diffusive and dissipative operators are of second order. 
+Observations show that lateral mixing induced by mesoscale turbulence tends 
+to be along isopycnal surfaces (or more precisely neutral surfaces \cite{McDougall1987}) rather than across them. 
+As the slope of neutral surfaces is small in the ocean, a common 
+approximation is to assume that the `lateral' direction is the horizontal, 
+$i.e.$ the lateral mixing is performed along geopotential surfaces. This leads 
+to a geopotential second order operator for lateral subgrid scale physics. 
+This assumption can be relaxed: the eddy-induced turbulent fluxes can be 
+better approached by assuming that they depend linearly on the gradients of 
+large-scale quantities computed along neutral surfaces. In such a case, 
+the diffusive operator is an isoneutral second order operator and it has 
+components in the three space directions. However, both horizontal and 
+isoneutral operators have no effect on mean ($i.e.$ large scale) potential 
+energy whereas potential energy is a main source of turbulence (through 
+baroclinic instabilities). \citet{Gent1990} have proposed a 
+parameterization of mesoscale eddy-induced turbulence which associates an 
+eddy-induced velocity to the isoneutral diffusion. Its mean effect is to 
+reduce the mean potential energy of the ocean. This leads to a formulation 
+of lateral subgrid-scale physics made up of an isoneutral second order 
+operator and an eddy induced advective part. In all these lateral diffusive 
+formulations, the specification of the lateral eddy coefficients remains the 
+problematic point as there is no really satisfactory formulation of these 
+coefficients as a function of large-scale features.
+
+In eddy-resolving configurations, a second order operator can be used, but 
+usually a more scale selective one (biharmonic operator) is preferred as the 
+grid-spacing is usually not small enough compared to the scale of the 
+eddies. The role devoted to the subgrid-scale physics is to dissipate the 
+energy that cascades toward the grid scale and thus ensures the stability of 
+the model while not interfering with the solved mesoscale activity. Another approach 
+is becoming more and more popular: instead of specifying explicitly a sub-grid scale 
+term in the momentum and tracer time evolution equations, one uses a advective scheme which is diffusive enough to maintain the model stability. It must be emphasised
+that then, all the sub-grid scale physics is in this case include in the formulation of the
+advection scheme. 
+
+All these parameterizations of subgrid scale physics present advantages and 
+drawbacks. There are not all available in \NEMO. In the $z$-coordinate 
+formulation, five options are offered for active tracers (temperature and 
+salinity): second order geopotential operator, second order isoneutral 
+operator, \citet{Gent1990} parameterization, fourth order 
+geopotential operator, and various slightly diffusive advection schemes. The same options are available for momentum, except 
+\citet{Gent1990} parameterization which only involves tracers. In the
+$s$-coordinate formulation, additional options are offered for tracers: second 
+order operator acting along $s-$surfaces, and for momentum: fourth order 
+operator acting along $s-$surfaces (see \S\ref{LDF}).
+
+\subsubsection{lateral second order tracer diffusive operator}
+
+The lateral second order tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):
+\begin{equation} \label{Eq_PE_iso_tensor}
+D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
+\mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill & {-r_1 } \hfill \\
+ 0 \hfill & 1 \hfill & {-r_2 } \hfill \\
+ {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\
+\end{array} }} \right)
+\end{equation}
+where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along 
+which the diffusive operator acts and the model level ($e. g.$ $z$- or 
+$s$-surfaces). Note that the formulation \eqref{Eq_PE_iso_tensor} is exact for the 
+rotation between geopotential and $s$-surfaces, while it is only an approximation 
+for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter 
+case, two assumptions are made to simplify  \eqref{Eq_PE_iso_tensor} \citep{Cox1987}. First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity. Second, the two isoneutral directions of diffusion are assumed to be independent since the slopes are generally less than $10^{-2}$ in the ocean (see Appendix~\ref{Apdx_B}).
+
+For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the 
+geopotential and computational surfaces: in $z$-coordinates they are zero ($r_1 = r_2 = 0$) while in $s$-coordinate (including $\textit{z*}$ case) they are equal to $\sigma _1$ and $\sigma _2$, respectively (see \eqref{Eq_PE_sco_slope} ).
+
+For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces. Therefore, they have a same expression in $z$- and $s$-coordinates:
+\begin{equation} \label{Eq_PE_iso_slopes}
+r_1 =\frac{e_3 }{e_1 }	\left( {\frac{\partial \rho }{\partial i}} \right)
+						\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \ , \quad
+r_1 =\frac{e_3 }{e_1 }	\left( {\frac{\partial \rho }{\partial i}} \right)
+						\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
+\end{equation}
+
+When the \textit{Eddy Induced Velocity} parametrisation (eiv) \citep{Gent1990} is used, an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers:
+\begin{equation} \label{Eq_PE_iso+eiv}
+D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right)
+           +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right)
+\end{equation}
+where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent, eddy-induced transport velocity. This velocity field is defined by:
+\begin{equation} \label{Eq_PE_eiv}
+   \begin{split}
+ u^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\ 
+ v^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\ 
+ w^\ast &=  -\frac{1}{e_1 e_2 }\left[ 
+                      \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right)
+                    +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right)      \right]
+   \end{split}
+\end{equation}
+where $A^{eiv}$ is the eddy induced velocity coefficient (or equivalently the isoneutral thickness diffusivity coefficient), and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes between isoneutral and \emph{geopotential} surfaces and thus depends on the coordinate considered: 
+\begin{align} \label{Eq_PE_slopes_eiv}
+\tilde{r}_n = \begin{cases}
+   r_n                  &      \text{in $z$-coordinate}    \\
+   r_n + \sigma_n &      \text{in \textit{z*} and $s$-coordinates}  
+                   \end{cases}
+\quad \text{where } n=1,2
+\end{align}
+
+The normal component of the eddy induced velocity is zero at all the boundaries. this can be achieved in a model by tapering either the eddy coefficient or the slopes to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}).
+
+\subsubsection{lateral fourth order tracer diffusive operator}
+
+The lateral fourth order tracer diffusive operator is defined by:
+\begin{equation} \label{Eq_PE_bilapT}
+D^{lT}=\Delta \left( {A^{lT}\;\Delta T} \right) 
+\qquad \text{where} \  D^{lT}=\Delta \left( {A^{lT}\;\Delta T} \right)
+ \end{equation}
+
+It is the second order operator given by \eqref{Eq_PE_iso_tensor} applied twice with the eddy diffusion coefficient correctly placed. 
+
+
+\subsubsection{lateral second order momentum diffusive operator}
+
+The second order momentum diffusive operator along $z$- or $s$-surfaces is found by 
+applying \eqref{Eq_PE_lap_vector} to the horizontal velocity vector (see Appendix~\ref{Apdx_B}):
+\begin{equation} \label{Eq_PE_lapU}
+\begin{split}
+{\rm {\bf D}}^{l{\rm {\bf U}}} 
+&= \quad \  \nabla _h \left( {A^{lm}\chi } \right)
+   \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right)     \\
+&=   \left(      \begin{aligned}
+             \frac{1}{e_1      } \frac{\partial \left( A^{lm} \chi          \right)}{\partial i} 
+         &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j}  \\
+             \frac{1}{e_2      }\frac{\partial \left( {A^{lm} \chi         } \right)}{\partial j}   
+         &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i}
+        \end{aligned}    \right)
+\end{split}
+\end{equation}
+
+Such a formulation ensures a complete separation between the vorticity and 
+horizontal divergence fields (see Appendix~\ref{Apdx_C}). Unfortunately, it is not 
+available for geopotential diffusion in $s-$coordinates and for isoneutral 
+diffusion in both $z$- and $s$-coordinates ($i.e.$ when a rotation is required). In these two cases, the $u$ and $v-$fields are considered as independent scalar fields, so that the diffusive operator is given by:
+\begin{equation} \label{Eq_PE_lapU_iso}
+\begin{split}
+ D_u^{l{\rm {\bf U}}} &= \nabla .\left( {\Re \;\nabla u} \right) \\ 
+ D_v^{l{\rm {\bf U}}} &= \nabla .\left( {\Re \;\nabla v} \right)
+ \end{split}
+ \end{equation}
+where $\Re$ is given by  \eqref{Eq_PE_iso_tensor}. It is the same expression as those used for diffusive operator on tracers. It must be emphasised that such a formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or $\beta-$plane, not on the sphere. It is also a very good approximation in vicinity of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}.
+
+\subsubsection{lateral fourth order momentum diffusive operator}
+
+As for tracers, the fourth order momentum diffusive operator along $z$ or $s$-surfaces is a re-entering second order operator \eqref{Eq_PE_lapU} or \eqref{Eq_PE_lapU} with the eddy viscosity coefficient correctly placed:
+
+geopotential diffusion in $z$-coordinate:
+\begin{equation} \label{Eq_PE_bilapU}
+\begin{split}
+{\rm {\bf D}}^{l{\rm {\bf U}}} &=\nabla _h \left\{ {\;\nabla _h {\rm {\bf 
+.}}\left[ {A^{lm}\,\nabla _h \left( \chi \right)} \right]\;} 
+\right\}\;   \\
+&+\nabla _h \times \left\{ {\;{\rm {\bf k}}\cdot \nabla \times 
+\left[ {A^{lm}\,\nabla _h \times \left( {\zeta \;{\rm {\bf k}}} \right)} 
+\right]\;} \right\}
+\end{split}
+\end{equation}
+
+\gmcomment{  change the position of the coefficient, both here and in the code}
+
+geopotential diffusion in $s$-coordinate:
+\begin{equation} \label{Eq_bilapU_iso}
+   \left\{   \begin{aligned}
+         D_u^{l{\rm {\bf U}}} =\Delta \left( {A^{lm}\;\Delta u} \right) \\ 
+         D_v^{l{\rm {\bf U}}} =\Delta \left( {A^{lm}\;\Delta v} \right)
+   \end{aligned}    \right.
+   \quad \text{where} \quad 
+   \Delta \left( \bullet \right) = \nabla \cdot \left( \Re \nabla(\bullet) \right) 
+\end{equation}
+
Index: trunk/DOC/TexFiles/Chapters/Chap_Model_Basics_zstar.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_Model_Basics_zstar.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_Model_Basics_zstar.tex	(revision 994)
@@ -0,0 +1,243 @@
+% ================================================================
+% Chapter 1 Ñ Model Basics
+% ================================================================
+% ================================================================
+% Curvilinear z*- s*-coordinate System
+% ================================================================
+\chapter{ essai z* s*}
+\section{Curvilinear \textit{z*}- or \textit{s*} coordinate System}
+
+% -------------------------------------------------------------------------------------------------------------
+% ????
+% -------------------------------------------------------------------------------------------------------------
+
+\colorbox{yellow}{ to be updated }
+
+In that case, the free surface equation is nonlinear, and the variations of 
+volume are fully taken into account. These coordinates systems is presented in a report 
+\citep{Levier2007} available on the \NEMO web site. 
+
+\colorbox{yellow}{  end of to be updated}
+\newline
+
+% from MOM4p1 documentation
+
+To overcome problems with vanishing surface and/or bottom cells, we consider the 
+zstar coordinate 
+\begin{equation} \label{PE_}
+	z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
+\end{equation}
+
+This coordinate is closely related to the "eta" coordinate used in many atmospheric 
+models (see Black (1994) for a review of eta coordinate atmospheric models). It 
+was originally used in ocean models by Stacey et al. (1995) for studies of tides 
+next to shelves, and it has been recently promoted by Adcroft and Campin (2004) 
+for global climate modelling.
+
+The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between 
+undulations of the bottom topography versus undulations in the surface height, it 
+is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \S\ref{PE_sco}. 
+Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an 
+unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in 
+the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure 
+gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using 
+the same techniques as in $z$-models (see Chapters 13-16 of Griffies (2004) for a 
+discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp} 
+in this document for treatment in \NEMO). 
+
+The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all 
+cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This 
+is a minor constraint relative to that encountered on the surface height when using 
+$s = z$ or $s = z - \eta$. 
+
+Because $z^\star$ has a time independent range, all grid cells have static increments 
+ds, and the sum of the ver tical increments yields the time independent ocean 
+depth %·k ds = H. 
+The $z^\star$ coordinate is therefore invisible to undulations of the 
+free surface, since it moves along with the free surface. This proper ty means that 
+no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the 
+motion of external gravity waves. Such spurious transpor t can be a problem in 
+z-models, especially those with tidal forcing. Quite generally, the time independent 
+range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly 
+arbitrary ver tical resolution even in the presence of large amplitude fluctuations of 
+the surface height, again so long as $\eta > -H$. 
+
+
+
+%%%
+%  essai update time splitting...
+%%%
+
+
+% ================================================================
+% Surface Pressure Gradient and Sea Surface Height
+% ================================================================
+\section{Surface pressure gradient and Sea Surface Heigth (\mdl{dynspg})}
+\label{DYN_hpg_spg}
+%-----------------------------------------nam_dynspg----------------------------------------------------
+\namdisplay{nam_dynspg} 
+%------------------------------------------------------------------------------------------------------------
+The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\S\ref{PE_free_surface}) and rigid lid (\S\ref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\S\ref{PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
+
+%-------------------------------------------------------------
+% Explicit
+%-------------------------------------------------------------
+\subsubsection{Explicit (\key{dynspg\_exp})}
+\label{DYN_spg_exp}
+
+In the explicit free surface formulation, the model time step is chosen small enough to describe the external gravity waves (typically a few ten seconds). The sea surface height is given by :
+\begin{equation} \label{Eq_dynspg_ssh}
+\frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} 
+e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u} 
+\right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)} 
+\end{equation}
+
+where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \S\ref{SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\ref{Eq_dynspg_ssh}) is centred in time (\textit{now} velocity). 
+
+The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by :
+\begin{equation} \label{Eq_dynspg_exp}
+\left\{ \begin{aligned}
+ - \frac{1}    						{e_{1u}} \;	\delta _{i+1/2} \left[  \,\eta\,  \right] 	\\
+ \\
+ - \frac{1}    						{e_{2v}} \;	\delta _{j+1/2} \left[  \,\eta\,  \right]  
+\end{aligned} \right.
+\end{equation} 
+
+Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\ref{Eq_dynspg_exp}). 
+
+%-------------------------------------------------------------
+% Split-explicit time-stepping
+%-------------------------------------------------------------
+\subsubsection{Split-explicit time-stepping (\key{dynspg\_ts})}
+\label{DYN_spg_ts}
+%--------------------------------------------namdom----------------------------------------------------
+\namdisplay{namdom} 
+%--------------------------------------------------------------------------------------------------------------
+The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}. The general idea is to solve the free surface equation with a small time step, while the three dimensional prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt} (Figure III.3). 
+
+%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
+\begin{figure}[!t] \label{Fig_DYN_dynspg_ts}
+\begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_DYN_dynspg_ts.pdf}
+\caption{Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. The curved line represents a leap-frog time step, and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. The vertically integrated forcing \textbf{M}(t) computed at baroclinic time step t represents the interaction between the barotropic and baroclinic motions. While keeping the total depth, tracer, and freshwater forcing fields fixed, a leap-frog integration carries the surface height and vertically integrated velocity from t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. Time averaging the barotropic fields over the N+1 time steps (endpoints included) centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of the time averaged vertically integrated velocity taken from baroclinic time step t. }
+\end{center}
+\end{figure}
+%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
+
+The split-explicit formulation has a damping effect on external gravity waves, which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. 
+
+%from griffies book: .....   copy past !
+
+\textbf{title: Time stepping the barotropic system }
+
+Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence, 
+we can update the surface height and vertically integrated velocity with a leap-frog 
+scheme using the small barotropic time step $\Delta t$. We have 
+
+\begin{equation} \label{DYN_spg_ts_eta}
+\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
+	= 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
+\end{equation}
+\begin{multline} \label{DYN_spg_ts_u}
+\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\
+	= 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) 
+	- H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
+\end{multline}
+\
+
+In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ 
+and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time 
+that sets the barotropic time steps via 
+\begin{equation} \label{DYN_spg_ts_t}
+t_n=\tau+n\Delta t   
+\end{equation}
+with $n$ an integer. The density scaled surface pressure is evaluated via 
+\begin{equation} \label{DYN_spg_ts_ps}
+p_s^{(b)}(\tau,t_{n}) = \begin{cases}
+	g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
+	g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case} 
+	\end{cases}
+\end{equation}
+To get started, we assume the following initial conditions 
+\begin{equation} \label{DYN_spg_ts_eta}
+\begin{split}
+\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
+\\
+\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 
+\end{split}
+\end{equation}
+with 
+\begin{equation} \label{DYN_spg_ts_etaF}
+ \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
+\end{equation}
+the time averaged surface height taken from the previous barotropic cycle. Likewise, 
+\begin{equation} \label{DYN_spg_ts_u}
+\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}	\\
+\\
+\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}  	
+\end{equation}
+with 
+\begin{equation} \label{DYN_spg_ts_u}
+ \overline{\textbf{U}^{(b)}(\tau)} 
+ 	= \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
+\end{equation}
+the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 
+
+Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 
+\begin{equation} \label{DYN_spg_ts_u}
+\textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 
+ 	= \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
+\end{equation}
+The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form 
+
+\begin{equation} \label{DYN_spg_ts_ssh}
+\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]  
+\end{equation}
+
+ The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 
+ 
+In general, some form of time filter is needed to maintain integrity of the surface 
+height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We 
+have tried various forms of such filtering, with the following method discussed in 
+Griffies et al. (2001) chosen due to its stability and reasonably good maintenance of 
+tracer conservation properties (see Section ??) 
+
+\begin{equation} \label{DYN_spg_ts_sshf}
+\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
+\end{equation}
+Another approach tried was 
+
+\begin{equation} \label{DYN_spg_ts_sshf2}
+\eta^{F}(\tau-\Delta) = \eta(\tau) 
+	+ (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
+				    + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right]
+\end{equation}
+
+which is useful since it isolates all the time filtering aspects into the term multiplied 
+by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when 
+eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. 
+
+
+
+
+
+%-------------------------------------------------------------
+% Filtered formulation 
+%-------------------------------------------------------------
+\subsubsection{Filtered formulation (\key{dynspg\_flt})}
+\label{DYN_spg_flt}
+
+The filtered formulation follows the \citet{Roullet2000} implementation. The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic solvers available in the code are 
+documented in \S\ref{MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000}
+
+\colorbox{red}{\np{rnu}=1 to be suppressed from namelist !}
+
+%-------------------------------------------------------------
+% Non-linear free surface formulation 
+%-------------------------------------------------------------
+\subsection{Non-linear free surface formulation (\key{vvl})}
+\label{DYN_spg_vvl}
+
+In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \S\ref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
+
+
Index: trunk/DOC/TexFiles/Chapters/Chap_SBC.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_SBC.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_SBC.tex	(revision 994)
@@ -0,0 +1,451 @@
+% ================================================================
+% Chapter Ñ Surface Boundary Condition (SBC) 
+% ================================================================
+\chapter{Surface Boundary Condition (SBC) }
+\label{SBC}
+\minitoc
+
+\newpage
+$\ $\newline    % force a new ligne
+%---------------------------------------namsbc--------------------------------------------------
+\namdisplay{namsbc}
+%--------------------------------------------------------------------------------------------------------------
+$\ $\newline    % force a new ligne
+
+The ocean needs six fields as surface boundary condition:
+\begin{itemize}
+\item the two components of the surface ocean stress $\left( {\tau _u \;,\;\tau _v} \right)$
+\item the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$
+\item the surface freshwater budget $\left( {\text{EMP},\;\text{EMP}_S } \right)$
+\end{itemize}
+
+Four different ways to provide those six fields to the ocean are available which 
+are controlled by namelist variables: an analytical formulation (\np{ln\_ana}=true), 
+a flux formulation (\np{ln\_flx}=true), a bulk formulae formulation (CORE 
+(\np{ln\_core}=true) or CLIO (\np{ln\_clio}=true) bulk formulae) and a coupled 
+formulation (exchanges with a atmospheric model via the OASIS coupler) 
+(\np{ln\_cpl}=true). The frequency at which the six fields have to be updated is
+the  \np{nf\_sbc} namelist parameter. 
+In addition, the resulting fields can be further modified using 
+several namelist options. These options control the addition of a surface restoring 
+term to observed SST and/or SSS (\np{ln\_ssr}=true), the modification of fluxes 
+below ice-covered areas (using observed ice-cover or a sea-ice model) 
+(\np{nn\_ice}=0,1, 2 or 3), the addition of river runoffs as surface freshwater 
+fluxes (\np{ln\_rnf}=true), the addition of a freshwater flux adjustment in 
+order to avoid a mean sea-level drift (\np{nn\_fwb}= 0, 1 or 2), and the 
+transformation of the solar radiation (if provided as daily mean) into a diurnal 
+cycle (\np{ln\_dm2dc}=true).
+
+In this chapter, we first discuss where the surface boundary condition 
+appears in the model equations. Then we present the four ways of providing 
+the surface boundary condition. Finally, the different options that further modify 
+the fluxes applied to the ocean are discussed.
+
+
+% ================================================================
+% Surface boundary condition for the ocean
+% ================================================================
+\section{Surface boundary condition for the ocean}
+\label{SBC_general}
+
+
+The surface ocean stress is the stress exerted by the wind and the sea-ice 
+on the ocean. The two components of stress are assumed to be interpolated 
+onto the ocean mesh, $i.e.$ resolved onto the model (\textbf{i},\textbf{j}) direction 
+at $u$- and $v$-points They are applied as a surface boundary condition of the 
+computation of the momentum vertical mixing trend (\mdl{dynzdf} module) :
+\begin{equation} \label{Eq_sbc_dynzdf}
+\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
+	 = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
+\end{equation}
+where $(\tau _u ,\;\tau _v )=(utau,vtau)$ are the two components of the wind 
+stress vector in the $(\textbf{i},\textbf{j})$ coordinate system.
+
+The surface heat flux is decomposed into two parts, a non solar and a solar heat 
+flux, $Q_{ns}$ and $Q_{sr}$, respectively. The former is the non penetrative part 
+of the heat flux ($i.e.$ the sum of sensible, latent and long wave heat fluxes). 
+It is applied as a surface boundary condition trend of the first level temperature 
+time evolution equation (\mdl{trasbc} module). 
+\begin{equation} \label{Eq_sbc_trasbc_q}
+\frac{\partial T}{\partial t}\equiv \cdots \;+\;\left. {\frac{Q_{ns} }{\rho 
+_o \;C_p \;e_{3T} }} \right|_{k=1} \quad
+\end{equation}
+$Q_{sr}$ is the penetrative part of the heat flux. It is applied as a 3D 
+trends of the temperature equation (\mdl{traqsr} module) when \np{ln\_traqsr}=True.
+
+\begin{equation} \label{Eq_sbc_traqsr}
+\frac{\partial T}{\partial t}\equiv \cdots \;+\frac{Q_{sr} }{\rho _o C_p 
+\,e_{3T} }\delta _k \left[ {I_w } \right]
+\end{equation}
+where $I_w$ is a non-dimensional function that describes the way the light 
+penetrates inside the water column. It is generally a sum of decreasing 
+exponentials (see \S\ref{TRA_qsr}).
+
+The surface freshwater budget is provided by fields: EMP and EMP$_S$ which 
+may or may not be identical. Indeed, a surface freshwater flux has two effects: 
+it changes the volume of the ocean and it changes the surface concentration of 
+salt (and other tracers). Therefore it appears in the sea surface height as a volume 
+flux, EMP (\textit{dynspg\_xxx} modules), and in the salinity time evolution equations 
+as a concentration/dilution effect, 
+EMP$_{S}$ (\mdl{trasbc} module). 
+\begin{equation} \label{Eq_trasbc_emp}
+\begin{aligned}
+&\frac{\partial \eta }{\partial t}\equiv \cdots \;+\;\text{EMP}\quad  \\ 
+\\
+ &\frac{\partial S}{\partial t}\equiv \cdots \;+\left. {\frac{\text{EMP}_S \;S}{e_{3T} }} \right|_{k=1} \\ 
+ \end{aligned}
+\end{equation} 
+
+In the real ocean, EMP$=$EMP$_S$ and the ocean salt content is conserved, 
+but it exist several numerical reasons why this equality should be broken. 
+For example:
+
+When rigid-lid assumption is made, the ocean volume becomes constant and 
+thus, EMP$=$0, not EMP$_{S }$.
+
+When the ocean is coupled to a sea-ice model, the water exchanged between ice and 
+ocean is slightly salty (mean sea-ice salinity is $\sim $\textit{4 psu}). In this case, 
+EMP$_{S}$ take into account both concentration/dilution effect associated with 
+freezing/melting and the salt flux between ice and ocean, while EMP is 
+only the volume flux. In addition, in the current version of \NEMO, the 
+sea-ice is assumed to be above the ocean. Freezing/melting does not change 
+the ocean volume (not impact on EMP) but it modifies the SSS.
+%gm  \colorbox{yellow}{(see {\S} on LIM sea-ice model)}.
+
+Note that SST can also be modified by a freshwater flux. Precipitation (in 
+particular solid precipitation) may have a temperature significantly different from 
+the SST. Due to the lack of information about the temperature of 
+precipitation, we assume it is equal to the SST. Therefore, no 
+concentration/dilution term appears in the temperature equation. It has to 
+be emphasised that this absence does not mean that there is no heat flux 
+associated with precipitation! Precipitation can change the ocean volume and thus the
+ocean heat content. It is therefore associated with a heat flux (not yet  
+diagnosed in the model) \citep{Roullet2000}).
+
+%\colorbox{yellow}{Miss: }
+%
+%A extensive description of all namsbc namelist (parameter that have to be 
+%created!)
+%
+%Especially the \np{nf\_sbc}, the \mdl{sbc\_oce} module (fluxes + mean sst sss ssu 
+%ssv) i.e. information required by flux computation or sea-ice
+%
+%\mdl{sbc\_oce} containt the definition in memory of the 7 fields (6+runoff), add 
+%a word on runoff: included in surface bc or add as lateral obc{\ldots}.
+%
+%Sbcmod manage the ``providing'' (fourniture) to the ocean the 7 fields
+%
+%Fluxes update only each nf{\_}sbc time step (namsbc) explain relation 
+%between nf{\_}sbc and nf{\_}ice, do we define nf{\_}blk??? ? only one 
+%nf{\_}sbc
+%
+%Explain here all the namlist namsbc variable{\ldots}.
+%
+%\colorbox{yellow}{End Miss }
+
+The ocean model provides the surface currents, temperature and salinity 
+averaged over \np{nf\_sbc} time-step (\ref{Tab_ssm}).The computation of the 
+mean is done in \mdl{sbcmod} module.
+
+%-------------------------------------------------TABLE---------------------------------------------------
+\begin{table}[tb]  \label{Tab_ssm}
+\begin{center}
+\begin{tabular}{|l|l|l|l|}
+\hline
+Variable description					& Model variable	& Units	& point \\	\hline
+i-component of the surface current	& ssu\_m	& $m.s^{-1}$	& U \\	\hline
+j-component of the surface current	& ssv\_m	& $m.s^{-1}$	& V \\	\hline
+Sea surface temperature				& sst\_m	& \r{}$K$		& T \\	\hline
+Sea surface salinty					& sss\_m	& $psu$			& T \\	\hline
+\end{tabular}
+\caption{Ocean variables provided by the ocean to the surface module (SBC). 
+The variable are averaged over nf{\_}sbc time step, $i.e.$ the frequency of 
+computation of surface fluxes.}
+\end{center}
+\end{table}
+%--------------------------------------------------------------------------------------------------------------
+
+
+
+%\colorbox{yellow}{Penser a} mettre dans le restant l'info nf{\_}sbc ET nf{\_}sbc*rdt de sorte de reinitialiser la moyenne si on change la frequence ou le pdt
+
+
+% ================================================================
+% Analytical formulation (sbcana module) 
+% ================================================================
+\section  [Analytical formulation (\textit{sbcana}) ]
+		{Analytical formulation (\mdl{sbcana} module) }
+\label{SBC_ana}
+
+%---------------------------------------namsbc_ana--------------------------------------------------
+\namdisplay{namsbc_ana}
+%--------------------------------------------------------------------------------------------------------------
+
+
+The analytical formulation of the surface boundary condition is the default scheme.
+In this case, all the six fluxes needed by the ocean are assumed to 
+be uniform in space. They take constant values given in the namelist 
+namsbc{\_}ana by the variables \np{rn\_utau0}, \np{rn\_vtau0}, \np{rn\_qns0}, 
+\np{rn\_qsr0}, and \np{rn\_emp0} (EMP$=$EMP$_S$). The runoff is set to zero. 
+In addition, the wind is allowed to reach its nominal value within a given number 
+of time steps (\np{nn\_tau000}).
+
+If a user wants to apply a different analytical forcing, the \mdl{sbcana} 
+module can be modified to use another scheme. As an example, 
+the \mdl{sbc\_ana\_gyre} routine provides the analytical forcing for the 
+GYRE configuration (see GYRE configuration manual, in preparation).
+
+
+% ================================================================
+% Flux formulation 
+% ================================================================
+\section  [Flux formulation (\textit{sbcflx}) ]
+		{Flux formulation (\mdl{sbcflx} module) }
+\label{SBC_flx}
+%------------------------------------------namsbc_flx----------------------------------------------------
+\namdisplay{namsbc_flx} 
+%-------------------------------------------------------------------------------------------------------------
+
+In the flux formulation (\np{ln\_flx}=true), the surface boundary 
+condition fields are directly read from input files. The user has to define 
+in the namelist namsbc{\_}flx the name of the file, the name of the variable 
+read in the file, the time frequency at which it is given (in hours), and a logical 
+setting whether a time interpolation to the model time step is required 
+for this field). (fld\_i namelist structure).
+
+\textbf{Caution}: when the frequency is set to --12, the data are monthly 
+values. These are assumed to be climatological values, so time interpolation 
+between December the 15$^{th}$ and January the 15$^{th}$ is done using 
+records 12 and 1
+
+When higher frequency is set and time interpolation is demanded, the model 
+will try to read the last (first) record of previous (next) year in a file 
+having the same name but a suffix {\_}prev{\_}year ({\_}next{\_}year) being 
+added (e.g. "{\_}1989"). These files must only contain a single record. If they don't exist, 
+the model assumes that the last record of the previous year is equal to the first 
+record of the current year, and similarly, that the first record of the 
+next year is equal to the last record of the current year. This will cause 
+the forcing to remain constant over the first and last half fld\_frequ hours.
+
+Note that in general, a flux formulation is used in associated with a 
+restoring term to observed SST and/or SSS. See \S\ref{SBC_ssr} for its 
+specification.
+
+
+% ================================================================
+% Bulk formulation
+% ================================================================
+\section  [Bulk formulation (\textit{sbcblk\_core} or \textit{sbcblk\_clio}) ]
+		{Bulk formulation \small{(\mdl{sbcblk\_core} or \mdl{sbcblk\_clio} module)} }
+\label{SBC_blk}
+
+In the bulk formulation, the surface boundary condition fields are computed 
+using bulk formulae and atmospheric fields and ocean (and ice) variables. 
+
+The atmospheric fields used depend on the bulk formulae used. Two bulk formulations 
+are available : the CORE and CLIO bulk formulea. The choice is made by setting to true
+one of the following namelist variable : \np{ln\_core} and \np{ln\_clio}.
+
+Note : in forced mode, when a sea-ice model is used, a bulk formulation have to be used. 
+Therefore the two bulk formulea provided include the computation of the fluxes over both 
+an ocean and an ice surface. 
+
+% -------------------------------------------------------------------------------------------------------------
+%        CORE Bulk formulea
+% -------------------------------------------------------------------------------------------------------------
+\subsection    [CORE Bulk formulea (\np{ln\_core}=true)]
+		      {CORE Bulk formulea (\np{ln\_core}=true, \mdl{sbcblk\_core})}
+\label{SBC_blk_core}
+%------------------------------------------namsbc_core----------------------------------------------------
+\namdisplay{namsbc_core} 
+%-------------------------------------------------------------------------------------------------------------
+
+The CORE bulk formulae have been developed by \citet{LargeYeager2004}. 
+They have been designed to handle the CORE forcing, a mixture of NCEP 
+reanalysis and satellite data. They use an inertial dissipative method to compute 
+the turbulent transfer coefficients (momentum, sensible heat and evaporation) 
+from the 10 metre wind speed, air temperature and specific humidity.
+
+Note that substituting ERA40 to NCEP reanalysis fields 
+does not require changes in the bulk formulea themself. 
+
+The required 8 input fields are:
+
+%--------------------------------------------------TABLE--------------------------------------------------
+\begin{table}[htbp]   \label{Tab_CORE}
+\begin{center}
+\begin{tabular}{|l|l|l|l|}
+\hline
+Variable desciption					& Model variable	& Units	 & point \\		\hline
+i-component of the 10m air velocity	& utau		& $m.s^{-1}$			& T  \\ 	\hline
+j-component of the 10m air velocity	& vtau		& $m.s^{-1}$			& T  \\	\hline
+10m air temperature					& tair		& \r{}$K$				& T 	\\	\hline
+Specific humidity					& humi		& \%					& T \\		\hline
+Incoming long wave radiation		& qlw		& $W.m^{-2}$			& T \\		\hline
+Incoming short wave radiation		& qsr		& $W.m^{-2}$			& T \\		\hline
+Total precipitation (liquid + solid)	& precip	& $Kg.m^{-2}.s^{-1}$	& T \\ 	\hline
+Solid precipitation					& snow		& $Kg.m^{-2}.s^{-1}$	& T \\	\hline
+\end{tabular}
+\end{center}
+\end{table}
+%--------------------------------------------------------------------------------------------------------------
+
+Note that the air velocity is provided at a tracer ocean point, not at a velocity ocean point ($u$- and $v$-points). It is simpler and faster (less fields to be read), but it is not the recommended method when the ocean grid
+size is the same or larger than the one of the input atmospheric fields.
+
+% -------------------------------------------------------------------------------------------------------------
+%        CLIO Bulk formulea
+% -------------------------------------------------------------------------------------------------------------
+\subsection    [CLIO Bulk formulea (\np{ln\_clio}=true)]
+		      {CLIO Bulk formulea (\np{ln\_clio}=true, \mdl{sbcblk\_clio})}
+\label{SBC_blk_clio}
+%------------------------------------------namsbc_clio----------------------------------------------------
+\namdisplay{namsbc_clio} 
+%-------------------------------------------------------------------------------------------------------------
+
+The CLIO bulk formulae were developed several years ago for the 
+Louvain-la-neuve coupled ice-ocean model (CLIO, \cite{Goosse_al_JGR99}). 
+They are simpler bulk formulae. They assume the stress to be known and 
+compute the radiative fluxes from a climatological cloud cover. 
+
+The required 7 input fields are:
+
+%--------------------------------------------------TABLE--------------------------------------------------
+\begin{table}[htbp]   \label{Tab_CLIO}
+\begin{center}
+\begin{tabular}{|l|l|l|l|}
+\hline
+Variable desciption				& Model variable	& Units				& point \\	\hline
+i-component of the ocean stress		& utau			& $N.m^{-2}$			& U \\	\hline
+j-component of the ocean stress		& vtau			& $N.m^{-2}$			& V \\	\hline
+Wind speed module					& vatm			& $m.s^{-1}$			& T \\	\hline
+10m air temperature					& tair			& \r{}$K$				& T \\	\hline
+Specific humidity						& humi			& \%					& T \\	\hline
+Cloud cover							& 				& \%					& T \\	\hline
+Total precipitation (liquid + solid)	& precip		& $Kg.m^{-2}.s^{-1}$	& T \\	\hline
+Solid precipitation 					& snow			& $Kg.m^{-2}.s^{-1}$	& T \\	\hline
+\end{tabular}
+\end{center}
+\end{table}
+%--------------------------------------------------------------------------------------------------------------
+
+As for the flux formulation, information about the input data required by the 
+model is provided in the namsbc\_blk\_core or namsbc\_blk\_clio 
+namelist (via the structure fld\_i). The first and last record assumption is also made 
+(see \S\ref{SBC_flx})
+
+% ================================================================
+% Coupled formulation
+% ================================================================
+\section  [Coupled formulation (\textit{sbccpl}) ]
+		{Coupled formulation (\mdl{sbccpl} module)}
+\label{SBC_cpl}
+%------------------------------------------namsbc_cpl----------------------------------------------------
+\namdisplay{namsbc_cpl} 
+%-------------------------------------------------------------------------------------------------------------
+
+In the coupled formulation of the surface boundary condition, the fluxes are 
+provided by the OASIS coupler at each \np{nf\_cpl} time-step, while sea and ice 
+surface temperature, ocean and ice albedo, and ocean currents are sent to 
+the atmospheric component.
+
+The generalised coupled interface is under development. It should be available
+in summer 2008. It will include the ocean interface for most of the European 
+atmospheric GCM (ARPEGE, ECHAM, ECMWF, HadAM, LMDz).
+
+
+% ================================================================
+% Miscellanea options
+% ================================================================
+\section{Miscellaneous options}
+\label{SBC_misc}
+
+% -------------------------------------------------------------------------------------------------------------
+%        Surface restoring to observed SST and/or SSS
+% -------------------------------------------------------------------------------------------------------------
+\subsection    [Surface restoring to observed SST and/or SSS (\textit{sbcssr})]
+			{Surface restoring to observed SST and/or SSS (\mdl{sbcssr})}
+\label{SBC_ssr}
+%------------------------------------------namsbc_ssr----------------------------------------------------
+\namdisplay{namsbc_ssr} 
+%-------------------------------------------------------------------------------------------------------------
+
+In forced mode using a flux formulation (default option or \key{flx} defined), a 
+feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$:
+\begin{equation} \label{Eq_sbc_dmp_q}
+Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right)
+\end{equation}
+where SST is a sea surface temperature field (observed or climatological), $T$ is 
+the model surface layer temperature and $\frac{dQ}{dT}$ is a negative feedback 
+coefficient usually taken equal to $-40~W/m^2/K$. For a $50~m$ 
+mixed-layer depth, this value corresponds to a relaxation time scale of two months. 
+This term ensures that if $T$ perfectly matches the supplied SST, then $Q$ is 
+equal to $Q_o$. 
+
+In the fresh water budget, a feedback term can also be added. Converted into an 
+equivalent freshwater flux, it takes the following expression :
+
+\begin{equation} \label{Eq_sbc_dmp_emp}
+EMP = EMP_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)}
+												         {\left.S\right|_{k=1}}
+\end{equation}
+
+where EMP$_{o }$ is a net surface fresh water flux (observed, climatological or an
+atmospheric model product), \textit{SSS}$_{Obs}$ is a sea surface salinity (usually a time 
+interpolation of the monthly mean Polar Hydrographic Climatology \citep{Steele2001}), 
+$\left.S\right|_{k=1}$ is the model surface layer salinity and $\gamma_s$ is a negative 
+feedback coefficient which is provided as a namelist parameter. Unlike heat flux, there is no 
+physical justification for the feedback term in \ref{Eq_sbc_dmp_emp} as the atmosphere 
+does not care about ocean surface salinity \citep{Madec1997}. The SSS restoring 
+term should be viewed as a flux correction on freshwater fluxes to reduce the 
+uncertainties we have on the observed freshwater budget.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Handling of ice-covered area
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Handling of ice-covered area}
+\label{SBC_ice-cover}
+
+The presence at the sea surface of an ice covered area modifies all the fluxes 
+transmitted to the ocean. There are several way to handle sea-ice in the system depending on the value of the \np{nn{\_}ice} namelist parameter.  
+\begin{description}
+\item[nn{\_}ice = 0]  there will never be sea-ice in the computational domain. This is a typical namelist value used for tropical ocean domain. The surface fluxes are simply specified for an ice-free ocean. No specific things are done for sea-ice.
+\item[nn{\_}ice = 1]  sea-ice can exist in the computational domain, but no sea-ice model is used. An observed ice covered area is read in a file. Below this area, the SST is restored to the freezing point and the heat fluxes are set to $-4~W/m^2$ ($-2~W/m^2$) in the northern (southern) hemisphere. The associated modification of the freshwater fluxes are done in such a way that the change in buoyancy fluxes remains zero. This prevents deep convection to occur when trying to reach the freezing point (and so ice covered area condition) while the SSS is too large. This manner of managing sea-ice area, just by using si IF case, is usually referred as the \textit{ice-if} model. It can be found in the \mdl{sbcice{\_}if} module.
+\item[nn{\_}ice = 2 or more]  A full sea ice model is used. This model computes the ice-ocean fluxes, that are combined with the air-sea fluxes using the ice fraction of each model cell to provide the surface ocean fluxes. Note that the activation of a sea-ice model is is done by defining a CPP key (\key{lim2} or \key{lim3}). The activation automatically ovewrite the read value of nn{\_}ice to its appropriate value ($i.e.$ $2$ for LIM-2 and $3$ for LIM-3).
+\end{description}
+
+% {Description of Ice-ocean interface to be added here or in LIM 2 and 3 doc ?}
+
+% -------------------------------------------------------------------------------------------------------------
+%        Addition of river runoffs
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Addition of river runoffs (\textit{sbcrnf})]
+			{Addition of river runoffs (\mdl{sbcrnf})}
+\label{SBC_rnf}
+%------------------------------------------namsbc_rnf----------------------------------------------------
+\namdisplay{namsbc_rnf} 
+%-------------------------------------------------------------------------------------------------------------
+
+It is convenient to introduce the river runoff in the model as a surface 
+fresh water flux. 
+
+\colorbox{yellow}{Nevertheless, Pb of vertical resolution and increase of Kz in vicinity of }
+
+\colorbox{yellow}{river mouths{\ldots}}
+
+Control of the mean sea level
+
+% -------------------------------------------------------------------------------------------------------------
+%        Freshwater budget control 
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Freshwater budget control (\textit{sbcfwb})]
+			{Freshwater budget control (\mdl{sbcfwb})}
+\label{SBC_fwb}
+
+To be written later...
+
+\gmcomment{The descrition of the technique used to control the freshwater budget has to be added here}
+
+
+
+
Index: trunk/DOC/TexFiles/Chapters/Chap_TRA.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_TRA.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_TRA.tex	(revision 994)
@@ -0,0 +1,1296 @@
+% ================================================================
+% Chapter 1 Ñ Ocean Tracers (TRA)
+% ================================================================
+\chapter{Ocean Tracers (TRA)}
+\label{TRA}
+\minitoc
+
+% missing/update 
+% traqsr: need to coordinate with SBC module
+% trabbl : advective case to be discussed
+%			diffusive case : add : only the bottom ocean cell is concerned
+%			==> addfigure on bbl
+
+%STEVEN :  is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below
+
+\newpage
+$\ $\newline    % force a new ligne
+
+Using the representation described in Chap.~\ref{DOM}, several semi-discrete 
+space forms of the tracer equations are available depending on the vertical 
+coordinate used and on the physics used. In all the equations presented 
+here, the masking has been omitted for simplicity. One must be aware that 
+all the quantities are masked fields and that each time a mean or difference 
+operator is used, the resulting field is multiplied by a mask.
+
+The two active tracers are potential temperature and salinity. Their prognostic 
+equations can be summarized as follows:
+\begin{equation*}
+\text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC}
+                   \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP})
+\end{equation*}
+
+NXT stands for next, referring to the time-stepping. From left to right, the terms 
+on the rhs of the tracer equations are the advection (ADV), the lateral diffusion 
+(LDF), the vertical diffusion (ZDF), the contributions from the external forcings 
+(SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: 
+Bottom Boundary Condition), the contribution from the bottom boundary Layer 
+(BBL) parametrisation, and an internal damping (DMP) term. The terms QSR, 
+BBC, BBL and DMP are optional. The external forcings and parameterizations 
+require complex inputs and complex calculations (e.g. bulk formulae, estimation 
+of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and 
+described in chapters \S\ref{SBC}, \S\ref{LDF} and  \S\ref{ZDF}, respectively. 
+Note that \mdl{tranpc}, the non-penetrative convection module,  although 
+(temporarily) located in the NEMO/OPA/TRA directory, is described with the 
+model vertical physics (ZDF).
+%%%
+\gmcomment{change the position of eosbn2 in the reference code}
+%%%
+
+In the present chapter we also describe the diagnostic equations used to compute 
+the sea-water properties (density, Brunt-Vais\"{a}l\"{a} frequency, specific heat and 
+freezing point) although the associated modules ($i.e.$ \mdl{eosbn2}, \mdl{ocfzpt} 
+and \mdl{phycst}) are (temporarily) located in the NEMO/OPA directory.
+
+The different options available to the user are managed by namelist logical or 
+CPP keys. For each equation term ttt, the namelist logicals are \textit{ln\_trattt\_xxx}, 
+where \textit{xxx} is a 3 or 4 letter acronym accounting for each optional scheme. 
+The CPP key (when it exists) is \textbf{key\_trattt}. The corresponding code can be 
+found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory.
+
+The user has the option of extracting each tendency term on the rhs of the tracer 
+equation for output (\key{trdtra} is defined), as described in Chap.~\ref{MISC}.
+
+% ================================================================
+% Tracer Advection
+% ================================================================
+\section  [Tracer Advection (\textit{traadv})]
+		{Tracer Advection (\mdl{traadv})}
+\label{TRA_adv}
+%------------------------------------------nam_traadv-----------------------------------------------------
+\namdisplay{nam_traadv}
+%-------------------------------------------------------------------------------------------------------------
+
+The advection tendency of a tracer in flux form is the divergence of the advective 
+fluxes. Its discrete expression is given by :
+\begin{equation} \label{Eq_tra_adv}
+ADV_\tau =-\frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}e_{3T} }\left( 
+{\;\delta _i \left[ {e_{2u} {\kern 1pt}e_{3u} {\kern 1pt}\;u\;\tau _u } 
+\right]+\delta _j \left[ {e_{1v} {\kern 1pt}e_{3v} {\kern 1pt}v\;\tau _v } 
+\right]\;} \right)-\frac{1}{\mathop e\nolimits_{3T} }\delta _k \left[ 
+{w\;\tau _w } \right]
+\end{equation}
+where $\tau$ is either T or S. In pure $z$-coordinate (\key{zco} is defined), 
+it reduces to :
+\begin{equation} \label{Eq_tra_adv_zco}
+ADV_\tau =-\frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}}\left( {\;\delta _i 
+\left[ {e_{2u} {\kern 1pt}{\kern 1pt}\;u\;\tau _u } \right]+\delta _j \left[ 
+{e_{1v} {\kern 1pt}v\;\tau _v } \right]\;} \right)-\frac{1}{\mathop 
+e\nolimits_{3T} }\delta _k \left[ {w\;\tau _w } \right]
+\end{equation}
+since the vertical scale factors are functions of $k$ only, and thus $e_{3u} 
+=e_{3v} =e_{3T} $.
+
+The flux form in \eqref{Eq_tra_adv} requires implicitly the use of the continuity equation: 
+$\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 
+(using $\nabla \cdot \vect{U}=0)$ or $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$
+ in variable volume case ($i.e.$ \key{vvl} defined). Therefore it is of 
+paramount importance to design the discrete analogue of the advection 
+tendency so that it is consistent with the continuity equation in order to 
+enforce the conservation properties of the continuous equations. In other words, 
+by substituting $\tau$ by 1 in (\ref{Eq_tra_adv}) we recover the discrete form of 
+the continuity equation which is used to calculate the vertical velocity.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_adv_scheme}  \begin{center}
+\includegraphics[width=0.9\textwidth]{./Figures/Fig_adv_scheme.pdf}
+\caption{Schematic representation of some ways used to evaluate the tracer value 
+at $u$-point and the amount of tracer exchanged between two neighbouring grid 
+points. Upsteam biased scheme (ups): the upstream value is used and the black 
+area is exchanged. Piecewise parabolic method (ppm): a parabolic interpolation 
+is used and the black and dark grey areas are exchanged. Monotonic upstream 
+scheme for conservative laws (muscl):  a parabolic interpolation is used and black, 
+dark grey and grey areas are exchanged. Second order scheme (cen2): the mean 
+value is used and black, dark grey, grey and light grey areas are exchanged. Note 
+that this illustration does not include the flux limiter used in ppm and muscl schemes.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The key difference between the advection schemes used in \NEMO is the choice 
+made in space and time interpolation to define the value of the tracer at the 
+velocity points (Fig.~\ref{Fig_adv_scheme}). 
+
+Along solid lateral and bottom boundaries a zero tracer flux is naturally 
+specified, since the normal velocity is zero there. At the sea surface the 
+boundary condition depends on the type of sea surface chosen: 
+\begin{description}
+\item  [rigid-lid formulation:] $w=0$ at the surface, so the advective 
+fluxes through the surface are zero.
+\item [linear free surface:] the first level thickness is constant in time: 
+the vertical boundary condition is applied at the fixed surface $z=0$ 
+rather than on the moving surface $z=\eta$. There is a non-zero advective 
+flux which is set for all advection schemes as the product of surface 
+velocity (at $z=0$) by the first level tracer value: 
+$\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $. 
+\item [non-linear free surface:] (\key{vvl} is defined) 
+convergence/divergence in the first ocean level moves the free surface 
+up/down. There is no tracer advection through it so that the advective 
+fluxes through the surface are also zero 
+\end{description}
+In all cases, this boundary condition retains local conservation of tracer. 
+Global conservation is obtained in both rigid-lid and non-linear free surface 
+cases, but not in the linear free surface case. Nevertheless, in the latter
+case, it is achieved to a good approximation since the non-conservative 
+term is the product of the time derivative of the tracer and the free surface 
+height, two quantities that are not correlated (see \S\ref{PE_free_surface}, 
+and also \citet{Roullet2000,Griffies2001,Campin2004}).
+
+The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco}) 
+is the centred (\textit{now}) \textit{eulerian} ocean velocity (see Chap.~\ref{DYN}). 
+When advective bottom boundary layer (\textit{bbl}) and/or eddy induced velocity 
+(\textit{eiv}) parameterisations are used it is the \textit{now} \textit{effective} 
+velocity ($i.e.$ the sum of the eulerian, the bbl and/or the eiv velocities) which is used.
+
+The choice of an advection scheme is made in the \np{nam\_traadv} namelist, by 
+setting to \textit{true} one and only one of the logicals \textit{ln\_traadv\_xxx}. The 
+corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 
+\textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. Details 
+of the advection schemes are given below. The choice of an advection scheme 
+is a complex matter which depends on the model physics, model resolution, 
+type of tracer, as well as the issue of numerical cost. 
+
+Note that 
+(1) cen2, cen4 and TVD schemes require an explicit diffusion 
+operator while the other schemes are diffusive enough so that they do not 
+require additional diffusion ; 
+(2) cen2, cen4, MUSCL2, and UBS are not \textit{positive} schemes
+\footnote{negative values can appear in an initially strictly positive tracer field 
+which is advected}
+, implying that false extrema are permitted. Their use is not recommended on passive tracers ; 
+(3) It is highly recommended that the same advection-diffusion scheme is 
+used on both active and passive tracers. Indeed, if a source or sink of a 
+passive tracer depends on an active one, the difference of treatment of 
+active and passive tracers can create very nice-looking frontal structures 
+that are pure numerical artefacts. 
+
+% -------------------------------------------------------------------------------------------------------------
+%        2nd order centred scheme  
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2})]
+			{$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=.true.)}
+\label{TRA_adv_cen2}
+
+In the centred second order formulation, the tracer at velocity points is 
+evaluated as the mean of the two neighbouring $T$-point values. 
+For example, in the $i$-direction :
+\begin{equation} \label{Eq_tra_adv_cen2}
+\tau _u^{cen2} =\overline T ^{i+1/2}
+\end{equation}
+
+The scheme is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ 
+but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously 
+noisy and must be used in conjunction with an explicit diffusion operator to 
+produce a sensible solution. The associated time-stepping is performed using 
+a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in 
+(\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value.
+
+Note that using the cen2 scheme, the overall tracer advection is of second 
+order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) 
+have this order of accuracy.
+
+% -------------------------------------------------------------------------------------------------------------
+%        4nd order centred scheme  
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4})]
+		     {$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=.true.)}
+\label{TRA_adv_cen4}
+
+In the $4^{th}$ order formulation (to be implemented), tracer values are 
+evaluated at velocity points as a $4^{th}$ order interpolation, and thus uses 
+the four neighbouring $T$-points. For example, in the $i$-direction:
+\begin{equation} \label{Eq_tra_adv_cen4}
+\tau _u^{cen4} 
+=\overline{   T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2}
+\end{equation}
+
+Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme 
+but a $4^{th}$ order evaluation of advective fluxes, since the divergence of 
+advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. The phrase ``$4^{th}$ 
+order scheme'' used in oceanographic literature is usually associated 
+with the scheme presented here. Introducing a \textit{true} $4^{th}$ order advection 
+scheme is feasible but, for consistency reasons, it requires changes in the 
+discretisation of the tracer advection together with changes in both the 
+continuity equation and the momentum advection terms.  
+
+A direct consequence of the pseudo-fourth order nature of the scheme is that 
+it is not non-diffusive, i.e. the global variance of a tracer is not 
+preserved using \textit{cen4}. Furthermore, it must be used in conjunction with an 
+explicit diffusion operator to produce a sensible solution. The 
+time-stepping is also performed using a leapfrog scheme in conjunction with 
+an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.
+
+At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an 
+additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This 
+hypothesis usually reduces the order of the scheme. Here we choose to set 
+the gradient of $T$ across the boundary to zero. Alternative conditions can be 
+specified, such as a reduction to a second order scheme for these near boundary 
+grid points.
+
+% -------------------------------------------------------------------------------------------------------------
+%        TVD scheme  
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd})]
+			{Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=.true.)}
+\label{TRA_adv_tvd}
+
+In the Total Variance Dissipation (TVD) formulation, the tracer at velocity 
+points is evaluated using a combination of an upstream and a centred scheme. For 
+example, in the $i$-direction :
+\begin{equation} \label{Eq_tra_adv_tvd}
+\begin{split}
+\tau _u^{ups}&= \begin{cases}
+ 					T_{i+1} 	& \text{if $\ u_{i+1/2} <     0$} \hfill \\
+ 					T_i   		& \text{if $\ u_{i+1/2} \geq 0$} \hfill \\
+				  \end{cases}     \\
+\\
+\tau _u^{tvd}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen2} -\tau _u^{ups} } \right)
+\end{split}
+\end{equation}
+where $c_u$ is a flux limiter function taking values between 0 and 1. There 
+exist many ways to define $c_u$, each correcponding to a different total 
+variance decreasing scheme. The one chosen in \NEMO is described in \citet{Zalesak1979}. $c_u$ only departs from $1$ when the advective term 
+produces a local extremum in the tracer field. The resulting scheme is quite 
+expensive but \emph{positive}. It can be used on both active and passive tracers. 
+This scheme is tested and compared with MUSCL and the MPDATA scheme in 
+\citet{Levy2001}; note that in this paper it is referred to as "FCT" (Flux corrected 
+transport) rather than TVD.
+
+For stability reasons (see \S\ref{DOM_nxt}), in (\ref{Eq_tra_adv_tvd}) 
+$\tau _u^{cen2}$ is evaluated using the \textit{now} tracer while $\tau _u^{ups}$ 
+is evaluated using the \textit{before} tracer. In other words, the advective part of 
+the scheme is time stepped with a leap-frog scheme while a forward scheme is 
+used for the diffusive part. 
+
+% -------------------------------------------------------------------------------------------------------------
+%        MUSCL scheme  
+% -------------------------------------------------------------------------------------------------------------
+\subsection[MUSCL scheme  (\np{ln\_traadv\_muscl})]
+	{Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_muscl}=T)}
+\label{TRA_adv_muscl}
+
+The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been 
+implemented by \citet{Levy2001}. In its formulation, the tracer at velocity points 
+is evaluated assuming a linear tracer variation between two $T$-points 
+(Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction :
+\begin{equation} \label{Eq_tra_adv_muscl}
+   \tau _u^{mus} = \left\{      \begin{aligned}
+         &\tau _i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\Delta t}{e_{1u}} \right)
+         &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\
+         &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\Delta t}{e_{1u} } \right)
+         &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0
+   \end{aligned}    \right.
+\end{equation}
+where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation 
+is imposed to ensure the \textit{positive} character of the scheme.
+
+The time stepping is performed using a forward scheme, that is the \textit{before} 
+tracer field is used to evaluate $\tau _u^{mus}$.
+
+For an ocean grid point adjacent to land and where the ocean velocity is 
+directed toward land, two choices are available: an upstream flux 
+(\np{ln\_traadv\_muscl}=.true.) or a second order flux 
+(\np{ln\_traadv\_muscl2}=.true.). Note that the latter choice does not ensure 
+the \textit{positive} character of the scheme. Only the former can be used 
+on both active and passive tracers.
+
+% -------------------------------------------------------------------------------------------------------------
+%        UBS scheme  
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs})]
+			{Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=.true.)}
+\label{TRA_adv_ubs}
+
+The UBS advection scheme is an upstream-biased third order scheme based on 
+an upstream-biased parabolic interpolation. It is also known as the Cell 
+Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective 
+Kinematics). For example, in the $i$-direction :
+\begin{equation} \label{Eq_tra_adv_ubs}
+   \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{      
+   \begin{aligned}
+         &\tau"_i        	& \quad \text{if }\ u_{i+1/2} \geqslant 0      \\
+         &\tau"_{i+1}	& \quad \text{if }\ u_{i+1/2}       <       0
+   \end{aligned}    \right.
+\end{equation}
+where $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$.
+
+This results in a dissipatively dominant (i.e. hyper-diffusive) truncation 
+error \citep{Sacha2005}. The overall performance of the 
+advection scheme is similar to that reported in \cite{Farrow1995}. 
+It is a relatively good compromise between accuracy and smoothness. It is 
+not a \emph{positive} scheme, meaning that false extrema are permitted, but the 
+amplitude of such are significantly reduced over the centred second order 
+method. Nevertheless it is not recommended that it should be applied to a passive 
+tracer that requires positivity. 
+
+The intrinsic diffusion of UBS makes its use risky in the vertical direction 
+where the control of artificial diapycnal fluxes is of paramount importance. 
+Therefore the vertical flux is evaluated using the TVD 
+scheme when \np{ln\_traadv\_ubs}=.true..
+
+For stability reasons  (see \S\ref{DOM_nxt}), in \eqref{Eq_tra_adv_ubs}, 
+the first term (which corresponds to a second order centred scheme) 
+is evaluated using the \textit{now} tracer (centred in time) while the 
+second term (which is the diffusive part of the scheme), is 
+evaluated using the \textit{before} tracer (forward in time). 
+This is discussed by \citet{Webb1998} in the context of the Quick 
+advection scheme. UBS and QUICK 
+schemes only differ by one coefficient. Replacing 1/6 with 1/8 in 
+\eqref{Eq_tra_adv_ubs} leads to the QUICK advection scheme 
+\citep{Webb1998}. This option is not available through a namelist 
+parameter, since the 1/6 coefficient is hard coded. Nevertheless 
+it is quite easy to make the substitution in the \mdl{traadv\_ubs} module 
+and obtain a QUICK scheme.
+
+Note that :
+
+(1): When a high vertical resolution $O(1m)$ is used, the model stability can 
+be controlled by vertical advection (not vertical diffusion which is usually 
+solved using an implicit scheme). Computer time can be saved by using a 
+time-splitting technique on vertical advection. This case has been 
+implemented and validated in ORCA05 with 301 levels. It is not available in the 
+current reference version. 
+
+(2) : In a forthcoming release four options will be available for the vertical 
+component used in the UBS scheme. $\tau _w^{ubs}$ will be evaluated 
+using either \textit{(a)} a centred $2^{nd}$ order scheme , or  \textit{(b)} 
+a TVD scheme, or  \textit{(c)} an interpolation based on conservative 
+parabolic splines following the \citet{Sacha2005} implementation of UBS 
+in ROMS, or  \textit{(d)} a UBS. The $3^{rd}$ case has dispersion properties 
+similar to an eighth-order accurate conventional scheme.
+
+following \citet{Sacha2005} implementation of UBS in ROMS, or  \textit{(d)} 
+an UBS. The $3^{rd}$ case has dispersion properties similar to an 
+eight-order accurate conventional scheme.
+
+(3) : It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\tau _u^{ubs} = \left\{	 \begin{aligned}
+	& \tau _u^{cen4} + \frac{1}{12} \tau"_i		& \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
+	& \tau _u^{cen4} - \frac{1}{12} \tau"_{i+1}	& \quad \text{if }\ u_{i+1/2}       <       0
+  						 \end{aligned}    \right.
+\end{equation}
+or equivalently 
+\begin{equation} \label{Eq_tra_adv_ubs2b}
+u_{i+1/2} \ \tau _u^{ubs} 
+=u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2}
+- \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+\end{equation}
+\eqref{Eq_tra_adv_ubs2} has several advantages. Firstly, it clearly reveals 
+that the UBS scheme is based on the fourth order scheme to which an 
+upstream-biased diffusion term is added. Secondly, this emphasises that the 
+$4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has 
+to be evaluated at the \emph{now} time step using \eqref{Eq_tra_adv_ubs}. 
+Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 
+coefficient which is simply proportional to the velocity:
+ $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v2.3 still uses 
+ \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_tra_adv_ubs2}. This should be 
+ changed in forthcoming release.
+ %%%
+ \gmcomment{the change in UBS scheme has to be done}
+ %%%
+
+% -------------------------------------------------------------------------------------------------------------
+%        QCK scheme  
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [QUICKEST scheme (QCK) (\np{ln\_traadv\_qck})]
+			{QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=.true.)}
+\label{TRA_adv_qck}
+
+The Quadratic Upstream Interpolation for Convective Kinematics with 
+Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} 
+is the third order Godunov scheme. It is associated with the ULTIMATE QUICKEST 
+limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray 
+(MERCATOR-ocean). 
+The resulting scheme is quite expensive but \emph{positive}. It can be used on both active and passive tracers. Nevertheless, the intrinsic diffusion of QCK makes its use 
+risky in the vertical direction where the control of artificial diapycnal fluxes is of 
+paramount importance. Therefore the vertical flux is evaluated using the CEN2 
+scheme. This no more ensure the positivity of the scheme. The use of TVD in the 
+vertical direction as for the UBS case should be implemented to maintain the property.
+
+
+% -------------------------------------------------------------------------------------------------------------
+%        PPM scheme  
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm})]
+			{Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=.true.)}
+\label{TRA_adv_ppm}
+
+The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984) 
+is based on a quadradic piecewise rebuilding. Like the QCK scheme, it is associated 
+with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented 
+in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference 
+version 2.3.
+
+% ================================================================
+% Tracer Lateral Diffusion
+% ================================================================
+\section  [Tracer Lateral Diffusion (\textit{traldf})]
+		{Tracer Lateral Diffusion (\mdl{traldf})}
+\label{TRA_ldf}
+%-----------------------------------------nam_traldf------------------------------------------------------
+\namdisplay{nam_traldf}
+%-------------------------------------------------------------------------------------------------------------
+ 
+The options available for lateral diffusion are a laplacian (rotated or not) 
+or a biharmonic operator, the latter being more scale-selective (more 
+diffusive at small scales). The specification of eddy diffusivity 
+coefficients (either constant or variable in space and time) as well as the 
+computation of the slope along which the operators act, are performed in the 
+\mdl{ldftra} and \mdl{ldfslp} modules, respectively. This is described in Chap.~\ref{LDF}. The lateral diffusion of tracers is evaluated using a forward scheme, 
+$i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time, 
+except for the pure vertical component that appears when a rotation tensor 
+is used. This latter term is solved implicitly together with the 
+vertical diffusion term (see \S\ref{DOM_nxt}).
+
+% -------------------------------------------------------------------------------------------------------------
+%        Iso-level laplacian operator
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Iso-level laplacian operator (\textit{traldf\_lap} - \np{ln\_traldf\_lap})]
+			{Iso-level laplacian operator (\mdl{traldf\_lap} - \np{ln\_traldf\_lap}=.true.) }
+\label{TRA_ldf_lap}
+
+A laplacian diffusion operator (i.e. a harmonic operator) acting along the model 
+surfaces is given by: 
+\begin{equation} \label{Eq_tra_ldf_lap}
+\begin{split}
+D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta _i 
+\left[ {A_u^{lT} \left( {\frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} 
+\left[ T \right]} \right)} \right]} \right.
+\\
+&\ \left. {+\; \delta _j \left[ 
+{A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta _{j+1/2} \left[ T 
+\right]} \right)} \right]\quad } \right]
+\end{split}
+\end{equation}
+
+This lateral operator is a \emph{horizontal} one ($i.e.$ acting along 
+geopotential surfaces) in the $z$-coordinate with or without partial step, 
+but is simply an iso-level operator in the $s$-coordinate. 
+It is thus used when, in addition to \np{ln\_traldf\_lap}=.true., we have 
+\np{ln\_traldf\_level}=.true., or both \np{ln\_traldf\_hor}=.true. and 
+\np{ln\_zco}=.false.. In both cases, it significantly contributes to 
+diapycnal mixing. It is therefore not recommended.
+
+Note that 
+(1) In pure $z$-coordinate (\key{zco} is defined), $e_{3u}=e_{3v}=e_{3T}$, so 
+that the vertical scale factors disappear from (\ref{Eq_tra_ldf_lap}). 
+(2) In partial step $z$-coordinate (\np{ln\_zps}=.true.), tracers in horizontally 
+adjacent cells are located at different depths in the vicinity of the bottom. 
+In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level 
+require a specific treatment. They are calculated in the \mdl{zpshde} module, 
+described in \S\ref{TRA_zpshde}.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Rotated laplacian operator
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Rotated laplacian operator (\textit{traldf\_iso} - \np{ln\_traldf\_lap})]
+			{Rotated laplacian operator (\mdl{traldf\_iso} - \np{ln\_traldf\_lap}=.true.)}
+\label{TRA_ldf_iso}
+
+The general form of the second order lateral tracer subgrid scale physics 
+(\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and 
+$s$-coordinates:
+
+\begin{equation} \label{Eq_tra_ldf_iso}
+\begin{split}
+ D_T^{lT} =& \frac{1}{e_{1T}\,e_{2T}\,e_{3T} }
+ \\
+& \left\{ {\delta _i \left[ {A_u^{lT}  \left( 
+	 {\frac{e_{2u} \; e_{3u} }{e_{1u} } \,\delta _{i+1/2}[T]
+	-e_{2u} \; r_{1u} \,\overline{\overline {\delta _{k+1/2}[T]}}^{\,i+1/2,k}}
+ \right)} \right]} \right. 
+\\ 
+& +\delta 
+_j \left[ {A_v^{lT} \left( {\frac{e_{1v}\,e_{3v} }{e_{2v} 
+}\,\delta _{j+1/2} \left[ T \right]-e_{1v}\,r_{2v} 
+\,\overline{\overline {\delta _{k+1/2} \left[ T \right]}} ^{\,j+1/2,k}} 
+\right)} \right] 
+\\ 
+& +\delta 
+_k \left[ {A_w^{lT} \left( 
+-e_{2w}\,r_{1w} \,\overline{\overline {\delta _{i+1/2} \left[ T \right]}} ^{\,i,k+1/2}
+\right.} \right. 
+\\ 
+& \qquad \qquad \quad 
+-e_{1w}\,r_{2w} \,\overline{\overline {\delta _{j+1/2} \left[ T \right]}} ^{\,j,k+1/2}
+\\
+& \left. {\left. { 
+ \quad \quad \quad \left. {{\kern 
+1pt}+\frac{e_{1w}\,e_{2w} }{e_{3w} }\,\left( {r_{1w} ^2+r_{2w} ^2} 
+\right)\,\delta _{k+1/2} \left[ T \right]} \right)} \right]\;\;\;} \right\} 
+ \end{split}
+ \end{equation}
+where $r_1$ and $r_2$ are the slopes between the surface of computation 
+($z$- or $s$-surfaces) and the surface along which the diffusion operator 
+acts ($i.e.$ horizontal or iso-neutral surfaces).  It is thus used when, 
+in addition to \np{ln\_traldf\_lap}=.true., we have \np{ln\_traldf\_iso}=.true., 
+or both \np{ln\_traldf\_hor}=.true. and \np{ln\_zco}=.true.. The way these 
+slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom 
+and lateral boundaries, the turbulent fluxes of heat and salt are set to zero 
+using the mask technique (see \S\ref{LBC_coast}). 
+
+The operator in \eqref{Eq_tra_ldf_iso} involves both lateral and vertical 
+derivatives. For numerical stability, the vertical second derivative must 
+be solved using the same implicit time scheme as that used in the vertical 
+physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term 
+is not computed in the \mdl{traldf} module, but in the \mdl{trazdf} module 
+where, if iso-neutral mixing is used, the vertical mixing coefficient is simply 
+increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$. 
+
+This formulation conserves the tracer but does not ensure the decrease 
+of the tracer variance. Nevertheless the treatment performed on the slopes 
+(see \S\ref{LDF}) allows the model to run safely without any additional 
+background horizontal diffusion \citep{Guily2001}. An alternative scheme 
+\citep{Griffies1998} which preserves both tracer and its variance is currently 
+been tested in \NEMO. 
+
+Note that in the partial step $z$-coordinate (\np{ln\_zps}=.true.), the horizontal 
+derivatives at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific 
+treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Iso-level bilaplacian operator
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Iso-level bilaplacian operator (\textit{traldf\_bilap} - \np{ln\_traldf\_bilap})]
+			{Iso-level bilaplacian operator (\mdl{traldf\_bilap} - \np{ln\_traldf\_bilap}=.true.)}
+\label{TRA_ldf_bilap}
+
+The lateral fourth order bilaplacian operator on tracers is obtained by 
+applying (\ref{Eq_tra_ldf_lap}) twice. It requires an additional assumption 
+on boundary conditions: the first and third derivative terms normal to the 
+coast are set to zero.
+
+It is used when, in addition to \np{ln\_traldf\_bilap}=.true., we have 
+\np{ln\_traldf\_level}=.true., or both \np{ln\_traldf\_hor}=.true. and 
+\np{ln\_zco}=.false.. In both cases, it can contribute diapycnal mixing, 
+although less than in the laplacian case. It is therefore not recommended.
+
+Note that in the code, the bilaplacian routine does not call the laplacian 
+routine twice but is rather a separate routine. This is due to the fact that we 
+introduce the eddy diffusivity coefficient, A, in the operator as: $\nabla 
+\cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$, instead of 
+$-\nabla \cdot a\nabla \left( {\nabla \cdot a\nabla T} \right)$ where 
+$a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations 
+ensure the total variance decrease, but the former requires a larger number 
+of code-lines. It will be corrected in a forthcoming release.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Rotated bilaplacian operator
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Rotated bilaplacian operator (\textit{traldf\_bilapg} - \np{ln\_traldf\_bilap})]
+			{Rotated bilaplacian operator (\mdl{traldf\_bilapg} - \np{ln\_traldf\_bilap}=.true.)}
+\label{TRA_ldf_bilapg}
+
+The lateral fourth order operator formulation on tracers is obtained by 
+applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption 
+on boundary conditions: first and third derivative terms normal to the 
+coast, the bottom and the surface are set to zero.
+
+It is used when, in addition to \np{ln\_traldf\_bilap}=T, we have 
+\np{ln\_traldf\_iso}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=T. 
+Nevertheless, this rotated bilaplacian operator has never been seriously 
+tested. No warranties that it is neither free of bugs or correctly formulated. 
+Moreover, the stability range of such an operator will be probably quite 
+narrow, requiring a significantly smaller time-step than the one used on 
+unrotated operator.
+
+% ================================================================
+% Tracer Vertical Diffusion
+% ================================================================
+\section  [Tracer Vertical Diffusion (\textit{trazdf})]
+		{Tracer Vertical Diffusion (\mdl{trazdf})}
+\label{TRA_zdf}
+%--------------------------------------------namzdf---------------------------------------------------------
+\namdisplay{namzdf}
+%--------------------------------------------------------------------------------------------------------------
+
+The formulation of the vertical subgrid scale tracer physics is the same 
+for all the vertical coordinates, and is based on a laplacian operator. 
+The vertical diffusion operator given by (\ref{Eq_PE_zdf}) takes the 
+following semi-discrete space form:
+(\ref{Eq_PE_zdf}) takes the following semi-discrete space form:
+\begin{equation} \label{Eq_tra_zdf}
+\begin{split}
+D^{vT}_T &= \frac{1}{e_{3T}} \; \delta_k \left[ 
+\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T] 	\right] 
+\\
+D^{vS}_T &= \frac{1}{e_{3T}} \; \delta_k \left[ 
+\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S] 	\right] 
+\end{split}
+\end{equation}
+
+where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity 
+coefficients on Temperature and Salinity, respectively. Generally, 
+$A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is 
+parameterised (\key{zdfddm} is defined). The way these coefficients 
+are evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when 
+iso-neutral mixing is used, both mixing coefficients are increased 
+by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ 
+to account for the vertical second derivative of \eqref{Eq_tra_ldf_iso}. 
+
+At the surface and bottom boundaries, the turbulent fluxes of 
+momentum, heat and salt must be specified. At the surface they 
+are prescribed from the surface forcing (see \S\ref{TRA_sbc}), 
+whilst at the bottom they are set to zero for heat and salt unless 
+a geothermal flux forcing is prescribed as a bottom boundary 
+condition (\S\ref{TRA_bbc}). 
+
+The large eddy coefficient found in the mixed layer together with high 
+vertical resolution implies that in the case of explicit time stepping 
+(\np{ln\_zdfexp}=.true.) there would be too restrictive a constraint on 
+the time step. Therefore, the default implicit time stepping is preferred 
+for the vertical diffusion since it overcomes the stability constraint. 
+A forward time differencing scheme (\np{ln\_zdfexp}=.true.) using a time 
+splitting technique (\np{n\_zdfexp} $> 1$) is provided as an alternative. 
+Namelist variables \np{ln\_zdfexp} and \np{n\_zdfexp} apply to both 
+tracers and dynamics. 
+
+% ================================================================
+% External Forcing
+% ================================================================
+\section{External Forcing}
+\label{TRA_sbc_qsr_bbc}
+
+% -------------------------------------------------------------------------------------------------------------
+%        surface boundary condition
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Surface boundary condition (\textit{trasbc})]
+			{Surface boundary condition (\mdl{trasbc})}
+\label{TRA_sbc}
+
+The surface boundary condition for tracers is implemented in a separate 
+module (\mdl{trasbc}) instead of entering as a boundary condition on the vertical 
+diffusion operator (as in the case of momentum). This has been found to 
+enhance readability of the code. The two formulations are completely 
+equivalent; the forcing terms in trasbc are the surface fluxes divided by 
+the thickness of the top model layer. Following \citet{Roullet2000} the 
+forcing on an ocean tracer, $c$, can be split into two parts: $F_{ext}$, the 
+flux of tracer crossing the sea surface and not linked with the water 
+exchange with the atmosphere, $F_{wf}^d$, and $F_{wf}^i$ the forcing 
+on the tracer concentration associated with this water exchange. 
+The latter forcing has two components: a direct effect of change 
+in concentration associated with the tracer carried by the water flux, 
+and an indirect concentration/dilution effect :
+\begin{equation*}
+\begin{split}
+ F^C &= F_{ext} + F_{wf}^d                                          +F_{wf}^i  	\\
+        &= F_{ext} - \left( c_E \, E - c_p \,P - c_R \,R \right) +c\left( E-P-R \right)
+\end{split}
+\end{equation*} 
+
+\gmcomment{add here a description of the variable names used in the above equation}
+
+Two cases must be distinguished, the nonlinear free surface case 
+(\key{vvl} is defined) and the linear free surface case. The first case 
+is simpler, because the indirect concentration/dilution effect is naturally 
+taken into account by letting the vertical scale factors vary in time. 
+The salinity of water exchanged at the surface is assumed to be zero, 
+so there is no salt flux at the free surface, except in the presence of sea ice. 
+The heat flux at the free surface is the sum of $F_{ext}$, the direct 
+heating/cooling (by the total non-penetrative heat flux) and $F_{wf}^e$ 
+the heat carried by the water exchanged through the surface (evaporation, 
+precipitation, runoff). The temperature of precipitation is not well known. 
+In the model we assume that this water has the same temperature as 
+the sea surface temperature. The resulting forcing terms for temperature 
+T and salinity S are: 
+\begin{equation} \label{Eq_tra_forcing}
+\begin{aligned}
+ &F^T =\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} }-\frac{\text{EMP}\;\left. T 
+\right|_{k=1} }{e_{3T} }  & \\ 
+\\
+& F^S =\frac{\text{EMP}_S\;\left. S \right|_{k=1} }{e_{3T} }   &
+ \end{aligned}
+\end{equation} 
+
+where EMP is the freshwater budget (evaporation minus precipitation 
+minus river runoff) which forces the ocean volume, $Q_{ns}$ is the 
+non-penetrative part of the net surface heat flux (difference between 
+the total surface heat flux and the fraction of the short wave flux that 
+penetrates into the water column), the product EMP$_S\;.\left. S \right|_{k=1}$ 
+is  the ice-ocean salt flux, and $\left. S\right|_{k=1}$ is the sea surface 
+salinity (\textit{SSS}). The total salt content is conserved in this formulation 
+(except for the effect of the Asselin filter).
+
+%AMT note: the ice-ocean flux had been forgotten in the first release of the key_vvl option, has this been corrected in the code? 
+
+In the second case (linear free surface), the vertical scale factors are 
+fixed in time so that the concentration/dilution effect must be added in 
+the \mdl{trasbc} module. Because of the hypothesis made for the 
+temperature of precipitation and runoffs, $F_{wf}^e +F_{wf}^i =0$ 
+for temperature. The resulting forcing term for temperature is: 
+
+\begin{equation} \label{Eq_tra_forcing_q}
+F^T=\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} }
+\end{equation} 
+
+The salinity forcing is still given by \eqref{Eq_tra_forcing} but the 
+definition of EMP$_S$ is different: it is the total surface freshwater 
+budget (evaporation minus precipitation minus river runoff plus 
+the rate of change of the sea ice thickness). The total salt content 
+is not exactly conserved (\citet{Roullet2000}. 
+See also \S\ref{PE_free_surface}).
+
+In the case of the rigid lid approximation, the surface salinity forcing $F^s$ 
+is also expressed by \eqref{Eq_tra_forcing}, but now the global integral of 
+the product of EMP and S, is not compensated by the advection of fluid 
+through the top level: this is because in the rigid lid case \textit{w(k=1) = 0} 
+(in contrast to the linear free surface case). As a result, even if the budget 
+of \textit{EMP} is zero on average over the whole ocean domain, the 
+associated salt flux is not, since sea-surface salinity and \textit{EMP} are 
+intrinsically correlated (high \textit{SSS} are found where evaporation is 
+strong whilst low \textit{SSS} is usually associated with high precipitation 
+or river runoff).
+
+The $Q_{ns} $ and \textit{EMP} fields are defined and updated in the 
+\mdl{sbcmod} module (see \S\ref{SBC}).
+
+% -------------------------------------------------------------------------------------------------------------
+%        Solar Radiation Penetration 
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Solar Radiation Penetration (\textit{traqsr})]
+			{Solar Radiation Penetration (\mdl{traqsr})}
+\label{TRA_qsr}
+%--------------------------------------------namqsr--------------------------------------------------------
+\namdisplay{namqsr}
+%--------------------------------------------------------------------------------------------------------------
+
+When the penetrative solar radiation option is used (\np{ln\_flxqsr}=.true.), 
+the solar radiation penetrates the top few meters of the ocean, otherwise 
+all the heat flux is absorbed in the first ocean level (\np{ln\_flxqsr}=.false.). 
+Thus, in the former case a term is added to the time evolution equation of 
+temperature \eqref{Eq_PE_tra_T} whilst the surface boundary condition is 
+modified to take into account only the non-penetrative part of the surface 
+heat flux:
+\begin{equation} \label{Eq_PE_qsr}
+\begin{split}
+\frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}	\\
+Q_{ns} &= Q_\text{Total} - Q_{sr}
+\end{split}
+\end{equation}
+
+where $I$ is the downward irradiance. The additional term in \eqref{Eq_PE_qsr} is discretized as follows:
+\begin{equation} \label{Eq_tra_qsr}
+\frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3T}} \delta_k \left[ I_w \right]
+\end{equation}
+
+A formulation involving two extinction coefficients is assumed for the 
+downward irradiance $I$ \citep{Paulson1977}:
+\begin{equation} \label{Eq_traqsr_iradiance}
+I(z) = Q_{sr} \left[Re^{-z / \xi_1} + \left( 1-R\right) e^{-z / \xi_2} \right]
+\end{equation}
+where $Q_{sr}$ is the penetrative part of the surface heat flux, 
+$\xi_1$ and $\xi_2$ are two extinction length scales and $R$ 
+determines the relative contribution of the two terms. 
+The default values used correspond to a Type I water in Jerlov's [1968] 
+%
+\gmcomment : Jerlov reference to be added
+%
+classification: $\xi_1 = 0.35m$, $\xi_2 = 0.23m$ and $R = 0.58$ 
+(corresponding to \np{xsi1}, \np{xsi2} and \np{rabs} namelist parameters, 
+respectively). $I$ is masked (no flux through the ocean bottom), 
+so all the solar radiation that reaches the last ocean level is absorbed 
+in that level. The trend in \eqref{Eq_tra_qsr} associated with the 
+penetration of the solar radiation is added to the temperature trend, 
+and the surface heat flux is modified in routine \mdl{traqsr}. 
+Note that in the $z$-coordinate, the depth of $T-$levels depends 
+on the single variable $k$. A one dimensional array of the coefficients 
+$gdsr(k) = Re^{-z_w (k)/\xi_1} + (1-R)e^{-z_w (k)/\xi_2}$ can then 
+be computed once and saved in memory. Moreover \textit{nksr}, 
+the level at which $gdrs$ becomes negligible (less than the 
+computer precision) is computed once, and the trend associated 
+with the penetration of the solar radiation is only added until that level. 
+Finally, note that when the ocean is shallow (< 200~m), part of the 
+solar radiation can reach the ocean floor. In this case, we have 
+chosen that all remaining radiation is absorbed in the last ocean 
+level ($i.e.$ $I_w$ is masked). 
+
+When coupling with a biological model (for example PISCES or LOBSTER), 
+it is possible to calculate the light attenuation using information from 
+the biology model. Without biological model, it is still possible to introduce 
+a horizontal variation of the light attenuation by using the observed ocean 
+surface color. At the time of writing, the latter has not been implemented
+ in the reference version.
+%
+\gmcomment{  {yellow}{case 4 bands and bio-coupling to add !!!}  }
+%
+
+% -------------------------------------------------------------------------------------------------------------
+%        Bottom Boundary Condition
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Bottom Boundary Condition (\textit{trabbc} - \key{bbc})]
+			{Bottom Boundary Condition (\mdl{trabbc} - \key{bbc})}
+\label{TRA_bbc}
+%--------------------------------------------nambbc--------------------------------------------------------
+\namdisplay{nambbc}
+%--------------------------------------------------------------------------------------------------------------
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_geothermal}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./Figures/Fig_TRA_geoth.pdf}
+\caption{Geothermal Heat flux (in $mW.m^{-2}$) as inferred from the age 
+of the sea floor and the formulae of \citet{Stein1992}.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Usually it is assumed that there is no exchange of heat or salt through 
+the ocean bottom, $i.e.$ a no flux boundary condition is applied on active 
+tracers at the bottom. This is the default option in \NEMO, and it is 
+implemented using the masking technique. Hoever, there is a 
+non-zero heat flux across the seafloor that is associated with solid 
+earth cooling. This flux is weak compared to surface fluxes (a mean 
+global value of $\sim0.1\;W/m^2$ \citep{Stein1992}), but it is 
+systematically positive and acts on the densest water masses. Taking 
+this flux into account in a global ocean model increases
+the deepest overturning cell (i.e. the one associated with the Antarctic 
+Bottom Water) by a few Sverdrups. 
+
+The presence or not of geothermal heating is controlled by the namelist 
+parameter  \np{ngeo\_flux}. If this parameter is set to 1, a constant 
+geothermal heating is introduced whose value is given by the 
+\np{ngeo\_flux\_const}, which is also a namelist parameter. If it is set to 2, 
+a spatially varying geothermal heat flux is introduced which is provided 
+in the geothermal\_heating.nc NetCDF file (Fig.\ref{Fig_geothermal}).
+
+% ================================================================
+% Bottom Boundary Layer
+% ================================================================
+\section  [Bottom Boundary Layer (\textit{trabbl}, \textit{trabbl\_adv} )]
+		{Bottom Boundary Layer (\mdl{trabbl}, \mdl{trabbl\_adv})}
+\label{TRA_bbl}
+%--------------------------------------------nambbl---------------------------------------------------------
+\namdisplay{nambbl}
+%--------------------------------------------------------------------------------------------------------------
+
+In a $z$-coordinate configuration, the bottom topography is represented by a 
+series of discrete steps. This is not adequate to represent gravity driven 
+downslope flows. Such flows arise downstream of sills such as the Strait of 
+Gibraltar, Bab El Mandeb, or Denmark Strait, where dense water formed in 
+marginal seas flows into a basin filled with less dense water. The amount of 
+entrainment that occurs in these gravity plumes is critical in determining the 
+density and volume flux of the densest waters of the ocean, such as 
+Antarctic Bottom Water, or North Atlantic Deep Water. $z$-coordinate 
+models tend to overestimate the entrainment, because the gravity flow is 
+mixed down vertically by convection as it goes ``downstairs'' following the 
+step topography, sometimes over a thickness much larger than the thickness 
+of the observed gravity plume. A similar problem occurs in the $s$-coordinate when 
+the thickness of the bottom level varies in large proportions downstream of 
+a sill \citep{Willebrand2001}, and the thickness of the plume is not resolved. 
+
+The idea of the bottom boundary layer (BBL) parameterization first introduced by 
+\citet{BeckDos1998} is to allow a direct communication between 
+two adjacent bottom cells at different levels, whenever the densest water is 
+located above the less dense water. The communication can be by a diffusive 
+(diffusive BBL), advective fluxes (advective BBL), or both. In the current 
+implementation of the BBL, only the tracers are modified, not the velocities. 
+Furthermore, it only connects ocean bottom cells, and therefore does not include 
+the improvment proposed by \citet{Campin_Goosse_Tel99}.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Diffusive BBL
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Diffusive Bottom Boundary layer (\key{bbl\_diff})}
+\label{TRA_bbl_diff}
+
+When applying sigma-diffusion (\key{trabbl} is defined), the diffusive flux between 
+two adjacent cells living at the ocean bottom is given by 
+\begin{equation} \label{Eq_tra_bbl_diff}
+{\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T
+\end{equation} 
+with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
+and  $A_l^\sigma $ the lateral diffusivity in the BBL. Following \citet{BeckDos1998}, 
+the latter is prescribed with a spatial dependence, $e.g.$ in the conditional form
+\begin{equation} \label{Eq_tra_bbl_coef}
+A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
+ A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\ 
+ \\
+ 0\quad \quad \;\,\mbox{otherwise} \\ 
+ \end{array}} \right.
+\end{equation} 
+where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist 
+parameter \np{atrbbl}. $A_{bbl}$ is usually set to a value much larger 
+than the one used on lateral mixing in open ocean. 
+Note that in practice, \eqref{Eq_tra_bbl_coef} constraint is applied 
+separately in the two horizontal directions, and the density gradient in 
+\eqref{Eq_tra_bbl_coef} is evaluated at $\overline{H}^i$ ($\overline{H}^j$) 
+using the along bottom mean temperature and salinity. 
+
+% -------------------------------------------------------------------------------------------------------------
+%        Advective BBL
+% -------------------------------------------------------------------------------------------------------------
+\subsection   {Advective Bottom Boundary Layer (\key{bbl\_adv})}
+\label{TRA_bbl_adv}
+
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_bbl}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./Figures/Fig_BBL_adv.pdf}
+\caption{Advective Bottom Boundary Layer.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+%%%gmcomment   :  this section has to be really written
+
+The advective BBL is in fact not only an advective one but include a diffusive 
+component as we chose an upstream scheme to perform the advection within 
+the BBL. The associated diffusion only act in the stream direction and is 
+proportional to the velocity.
+
+When applying sigma-advection (\key{trabbl\_adv} defined), the advective 
+flux between two adjacent cells living at the ocean bottom is given by 
+\begin{equation} \label{Eq_tra_bbl_fadv}
+{\rm {\bf F}}_\sigma={\rm {\bf U}}_h^\sigma \; \overline{T}^\sigma
+\end{equation} 
+with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
+and  $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{BeckDos1998}, 
+the latter is prescribed with a spatial dependence, $e.g.$ in the conditional form
+\begin{equation} \label{Eq_tra_bbl_Aadv}
+A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
+ A_{bbl} \quad \quad \mbox{if}     \quad    \nabla_\sigma \rho \cdot \nabla H<0 
+              \quad \quad \mbox{and} \quad         {\rm {\bf U}}_h  \cdot \nabla H<0 \\ 
+ \\
+ 0\quad \quad \;\,\mbox{otherwise} \\ 
+ \end{array}} \right.
+\end{equation} 
+
+% ================================================================
+% Tracer damping
+% ================================================================
+\section  [Tracer damping (\textit{tradmp})]
+		{Tracer damping (\mdl{tradmp})}
+\label{TRA_dmp}
+%--------------------------------------------namdmp-----------------------------------------------------
+\namdisplay{namdmp}
+%--------------------------------------------------------------------------------------------------------------
+
+In some applications it can be useful to add a Newtonian damping term 
+into the temperature and salinity equations:
+\begin{equation} \label{Eq_tra_dmp}
+\begin{split}
+ \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right)  \\
+\\ 
+ \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) 
+ \end{split}
+ \end{equation} 
+where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ 
+are given temperature and salinity fields (usually a climatology). 
+The restoring term is added when \key{tradmp} is defined. 
+It also requires that both \key{temdta} and \key{saldta} are defined 
+($i.e.$ that $T_o$ and $S_o$ are read). The restoring coefficient 
+$S_o$ is a three-dimensional array initialized by the user in routine 
+\rou{dtacof} also located in module \mdl{tradmp}. 
+
+The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} 
+the specification of the boundary conditions along artificial walls of a 
+limited domain basin and \textit{(b)} the computation of the velocity 
+field associated with a given $T$-$S$ field (for example to build the 
+initial state of a prognostic simulation, or to use the resulting velocity 
+field for a passive tracer study). The first case applies to regional 
+models that have artificial walls instead of open boundaries. 
+In the vicinity of these walls, $S_o$ takes large values (equivalent to 
+a time scale of a few days) whereas it is zero in the interior of the 
+model domain. The second case corresponds to the use of the robust 
+diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity 
+field consistent with the model dynamics whilst having a $T$-$S$ field 
+close to a given climatological field ($T_o -S_o$). The time scale 
+associated with $S_o$ is generally not a constant but spatially varying 
+in order to respect other properties. For example, it is usually set to zero 
+in the mixed layer (defined either on a density or $S_o$ criterion) 
+\citep{Madec1996} and in the equatorial region 
+\citep{Reverdin1991, Fujio1991, MartiTh1992} since these two regions 
+have a short time scale of adjustment; while smaller $S_o$ are used 
+in the deep ocean where the typical time scale is long \citep{Sarmiento1982}. 
+In addition the time scale is reduced (even to zero) along the western 
+boundary to allow the model to reconstruct its own western boundary 
+structure in equilibrium with its physics. The choice of a 
+Newtonian damping acting in the mixed layer or not is controlled by
+namelist parameter \np{nmldmp}.
+
+The robust diagnostic method is very efficient in preventing temperature 
+drift in intermediate waters but it produces artificial sources of heat and salt 
+within the ocean. It also has undesirable effects on the ocean convection. 
+It tends to prevent deep convection and subsequent deep-water formation, 
+by stabilising the water column too much.
+
+An example of the computation of $S_o$ for robust diagnostic experiments 
+with the ORCA2 model is provided in the \mdl{tradmp} module 
+(subroutines \rou{dtacof} and \rou{cofdis} which compute the coefficient 
+and the distance to the bathymetry, respectively). These routines are 
+provided as examples and can be customised by the user. 
+
+% ================================================================
+% Tracer time evolution
+% ================================================================
+\section  [Tracer time evolution (\textit{tranxt})]
+		{Tracer time evolution (\mdl{tranxt})}
+\label{TRA_nxt}
+%--------------------------------------------namdom-----------------------------------------------------
+\namdisplay{namdom}
+%--------------------------------------------------------------------------------------------------------------
+
+The general framework for tracer time stepping is a leap-frog scheme, 
+$i.e.$ a three level centred time scheme associated with a Asselin time 
+filter (cf. \S\ref{DOM_nxt}):
+\begin{equation} \label{Eq_tra_nxt}
+\begin{split}
+T^{t+\Delta t} &= T^{t-\Delta t} + 2 \, \Delta t  \ \text{RHS}_T^t 	\\
+\\
+T_f^t  \;\ \quad &= T^t \;\quad +\gamma \,\left[ {T_f^{t-\Delta t} -2T^t+T^{t+\Delta t}} \right]
+\end{split}
+\end{equation} 
+
+where $\text{RHS}_T$ is the right hand side of the temperature equation, 
+the subscript $f$ denotes filtered values and $\gamma$ is the Asselin 
+coefficient. $\gamma$ is initialized as \np{atfp} (\textbf{namelist} parameter). 
+Its default value is \np{atfp=0.1}. 
+
+When the vertical mixing is solved implicitly, the update of the \textit{next} tracer 
+fields is done in module \mdl{trazdf}. In this case only the swapping of arrays 
+and the Asselin filtering is done in the \mdl{tranxt} module.
+
+In order to prepare for the computation of the \textit{next} time step, 
+a swap of tracer arrays is performed: $T^{t-\Delta t} = T^t$ and $T^t = T_f$. 
+
+% ================================================================
+% Equation of State (eosbn2) 
+% ================================================================
+\section  [Equation of State (\textit{eosbn2}) ]
+		{Equation of State (\mdl{eosbn2}) }
+\label{TRA_eosbn2}
+%--------------------------------------------nameos-----------------------------------------------------
+\namdisplay{nameos}
+%--------------------------------------------------------------------------------------------------------------
+
+% -------------------------------------------------------------------------------------------------------------
+%        Equation of State
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Equation of State (\np{neos} = 0, 1 or 2)}
+\label{TRA_eos}
+
+It is necessary to know the equation of state for the ocean very accurately 
+to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), 
+particularly in the deep ocean. The ocean density is a non linear empirical 
+function of \textit{in situ }temperature, salinity and pressure. The reference 
+equation of state is that defined by the Joint Panel on Oceanographic Tables 
+and Standards \citep{UNESCO1983}. It was the standard equation of state 
+used in early releases of OPA. However, even though this computation is 
+fully vectorised, it is quite time consuming ($15$ to $20${\%} of the total 
+CPU time) since it requires the prior computation of the \textit{in situ} 
+temperature from the model \textit{potential} temperature using the 
+\citep{Bryden1973} polynomial for adiabatic lapse rate and a $4^th$ order 
+Runge-Kutta integration scheme. Since OPA6, we have used the 
+\citet{JackMcD1995} equation of state for seawater instead. It allows the 
+computation of the \textit{in situ} ocean density directly as a function of 
+\textit{potential} temperature relative to the surface (an \NEMO variable), 
+the practical salinity (another \NEMO variable) and the pressure (assuming no 
+pressure variation along geopotential surfaces, i.e. the pressure in decibars is 
+approximated by the depth in meters). Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state have exactly the same except that 
+the values of the various coefficients have been adjusted by \citet{JackMcD1995} 
+in order to directly use the \textit{potential} temperature instead of the 
+\textit{in situ} one. This reduces the CPU time of the in situ density computation 
+to about $3${\%} of the total CPU time, while maintaining a quite accurate 
+equation of state.
+
+In the computer code, a \textit{true} density $d$ is computed, $i.e.$ the ratio 
+of seawater volumic mass to $\rho_o$, a reference volumic mass (\textit{rau0} 
+defined in \mdl{phycst}, usually $rau0= 1,020~Kg/m^3$). The default option 
+(namelist prameter \np{neos}=0) is the \citet{JackMcD1995} equation of state. 
+Its use is highly recommended. However, for process studies, it is often 
+convenient to use a linear approximation of the density$^{\ast}$
+\footnote{$^{\ast }$ With the linear equation of state there is no longer 
+a distinction between \textit{in situ} and \textit{potential} density. Cabling 
+and thermobaric effects are also removed.}. 
+Two linear formulations are available: a function of $T$ only (\np{neos}=1) 
+and a function of both $T$ and $S$ (\np{neos}=2):
+\begin{equation} \label{Eq_tra_eos_linear}
+\begin{aligned}
+ d(T)    &= {\rho (T)} / {\rho _0 } &&= 1.028 - \alpha \;T     \\ 
+ d(T,S) &= {\rho (T,S)}                &&= \ \ \ \beta \;S - \alpha \;T 
+\end{aligned}
+\end{equation} 
+where $\alpha$ and $\beta$ are the thermal and haline expansion 
+coefficients, and $\rho_o$, the reference volumic mass, $rau0$. 
+($\alpha$ and $\beta$ can be modified through the \np{ralpha} and 
+\np{rbeta} namelist parameters). Note that when $d$ is a function 
+of $T$ only (\np{neos}=1), the salinity is a passive tracer and can be 
+used as such.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Brunt-Vais\"{a}l\"{a} Frequency
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{neos} = 0, 1 or 2)}
+\label{TRA_bn2}
+
+An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a}
+ frequency) is of paramount importance as it is used in several ocean 
+ parameterisations (namely TKE, KPP, Richardson number dependent 
+ vertical diffusion, enhanced vertical diffusion, non-penetrative convection, 
+ iso-neutral diffusion). In particular, one must be aware that $N^2$ has to 
+ be computed with an \textit{in situ} reference. The expression for $N^2$ 
+ depends on the type of equation of state used (\np{neos} namelist parameter).
+
+For \np{neos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} 
+polynomial expression is used (with the pressure in decibar approximated by 
+the depth in meters): 
+\begin{equation} \label{Eq_tra_bn2}
+N^2 = \frac{g}{e_{3w}} \; \beta   \ 
+	   \left(  \alpha / \beta \ \delta_{k+1/2}[T]     - \delta_{k+1/2}[S]   \right) 
+\end{equation} 
+where $\alpha$ ($\beta$) is the thermal (haline) expansion coefficient. 
+They are a function of  
+$\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$, 
+and  $z_w$, with $T$ the \textit{potential} temperature and 
+$\widetilde{S}$ a salinity anomaly. 
+Note that both $\alpha$ and $\beta$ depend on \textit{potential} 
+temperature and salinity which are averaged at $w$-points prior 
+to the computation instead of being computed at $T$-points and 
+then averaged to $w$-points.
+
+When a linear equation of state is used (\np{neos}=1 or 2, 
+\eqref{Eq_tra_bn2} reduces to:
+\begin{equation} \label{Eq_tra_bn2_linear}
+N^2 =	\frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right)
+\end{equation} 
+where $\alpha$ and $\beta $ are the constant coefficients used to 
+defined the linear equation of state \eqref{Eq_tra_eos_linear}.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Specific Heat
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Specific Heat (\textit{phycst})]
+			{Specific Heat (\mdl{phycst})}
+\label{TRA_adv_ldf}
+
+The specific heat of sea water, $C_p$, is a function of temperature, salinity 
+and pressure \citep{UNESCO1983}. It is only used in the model to convert 
+surface heat fluxes into surface temperature increase and so the pressure 
+dependence is neglected. The dependence on $T$ and $S$ is weak. 
+For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ 
+when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has 
+been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$. 
+Its value is set in \mdl{phycst} module. 
+
+%%%
+\gmcomment{ STEVEN:  consistency, no other computer variable names are 
+supplied, so why this one}
+%%%
+
+% -------------------------------------------------------------------------------------------------------------
+%        Freezing Point of Seawater
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Freezing Point of Seawater (\textit{ocfzpt})]
+			{Freezing Point of Seawater (\mdl{ocfzpt})}
+\label{TRA_fzp}
+
+The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}:
+\begin{equation} \label{Eq_tra_eos_fzp}
+   \begin{split}
+T_f (S,p) &= \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} 
+						     -  2.154996 \;10^{-4} \,S  \right) \ S    \\
+               & - 7.53\,10^{-3}\,p 
+   \end{split}
+\end{equation}
+
+\eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of 
+sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent 
+terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The \textit{before} 
+and \textit{now} surface freezing point is introduced in the code as $fzptb$ and 
+$fzptn$ 2D arrays together with a  \textit{now} mask (\textit{freezn}) which takes 
+the value 0 or 1 depending on whether the ocean temperature is above or at the 
+freezing point. Caution: do not confuse \textit{freezn} with the fraction of lead 
+(\textit{frld}) defined in LIM.  
+
+%%%
+\gmcomment{STEVEN: consistency, not many computer variable names are supplied, so why these    ===>  gm  I agree   this should evolve both here and in the code itself}
+%%%
+
+% ================================================================
+% Horizontal Derivative in zps-coordinate 
+% ================================================================
+\section  [Horizontal Derivative in \textit{zps}-coordinate (\textit{zpshde})]
+		{Horizontal Derivative in \textit{zps}-coordinate (\mdl{zpshde})}
+\label{TRA_zpshde}
+
+\gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"}
+
+With partial bottom cells (\np{ln\_zps}=.true.), in general, tracers in horizontally 
+adjacent cells live at different depths. Horizontal gradients of tracers are needed 
+for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure 
+gradient (\mdl{dynhpg} module) to be active. 
+\gmcomment{STEVEN from gm : question: not sure of  what -to be active- means}
+Before taking horizontal gradients between the tracers next to the bottom, a linear 
+interpolation in the vertical is used to approximate the deeper tracer as if it actually 
+lived at the depth of the shallower tracer point (Fig.~\ref{Fig_Partial_step_scheme}). 
+For example, for temperature in the $i$-direction the needed interpolated 
+temperature, $\widetilde{T}$, is:
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!p] \label{Fig_Partial_step_scheme}  \begin{center}
+\includegraphics[width=0.9\textwidth]{./Figures/Partial_step_scheme.pdf}
+\caption{ Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate (\np{ln\_zps}=.true.) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$. A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. The horizontal difference is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$.  }
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{equation*}
+\widetilde{T}= \left\{  \begin{aligned}  
+&T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta _k T^{i+1}	
+ 								&& \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   } 	\\
+										\\
+&T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta _k T^{i+1}
+			 					&& \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
+            \end{aligned}   \right.
+\end{equation*}
+and the resulting forms for the horizontal difference and the horizontal average 
+value of $T$ at a $U$-point are: 
+\begin{equation} \label{Eq_zps_hde}
+\begin{aligned}
+ \delta _{i+1/2} T= 	\begin{cases}
+\ \ \ \widetilde {T}\quad\ -T^i	 	& \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\
+										\\
+\ \ \ T^{\,i+1}-\widetilde{T}		& \ \ \quad\quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
+            		\end{cases}     \\
+\\
+\overline {T}^{\,i+1/2} \ = 	\begin{cases}
+( \widetilde {T}\ \ \;\,-T^{\,i})	 / 2	& \;\ \ \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\
+										\\
+( T^{\,i+1}-\widetilde{T} ) / 2		& \;\ \ \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
+            \end{cases}
+\end{aligned}
+\end{equation}
+
+The computation of horizontal derivative of tracers as well as of density is 
+performed once for all at each time step in \mdl{zpshde} module and stored 
+in shared arrays to be used when needed. It has to be emphasized that the 
+procedure used to compute the interpolated density, $\widetilde{\rho}$, is not 
+the same as that used for $T$ and $S$. Instead of forming a linear approximation 
+of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ 
+and $S$, and the pressure at a $u$-point (in the equation of state pressure is 
+approximated by depth, see \S\ref{TRA_eos} ) : 
+\begin{equation} \label{Eq_zps_hde_rho}
+\widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) 
+\quad \text{where }\  z_u = \min \left( {z_T^{i+1} ,z_T^i } \right)
+\end{equation} 
+
+This is a much better approximation as the variation of $\rho$ with depth (and 
+thus pressure) is highly non-linear with a true equation of state and thus is badly 
+approximated with a linear interpolation. This approximation is used to compute 
+both the horizontal pressure gradient (\S\ref{DYN_hpg}) and the slopes of neutral 
+surfaces (\S\ref{LDF_slp})
+
+Note that in almost all the advection schemes presented in this Chapter, both 
+averaging and differencing operators appear. Yet \eqref{Eq_zps_hde} has not 
+been used in these schemes: in contrast to diffusion and pressure gradient 
+computations, no correction for partial steps is applied for advection. The main 
+motivation is to preserve the domain averaged mean variance of the advected 
+field when using the $2^{nd}$ order centred scheme. Sensitivity of the advection 
+schemes to the way horizontal averages are performed in the vicinity of partial 
+cells should be further investigated in the near future.
+%%%
+\gmcomment{gm :   this last remark has to be done}
+%%%
Index: trunk/DOC/TexFiles/Chapters/Chap_ZDF.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Chap_ZDF.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Chap_ZDF.tex	(revision 994)
@@ -0,0 +1,603 @@
+% ================================================================
+% Chapter Ñ Vertical Ocean Physics (ZDF)
+% ================================================================
+\chapter{Vertical Ocean Physics (ZDF)}
+\label{ZDF}
+\minitoc
+
+%gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN.
+\gmcomment{Steven remark : problem here with turbulent vs turbulence.  I've changed "turbulent closure" to "turbulence closure", "turbulent mixing length" to "turbulence mixing length", but I've left "turbulent kinetic energy" alone - though I think it is an historical abberation!
+Gurvan :  I kept "turbulent closure"...}
+\gmcomment{Steven bis : parameterization is the american spelling, parameterisation is the british}
+
+
+% ================================================================
+% Vertical Mixing
+% ================================================================
+\section{Vertical Mixing}
+\label{ZDF_zdf}
+
+The discrete form of the ocean subgrid scale physics has been presented in 
+\S\ref{TRA_zdf} and \S\ref{DYN_zdf}. At the surface and bottom boundaries, 
+the turbulent fluxes of momentum, heat and salt have to be defined. At the 
+surface they are prescribed from the surface forcing (see Chap.~\ref{SBC}), 
+while at the bottom they are set to zero for heat and salt, unless a geothermal 
+flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl} 
+defined, see \S\ref{TRA_bbc}), and specified through a bottom friction 
+parameterization for momentum (see \S\ref{ZDF_bfr}).
+
+In this section we briefly discuss the various choices offered to compute 
+the vertical eddy viscosity and diffusivity coefficients, $A_u^{vm}$ , 
+$A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- 
+points, respectively (see \S\ref{TRA_zdf} and \S\ref{DYN_zdf}). These 
+coefficients can be assumed to be either constant, or a function of the local 
+Richardson number, or computed from a turbulent closure model (either 
+TKE or KPP formulation). The computation of these coefficients is initialized 
+in the \mdl{zdfini} module and performed in the \mdl{zdfric}, \mdl{zdftke} or 
+\mdl{zdfkpp} modules. The trends due to the vertical momentum and tracer 
+diffusion, including the surface forcing, are computed and added to the 
+general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively. 
+These trends can be computed using either a forward time stepping scheme 
+(namelist parameter \np{np\_zdfexp}=true) or a backward time stepping 
+scheme (\np{np\_zdfexp}=false) depending on the magnitude of the mixing 
+coefficients, and thus of the formulation used (see \S\ref{DOM_nxt}).
+
+% -------------------------------------------------------------------------------------------------------------
+%        Constant 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Constant (\key{zdfcst})}
+\label{ZDF_cst}
+%--------------------------------------------namzdf---------------------------------------------------------
+\namdisplay{namzdf}
+%--------------------------------------------------------------------------------------------------------------
+
+When \key{zdfcst} is defined, the momentum and tracer vertical eddy coefficients 
+are set to constant values over the whole ocean. This is the crudest way to define 
+the vertical ocean physics. It is recommended that this option is only used in 
+process studies, not in basin scale simulations. Typical values used in this case are:
+\begin{align*} 
+A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1} 	\\
+\\
+A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1}
+\end{align*}
+
+These values are set through the \np{avm0} and \np{avt0} namelist parameters. 
+In all cases, do not use values smaller that those associated with the molecular 
+viscosity and diffusivity, that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, 
+$\sim10^{-7}~m^2.s^{-1}$ for temperature and $\sim10^{-9}~m^2.s^{-1}$ for salinity.
+
+
+% -------------------------------------------------------------------------------------------------------------
+%        Richardson Number Dependent
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Richardson Number Dependent (\key{zdfric})}
+\label{ZDF_ric}
+
+%--------------------------------------------namric---------------------------------------------------------
+\namdisplay{namric}
+%--------------------------------------------------------------------------------------------------------------
+
+When \key{zdfric} is defined, a local Richardson number dependent formulation 
+for the vertical momentum and tracer eddy coefficients is set. The vertical mixing 
+coefficients are diagnosed from the large scale variables computed by the model. 
+\textit{In situ} measurements have been used to link vertical turbulent activity to 
+large scale ocean structures. The hypothesis of a mixing mainly maintained by the 
+growth of Kelvin-Helmholtz like instabilities leads to a dependency between the 
+vertical turbulence eddy coefficients and the local Richardson number ($i.e.$ the 
+ratio of stratification to vertical shear). Following \citet{PacPhil1981}, the following 
+formulation has been implemented:
+\begin{equation} \label{Eq_zdfric}
+   \left\{      \begin{aligned}
+         A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\
+         \\
+         A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm}
+   \end{aligned}    \right.
+\end{equation}
+where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson 
+number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}), 
+$A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the 
+constant case (see \S\ref{ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ 
+is the maximum value that can be reached by the coefficient when $Ri\leq 0$, 
+$a=5$ and $n=2$. The last three values can be modified by setting the 
+\np{avmri}, \np{alp} and \np{nric} namelist parameters, respectively.
+
+% -------------------------------------------------------------------------------------------------------------
+%        TKE Turbulent Closure Scheme 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{TKE Turbulent Closure Scheme (\key{zdftke})}
+\label{ZDF_tke}
+
+%--------------------------------------------namtke---------------------------------------------------------
+\namdisplay{nam_tke}
+%--------------------------------------------------------------------------------------------------------------
+
+The vertical eddy viscosity and diffusivity coefficients are computed from a TKE 
+turbulent closure model based on a prognostic equation for $\bar {e}$, the turbulent 
+kinetic energy, and a closure assumption for the turbulence length scales. This 
+turbulent closure model has been developed by \citet{Bougeault1989} in the 
+atmospheric case, adapted by \citet{Gaspar1990} for the oceanic case, and 
+embedded in OPA by \citet{Blanke1993} for equatorial Atlantic simulations. Since 
+then, significant modifications have been introduced by \citet{Madec1998} in both 
+the implementation and the formulation of the mixing length scale. The time 
+evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical 
+shear, its destruction through stratification, its vertical diffusion, and its dissipation 
+of \citet{Kolmogorov1942} type:
+\begin{equation} \label{Eq_zdftke_e}
+\frac{\partial \bar{e}}{\partial t} = 
+\frac{A^{vm}}{e_3 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
+							+\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
+-A^{vT}\,N^2
++\frac{1}{e_3}	\;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
+				\;\frac{\partial \bar{e}}{\partial k}} \right]
+- c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon }
+\end{equation}
+\begin{equation} \label{Eq_zdftke_kz}
+   \begin{split}
+         A^{vm} &= C_k\  l_k\  \sqrt {\bar{e}}  	\\
+         A^{vT} &= A^{vm} / P_{rt}
+   \end{split}
+\end{equation}
+where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}), 
+$l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales, 
+$P_{rt} $ is the Prandtl number. The constants $C_k = \sqrt {2} /2$ and 
+$C_\epsilon = 0.1$ are designed to deal with vertical mixing at any depth 
+\citep{Gaspar1990}. They are set through namelist parameters \np{ediff} 
+and \np{ediss}. $P_{rt} $ can be set to unity or, following \citet{Blanke1993}, 
+be a function of the local Richardson number, $R_i $:
+\begin{align*} \label{Eq_prt}
+P_{rt} = \begin{cases}
+                    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$} 	\\
+                    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$} 	\\
+                    \ \ 10 &      \text{if $\ 2 \leq R_i$} 
+            \end{cases}
+\end{align*}
+Note that a horizontal Shapiro filter can optionally be applied to $R_i$. 
+However it is an obsolescent option that is not recommended.  
+The choice of $P_{rt} $ is controlled by the \np{npdl} namelist parameter.
+
+For computational efficiency, the original formulation of the turbulence length 
+scales proposed by \citet{Gaspar1990} has been simplified. Four formulations 
+are proposed, the choice of which is controlled by the \np{nmxl} namelist 
+parameter. The first two are based on the following first order approximation 
+\citep{Blanke1993}:
+\begin{equation} \label{Eq_tke_mxl0_1}
+l_k = l_\epsilon = \sqrt {2 \bar e} / N
+\end{equation}
+which is valid in a stable stratified region with constant values of the brunt-
+Vais\"{a}l\"{a} frequency. The resulting length scale is bounded by the distance 
+to the surface or to the bottom (\np{nmxl}=0) or by the local vertical scale factor (\np{nmxl}=1). \citet{Blanke1993} notice that this simplification has two major 
+drawbacks: it makes no sense for locally unstable stratification and the 
+computation no longer uses all the information contained in the vertical density 
+profile. To overcome these drawbacks, \citet{Madec1998} introduces the 
+\np{nmxl}=2 or 3 cases, which add an extra assumption concerning the vertical 
+gradient of the computed length scale. So, the length scales are first evaluated 
+as in \eqref{Eq_tke_mxl0_1} and then bounded such that:
+\begin{equation} \label{Eq_tke_mxl_constraint}
+\frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
+\qquad \text{with }\  l =  l_k = l_\epsilon
+\end{equation}
+\eqref{Eq_tke_mxl_constraint} means that the vertical variations of the length 
+scale cannot be larger than the variations of depth. It provides a better 
+approximation of the \citet{Gaspar1990} formulation while being much less 
+time consuming. In particular, it allows the length scale to be limited not only 
+by the distance to the surface or to the ocean bottom but also by the distance 
+to a strongly stratified portion of the water column such as the thermocline 
+(Fig.~\ref{Fig_mixing_length}). In order to impose the \eqref{Eq_tke_mxl_constraint} 
+constraint, we introduce two additional length scales: $l_{up}$ and $l_{dwn}$, 
+the upward and downward length scales, and evaluate the dissipation and 
+mixing turbulence length scales as (and note that here we use numerical 
+indexing):
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_mixing_length}  \begin{center}
+\includegraphics[width=1.00\textwidth]{./Figures/Fig_mixing_length.pdf}
+\caption {Illustration of the mixing length computation. }
+\end{center}  
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{equation} \label{Eq_tke_mxl2}
+\begin{aligned}
+ l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3T}^{(k)}\ \ \ \;  \right)
+    \quad &\text{ from $k=1$ to $jpk$ }\ \\
+ l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3T}^{(k-1)}  \right)   
+    \quad &\text{ from $k=jpk$ to $1$ }\ \\
+\end{aligned}
+\end{equation}
+where $l^{(k)}$ is computed using \eqref{Eq_tke_mxl0_1}, 
+$i.e.$ $l^{(k)} = \sqrt {2 \bar e^{(k)} / N^{(k)} }$.
+
+In the \np{nmxl}=2 case, the dissipation and mixing length scales take the same 
+value: $ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the 
+\np{nmxl}=2 case, the dissipation and mixing turbulence length scales are give 
+as in \citet{Gaspar1990}:
+\begin{equation} \label{Eq_tke_mxl_gaspar}
+\begin{aligned}
+& l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }  	\\
+& l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right) 
+\end{aligned}
+\end{equation}
+
+At the sea surface the value of $\bar{e}$ is prescribed from the wind 
+stress field: $\bar{e}=ebb\;\left| \tau \right|$ ($ebb=60$ by default) 
+with a minimal threshold of $emin0=10^{-4}~m^2.s^{-2}$ (namelist
+parameters). Its value at the bottom of the ocean is assumed to be 
+equal to the value of the level just above. The time integration of the 
+$\bar{e}$ equation may formally lead to negative values because the 
+numerical scheme does not ensure its positivity. To overcome this 
+problem, a cut-off in the minimum value of $\bar{e}$ is used. Following 
+\citet{Gaspar1990}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. 
+This allows the subsequent formulations to match that of\citet{Gargett1984} 
+for the diffusion in the thermocline and deep ocean :  $(A^{vT} = 10^{-3} / N)$. 
+In addition, a cut-off is applied on $A^{vm}$ and $A^{vT}$ to avoid numerical 
+instabilities associated with too weak vertical diffusion. They must be 
+specified at least larger than the molecular values, and are set through 
+\textit{avm0} and \textit{avt0} (\textbf{namelist} parameters).
+
+% -------------------------------------------------------------------------------------------------------------
+%        K Profile Parametrisation (KPP) 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{K Profile Parametrisation (KPP) (\key{zdfkpp}) }
+\label{ZDF_kpp}
+
+%--------------------------------------------namkpp--------------------------------------------------------
+\namdisplay{namkpp}
+%--------------------------------------------------------------------------------------------------------------
+
+The KKP scheme has been implemented by J. Chanut ...
+
+\colorbox{yellow}{Add a description of KPP here.}
+
+
+% ================================================================
+% Convection
+% ================================================================
+\section{Convection}
+\label{ZDF_conv}
+
+%--------------------------------------------namzdf--------------------------------------------------------
+\namdisplay{namzdf}
+%--------------------------------------------------------------------------------------------------------------
+
+Static instabilities (i.e. light potential densities under heavy ones) may 
+occur at particular ocean grid points. In nature, convective processes 
+quickly re-establish the static stability of the water column. These 
+processes have been removed from the model via the hydrostatic 
+assumption so they must be parameterized. Three parameterizations 
+are available to deal with convective processes: a non-penetrative 
+convective adjustment or an enhanced vertical diffusion, or/and the 
+use of a turbulent closure scheme.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Non-Penetrative Convective Adjustment 
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Non-Penetrative Convective Adjustment (\np{ln\_tranpc}) ]
+			{Non-Penetrative Convective Adjustment (\np{ln\_tranpc}=.true.) }
+\label{ZDF_npc}
+
+%--------------------------------------------namnpc--------------------------------------------------------
+\namdisplay{namnpc}
+%--------------------------------------------------------------------------------------------------------------
+
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!htb] \label{Fig_npc}  	\begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_npc.pdf}
+\caption {Example of an unstable density profile treated by the non penetrative 
+convective adjustment algorithm. $1^{st}$ step: the initial profile is checked from 
+the surface to the bottom. It is found to be unstable between levels 3 and 4. 
+They are mixed. The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 
+are mixed. The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are 
+mixed. The $1^{st}$ step ends since the density profile is then stable below 
+the level 3. $2^{nd}$ step: the new $\rho$ profile is checked following the same 
+procedure as in $1^{st}$ step: levels 2 to 5 are mixed. The new density profile 
+is checked. It is found stable: end of algorithm.}
+\end{center}  	\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}=true. 
+It is applied at each \np{nnpc1} time step and mixes downwards instantaneously 
+the statically unstable portion of the water column, but only until the density 
+structure becomes neutrally stable ($i.e.$ until the mixed portion of the water 
+column has \textit{exactly} the density of the water just below) \citep{Madec1991a}. 
+The associated algorithm is an iterative process used in the following way 
+(Fig. \ref{Fig_npc}): starting from the top of the ocean, the first instability is 
+found. Assume in the following that the instability is located between levels 
+$k$ and $k+1$. The potential temperature and salinity in the two levels are 
+vertically mixed, conserving the heat and salt contents of the water column. 
+The new density is then computed by a linear approximation. If the new 
+density profile is still unstable between levels $k+1$ and $k+2$, levels $k$, 
+$k+1$ and $k+2$ are then mixed. This process is repeated until stability is 
+established below the level $k$ (the mixing process can go down to the 
+ocean bottom). The algorithm is repeated to check if the density profile 
+between level $k-1$ and $k$ is unstable and/or if there is no deeper instability.
+
+This algorithm is significantly different from mixing statically unstable levels 
+two by two. The latter procedure cannot converge with a finite number 
+of iterations for some vertical profiles while the algorithm used in \NEMO 
+converges for any profile in a number of iterations which is less than the 
+number of vertical levels. This property is of paramount importance as 
+pointed out by \citet{Killworth1989}: it avoids the existence of permanent 
+and unrealistic static instabilities at the sea surface. This non-penetrative 
+convective algorithm has been proved successful in studies of the deep 
+water formation in the north-western Mediterranean Sea 
+\citep{Madec1991a, Madec1991b, Madec1991c}.
+
+Note that in the current implementation of this algorithm presents several 
+limitations. First, potential density referenced to the sea surface is used to 
+check whether the density profile is stable or not. This is a strong 
+simplification which leads to large errors for realistic ocean simulations. 
+Indeed, many water masses of the world ocean, especially Antarctic Bottom
+Water, are unstable when represented in surface-referenced potential density. 
+The scheme will erroneously mix them up. Second, the mixing of potential 
+density is assumed to be linear. This assures the convergence of the algorithm 
+even when the equation of state is non-linear. Small static instabilities can thus 
+persist due to cabbeling: they will be treated at the next time step. 
+Third, temperature and salinity, and thus density, are mixed, but the 
+corresponding velocity fields remain unchanged. When using a Richardson 
+Number dependent eddy viscosity, the mixing of momentum is done through 
+the vertical diffusion: after a static adjustment, the Richardson Number is zero 
+and thus the eddy viscosity coefficient is at a maximum. When this convective 
+adjustment algorithm is used with constant vertical eddy viscosity, spurious 
+solutions can occur since the vertical momentum diffusion remains small even 
+after a static adjustment. In that case, we recommend the addition of momentum 
+mixing in a manner that mimics the mixing in temperature and salinity 
+\citep{Speich1992, Speich1996}.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Enhanced Vertical Diffusion 
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Enhanced Vertical Diffusion (\np{ln\_zdfevd})]
+			{Enhanced Vertical Diffusion (\np{ln\_zdfevd}=.true.)}
+\label{ZDF_evd}
+
+%--------------------------------------------namzdf--------------------------------------------------------
+\namdisplay{namzdf}
+%--------------------------------------------------------------------------------------------------------------
+
+The enhanced vertical diffusion parameterization is used when \np{ln\_zdfevd}=true. 
+In this case, the vertical eddy mixing coefficients are assigned very large values 
+(a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable 
+($i.e.$ when the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{Lazar1997, Lazar1999}. 
+This is done either on tracers only (\np{n\_evdm}=0) or on both momentum and 
+tracers (\np{n\_evdm}=1).
+
+In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and 
+if \np{n\_evdm}=1, the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ 
+values also, are set equal to the namelist parameter \np{avevd}. A typical value 
+for $avevd$ is between 1 and $100~m^2.s^{-1}$. This parameterization of 
+convective processes is less time consuming than the convective adjustment 
+algorithm presented above when mixing both tracers and momentum in the 
+case of static instabilities. It requires the use of an implicit time stepping on 
+vertical diffusion terms (i.e. \np{ln\_zdfexp}=false). 
+
+% -------------------------------------------------------------------------------------------------------------
+%       Turbulent Closure Scheme 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Turbulent Closure Scheme (\key{zdftke})}
+\label{ZDF_tcs}
+
+The TKE turbulent closure scheme presented in \S\ref{ZDF_tke} and used 
+when the \key{zdftke} is defined, in theory solves the problem of statically 
+unstable density profiles. In such a case, the term corresponding to the 
+destruction of turbulent kinetic energy through stratification in \eqref{Eq_zdftke_e} 
+becomes a source term, since $N^2$ is negative. It results in large values of 
+$A_T^{vT}$ and  $A_T^{vT}$, and also the four neighbouring 
+$A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1})$. These large values 
+restore the static stability of the water column in a way similar to that of the 
+enhanced vertical diffusion parameterization (\S\ref{ZDF_evd}). However, 
+in the vicinity of the sea surface (first ocean layer), the eddy coefficients 
+computed by the turbulence scheme do not usually exceed $10^{-2}m.s^{-1}$, 
+because the mixing length scale is bounded by the distance to the sea surface 
+(see \S\ref{ZDF_tke}). It can thus be useful to combine the enhanced vertical 
+diffusion with the turbulent closure scheme, $i.e.$ setting the \np{ln\_zdfnpc} 
+namelist parameter to true and defining the \key{zdftke} CPP key all together.
+
+The KPP turbulent closure scheme already includes enhanced vertical diffusion 
+in the case of convection, as governed by the variables $bvsqcon$ and $difcon$ 
+found in \mdl{zdfkpp}, therefore \np{np\_zdfevd} should not be used with the KPP 
+scheme. %gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
+
+% ================================================================
+% Double Diffusion Mixing
+% ================================================================
+\section  [Double Diffusion Mixing (\textit{zdfddm} - \key{zdfddm})]
+		{Double Diffusion Mixing (\mdl{zdfddm} module - \key{zdfddm})}
+\label{ZDF_ddm}
+
+%-------------------------------------------namddm--------------------------------------------------------
+\namdisplay{namddm}
+%--------------------------------------------------------------------------------------------------------------
+
+Double diffusion occurs when relatively warm, salty water overlies cooler, fresher 
+water, or vice versa. The former condition leads to salt fingering and the latter 
+to diffusive convection. Double-diffusive phenomena contribute to diapycnal 
+mixing in extensive regions of the ocean.  \citet{Merryfield1999} include a 
+parameterization of such phenomena in a global ocean model and show that 
+it leads to relatively minor changes in circulation but exerts significant regional 
+influences on temperature and salinity. 
+
+Diapycnal mixing of S and T are described by diapycnal diffusion coefficients 
+\begin{align*} % \label{Eq_zdfddm_Kz}
+    &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT}	\\
+    &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
+\end{align*}
+where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection, 
+and $o$ by processes other than double diffusion. The rates of double-diffusive mixing depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$, 
+where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline 
+contraction (see \S\ref{TRA_eos}). To represent mixing of $S$ and $T$ by salt 
+fingering, we adopt the diapycnal diffusivities suggested by Schmitt (1981):
+\begin{align} \label{Eq_zdfddm_f}
+A_f^{vS} &= 	\begin{cases}
+	\frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
+	0 				  					    &\text{otherwise} 
+				\end{cases}   
+\\ 		    \label{Eq_zdfddm_f_T}
+A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho 
+\end{align}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_zdfddm}  \begin{center}
+\includegraphics[width=0.99\textwidth]{./Figures/Fig_zdfddm.pdf}
+\caption {From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ 
+and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy 
+curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves 
+$A^{\ast v} = 10^{-4}~m^2.s^{-1}$ ; (b) diapycnal diffusivities $A_d^{vT}$ and 
+$A_d^{vS}$ for temperature and salt in regions of diffusive convection. Heavy 
+curves denote the Federov parameterization and thin curves the Kelley 
+parameterization. The latter is not implemented in \NEMO. }
+\end{center}    \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The factor 0.7 in \eqref{Eq_zdfddm_f_T} reflects the measured ratio 
+$\alpha F_T /\beta F_S \approx  0.7$ of buoyancy flux of heat to buoyancy 
+flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}). Following  \citet{Merryfield1999}, 
+we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
+
+To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by Federov (1988) is used: 
+\begin{align} 	\label{Eq_zdfddm_d}
+A_d^{vT} &= 	\begin{cases}
+	1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
+									&\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
+	0 								&\text{otherwise} 
+				\end{cases}   
+\\   			\label{Eq_zdfddm_d_S}
+A_d^{vS} &= 	\begin{cases}
+	A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right)
+									&\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
+	A_d^{vT} \ 0.15 \ R_\rho
+									&\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
+	0 								&\text{otherwise} 
+				\end{cases}   
+\end{align}
+
+The dependencies of \eqref{Eq_zdfddm_f} to \eqref{Eq_zdfddm_d_S} on $R_\rho$ 
+are illustrated in Fig.~\ref{Fig_zdfddm}. Implementing this requires computing 
+$R_\rho$ at each grid point on every time step. This is done in \mdl{eosbn2} at the 
+same time as $N^2$ is computed. This avoids duplication in the computation of 
+$\alpha$ and $\beta$ (which is usually quite expensive).
+
+% ================================================================
+% Bottom Friction
+% ================================================================
+\section  [Bottom Friction (\textit{zdfbfr})]
+		{Bottom Friction (\mdl{zdfbfr} module)}
+\label{ZDF_bfr}
+
+%--------------------------------------------nambfr--------------------------------------------------------
+\namdisplay{nambfr}
+%--------------------------------------------------------------------------------------------------------------
+
+Both the surface momentum flux (wind stress) and the bottom momentum 
+flux (bottom friction) enter the equations as a condition on the vertical 
+diffusive flux. For the bottom boundary layer, one has:
+\begin{equation} \label{Eq_zdfbfr_flux}
+A^{vm} \left( \partial \textbf{U}_h / \partial z \right) = \textbf{F}_h
+\end{equation}
+where $\textbf{F}_h$ is supposed to represent the horizontal momentum flux 
+outside the logarithmic turbulent boundary layer (thickness of the order of 
+1~m in the ocean). How $\textbf{F}_h$ influences the interior depends on the 
+vertical resolution of the model near the bottom relative to the Ekman layer 
+depth. For example, in order to obtain an Ekman layer depth 
+$d = \sqrt{2\;A^{vm}} / f = 50$~m, one needs a vertical diffusion coefficient 
+$A^{vm} = 0.125$~m$^2$s$^{-1}$ (for a Coriolis frequency 
+$f = 10^{-4}$~m$^2$s$^{-1}$). With a background diffusion coefficient 
+$A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m. 
+When the vertical mixing coefficient is this small, using a flux condition is 
+equivalent to entering the viscous forces (either wind stress or bottom friction) 
+as a body force over the depth of the top or bottom model layer. To illustrate 
+this, consider the equation for $u$ at $k$, the last ocean level:
+\begin{equation} \label{Eq_zdfbfr_flux2}
+\frac{\partial u \; (k)}{\partial t} = \frac{1}{e_{3u}} \left[ A^{vm} \; (k) \frac{U \; (k-1) - U \; (k)}{e_{3uw} \; (k-1)} - F_u \right] \approx - \frac{F_u}{e_{3u}}
+\end{equation}
+For example, if the bottom layer thickness is 200~m, the Ekman transport will 
+be distributed over that depth. On the other hand, if the vertical resolution 
+is high (1~m or less) and a turbulent closure model is used, the turbulent 
+Ekman layer will be represented explicitly by the model. However, the 
+logarithmic layer is never represented in current primitive equation model 
+applications: it is \emph{necessary} to parameterize the flux $\textbf{F}_h $. 
+Two choices are available in \NEMO: a linear and a quadratic bottom friction. 
+Note that in both cases, the rotation between the interior velocity and the 
+bottom friction is neglected in the present release of \NEMO.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Linear Bottom Friction
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Linear Bottom Friction (\np{nbotfr} = 1) }
+\label{ZDF_bfr_linear}
+
+The linear bottom friction parameterization assumes that the bottom friction 
+is proportional to the interior velocity (i.e. the velocity of the last model level):
+\begin{equation} \label{Eq_zdfbfr_linear}
+\textbf{F}_h = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
+\end{equation}
+where $\textbf{U}_h^b$ is the horizontal velocity vector of the bottom ocean 
+layer and $r$ is a friction coefficient expressed in m.s$^{-1}$. This coefficient 
+is generally estimated by setting a typical decay time $\tau$ in the deep ocean, 
+and setting $r = H / \tau$, where $H$ is the ocean depth. Commonly accepted 
+values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly1984}. 
+A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used 
+in quasi-geostrophic models. One may consider the linear friction as an 
+approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ (\citet{Gill1982}, 
+Eq. 9.6.6). For example, with a drag coefficient $C_D = 0.002$, a typical speed 
+of tidal currents of $U_{av} =0.1$~m.s$^{-1}$, and assuming an ocean depth 
+$H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m.s$^{-1}$. 
+This is the default value used in \NEMO. It corresponds to a decay time scale 
+of 115~days. It can be changed by specifying \np{bfric1} (namelist parameter).
+
+In the code, the bottom friction is imposed by updating the value of the 
+vertical eddy coefficient at the bottom level. Indeed, the discrete formulation 
+of (\ref{Eq_zdfbfr_linear}) at the last ocean $T-$level, using the fact that 
+$\textbf {U}_h =0$ below the ocean floor, leads to
+\begin{equation} \label{Eq_zdfbfr_linKz}
+\begin{split}
+A_u^{vm} &= r\;e_{3uw}\\
+A_v^{vm} &= r\;e_{3vw}\\
+\end{split}
+\end{equation}
+
+This update is done in \mdl{zdfbfr} when \np{nbotfr}=1. The value of $r$ 
+used is \np{bfric1}. Setting \np{nbotfr}=3 is equivalent to setting $r=0$ and 
+leads to a free-slip bottom boundary condition. Setting \np{nbotfr}=0 sets 
+$r=2\;A_{vb}^{\rm {\bf U}} $, where $A_{vb}^{\rm {\bf U}} $ is the background 
+vertical eddy coefficient, and a no-slip boundary condition is imposed. 
+Note that this latter choice generally leads to an underestimation of the 
+bottom friction: for example with a deepest level thickness of $200~m$ 
+and $A_{vb}^{\rm {\bf U}} =10^{-4}$m$^2$.s$^{-1}$, the friction coefficient 
+is only $r=10^{-6}$m.s$^{-1}$.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Non-Linear Bottom Friction
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Non-Linear Bottom Friction (\np{nbotfr} = 2)}
+\label{ZDF_bfr_nonlinear}
+
+The non-linear bottom friction parameterization assumes that the bottom 
+friction is quadratic: 
+\begin{equation} \label{Eq_zdfbfr_nonlinear}
+\textbf {F}_h = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 
+}{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b 
+\end{equation}
+
+with $\textbf{U}_h^b = (u_b\;,\;v_b)$ the horizontal interior velocity ($i.e.$ 
+the horizontal velocity of the bottom ocean layer), $C_D$ a drag coefficient, 
+and $e_b $ a bottom turbulent kinetic energy due to tides, internal waves 
+breaking and other short time scale currents. A typical value of the drag 
+coefficient is $C_D = 10^{-3} $. As an example, the CME experiment 
+\citep{Treguier1992} uses $C_D = 10^{-3}$ and $e_b = 2.5\;10^{-3}$m$^2$.s$^{-2}$, 
+while the FRAM experiment \citep{Killworth1992} uses $e_b =0$ 
+and $e_b =2.5\;\;10^{-3}$m$^2$.s$^{-2}$. The FRAM choices have been 
+set as default values (\np{bfric2} and \np{bfeb2} namelist parameters).
+
+As for the linear case, the bottom friction is imposed in the code by 
+updating the value of the vertical eddy coefficient at the bottom level:
+\begin{equation} \label{Eq_zdfbfr_nonlinKz}
+\begin{split}
+A_u^{vm} &=C_D\; e_{3uw} \left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^
+{1/2}\\
+A_v^{vm} &=C_D\; e_{3uw} \left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\
+\end{split}
+\end{equation}
+
+This update is done in \mdl{zdfbfr}. The coefficients that control the strength of the 
+non-linear bottom friction are initialized as namelist parameters: $C_D$= \np{bfri2}, and $e_b$ =\np{bfeb2}.
+
+% ================================================================
Index: trunk/DOC/TexFiles/Chapters/Introduction.tex
===================================================================
--- trunk/DOC/TexFiles/Chapters/Introduction.tex	(revision 994)
+++ trunk/DOC/TexFiles/Chapters/Introduction.tex	(revision 994)
@@ -0,0 +1,142 @@
+
+% ================================================================
+% INTRODUCTION
+% ================================================================
+
+\chapter{Introduction}
+
+The Nucleus for European Modelling of the Ocean (\NEMO) is a framework of ocean 
+related engines, namely OPA\footnote{OPA = Oc\'{e}an PArall\'{e}lis\'{e}} for the 
+ocean dynamics and thermodynamics, LIM\footnote{LIM= Louvain)la-neuve Ice 
+Model} for the sea-ice dynamics and thermodynamics, TOP\footnote{TOP = Tracer 
+in the Ocean Paradigm} for the biogeochemistry (both transport (TRP) and sources 
+minus sinks (LOBSTER, PISCES)\footnote{Both LOBSTER and PISCES are not 
+acronyms just name}. It is intended to be a flexible tool for studying the ocean and 
+its interactions with the other components of the earth climate system (atmosphere, 
+sea-ice, biogeochemical tracers, ...) over a wide range of space and time scales. 
+This documentation provides information about the physics represented by the ocean component of \NEMO and the rationale for the choice of numerical schemes and 
+the model design. More specific information about running the model on different 
+computers, or how to set up a configuration, are found on the \NEMO web site 
+(www.locean-ipsl.upmc.fr/NEMO). 
+
+The ocean component of \NEMO has been developed from the OPA model, release 8.2, described in \citet{Madec1998}. This model has been used for a wide range of applications, both regional or global, as a forced ocean model and as a model coupled with the atmosphere. A complete list of references is found on the \NEMO web site. 
+
+This manual is organised in as follows. Chapter~\ref{PE} presents the model basics, 
+$i.e.$ the equations and their assumptions, the vertical coordinates used, and the 
+subgrid scale physics. This part deals with the continuous equations of the model 
+(primitive equations, with potential temperature, salinity and an equation of state). The 
+equations are written in a curvilinear coordinate system, with a choice of vertical 
+coordinates ($z$ or $s$, with the rescaled height coordinate formulation \textit{z*}, or  
+\textit{s*}). Momentum equations are formulated in the vector invariant form or in the 
+flux form. Dimensional units in the meter, kilogram, second (MKS) international system 
+are used throughout.
+
+The following chapters deal with the discrete equations. Chapter~\ref{DOM} presents the 
+space and time domain. The model is discretised on a staggered grid (Arakawa C grid) 
+with masking of land areas and uses a Leap-frog environment for time-stepping. Vertical 
+discretisation used depends on both how the bottom topography is represented and 
+whether the free surface is linear or not. Full step or partial step $z$-coordinate or 
+$s$- (terrain-following) coordinate is used with linear free surface (level position are then 
+fixed in time). In non-linear free surface, the corresponding rescaled height coordinate 
+formulation (\textit{z*} or \textit{s*}) is used (the level position then vary in time as a 
+function of the sea surface heigh). The following two chapters (\ref{TRA} and \ref{DYN}) 
+describe the discretisation of the prognostic equations for the active tracers and the 
+momentum. Explicit, split-explicit and implicit free surface formulations are implemented 
+as well as rigid-lid case. A number of numerical schemes are available for momentum 
+advection, for the computation of the pressure gradients, as well as for the advection of 
+tracers (second or higher order advection schemes, including positive ones).
+
+Surface boundary conditions (chapter~\ref{SBC}) can be implemented as prescribed
+fluxes, or bulk formulations for the surface fluxes (wind stress, heat, freshwater). The 
+model allows penetration of solar radiation  There is an optional geothermal heating at 
+the ocean bottom. Within the \NEMO system the ocean model is interactively coupled 
+with a sea ice model (LIM) and with biogeochemistry models (PISCES, LOBSTER). 
+Interactive coupling to Atmospheric models is possible via the OASIS coupler 
+\citep{OASIS2006}. 
+
+Other model characteristics are the lateral boundary conditions (chapter~\ref{LBC}).  
+Global configurations of the model make use of the ORCA tripolar grid, with special north 
+fold boundary condition. Free-slip or no-slip boundary conditions are allowed at land 
+boundaries. Closed basin geometries as well as periodic domains and open boundary 
+conditions are possible. 
+
+Physical parameterisations are described in chapters~\ref{LDF} and \ref{ZDF}. The 
+model includes an implicit treatment of vertical viscosity and diffusivity. The lateral 
+Laplacian and biharmonic viscosity and diffusion can be rotated following a geopotential 
+or neutral direction. There is an optional eddy induced velocity \citep{Gent1990} with a 
+space and time variable coefficient \citet{Treguier1997}. The model has vertical harmonic 
+viscosity and diffusion with a space and time variable coefficient, with options to compute 
+the coefficients with \citet{Blanke1993}, \citet{Large1994}, or \citet{PacPhil1981} mixing 
+schemes.
+
+Specific online diagnostics (not documented yet) are available in the model: output of all 
+the tendencies of the momentum and tracers equations, output of tracers tendencies 
+averaged over the time evolving mixed layer.
+
+The model is implemented in \textsc{Fortran 90}, with preprocessing (C-pre-processor). 
+It runs under UNIX. It is optimized for vector computers and parallelised by domain  
+decomposition with MPI. All input and output is done in NetCDF (Network Common Data 
+Format) with a optional direct access format for output. To ensure the clarity and 
+readability of the code it is necessary to follow coding rules. The coding rules for OPA 
+include conventions for naming variables, with different starting letters for different types 
+of variables (real, integer, parameter\ldots). Those rules are presented in a document 
+available on the \NEMO web site.
+
+The model is organized with a high internal modularity based on physics. For example, 
+each trend ($i.e.$, a term in the RHS of the prognostic equation) for momentum and 
+tracers is computed in a dedicated module.  To make it easier for the user to find his way 
+around the code, the module names follow a three-letter rule. For example, \mdl{tradmp} 
+is a module related to the TRAcers equation, computing the DaMPing. The complete list 
+of module names is presented in \colorbox{yellow}{annex}. Furthermore, modules are  
+organized in a few directories that correspond to their category, as indicated by the first 
+three letters of their name. 
+
+The manual mirrors the organization of the model. After the presentation of the 
+continuous equations (Chapter \ref{PE}), the following chapters refer to specific terms of 
+the equations each associated with a group of modules .
+
+
+\begin{table}[htbp] \label{tab1}
+%\begin{center} \begin{tabular}{|p{143pt}|l|l|}	\hline
+\begin{center} \begin{tabular}{|l|l|l|}	\hline
+Chapter \ref{DOM}	& DOM		& model DOMain \\		\hline
+Chapter \ref{TRA}	& TRA		& TRAcer equations (potential temperature and salinity) \\   \hline
+Chapter \ref{DYN}	& DYN		& DYNamic equations (momentum) \\		\hline
+Chapter \ref{SBC} 	& SBC		& Surface Boundary Conditions \\			\hline
+Chapter \ref{LBC}	& LBC		& Lateral Boundary Conditions  \\ 		\hline
+Chapter \ref{LDF}	& LDF		& Lateral DiFfusion (parameterisations) \\	\hline
+Chapter \ref{ZDF}	& ZDF		& Vertical DiFfusion  \\		\hline
+Chapter \ref{MISC}	& ...		& Miscellaneous  topics	 \\			\hline
+\end{tabular} 	\end{center}
+\end{table}
+
+In the current release (v2.3), LBC directory (see Chap.~\ref{LBC}) does not yet exist. 
+When created LBC will contain the OBC directory (Open Boundary Condition), and the
+\mdl{lbclnk}, \mdl{mppini} and \mdl{lib\_mpp} modules. 
+
+ \vspace{1cm}   Nota Bene : \vspace{0.25cm}
+
+OPA, like all research tools, is in perpetual evolution. The present document describes 
+the OPA version include in the release 2.3 of NEMO. This release differs significantly
+from version 8, documented in \citet{Madec1998}. The main modifications are :\\
+(1) transition to full native \textsc{Fortran} 90, deep code restructuring and drastic reduction of CPP keys; \\
+(2) introduction of partial step representation of bottom topography \citep{Barnier_al_OD06}; \\
+(3) partial reactivation of a terrain-following vertical coordinate ($s$- and hybrid $s$-$z$) with the addition of several options for pressure gradient computation \footnote{Partial support of $s$-coordinate: there is presently no support for neutral physics in $s$-coordinate and for the new options for horizontal pressure gradient computation with a non-linear equation of state.}; \\ 
+(4) more choices for the treatment of the free surface: full explicit, split-explicit , filtered and rigid-lid; \\
+(5) non linear free surface option (associated with the rescaled height coordinate  \textit{z*} or  \textit{s*}); \\
+(6) additional schemes for vector and flux forms of the momentum  advection; \\
+(7) additional advection schemes for tracers; \\
+(8) implementation of the AGRIF package (Adaptative Grid Refinement in \textsc{Fortran}) \citep{Debreu_al_CG2008}; \\
+(9) online diagnostics : tracers trend in the mixed layer and vorticity balance; \\
+(10) rewriting of the I/O management; \\
+(11) OASIS 3 and 4 couplers interfacing with atmospheric global circulation models. 
+
+In addition, several minor modifications in the coding have been introduced with the constant concern of improving performance on both scalar and vector computers. 
+
+At the time of this writing, the current release is NEMO 2.3. The new surface module described in this document is not yet part of the current distribution.
+
+\vspace{1cm}  
+\colorbox{red}{Red color}: not in the current reference version (v2.3) but expected soon.
+ 
+\colorbox{yellow}{Yellow color}: missing references, text to be updated.
+
Index: trunk/DOC/TexFiles/NEMO.ist
===================================================================
--- trunk/DOC/TexFiles/NEMO.ist	(revision 994)
+++ trunk/DOC/TexFiles/NEMO.ist	(revision 994)
@@ -0,0 +1,3 @@
+delim_0 "\\dotfill"
+delim_1 "\\dotfill"
+delim_2 "\\dotfill"
Index: trunk/DOC/TexFiles/NEMO_book.tex
===================================================================
--- trunk/DOC/TexFiles/NEMO_book.tex	(revision 994)
+++ trunk/DOC/TexFiles/NEMO_book.tex	(revision 994)
@@ -0,0 +1,303 @@
+%description: Book template
+
+% template of document for LaTeX
+% (C) Xavier Perseguers 2002 - xavier.perseguers@epfl.ch
+
+\documentclass[a4paper,12pt]{book}
+
+%  makeindex NEMO_book     <== to regenerate the index
+%  bibtex         NEMO_book	<== to generate  the bibliography
+
+% ================================================================
+% HEADERS DEFINITION
+% ================================================================
+
+\usepackage[french]{babel}
+\usepackage{fancyhdr}
+
+\usepackage{alltt}  		%%  alltt for namelist
+\usepackage{verbatim}  	%%  alltt for namelist
+
+%hyperref  ===>>>  don't work.....
+%\usepackage[a4paper=true,ps2pdf=true,pagebackref=true,breaklinks=true]{hyperref}
+
+%\usepackage{amssymb}
+\pagestyle{fancy}
+% with this we ensure that the chapter and section
+% headings are in lowercase.
+\renewcommand{\chaptermark}[1]{\markboth{#1}{}}
+\renewcommand{\sectionmark}[1]{\markright{\thesection.\ #1}}
+\fancyhf{}             % delete current setting for header and footer
+\fancyhead[LE,RO]{\bfseries\thepage}
+\fancyhead[LO]{\bfseries\hspace{-0em}\rightmark}
+\fancyhead[RE]{\bfseries\leftmark}
+\renewcommand{\headrulewidth}{0.5pt}
+\renewcommand{\footrulewidth}{0pt}
+\addtolength{\headheight}{2.6pt}   % make space for the rule
+%\addtolength{\headheight}{1.6pt}   % make space for the rule
+\fancypagestyle{plain}{
+  \fancyhead{}         % get rid of headers on plain pages
+  \renewcommand{\headrulewidth}{0pt}  % and the line
+}
+
+% Leave blank pages completely empty, w/o header
+\makeatletter
+\def\cleardoublepage{\clearpage\if@twoside \ifodd\c@page\else
+  \hbox{}
+  \vspace*{\fill}
+  \vspace{\fill}
+  \thispagestyle{empty}
+  \newpage
+  \if@twocolumn\hbox{}\newpage\fi\fi\fi}
+\makeatother
+
+\usepackage{makeidx}
+
+\usepackage{minitoc}				%in french : \usepackage[french]{minitoc}
+
+%\usepackage{mtcoff}				% invalidate the use of minitocs
+\usepackage[latin1]{inputenc}
+\usepackage{graphics}				% allows insertion of pictures
+\usepackage{times}					% use vector fonts instead of bitmap
+\usepackage{fancybox}
+\usepackage{color}
+\usepackage{colortbl}
+
+%%%% add by Word2Tex
+%\documentclass{amsart}
+\usepackage{latexsym}
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage{graphicx}
+%%%%
+
+%%%%
+%\usepackage{mathtime}      % font for illustrator to work (belleek fonts )
+%%%%
+
+%\usepackage{showidx}				% show the index entry
+
+
+
+%%%% define the style of title for chapter  %%%
+\usepackage{pstricks}
+\makeatletter
+\def\LigneVerticale{\vrule height 5cm depth 2cm\hspace{0.1cm}\relax}
+\def\LignesVerticales{%
+  \let\LV\LigneVerticale\LV\LV\LV\LV\LV\LV\LV\LV\LV\LV}
+\def\GrosCarreAvecUnChiffre#1{%
+  \rlap{\vrule height 0.8cm width 1cm depth 0.2cm}%
+  \rlap{\hbox to 1cm{\hss\mbox{\white #1}\hss}}%
+  \vrule height 0pt width 1cm depth 0pt}
+\def\GrosCarreAvecTroisChiffre#1{%
+  \rlap{\vrule height 0.8cm width 1.6cm depth 0.2cm}%
+  \rlap{\hbox to 1.5cm{\hss\mbox{\white #1}\hss}}%
+  \vrule height 0pt width 1cm depth 0pt}
+
+\def\@makechapterhead#1{\hbox{%
+    \huge 
+    \LignesVerticales
+    \hspace{-0.5cm}%
+    \GrosCarreAvecUnChiffre{\thechapter}
+    \hspace{0.2cm}\hbox{#1}%
+%    \GrosCarreAvecTroisChiffre{\thechapter}
+%    \hspace{1cm}\hbox{#1}%
+%}\par\vskip 2cm}
+}\par\vskip 1cm}
+\def\@makeschapterhead#1{\hbox{%
+    \huge 
+    \LignesVerticales
+    %\hspace{0.5cm}%
+    \hbox{#1}%
+}\par\vskip 2cm}
+
+%\def\thechapter{\Roman{chapter}}   	% chapter number to be Roman
+
+%%%% end style chapter %%%
+
+%%% eqution shortcut:
+\newcommand{\vect} [1] { \rm{\textbf{#1}} }    % vector style: non-italic bold
+
+%%% essai commande
+\newcommand{\nl} [1] {\texttt{\small {\textcolor{blue}{#1}} } } 
+
+%essai commande  
+\newcommand{\mdl} [1] {\textit{#1.F90}\index{Modules!#1}}			%module (mdl)
+\newcommand{\rou} [1] {\textit{#1}\index{Routines!#1}}			%module (routine)
+\newcommand{\hf} [1] {\textit{#1.h90}\index{h90 file!#1}}			%module (h90 files)
+\newcommand{\np} [1] {\textit{#1}\index{Namelist parameters!#1}}%namelist parameter (nampar)
+\newcommand{\jp} [1] {\textit{#1}\index{Model parameters!#1}}%model parameter (jp)
+\newcommand{\pp} [1] {\textit{#1}\index{Model parameters!#1}}%namelist parameter (pp)
+\newcommand{\key} [1] {\textbf{key\_#1}\index{CPP keys!key\_#1}}	%key_cpp (key)
+\newcommand{\NEMO} {\textit{NEMO }}						%NEMO (nemo)
+\newcommand{\amtcomment}[1]{#1}   	% command to allow "commented out" portions of 
+\newcommand{\gmcomment}[1]{}   	% command to allow "commented out" portions of 
+%                                    				% text that span line breaks
+
+%--------------------------------------------namlist---------------------------------------------------------
+%\newcommand{\namdisplay} [1] {	\vspace{-25pt}
+%	\begin{alltt}  \textcolor{blue}{{\scriptsize \verbatiminput{./Namelist/#1}}}  \end{alltt}
+%									\vspace{-20pt}	}
+\newcommand{\namdisplay} [1] { 
+	\begin{alltt}  {{\tiny \verbatiminput{./Namelist/#1}}}  \end{alltt} 
+                               \vspace{-10pt}	}
+%--------------------------------------------------------------------------------------------------------------
+
+%Red (NR) or Yellow(WARN)
+%\newcommand{\NR} {\colorbox{red}{#1}}
+%\newcommand{\WARN} {{ \colorbox{yellow}{#1}} }
+
+%---------------------%%%%%  Section number in Margin  %%%%%-----------------------------
+% typeset the number of each section in the left margin, with the start of each instance of
+% sectional heading text aligned with the left hand edge of  the body text.
+\makeatletter 
+\def\@seccntformat#1{\protect\makebox[0pt][r]{\csname the#1\endcsname\quad}} 
+\makeatother 
+%--------------------------------------------------------------------------------------------------------------
+
+%----------------------------%%%%%   Bilbiography   %%%%%%-------------------------------%
+\usepackage[nottoc, notlof, notlot]{tocbibind}
+\usepackage[square, comma]{natbib}
+\bibpunct{[}{]}{,}{a}{}{;}                           %suppress "," after "et al."
+\newcommand{\bibfont}{\small}
+%--------------------------------------------------------------------------------------------------------------
+
+% ================================================================
+% FRONT PAGE
+% ================================================================
+
+\title{ 
+\includegraphics[width=0.2\textwidth]{./Figures/logo_CNRS.pdf} 
+\quad
+\includegraphics[width=0.2\textwidth]{./Figures/logo_MERCATOR.pdf} 
+\quad
+\includegraphics[width=0.1\textwidth]{./Figures/logo_UKMO.pdf} 
+\quad
+\includegraphics[width=0.2\textwidth]{./Figures/logo_NERC.pdf} \\
+\vspace{3.0cm}
+\includegraphics[width=0.3\textwidth]{./Figures/NEMO_logo_Black.pdf} \\
+\vspace{1.0cm}
+\rule{345pt}{1.5pt} \\
+\vspace{0.45cm}
+ {\Huge NEMO ocean engine} 
+\rule{345pt}{1.5pt} \\ 
+%{\small NEMO v2.3} \\
+{\small  -- version 3.0 --}   }
+%{ -- Draft --}   }
+%\date{\today}
+\date{
+May 2008  \\
+\textit{\small Note du P\^ole de mod\'{e}lisation de l'Institut Pierre-Simon Laplace No 27 }\\
+\vspace{0.45cm}
+{ ISSN No 1288-1619.}
+}
+
+
+%%\includegraphics[width=0.2\textwidth]{./Figures/logo_NERC.pdf} \\
+
+%%    \hbox{\mbox{%
+%%       \hspace{4pt}%
+%%         \fbox{\includegraphics[width=3em]{./Figures/logo_CNRS.pdf}}%
+%%        \hspace{4pt}
+%%        }  }%
+
+\author{ \Large Gurvan Madec \\
+ \texttt{\small gurvan.madec@locean-ipsl.umpc.fr} \\
+{\small Laboratoire d'Oc\'{e}anographie  et du Climat: Exp\'{e}rimentation et Approches Num\'{e}riques}
+}
+
+\dominitoc
+\makeindex  		%type this first :     makeindex -s NEMO.ist NEMO_book.idx
+
+% ================================================================
+% ================================================================
+%      Include ONLY order
+% ================================================================
+
+%\includeonly{./Chapters/Chap_MISC}
+%\includeonly{./Chapters/Annex_E}
+%\includeonly{./Chapters/Chap_LBC,./Chapters/Chap_MISC}
+%\includeonly{./Chapters/Chap_Model_Basics_zstar}
+%\includeonly{./Chapters/Annex_A,./Chapters/Annex_B,./Chapters/Annex_C,./Chapters/Annex_D}
+
+% ================================================================
+% ================================================================
+\begin{document}
+
+\maketitle						% generate the title
+
+\frontmatter
+
+\tableofcontents					% generate a table of content
+%\listoffigures					% generate a table of content
+%\listoftables					% generate a table of content
+
+\mainmatter
+
+% ================================================================
+% Abstract - Foreword
+% ================================================================
+
+\include{./Chapters/Abstracts_Foreword}
+
+% ================================================================
+% INTRODUCTION
+% ================================================================
+
+\include{./Chapters/Introduction}
+
+% ================================================================
+% CHAPTERS
+% ================================================================
+
+\include{./Chapters/Chap_Model_Basics}
+
+%\include{./Chapters/Chap_Model_Basics_zstar}	% attent to describe z*-s*
+
+\include{./Chapters/Chap_DOM}
+
+\include{./Chapters/Chap_TRA}
+
+\include{./Chapters/Chap_DYN}
+
+\include{./Chapters/Chap_SBC}
+
+\include{./Chapters/Chap_LBC}
+
+\include{./Chapters/Chap_LDF}
+
+\include{./Chapters/Chap_ZDF}
+
+\include{./Chapters/Chap_MISC}
+
+% ================================================================
+% APPENDIX
+% ================================================================
+
+\appendix
+
+%\include{./Chapters/Chap_Conservation}
+\include{./Chapters/Annex_A}
+\include{./Chapters/Annex_B}
+\include{./Chapters/Annex_C}
+%\include{./Chapters/Annex_D}
+%\include{./Chapters/Annex_E}
+
+%%%%\include{./Chapters/0_page_de_garde_old}
+% ================================================================
+% INDEX
+% ================================================================
+
+\addcontentsline{toc}{chapter}{Index}
+\printindex
+
+% ================================================================
+% BIBLIOGRAPHY
+% ================================================================
+
+%%\bibliographystyle{plainnat}
+\bibliographystyle{./ametsoc}			% AMS biblio style (JPO)
+\bibliography{./Biblio/Biblio}
+
+% ================================================================
+\end{document}
Index: trunk/DOC/TexFiles/Namelist/nam_ctl
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_ctl	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_ctl	(revision 994)
@@ -0,0 +1,16 @@
+!-----------------------------------------------------------------------
+&nam_ctl      !   Control prints & Benchmark
+!-----------------------------------------------------------------------
+   ln_ctl =  .false.       !  trends control print (expensive!)
+   nprint =       0        !  level of print (0 no print)
+   nictls =       0        !  start i indice to make the control SUM (very usefull to compare mono-
+   nictle =       0        !  end   i indice to make the control SUM (-versus multi processor runs)
+   njctls =       0        !  start j indice to make the control SUM (very usefull to compare mono-
+   njctle =       0        !  end   j indice to make the control SUM (-versus multi processor runs)
+   isplt  =       1        !  number of processors following i
+   jsplt  =       1        !  number of processors following j
+   nbench =       0        !  = 1 Bench run (no physical meaning)
+                           !  = 0 Standard simulation
+   nbit_cmp =     0        !  = 1 enables bit comparison between mono and mpp runs
+                           !  = 0 faster mpp run 
+/
Index: trunk/DOC/TexFiles/Namelist/nam_dynadv
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_dynadv	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_dynadv	(revision 994)
@@ -0,0 +1,7 @@
+!-----------------------------------------------------------------------
+&nam_dynadv  !   option of physics/algorithm (not control by CPP keys)
+!----------------------------------------------------------------------- 
+   ln_dynadv_vec   = .TRUE.    !  vector form (T) flux form (F)
+   ln_dynadv_cen2  = .FALSE.   !  flux form - 2nd order centered scheme
+   ln_dynadv_ubs   = .FALSE.   !  flux form - 3rd order UBS scheme   
+/  
Index: trunk/DOC/TexFiles/Namelist/nam_dynhpg
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_dynhpg	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_dynhpg	(revision 994)
@@ -0,0 +1,12 @@
+!-----------------------------------------------------------------------
+&nam_dynhpg   !   Hydrostatic pressure gradient option
+!-----------------------------------------------------------------------
+   ln_hpg_zco = .false.       !  z-coordinate - full steps                   
+   ln_hpg_zps = .true.        !  z-coordinate - partial steps (interpolation)
+   ln_hpg_sco = .false.       !  s-coordinate (standard jacobian formulation)
+   ln_hpg_hel = .false.       !  s-coordinate (helsinki modification)
+   ln_hpg_wdj = .false.       !  s-coordinate (weighted density jacobian)
+   ln_hpg_djc = .false.       !  s-coordinate (Density Jacobian with Cubic polynomial)
+   ln_hpg_rot = .false.       !  s-coordinate (ROTated axes scheme)
+   gamm       = 0.e0          !  weighting coefficient (wdj scheme)
+/
Index: trunk/DOC/TexFiles/Namelist/nam_dynldf
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_dynldf	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_dynldf	(revision 994)
@@ -0,0 +1,17 @@
+!-----------------------------------------------------------------------
+&nam_dynldf   !   lateral diffusion on momentum
+!-----------------------------------------------------------------------
+   !                             !  Type of the operator : 
+   ln_dynldf_lap    =  .true.    !     laplacian operator          
+   ln_dynldf_bilap  =  .false.   !     bilaplacian operator       
+                                 !  Direction of action  : 
+   ln_dynldf_level  =  .false.   !     iso-level                  
+   ln_dynldf_hor    =  .true.    !     horizontal  (+ "key_ldfslp" if ln_sco=T)
+   ln_dynldf_iso    =  .false.   !     iso-neutral (+ "key_ldfslp")
+   !                             !  Coefficient
+   !  "key_ldfdyn_c1d"           !     Ahm = F(k)
+   !  "key_ldfdyn_c2d"           !     Ahm = F(i,j)
+   !  "key_ldfdyn_c3d"           !     Ahm = F(i,j,k)
+   ahm0    = 40000.              !     horizontal eddy viscosity for the dynamics (m2/s)
+   ahmb0   =     0.              !     background eddy viscosity for isopycnal diffusion (m2/s)
+/
Index: trunk/DOC/TexFiles/Namelist/nam_dynspg
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_dynspg	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_dynspg	(revision 994)
@@ -0,0 +1,9 @@
+!-----------------------------------------------------------------------
+&nam_dynspg   !   surface pressure gradient   (CPP key only)
+!-----------------------------------------------------------------------
+!  "key_vvl"                     !  Activate the variable volume level
+!  "key_dynspg_exp"              !  explicit free surface              
+!  "key_dynspg_flt"              !  filtered free surface           
+!  "key_dynspg_ts"               !  split-explicit free surface    
+!  "key_dynspg_rl"               !  rigid-lid                  
+/
Index: trunk/DOC/TexFiles/Namelist/nam_dynvor
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_dynvor	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_dynvor	(revision 994)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&nam_dynvor   !   option of physics/algorithm 
+!-----------------------------------------------------------------------
+   ln_dynvor_ene = .false.    !  enstrophy conserving scheme 
+   ln_dynvor_ens = .true.     !  energy conserving scheme   
+   ln_dynvor_mix = .false.    !  mixed energy/enstrophy conserving scheme               
+   ln_dynvor_een = .false.    !  energy and enstrophy scheme   
+/
Index: trunk/DOC/TexFiles/Namelist/nam_lbc
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_lbc	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_lbc	(revision 994)
@@ -0,0 +1,9 @@
+!-----------------------------------------------------------------------
+&namlbc   !   lateral momentum boundary condition
+!-----------------------------------------------------------------------
+! "key_noslip_accurate"    ! Activate accurate no-slip boundary condition
+   shlat  =      2.        !      shlat = 0 :           free slip
+                           !  0 < shlat < 2 : "partial" free-slip
+                           !      shlat = 2 :           no slip
+                           !  2 < shlat     : "strong"  no-slip
+/
Index: trunk/DOC/TexFiles/Namelist/nam_mpp
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_mpp	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_mpp	(revision 994)
@@ -0,0 +1,6 @@
+!-----------------------------------------------------------------------
+&nam_mpp      !   Massively Parallel Processing
+!-----------------------------------------------------------------------
+   c_mpi_send =  'S'       !  mpi send/recieve type   ='S', 'B', or 'I' for standard send,
+                           !  buffer blocking send or immediate non-blocking sends, resp.
+/
Index: trunk/DOC/TexFiles/Namelist/nam_tke
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_tke	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_tke	(revision 994)
@@ -0,0 +1,25 @@
+!-----------------------------------------------------------------------
+&namtke   !   turbulent eddy kinetic dependent vertical diffusion       
+!-----------------------------------------------------------------------
+!  "key_zdftke"               !  Activate the TKE physics
+   ln_rstke = .false.         !  restart with tke from a run without tke
+   nitke =        50          !  number of restart iterative loops
+   ediff =       0.1          !  coef. for avt (avt=ediff*mxl*sqrt(e))
+   ediss =       0.7          !  coef. for Kolmogoroff dissipation
+   ebb   =      3.75          !  coef. for surface input of tke
+   efave =        1.          !  coef. for enhance Kz on tke (avtke=efave*avm)
+   emin  =     1.e-6          !  background value of tke (m^2/s^2)
+   emin0 =     1.e-4          !  surface minimum value of tke (m^2/s^2)
+!! ri_c  =         0.22222222 !  critic richardson number (default = 2/9)
+   nmxl  =         2          !  mixing length type  
+   !                          !    = 0 bounded by the distance to surface & bottom
+   !                          !    = 1 bounded by the local vertical scale factor
+   !                          !    = 2 abs( dz(mxl) /e3 ) < 1
+   !                          !    = 3 same as 2 with ldiss /= lmix
+   npdl  =         1          !  prandtl number 
+   !                          !    = 0 no vertical prandtl number (avt=avm)
+   !                          !    = 1 prandtl number = F(Ri) (avt=pdl*avm)
+   !                          !    = 2 same as =1 but with a shapiro filter on pdl
+   nave  =         1          !  horizontal filter on avt and amv (=1) or not (=0)
+   navb  =         0          !  constant (=0) or profile (=1) background on avt
+/
Index: trunk/DOC/TexFiles/Namelist/nam_traadv
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_traadv	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_traadv	(revision 994)
@@ -0,0 +1,9 @@
+!-----------------------------------------------------------------------
+&nam_traadv   !   advection scheme for tracer 
+!-----------------------------------------------------------------------
+   ln_traadv_cen2   =  .true.    !  2nd order centered scheme    
+   ln_traadv_tvd    =  .false.   !  TVD scheme                  
+   ln_traadv_muscl  =  .false.   !  MUSCL scheme               
+   ln_traadv_muscl2 =  .false.   !  MUSCL2 scheme              
+   ln_traadv_ubs    =  .false.   !  UBS scheme                
+/
Index: trunk/DOC/TexFiles/Namelist/nam_traldf
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_traldf	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_traldf	(revision 994)
@@ -0,0 +1,19 @@
+!-----------------------------------------------------------------------
+&nam_traldf   !   lateral diffusion scheme for tracer 
+!-----------------------------------------------------------------------
+                                 !  Type of the operator : 
+   ln_traldf_lap    =  .true.    !     laplacian operator        
+   ln_traldf_bilap  =  .false.   !     bilaplacian operator      
+                                 !  Direction of action  :
+   ln_traldf_level  =  .false.   !     iso-level                  
+   ln_traldf_hor    =  .false.   !     horizontal  (+ "key_ldfslp" if ln_sco=T)
+   ln_traldf_iso    =  .true.    !     iso-neutral (+ "key_ldfslp")
+                                 !  Coefficient
+   !  "key_ldftra_c1d"           !     Aht = F(k)
+   !  "key_ldftra_c2d"           !     Aht = F(i,j)
+   !  "key_ldftra_c3d"           !     Aht = F(i,j,k)
+   aht0    =  2000.              !     lateral eddy diffusivity coef. (m2/s)
+   ahtb0   =     0.              !     background coef. for isopycnal diffusion (m2/s)
+   aeiv0   =  2000.              !     eddy induced velocity coefficient (m2/s)  
+   !                             !     (+ "key_traldf_eiv")
+/
Index: trunk/DOC/TexFiles/Namelist/nam_zgr
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_zgr	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_zgr	(revision 994)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&nam_zgr       !   vertical coordinate
+!-----------------------------------------------------------------------
+   ln_zco   =  .false.     !   z-coordinate - full    steps  
+   !                       !       ("key_zco" may also be defined)
+   ln_zps   =  .true.      !  z-coordinate - partial steps   
+   ln_sco   =  .false.     !  s- or hybrid z-s-coordinate   
+/
Index: trunk/DOC/TexFiles/Namelist/nam_zgr_sco
===================================================================
--- trunk/DOC/TexFiles/Namelist/nam_zgr_sco	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nam_zgr_sco	(revision 994)
@@ -0,0 +1,10 @@
+!-----------------------------------------------------------------------
+&nam_zgr_sco   !   s-coordinate or hybrid z-s-coordinate
+!-----------------------------------------------------------------------
+   sbot_min =  300.        !  minimum depth of s-bottom surface (>0) (m)
+   sbot_max = 5250.        !  maximum depth of s-bottom surface (>0) (m)
+   !                       !  (= maximum ocean depth allowed) 
+   theta    =    6.0       !  surface control parameter (0<=theta<=20)
+   thetb    =    0.75      !  bottom control parameter  (0<=thetb<= 1)
+   r_max    =    0.15      !  maximum cut-off r-value allowed (0<r_max<1)
+/
Index: trunk/DOC/TexFiles/Namelist/nambbc
===================================================================
--- trunk/DOC/TexFiles/Namelist/nambbc	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nambbc	(revision 994)
@@ -0,0 +1,10 @@
+!-----------------------------------------------------------------------
+&nambbc   !   bottom boundary condition (on temperature only)                 
+!-----------------------------------------------------------------------
+!  "key_trabbc"               ! Activate geothermal forcing (bbc)
+   ngeo_flux =  2             !  = 0 no geothermal heat flux
+   !                          !  = 1 constant geothermal heat flux
+   !                          !  = 2 variable geothermal heat flux 
+   !                          !      (read in geothermal_heating.nc in mW/m2) 
+   ngeo_flux_const = 86.4e-3  !  Constant value of geothermal heat flux (W/m2)
+/
Index: trunk/DOC/TexFiles/Namelist/nambbl
===================================================================
--- trunk/DOC/TexFiles/Namelist/nambbl	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nambbl	(revision 994)
@@ -0,0 +1,7 @@
+!-----------------------------------------------------------------------
+&nam_trabbl   !   bottom boundary layer scheme                          
+!-----------------------------------------------------------------------
+!  "key_trabbl_dif"              !  Activate the diffusive bbl 
+!  "key_trabbl_adv"              !  Activate the advective bbl 
+   atrbbl = 10000.               !  lateral tracer coeff. for bbl scheme (m2/s)
+/
Index: trunk/DOC/TexFiles/Namelist/nambfr
===================================================================
--- trunk/DOC/TexFiles/Namelist/nambfr	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nambfr	(revision 994)
@@ -0,0 +1,10 @@
+!-----------------------------------------------------------------------
+&nambfr   !   bottom friction
+!-----------------------------------------------------------------------
+   nbotfr =       1        !  type of bottom friction 
+   !                       !   = 0 : no slip        ,  = 2 : nonlinear friction
+                           !   = 1 : linear friction,  = 3 : free slip
+   bfri1  =   4.e-4        !  bottom drag coefficient (linear case)
+   bfri2  =   1.e-3        !  bottom drag coefficient (non linear case)
+   bfeb2  =  2.5e-3        !  bottom tke background (m^2/s^2)
+/
Index: trunk/DOC/TexFiles/Namelist/namcpl
===================================================================
--- trunk/DOC/TexFiles/Namelist/namcpl	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namcpl	(revision 994)
@@ -0,0 +1,4 @@
+!-----------------------------------------------------------------------
+&namcpl   !   coupled ocean/atmosphere model                            ("key_coupled")
+!-----------------------------------------------------------------------
+/
Index: trunk/DOC/TexFiles/Namelist/namcro
===================================================================
--- trunk/DOC/TexFiles/Namelist/namcro	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namcro	(revision 994)
@@ -0,0 +1,6 @@
+!-----------------------------------------------------------------------
+&namcro   !   cross land advection
+!-----------------------------------------------------------------------
+!  n_cla   advection between 2 ocean pts separates by land 
+   n_cla   = 1
+/
Index: trunk/DOC/TexFiles/Namelist/namddm
===================================================================
--- trunk/DOC/TexFiles/Namelist/namddm	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namddm	(revision 994)
@@ -0,0 +1,7 @@
+!-----------------------------------------------------------------------
+&namddm   !   double diffusive mixing parameterization                  
+!-----------------------------------------------------------------------
+!     "key_zdfddm"            !  Activate ddm phisics
+      avts  = 1.e-4           !  maximum avs (vertical mixing on salinity)
+      hsbfr = 1.6             !  heat/salt buoyancy flux ratio
+/
Index: trunk/DOC/TexFiles/Namelist/namdmp
===================================================================
--- trunk/DOC/TexFiles/Namelist/namdmp	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namdmp	(revision 994)
@@ -0,0 +1,17 @@
+!-----------------------------------------------------------------------
+&namdmp   tracer newtonian damping ('key_tradmp')
+!-----------------------------------------------------------------------
+   ndmp   =   -1                 !  type of damping in temperature and salinity 
+   !                             !     ='latitude', damping poleward of 'ndmp' degrees 
+   !                             !       and function of the distance-to-coast. 
+   !                             !       Red and Med Seas as ndmp=-1
+   !                             !     =-1 damping only in Med and Red Seas
+   ndmpf  =    1                 !  =1 create a damping.coeff NetCDF file
+   nmldmp =    1                 !  type of damping in the mixed layer
+   !                             !     =0 damping throughout the water column
+   !                             !     =1 no damping in the mixing layer (avt >5cm2/s )
+   !                             !     =2 no damping in the mixed  layer (rho<rho(surf)+.01 )
+   sdmp   =  50.                 !  surface time scale for internal damping (days)
+   bdmp   = 360.                 !  bottom  time scale for internal damping (days)
+   hdmp   = 800.                 !  depth of transition between sdmp and bdmp (meters)
+/
Index: trunk/DOC/TexFiles/Namelist/namdom
===================================================================
--- trunk/DOC/TexFiles/Namelist/namdom	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namdom	(revision 994)
@@ -0,0 +1,21 @@
+!-----------------------------------------------------------------------
+&namdom   !   space and time domain (bathymetry, mesh, timestep)
+!-----------------------------------------------------------------------
+   ntopo     =     1       !  = 1 read the bathymetry_level 
+   e3zps_min =     5.      !  minimum thickness of the partial step is the min of
+   e3zps_rat =     0.1     !  e3zps_min and e3zps_rat * e3t  (with 0<e3zps_rat<1)
+   nmsh      =     0       !  =1 create a mesh file (coordinates, scale factors, masks)
+   nacc      =     0       !  the acceleration of convergence method
+   !                       !     = 0, no acceleration, rdt = rdttra
+   !                       !     = 1, acceleration used, rdt < rdttra(k)
+   atfp      =     0.1     !  asselin time filter parameter
+   rdt       =  5760.      !  time step for the dynamics (and tracer if nacc=0)
+   rdtmin    =  5760.      !  minimum time step on tracers
+   rdtmax    =  5760.      !  maximum time step on tracers
+   rdth      =   800.      !  depth variation of tracer time step
+   rdtbt     =    90.      !  barotropic time step (for the time splitting algorithm)
+   nfice     =     5       !  frequency of ice model call
+   nfbulk    =     5       !  frequency of bulk formulea call (not used if ice used)
+   nclosea   =     0       !  = 0 no closed sea in the model domain
+!                          !  = 1 closed sea (Caspian Sea, Great US Lakes...) 
+/
Index: trunk/DOC/TexFiles/Namelist/nameos
===================================================================
--- trunk/DOC/TexFiles/Namelist/nameos	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/nameos	(revision 994)
@@ -0,0 +1,10 @@
+!-----------------------------------------------------------------------
+&nameos   !   ocean physical parameters
+!-----------------------------------------------------------------------
+   neos   =      0         !  type of equation of state and Brunt-Vaisala frequency
+                           !     = 0, Jackett and McDougall (1994) and of McDougall (1987)
+                           !     = 1, linear: rho(T)   = rau0 * ( 1.028 - ralpha * T )
+                           !     = 2, linear: rho(T,S) = rau0 * ( rbeta * S - ralpha * T )
+   ralpha =  2.e-4         !  thermal expension coefficient (neos= 1 or 2)
+   rbeta  =  0.001         !  saline  expension coefficient (neos= 2)
+/
Index: trunk/DOC/TexFiles/Namelist/namflg
===================================================================
--- trunk/DOC/TexFiles/Namelist/namflg	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namflg	(revision 994)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&namflg   !   algorithm flags (algorithm not control by CPP keys)
+!-----------------------------------------------------------------------
+   ln_dynhpg_imp   =  .false. !  hydrostatic pressure gradient:
+   !                          !     = T : semi-implicit time scheme 
+   !                          !     = f : centered      time scheme
+   nn_dynhpg_rst   =  0       !  add dynhpg implicit variables in restart ot not (1/0)
+/
Index: trunk/DOC/TexFiles/Namelist/namflo
===================================================================
--- trunk/DOC/TexFiles/Namelist/namflo	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namflo	(revision 994)
@@ -0,0 +1,10 @@
+!-----------------------------------------------------------------------
+&namflo    !   float parameters                                         ("key_float")
+!-----------------------------------------------------------------------
+    ln_rstflo = .false.          !  boolean term for float restart (true or false)
+    nwritefl  =      75          !  frequency of float output file 
+    nstockfl  =    5475          !  frequency of float restart file 
+    ln_argo   = .false.          !  Argo type floats (stay at the surface each 10 days)
+    ln_flork4 = .false.          !  = T trajectories computed with a 4th order Runge-Kutta
+                                 !  = F  (default)   computed with Blanke' scheme
+/
Index: trunk/DOC/TexFiles/Namelist/namfwb
===================================================================
--- trunk/DOC/TexFiles/Namelist/namfwb	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namfwb	(revision 994)
@@ -0,0 +1,5 @@
+!-----------------------------------------------------------------------
+&namfwb   !   freshwater budget correction
+!-----------------------------------------------------------------------
+   ln_fwb    = .true.      !  flag for freshwater budget correction (0 annual mean)
+/
Index: trunk/DOC/TexFiles/Namelist/namgap
===================================================================
--- trunk/DOC/TexFiles/Namelist/namgap	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namgap	(revision 994)
@@ -0,0 +1,6 @@
+!-----------------------------------------------------------------------
+&namgap    !   level mean model-data gap                                ('key_diagap')
+!-----------------------------------------------------------------------
+   ngap =  15                    !  time-step frequency of model-data gap computation
+   nprg =  10                    !  time-step frequency of gap print in model output
+/
Index: trunk/DOC/TexFiles/Namelist/namkpp
===================================================================
--- trunk/DOC/TexFiles/Namelist/namkpp	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namkpp	(revision 994)
@@ -0,0 +1,16 @@
+!-----------------------------------------------------------------------
+&namkpp   !   K-Profile Parameterization dependent vertical mixing      
+!-----------------------------------------------------------------------
+!  "key_zdfkpp"               !  activate the KPP physics
+!  "key_kppcustom"            !  KPP option 1
+!  "key_kpplktb"              !  KPP option 2    
+   ln_kpprimix  = .true.      !  shear instability mixing  (default T)
+   difmiw       =  1.0e-04    !  constant internal wave viscosity (m2/s)
+   difsiw       =  0.1e-04    !  constant internal wave diffusivity (m2/s)
+   Riinfty      =  0.8        !  local Richardson Number limit for shear instability
+   difri        =  0.0050     !  maximum shear mixing at Rig = 0    (m2/s)
+   bvsqcon      = -0.01e-07   !  Brunt-Vaisala squared (1/s**2) for maximum convection
+   difcon       =  1.         !  maximum mixing in interior convection (m2/s)
+   navb         =  0          !  horizontal averaged (=1) or not (=0) on avt and amv  
+   nave         =  1          !  constant (=0) or profile (=1) background on avt
+/
Index: trunk/DOC/TexFiles/Namelist/namlbc
===================================================================
--- trunk/DOC/TexFiles/Namelist/namlbc	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namlbc	(revision 994)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&namlbc   !   lateral momentum boundary condition
+!-----------------------------------------------------------------------
+   shlat  =      2.        !      shlat = 0 : free slip
+                           !  0 < shlat < 2 : partial slip
+                           !      shlat = 2 : no slip
+                           !  2 < shlat     : strong slip
+/
Index: trunk/DOC/TexFiles/Namelist/namnpc
===================================================================
--- trunk/DOC/TexFiles/Namelist/namnpc	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namnpc	(revision 994)
@@ -0,0 +1,6 @@
+!-----------------------------------------------------------------------
+&namnpc   !   non penetrative convective adjustment
+!-----------------------------------------------------------------------
+   nnpc1  =      1            !  computation frequency    (time-step)
+   nnpc2  =    365            !  control print frequency  (time-step)
+/
Index: trunk/DOC/TexFiles/Namelist/namobc
===================================================================
--- trunk/DOC/TexFiles/Namelist/namobc	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namobc	(revision 994)
@@ -0,0 +1,19 @@
+!-----------------------------------------------------------------------
+&namobc   !   open boundaries parameters                            ("key_obc")
+!-----------------------------------------------------------------------
+    nobc_dta    =    0     !  = 0 the obc data are equal to the initial state
+                           !  = 1 the obc data are read in 'obc   .dta' files
+    rdpein      =    1.    !  ???
+    rdpwin      =    1.    !  ???
+    rdpnin      =   30.    !  ???
+    rdpsin      =    1.    !  ???
+    rdpeob      = 1500.    !  time relaxation (days) for the east  open boundary
+    rdpwob      =   15.    !    "        "           for the west  open boundary
+    rdpnob      =  150.    !    "        "           for the north open boundary
+    rdpsob      =   15.    !    "        "           for the south open boundary
+    zbsic1      =  140.e+6 !  barotropic stream function on first  isolated coastline
+    zbsic2      =    1.e+6 !    "                   "    on second isolated coastline
+    zbsic3      =    0.    !    "                   "    on thrid  isolated coastline
+    ln_obc_clim = .true.   !  climatological obc data files (default T)
+    ln_vol_cst  = .true.   !  total volume conserved
+/
Index: trunk/DOC/TexFiles/Namelist/namptr
===================================================================
--- trunk/DOC/TexFiles/Namelist/namptr	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namptr	(revision 994)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&namptr   !   Poleward Transport Diagnostic
+!-----------------------------------------------------------------------
+   ln_diaptr = .false.           !  logical flag for Poleward transport computation
+   ln_subbas = .false.           !  logical flag for Atlantic/Pacific/Indian basins computation
+                                 !  need input basins mask file named "subbasins.nc"
+   nf_ptr    =  15               !  Frequency of computation
+/
Index: trunk/DOC/TexFiles/Namelist/namqsr
===================================================================
--- trunk/DOC/TexFiles/Namelist/namqsr	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namqsr	(revision 994)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&namqsr   !   penetrative solar radiation
+!-----------------------------------------------------------------------
+   ln_traqsr = .true.      !  penetrative solar radiation (T) or not (F)
+   rabs     =   0.58       !  fraction of qsr associated with xsi1
+   xsi1     =   0.35       !  first depth of extinction
+   xsi2     =   23.0       !  second depth of extinction
+/
Index: trunk/DOC/TexFiles/Namelist/namric
===================================================================
--- trunk/DOC/TexFiles/Namelist/namric	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namric	(revision 994)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&namric   richardson number dependent vertical diffusion               
+!-----------------------------------------------------------------------
+!  "key_zdfric"               !  Activate Kz =function of Ri
+   avmri = 100.e-4            !  maximum value of the vertical viscosity
+   alp   =      5.            !  coefficient of the parameterization
+   nric  =       2            !  coefficient of the parameterization
+/
Index: trunk/DOC/TexFiles/Namelist/namrun
===================================================================
--- trunk/DOC/TexFiles/Namelist/namrun	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namrun	(revision 994)
@@ -0,0 +1,20 @@
+!-----------------------------------------------------------------------
+&namrun   !   parameters of the run
+!-----------------------------------------------------------------------
+   no         =       0    !  job number
+   cexper     =  "ORCA2"   !  experience name for vairmer format
+   ln_rstart  = .false.    !  boolean term for restart (true or false)
+   nrstdt     =       0    !  restart control = 0 restart, do not control nit000 in the restart file.
+                           !                  = 1 restart, control nit000 in the restart file. Do not
+                           !                      use the date in the restart file (use ndate0 in namelist)
+                           !                  = 2 restart, control nit000 in the restart file, use the date
+                           !                      in the restart file. ndate0 in the namelist is ignored.
+   nit000     =       1    !  number of the first time step
+   nitend     =    5475    !  number of the first time step
+   ndate0     =  010101    !  initial calendar date aammjj
+   nleapy     =       0    !  Leap year calendar (0/1)
+   ninist     =       0    !  initial state output flag (0/1)
+   nstock     =    5475    !  number of the first time step
+   nwrite     =    5475    !  frequency of OUTPUT file
+   nrunoff    =       2    !  = 0 no, 1 runoff, 2 runoff+river mouth ups adv
+/
Index: trunk/DOC/TexFiles/Namelist/namsbc
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc	(revision 994)
@@ -0,0 +1,23 @@
+!-----------------------------------------------------------------------
+&namsbc       Surface Boundary Condition (surface module)
+!-----------------------------------------------------------------------
+   nn_fsbc     = 5         ! frequency of surface boundary condition computation 
+                           ! (also the frequency of sea-ice model call)
+   ln_ana      = .false.   ! analytical formulation (fill namsbc_ana )
+   ln_flx      = .false.   ! flux       formulation (fill namsbc_flx ) 
+   ln_blk_clio = .true.    ! CLIO bulk  formulation (fill namsbc_core) 
+   ln_blk_core = .false.   ! CORE bulk  formulation (fill namsbc_clio)
+   ln_cpl      = .false.   ! coupled    formulation (fill namsbc_cpl ) 
+   nn_ice      = 2         ! ice areas : =0 no ice in the domain
+                           !             =1 specify observed ice-cover
+                           !             =2 use an ice-model            ("key_lim3" or "key_lim2)
+   nn_ico_cpl  = 0         ! ice-ocean stress : =0 LIM-3 old case 
+                           !                    =1 stresses computed using now ocean velocity
+                           !                    =2 combination of 0 and 1 cases
+   ln_dm2dc    = .false.   ! Daily mean converted into Diurnal Cycle for short wave (qsr)
+   ln_rnf      = .false.   ! river runoffs (fill namsbc_rnf)
+   ln_ssr      = .false.   ! Sea Surface Restoring on T and/or S (fill namsbc_ssr)
+   nn_fwb      = 0         ! FreshWater Budget : =0 no control                             
+                           !                     =1 annual global mean set to zero
+                           !                     =2 global mean instantaneously set to zero
+/
Index: trunk/DOC/TexFiles/Namelist/namsbc_alb
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc_alb	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc_alb	(revision 994)
@@ -0,0 +1,9 @@
+!-----------------------------------------------------------------------
+&namalb                     albedo parameters
+!-----------------------------------------------------------------------
+   cgren    =      0.06    ! correction of the snow or ice albedo to take into account
+   albice   =      0.5     ! albedo of melting ice in the arctic and antarctic
+   alphd    =      0.80    ! coefficients for linear interpolation used to compute
+   alphc    =      0.65    ! the albedo between two extremes values (Pyane, 1972)
+   alphdi   =      0.72    ! 
+/
Index: trunk/DOC/TexFiles/Namelist/namsbc_ana
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc_ana	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc_ana	(revision 994)
@@ -0,0 +1,10 @@
+!-----------------------------------------------------------------------
+&namsbc_ana         analytical surface boundary condition
+!-----------------------------------------------------------------------
+   nn_tau000  =   0        ! gently increase the stress over the first ntau_rst time-steps
+   rn_utau0   =   0.5      ! uniform value for the i-stress
+   rn_vtau0   =   0.e0     ! uniform value for the j-stress
+   rn_q0      =   0.e0     ! uniform value for the total heat flux
+   rn_qsr0    =   0.e0     ! uniform value for the solar radiation
+   rn_emp0    =   0.e0     ! uniform value for the freswater budget (E-P)
+/
Index: trunk/DOC/TexFiles/Namelist/namsbc_clio
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc_clio	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc_clio	(revision 994)
@@ -0,0 +1,16 @@
+!-----------------------------------------------------------------------
+&namsbc_clio              CLIO bulk formulea
+!-----------------------------------------------------------------------
+   cn_dir       = './'     ! directory in which the model is executed
+!
+!  THE ORDER OF THE FILES MATTER:
+!               !   file name        ! frequency (hours) ! variable   ! time interpolation !  clim  ! starting !
+!               !                    !   (if <0  months) !   name     !    (logical)       !  (0/1) !  record  !
+   sn_utau      =    'taux_1m'       ,      -12.         , 'sozotaux' ,     .false.        ,    1   ,    0    
+   sn_vtau      =    'tauy_1m'       ,      -12.         , 'sometauy' ,     .false.        ,    1   ,    0    
+   sn_wndm      =    'flx'           ,      -12.         , 'socliowi' ,     .false.        ,    1   ,    0      
+   sn_tair      =    'flx'           ,      -12.         , 'socliot1' ,     .false.        ,    1   ,    0     
+   sn_humi      =    'flx'           ,      -12.         , 'socliohu' ,     .false.        ,    1   ,    0    
+   sn_ccov      =    'flx'           ,      -12.         , 'socliocl' ,     .false.        ,    1   ,    0      
+   sn_prec      =    'flx'           ,      -12.         , 'socliopl' ,     .false.        ,    1   ,    0       
+/
Index: trunk/DOC/TexFiles/Namelist/namsbc_core
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc_core	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc_core	(revision 994)
@@ -0,0 +1,19 @@
+!-----------------------------------------------------------------------
+&namsbc_core               CORE bulk formulea
+!-----------------------------------------------------------------------
+   cn_dir       = './'     ! directory in which the model is executed
+   ln_2m        = .false.  ! logical flag to use air temp. and hum referenced at 2m instead 10m
+   alpha_precip = 1.       ! multiplication factor for precipitation (total & snow)
+!
+!  THE ORDER OF THE FILES MATTER:
+!               !   file name        ! frequency (hours) ! variable   ! time interpolation !  clim  ! starting !
+!               !                    !   (if <0  months) !   name     !    (logical)       !  (0/1) !  record  !
+   sn_wndi      =    'u10'           ,       24.         , 'U_10_MOD' ,     .false.        ,    1   ,    0    
+   sn_wndj      =    'v10'           ,       24.         , 'V_10_MOD' ,     .false.        ,    1   ,    0    
+   sn_qsr       =    'rad'           ,       24.         , 'SWDN_MOD' ,     .false.        ,    1   ,    0      
+   sn_qlw       =    'rad'           ,       24.         , 'LWDN_MOD' ,     .false.        ,    1   ,    0      
+   sn_tair      =    't10'           ,       24.         , 'T_10_MOD' ,     .false.        ,    1   ,    0     
+   sn_humi      =    'q10'           ,       24.         , 'Q_10_MOD' ,     .false.        ,    1   ,    0    
+   sn_prec      =    'precip'        ,      -12.         , 'RAIN'     ,     .false.        ,    1   ,    0      
+   sn_snow      =    'precip'        ,      -12.         , 'SNOW'     ,     .false.        ,    1   ,    0       
+/
Index: trunk/DOC/TexFiles/Namelist/namsbc_cpl
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc_cpl	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc_cpl	(revision 994)
@@ -0,0 +1,4 @@
+!-----------------------------------------------------------------------
+&namsbc_cpl           coupled ocean/atmosphere model                    
+!-----------------------------------------------------------------------
+/
Index: trunk/DOC/TexFiles/Namelist/namsbc_flx
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc_flx	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc_flx	(revision 994)
@@ -0,0 +1,14 @@
+!-----------------------------------------------------------------------
+&namsbc_flx      surface boundary condition : flux formulation      
+!-----------------------------------------------------------------------
+   cn_dir       = './'     ! directory in which the model is executed
+!
+!  THE ORDER OF THE FILES MATTER:
+!               !      file name     ! frequency (hours) ! variable ! time interpolation !  clim  ! starting !
+!               !                    !   (if <0  months) !   name   !     (logical)      !  (0/1) !  record  !
+   sn_utau      =      'utau.nc'     ,        24.        ,  'utau'  ,      .false.       ,    0   ,     0
+   sn_vtau      =      'vtau.nc'     ,        24.        ,  'vtau'  ,      .false.       ,    0   ,     0
+   sn_qtot      =      'qtot.nc'     ,        24.        ,  'qtot'  ,      .false.       ,    0   ,     0
+   sn_qsr       =      'qsr.nc'      ,        24.        ,  'qsr'   ,      .false.       ,    0   ,     0
+   sn_emp       =      'emp.nc'      ,        24.        ,  'emp'   ,      .false.       ,    0   ,     0
+/      
Index: trunk/DOC/TexFiles/Namelist/namsbc_qsr
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc_qsr	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc_qsr	(revision 994)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&namqsr                penetrative solar radiation
+!-----------------------------------------------------------------------
+   ln_traqsr = .true.      ! penetrative solar radiation (T) or not (F)
+   rabs     =   0.58       ! fraction of qsr associated with xsi1
+   xsi1     =   0.35       ! first depth of extinction
+   xsi2     =   23.0       ! second depth of extinction
+/
Index: trunk/DOC/TexFiles/Namelist/namsbc_rnf
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc_rnf	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc_rnf	(revision 994)
@@ -0,0 +1,14 @@
+!-----------------------------------------------------------------------
+&namsbc_rnf        runoffs namelist surface boundary condition
+!-----------------------------------------------------------------------
+   cn_dir       = './'     ! directory in which the model is executed
+   nn_runoff  = 2          ! =0 no runoffs, 1 runoffs, 2 runoffs+river mouth ups adv
+   rn_hrnf    = 0.e0       ! runoffs, depth over which enhanced vertical mixing is used
+   rn_avt_rnf = 1.e-3      ! runoffs, value of the additional vertical mixing coef. [m2/s]
+!
+!  THE ORDER OF THE FILES MATTER:
+!               !   file name        ! frequency (hours) ! variable   ! time interpolation !  clim  ! starting !
+!               !                    !   (if <0  months) !   name     !     (logical)      !  (0/1) !  record  !
+   sn_rnf     = 'runoff_1m_nomask.nc' ,      -12.        ,  'sorunoff',      .true.        ,    1   ,    0     
+   sn_cnf     = 'runoff_1m_nomask.nc' ,        0.        ,  'socoefr' ,      .false.       ,    1   ,    0              
+/
Index: trunk/DOC/TexFiles/Namelist/namsbc_ssr
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsbc_ssr	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsbc_ssr	(revision 994)
@@ -0,0 +1,17 @@
+!-----------------------------------------------------------------------
+&namsbc_ssr     surface boundary condition : sea surface restoring
+!-----------------------------------------------------------------------
+   cn_dir       = './'     ! directory in which the model is executed
+   nn_sstr      = 0        ! =0/1   add a retroaction term in the surface heat flux
+   nn_sssr      = 0        ! =O/1/2 add a damping term in the surface freshwater flux
+                           !        (=1, salt flux, concentration/dillution emps only)
+                           !        (=2, volume flux, both emp and emps are updated) 
+   dqdt         = -40.     ! magnitude of the retroaction on temperature [W/m2/K]
+   deds         =  27.7    ! magnitude of the damping on salinity [mm/day]
+!      
+!  THE ORDER OF THE FILES MATTER:
+!               !   file name        ! frequency (hours) ! variable   ! time interpolation !  clim  ! starting !
+!               !                    !   (if <0  months) !   name     !     (logical)      !  (0/1) !  record  !
+   sn_sst       =  'sst_data.nc'     ,         24.       ,  'sst'     ,      .false.       ,    0   ,    0
+   sn_sss       =  'sss_data.nc'     ,        -12.       ,  'sss'     ,      .true.        ,    0   ,    0
+/      
Index: trunk/DOC/TexFiles/Namelist/namsol
===================================================================
--- trunk/DOC/TexFiles/Namelist/namsol	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namsol	(revision 994)
@@ -0,0 +1,19 @@
+!-----------------------------------------------------------------------
+&namsol   !   elliptic solver / island / free surface 
+!-----------------------------------------------------------------------
+   nsolv     =      1         !  type of elliptic solver  
+   !                          !     =1 preconditioned conjugate gradient (pcg)
+                              !     =2 successive-over-relaxation (sor)
+                              !     =3 FETI (fet) (+ "key_feti")
+                              !     =4 sor with extra outer halo
+   nsol_arp  =      0         !  absolute/relative (0/1) precision convergence test
+   nmin      =    300         !  minimum of iterations for the SOR solver
+   nmax      =    800         !  maximum of iterations for the SOR solver
+   nmod      =     10         !  frequency of test for the SOR solver
+   eps       =  1.E-6         !  absolute precision of the solver
+   resmax    = 1.E-10         !  absolute precision for the SOR solver
+   sor       =   1.92         !  optimal coefficient for SOR solver
+   epsisl    = 1.e-10         !  absolute precision on stream function solver
+   nmisl     =   4000         !  maximum pcg iterations for island ("key_islands")
+   rnu       =     1.         !  strength of the additional force ("key_dynspg_flt)
+/
Index: trunk/DOC/TexFiles/Namelist/namspr
===================================================================
--- trunk/DOC/TexFiles/Namelist/namspr	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namspr	(revision 994)
@@ -0,0 +1,7 @@
+!-----------------------------------------------------------------------
+&namspr  !   surface pressure diagnostic
+!-----------------------------------------------------------------------
+   nmaxp   =   1000              !  maximum of iterations for the solver
+   epsp    =  1.e-3              !  absolute precision of the solver
+   niterp  =    400              !  number of iteration done by the solver
+/
Index: trunk/DOC/TexFiles/Namelist/namtrd
===================================================================
--- trunk/DOC/TexFiles/Namelist/namtrd	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namtrd	(revision 994)
@@ -0,0 +1,11 @@
+!-----------------------------------------------------------------------
+&namtrd    !   diagnostics on dynamics and/or tracer trends             ("key_trddyn" and/or "key_trdtra")
+!                          or mixed-layer trends                        ('key_trdmld')
+!                          or barotropic vorticity                      ("key_trdvor")
+!-----------------------------------------------------------------------
+   ntrd  = 365                   !  time step frequency dynamics and tracers trends
+   nctls =   0                   !  control surface type in mixed-layer trends (0,1 or n<jpk)
+   ln_trdmld_restart = .false.   !  restart for ML diagnostics
+   ucf   =  1.                   !  unit conversion factor (=1 -> /seconds | =86400. -> /day)
+   ln_trdmld_instant = .false.   !  flag to diagnose trends of instantantaneous or mean ML T/S
+/
Index: trunk/DOC/TexFiles/Namelist/namzdf
===================================================================
--- trunk/DOC/TexFiles/Namelist/namzdf	(revision 994)
+++ trunk/DOC/TexFiles/Namelist/namzdf	(revision 994)
@@ -0,0 +1,15 @@
+!-----------------------------------------------------------------------
+&namzdf   !   vertical physics
+!-----------------------------------------------------------------------
+   ln_zdfnpc = .false.        !  Non-Penetrative Convection          
+   avm0      = 1.2e-4         !  Kz on momemtum (m2/s)   
+   !                          !  (background Kz if not "key_zdfcst")
+   avt0      = 1.2e-5         !  Kz for tracers (m2/s) 
+   !                          !  (background Kz if not "key_zdfcst")
+   ln_zdfevd = .true.         !  enhanced vertical diffusion        
+   avevd     =   100.         !  Kz for enhanced diffusion scheme (m2/s)
+   n_evdm    =     0          !  enhanced mixing Kz apply on tracer (=0) 
+   !                          !      or on both tracer and momentum (=1)
+   ln_zdfexp =  .false.       !  =T/F  split explicit / implicit 
+   n_zdfexp  =     3          !  number of sub-timestep for ln_zdfexp=T
+/
Index: trunk/DOC/TexFiles/ametsoc.bst
===================================================================
--- trunk/DOC/TexFiles/ametsoc.bst	(revision 994)
+++ trunk/DOC/TexFiles/ametsoc.bst	(revision 994)
@@ -0,0 +1,1662 @@
+%%
+%% This is file `ametsoc7.bst',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% merlin.mbs  (with options: `ay,nat,nm-rev1,ed-rev,jnrlst,nmdash,nmd-3,dt-beg,yr-com,yrp-col,thtit-a,vol-2bf,vnum-sp,volp-com,pgsep-s,num-xser,numser,ser-vol,bkpg-x,pg-bk,pre-edn,agu-doi,doi,edparc,bkedcap,edby,edbyy,blk-tit,in-x,pp,ed,abr,ednx,ord,jabr,eprint,url,url-nt,em-it,nfss,')
+%% ----------------------------------------
+%% *** Bibliography style file for ALL AMS Journals...version 1.0  ***
+%% *** Brian Papa - American Meteorological Society ***
+%% 
+%% Copyright 1994-2004 Patrick W Daly
+ % ===============================================================
+ % IMPORTANT NOTICE:
+ % This bibliographic style (bst) file has been generated from one or
+ % more master bibliographic style (mbs) files, listed above.
+ %
+ % This generated file can be redistributed and/or modified under the terms
+ % of the LaTeX Project Public License Distributed from CTAN
+ % archives in directory macros/latex/base/lppl.txt; either
+ % version 1 of the License, or any later version.
+ % ===============================================================
+ % Name and version information of the main mbs file:
+ % \ProvidesFile{merlin.mbs}[2004/02/09 4.13 (PWD, AO, DPC)]
+ %   For use with BibTeX version 0.99a or later
+ %-------------------------------------------------------------------
+ % This bibliography style file is intended for texts in ENGLISH
+ % This is an author-year citation style bibliography. As such, it is
+ % non-standard LaTeX, and requires a special package file to function properly.
+ % Such a package is    natbib.sty   by Patrick W. Daly
+ % The form of the \bibitem entries is
+ %   \bibitem[Jones et al.(1990)]{key}...
+ %   \bibitem[Jones et al.(1990)Jones, Baker, and Smith]{key}...
+ % The essential feature is that the label (the part in brackets) consists
+ % of the author names, as they should appear in the citation, with the year
+ % in parentheses following. There must be no space before the opening
+ % parenthesis!
+ % With natbib v5.3, a full list of authors may also follow the year.
+ % In natbib.sty, it is possible to define the type of enclosures that is
+ % really wanted (brackets or parentheses), but in either case, there must
+ % be parentheses in the label.
+ % The \cite command functions as follows:
+ %   \citet{key} ==>>                Jones et al. (1990)
+ %   \citet*{key} ==>>               Jones, Baker, and Smith (1990)
+ %   \citep{key} ==>>                (Jones et al., 1990)
+ %   \citep*{key} ==>>               (Jones, Baker, and Smith, 1990)
+ %   \citep[chap. 2]{key} ==>>       (Jones et al., 1990, chap. 2)
+ %   \citep[e.g.][]{key} ==>>        (e.g. Jones et al., 1990)
+ %   \citep[e.g.][p. 32]{key} ==>>   (e.g. Jones et al., p. 32)
+ %   \citeauthor{key} ==>>           Jones et al.
+ %   \citeauthor*{key} ==>>          Jones, Baker, and Smith
+ %   \citeyear{key} ==>>             1990
+ %---------------------------------------------------------------------
+
+ENTRY
+  { address
+    archive
+    author
+    booktitle
+    chapter
+    doi
+    edition
+    editor
+    eid
+    eprint
+    howpublished
+    department
+    institution
+    journal
+    key
+    month
+    note
+    number
+    organization
+    pages
+    publisher
+    school
+    series
+    title
+    type
+    url
+    volume
+    year
+  }
+  {}
+  { label extra.label sort.label short.list }
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+  #1 'mid.sentence :=
+  #2 'after.sentence :=
+  #3 'after.block :=
+}
+STRINGS { s t}
+FUNCTION {output.nonnull}
+{ 's :=
+  output.state mid.sentence =
+    { ", " * write$ }
+    { output.state after.block =
+        { add.period$ write$
+          newline$
+          "\newblock " write$
+        }
+        { output.state before.all =
+            'write$
+            { add.period$ " " * write$ }
+          if$
+        }
+      if$
+      mid.sentence 'output.state :=
+    }
+  if$
+  s
+}
+FUNCTION {output}
+{ duplicate$ empty$
+    'pop$
+    'output.nonnull
+  if$
+}
+FUNCTION {output.check}
+{ 't :=
+  duplicate$ empty$
+    { pop$ "empty " t * " in " * cite$ * warning$ }
+    'output.nonnull
+  if$
+}
+FUNCTION {fin.entry}
+{ add.period$
+  write$
+  newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+    'skip$
+    { after.block 'output.state := }
+  if$
+}
+FUNCTION {new.sentence}
+{ output.state after.block =
+    'skip$
+    { output.state before.all =
+        'skip$
+        { after.sentence 'output.state := }
+      if$
+    }
+  if$
+}
+FUNCTION {add.blank}
+{  " " * before.all 'output.state :=
+}
+
+FUNCTION {date.block}
+{
+  ":" *
+  add.blank
+}
+
+FUNCTION {not}
+{   { #0 }
+    { #1 }
+  if$
+}
+FUNCTION {and}
+{   'skip$
+    { pop$ #0 }
+  if$
+}
+FUNCTION {or}
+{   { pop$ #1 }
+    'skip$
+  if$
+}
+FUNCTION {new.block.checkb}
+{ empty$
+  swap$ empty$
+  and
+    'skip$
+    'new.block
+  if$
+}
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+    { pop$ "" }
+    'skip$
+  if$
+}
+FUNCTION {emphasize}
+{ duplicate$ empty$
+    { pop$ "" }
+    { "\textit{" swap$ * "}" * }
+  if$
+}
+FUNCTION {bolden}
+{ duplicate$ empty$
+    { pop$ "" }
+    { "\textbf{" swap$ * "}" * }
+  if$
+}
+FUNCTION {tie.or.space.prefix}
+{ duplicate$ text.length$ #3 <
+    { "~" }
+    { " " }
+  if$
+  swap$
+}
+
+FUNCTION {capitalize}
+{ "u" change.case$ "t" change.case$ }
+
+FUNCTION {space.word}
+{ " " swap$ * " " * }
+ % Here are the language-specific definitions for explicit words.
+ % Each function has a name bbl.xxx where xxx is the English word.
+ % The language selected here is ENGLISH
+FUNCTION {bbl.and}
+{ "and"}
+
+FUNCTION {bbl.etal}
+{ "et~al." }
+
+FUNCTION {bbl.editors}
+{ "Eds." }
+
+FUNCTION {bbl.editor}
+{ "Ed." }
+
+FUNCTION {bbl.edby}
+{ "edited by" }
+
+FUNCTION {bbl.edition}
+{ "ed." }
+
+FUNCTION {bbl.volume}
+{ "Vol." }
+
+FUNCTION {bbl.of}
+{ "of" }
+
+FUNCTION {bbl.number}
+{ "No." }
+
+FUNCTION {bbl.nr}
+{ "No." }
+
+FUNCTION {bbl.in}
+{ "in" }
+
+FUNCTION {bbl.pages}
+{ "pp." }
+
+FUNCTION {bbl.page}
+{ "p." }
+
+FUNCTION {bbl.chapter}
+{ "chap." }
+
+FUNCTION {bbl.techrep}
+{ "Tech. Rep." }
+
+FUNCTION {bbl.mthesis}
+{ "Master's thesis" }
+
+FUNCTION {bbl.phdthesis}
+{ "Ph.D. thesis" }
+
+FUNCTION {bbl.first}
+{ "1st" }
+
+FUNCTION {bbl.second}
+{ "2d" }
+
+FUNCTION {bbl.third}
+{ "3d" }
+
+FUNCTION {bbl.fourth}
+{ "4th" }
+
+FUNCTION {bbl.fifth}
+{ "5th" }
+
+FUNCTION {bbl.st}
+{ "st" }
+
+FUNCTION {bbl.nd}
+{ "d" }
+
+FUNCTION {bbl.rd}
+{ "d" }
+
+FUNCTION {bbl.th}
+{ "th" }
+
+MACRO {jan} {"Jan."}
+
+MACRO {feb} {"Feb."}
+
+MACRO {mar} {"Mar."}
+
+MACRO {apr} {"Apr."}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"Jun."}
+
+MACRO {jul} {"Jul."}
+
+MACRO {aug} {"Aug."}
+
+MACRO {sep} {"Sep."}
+
+MACRO {oct} {"Oct."}
+
+MACRO {nov} {"Nov."}
+
+MACRO {dec} {"Dec."}
+
+FUNCTION {eng.ord}
+{ duplicate$ "1" swap$ *
+  #-2 #1 substring$ "1" =
+     { bbl.th * }
+     { duplicate$ #-1 #1 substring$
+       duplicate$ "1" =
+         { pop$ bbl.st * }
+         { duplicate$ "2" =
+             { pop$ bbl.nd * }
+             { "3" =
+                 { bbl.rd * }
+                 { bbl.th * }
+               if$
+             }
+           if$
+          }
+       if$
+     }
+   if$
+}
+
+MACRO {acmcs} {"ACM Comput. Surv."}
+
+MACRO {acta} {"Acta Inf."}
+
+MACRO {cacm} {"Commun. ACM"}
+
+MACRO {ibmjrd} {"IBM J. Res. Dev."}
+
+MACRO {ibmsj} {"IBM Syst.~J."}
+
+MACRO {ieeese} {"IEEE Trans. Software Eng."}
+
+MACRO {ieeetc} {"IEEE Trans. Comput."}
+
+MACRO {ieeetcad}
+ {"IEEE Trans. Comput. Aid. Des."}
+
+MACRO {ipl} {"Inf. Process. Lett."}
+
+MACRO {jacm} {"J.~ACM"}
+
+MACRO {jcss} {"J.~Comput. Syst. Sci."}
+
+MACRO {scp} {"Sci. Comput. Program."}
+
+MACRO {sicomp} {"SIAM J. Comput."}
+
+MACRO {tocs} {"ACM Trans. Comput. Syst."}
+
+MACRO {tods} {"ACM Trans. Database Syst."}
+
+MACRO {tog} {"ACM Trans. Graphic."}
+
+MACRO {toms} {"ACM Trans. Math. Software"}
+
+MACRO {toois} {"ACM Trans. Office Inf. Syst."}
+
+MACRO {toplas} {"ACM Trans. Progr. Lang. Syst."}
+
+MACRO {tcs} {"Theor. Comput. Sci."}
+
+FUNCTION {bibinfo.check}
+{ swap$
+  duplicate$ missing$
+    {
+      pop$ pop$
+      ""
+    }
+    { duplicate$ empty$
+        {
+          swap$ pop$
+        }
+        { swap$
+          pop$
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {bibinfo.warn}
+{ swap$
+  duplicate$ missing$
+    {
+      swap$ "missing " swap$ * " in " * cite$ * warning$ pop$
+      ""
+    }
+    { duplicate$ empty$
+        {
+          swap$ "empty " swap$ * " in " * cite$ * warning$
+        }
+        { swap$
+          pop$
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.eprint}
+{ eprint duplicate$ empty$
+    'skip$
+    { "\eprint"
+      archive empty$
+        'skip$
+        { "[" * archive * "]" * }
+      if$
+      "{" * swap$ * "}" *
+    }
+  if$
+}
+STRINGS  { bibinfo}
+INTEGERS { nameptr namesleft numnames }
+
+FUNCTION {format.names}
+{ 'bibinfo :=
+  duplicate$ empty$ 'skip$ {
+  's :=
+  "" 't :=
+  #1 'nameptr :=
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr
+      duplicate$ #1 >
+        { "{f.~}{vv~}{ll}{, jj}" }
+        { "{vv~}{ll}{, f.}{, jj}" }
+      if$
+      format.name$
+      bibinfo bibinfo.check
+      't :=
+      nameptr #1 >
+        {
+          namesleft #1 >
+            { ", " * t * }
+            {
+              numnames #2 >
+                { "," * }
+                'skip$
+              if$
+              s nameptr "{ll}" format.name$ duplicate$ "others" =
+                { 't := }
+                { pop$ }
+              if$
+              t "others" =
+                {
+                  " " * bbl.etal *
+                }
+                {
+                  bbl.and
+                  space.word * t *
+                }
+              if$
+            }
+          if$
+        }
+        't
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+  } if$
+}
+FUNCTION {format.names.ed}
+{
+  format.names
+}
+FUNCTION {format.key}
+{ empty$
+    { key field.or.null }
+    { "" }
+  if$
+}
+
+FUNCTION {format.authors}
+{ author "author" format.names
+}
+FUNCTION {get.bbl.editor}
+{ editor num.names$ #1 > 'bbl.editors 'bbl.editor if$ }
+
+FUNCTION {format.editors}
+{ editor "editor" format.names duplicate$ empty$ 'skip$
+    {
+      "," *
+      " " *
+      get.bbl.editor
+      capitalize
+   "(" swap$ * ")" *
+      *
+    }
+  if$
+}
+FUNCTION {format.book.pages}
+{ pages "pages" bibinfo.check
+  duplicate$ empty$ 'skip$
+    { " " * bbl.pages * }
+  if$
+}
+FUNCTION {format.doi}
+{ doi "doi" bibinfo.check
+  duplicate$ empty$ 'skip$
+    {
+      "\doi{" swap$ * "}" *
+    }
+  if$
+}
+FUNCTION {format.note}
+{
+  url empty$
+    'skip$
+    { "\urlprefix\url{" url * "}" * output }
+  if$
+ note empty$
+    { "" }
+    { note #1 #1 substring$
+      duplicate$ "{" =
+        'skip$
+        { output.state mid.sentence =
+          { "l" }
+          { "u" }
+        if$
+        change.case$
+        }
+      if$
+      note #2 global.max$ substring$ * "note" bibinfo.check
+    }
+  if$
+}
+
+FUNCTION {format.title}
+{ title
+  duplicate$ empty$ 'skip$
+    { "t" change.case$ }
+  if$
+  "title" bibinfo.check
+}
+FUNCTION {format.full.names}
+{'s :=
+ "" 't :=
+  #1 'nameptr :=
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr
+      "{vv~}{ll}" format.name$
+      't :=
+      nameptr #1 >
+        {
+          namesleft #1 >
+            { ", " * t * }
+            {
+              s nameptr "{ll}" format.name$ duplicate$ "others" =
+                { 't := }
+                { pop$ }
+              if$
+              t "others" =
+                {
+                  " " * bbl.etal *
+                }
+                {
+                  numnames #2 >
+                    { "," * }
+                    'skip$
+                  if$
+                  bbl.and
+                  space.word * t *
+                }
+              if$
+            }
+          if$
+        }
+        't
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+}
+
+FUNCTION {author.editor.key.full}
+{ author empty$
+    { editor empty$
+        { key empty$
+            { cite$ #1 #3 substring$ }
+            'key
+          if$
+        }
+        { editor format.full.names }
+      if$
+    }
+    { author format.full.names }
+  if$
+}
+
+FUNCTION {author.key.full}
+{ author empty$
+    { key empty$
+         { cite$ #1 #3 substring$ }
+          'key
+      if$
+    }
+    { author format.full.names }
+  if$
+}
+
+FUNCTION {editor.key.full}
+{ editor empty$
+    { key empty$
+         { cite$ #1 #3 substring$ }
+          'key
+      if$
+    }
+    { editor format.full.names }
+  if$
+}
+
+FUNCTION {make.full.names}
+{ type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.key.full
+    { type$ "proceedings" =
+        'editor.key.full
+        'author.key.full
+      if$
+    }
+  if$
+}
+
+FUNCTION {output.bibitem}
+{ newline$
+  "\bibitem[{" write$
+  label write$
+  ")" make.full.names duplicate$ short.list =
+     { pop$ }
+     { * }
+   if$
+  "}]{" * write$
+  cite$ write$
+  "}" write$
+  newline$
+  ""
+  before.all 'output.state :=
+}
+
+FUNCTION {if.digit}
+{ duplicate$ "0" =
+  swap$ duplicate$ "1" =
+  swap$ duplicate$ "2" =
+  swap$ duplicate$ "3" =
+  swap$ duplicate$ "4" =
+  swap$ duplicate$ "5" =
+  swap$ duplicate$ "6" =
+  swap$ duplicate$ "7" =
+  swap$ duplicate$ "8" =
+  swap$ "9" = or or or or or or or or or
+}
+FUNCTION {n.separate}
+{ 't :=
+  ""
+  #0 'numnames :=
+  { t empty$ not }
+  { t #-1 #1 substring$ if.digit
+      { numnames #1 + 'numnames := }
+      { #0 'numnames := }
+    if$
+    t #-1 #1 substring$ swap$ *
+    t #-2 global.max$ substring$ 't :=
+    numnames #5 =
+      { duplicate$ #1 #2 substring$ swap$
+        #3 global.max$ substring$
+        "\," swap$ * *
+      }
+      'skip$
+    if$
+  }
+  while$
+}
+FUNCTION {n.dashify}
+{
+  n.separate
+  't :=
+  ""
+    { t empty$ not }
+    { t #1 #1 substring$ "-" =
+        { t #1 #2 substring$ "--" = not
+            { "--" *
+              t #2 global.max$ substring$ 't :=
+            }
+            {   { t #1 #1 substring$ "-" = }
+                { "-" *
+                  t #2 global.max$ substring$ 't :=
+                }
+              while$
+            }
+          if$
+        }
+        { t #1 #1 substring$ *
+          t #2 global.max$ substring$ 't :=
+        }
+      if$
+    }
+  while$
+}
+
+FUNCTION {word.in}
+{ "" }
+
+FUNCTION {format.date}
+{ year "year" bibinfo.check duplicate$ empty$
+    {
+      "empty year in " cite$ * "; set to ????" * warning$
+       pop$ "????"
+    }
+    'skip$
+  if$
+  extra.label *
+  before.all 'output.state :=
+  ", " swap$ *
+}
+FUNCTION {format.btitle}
+{ title "title" bibinfo.check
+  duplicate$ empty$ 'skip$
+    {
+      emphasize
+    }
+  if$
+}
+FUNCTION {either.or.check}
+{ empty$
+    'pop$
+    { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+  if$
+}
+FUNCTION {format.bvolume}
+{ volume empty$
+    { "" }
+    { bbl.volume volume tie.or.space.prefix
+      "volume" bibinfo.check * *
+      series "series" bibinfo.check
+      duplicate$ empty$ 'pop$
+%%% NO ITALIC SERIES        { emphasize ", " * swap$ * }
+      {", " * swap$ * }
+      if$
+      "volume and number" number either.or.check
+    }
+  if$
+}
+FUNCTION {format.number.series}
+{ volume empty$
+    { number empty$
+        { series field.or.null }
+        { series empty$
+            { number "number" bibinfo.check }
+            { output.state mid.sentence =
+                { bbl.number }
+                { bbl.number capitalize }
+              if$
+              number tie.or.space.prefix "number" bibinfo.check * *
+              bbl.in space.word *
+              series "series" bibinfo.check *
+            }
+          if$
+        }
+      if$
+    }
+    { "" }
+  if$
+}
+FUNCTION {is.num}
+{ chr.to.int$
+  duplicate$ "0" chr.to.int$ < not
+  swap$ "9" chr.to.int$ > not and
+}
+
+FUNCTION {extract.num}
+{ duplicate$ 't :=
+  "" 's :=
+  { t empty$ not }
+  { t #1 #1 substring$
+    t #2 global.max$ substring$ 't :=
+    duplicate$ is.num
+      { s swap$ * 's := }
+      { pop$ "" 't := }
+    if$
+  }
+  while$
+  s empty$
+    'skip$
+    { pop$ s }
+  if$
+}
+
+FUNCTION {convert.edition}
+{ extract.num "l" change.case$ 's :=
+  s "first" = s "1" = or
+    { bbl.first 't := }
+    { s "second" = s "2" = or
+        { bbl.second 't := }
+        { s "third" = s "3" = or
+            { bbl.third 't := }
+            { s "fourth" = s "4" = or
+                { bbl.fourth 't := }
+                { s "fifth" = s "5" = or
+                    { bbl.fifth 't := }
+                    { s #1 #1 substring$ is.num
+                        { s eng.ord 't := }
+                        { edition 't := }
+                      if$
+                    }
+                  if$
+                }
+              if$
+            }
+          if$
+        }
+      if$
+    }
+  if$
+  t
+}
+
+FUNCTION {format.edition}
+{ edition duplicate$ empty$ 'skip$
+    {
+      convert.edition
+      output.state mid.sentence =
+        { "l" }
+        { "t" }
+      if$ change.case$
+      "edition" bibinfo.check
+      " " * bbl.edition *
+    }
+  if$
+}
+INTEGERS { multiresult }
+FUNCTION {multi.page.check}
+{ 't :=
+  #0 'multiresult :=
+    { multiresult not
+      t empty$ not
+      and
+    }
+    { t #1 #1 substring$
+      duplicate$ "-" =
+      swap$ duplicate$ "," =
+      swap$ "+" =
+      or or
+        { #1 'multiresult := }
+        { t #2 global.max$ substring$ 't := }
+      if$
+    }
+  while$
+  multiresult
+}
+FUNCTION {format.pages}
+{ pages duplicate$ empty$ 'skip$
+    { duplicate$ multi.page.check
+        {
+          n.dashify
+        }
+        {
+        }
+      if$
+      "pages" bibinfo.check
+    }
+  if$
+}
+FUNCTION {format.journal.pages}
+{ pages duplicate$ empty$ 'pop$
+    { swap$ duplicate$ empty$
+        { pop$ pop$ format.pages }
+        {
+          ", " *
+          swap$
+          n.dashify
+          "pages" bibinfo.check
+          *
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.journal.eid}
+{ eid "eid" bibinfo.check
+  duplicate$ empty$ 'pop$
+    { swap$ duplicate$ empty$ 'skip$
+      {
+          ", " *
+      }
+      if$
+      swap$ *
+    }
+  if$
+}
+FUNCTION {format.vol.num.pages}
+{ volume field.or.null
+  duplicate$ empty$ 'skip$
+    {
+      "volume" bibinfo.check
+    }
+  if$
+  number "number" bibinfo.check duplicate$ empty$ 'skip$
+    {
+      swap$ duplicate$ empty$
+        { "there's a number but no volume in " cite$ * warning$ }
+        'skip$
+      if$
+      swap$
+      "~(" swap$ * ")" *
+    }
+  if$ *
+  bolden
+  eid empty$
+    { format.journal.pages }
+    { format.journal.eid }
+  if$
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+    'format.pages
+    { type empty$
+        { bbl.chapter }
+        { type "l" change.case$
+          "type" bibinfo.check
+        }
+      if$
+      chapter tie.or.space.prefix
+      "chapter" bibinfo.check
+      * *
+      pages empty$
+        'skip$
+        { ", " * format.pages * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.booktitle}
+{
+  booktitle "booktitle" bibinfo.check
+  emphasize
+}
+FUNCTION {format.in.ed.booktitle}
+{ format.booktitle duplicate$ empty$ 'skip$
+    {
+      editor "editor" format.names.ed duplicate$ empty$ 'pop$
+        {
+          get.bbl.editor
+          swap$ "," *
+          " " * swap$ *
+          swap$
+          "," *
+          " " * swap$
+          * }
+      if$
+      word.in swap$ *
+    }
+  if$
+}
+FUNCTION {format.thesis.type}
+{ type duplicate$ empty$
+    'pop$
+    { swap$ pop$
+      "t" change.case$ "type" bibinfo.check
+    }
+  if$
+}
+FUNCTION {format.tr.number}
+{ number "number" bibinfo.check
+  type duplicate$ empty$
+    { pop$ bbl.techrep }
+    'skip$
+  if$
+  "type" bibinfo.check
+  swap$ duplicate$ empty$
+    { pop$ "t" change.case$ }
+    { tie.or.space.prefix * * }
+  if$
+}
+FUNCTION {format.article.crossref}
+{
+  word.in
+  " \cite{" * crossref * "}" *
+}
+FUNCTION {format.book.crossref}
+{ volume duplicate$ empty$
+    { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+      pop$ word.in
+    }
+    { bbl.volume
+      capitalize
+      swap$ tie.or.space.prefix "volume" bibinfo.check * * bbl.of space.word *
+    }
+  if$
+  " \cite{" * crossref * "}" *
+}
+FUNCTION {format.incoll.inproc.crossref}
+{
+  word.in
+  " \cite{" * crossref * "}" *
+}
+FUNCTION {format.org.or.pub}
+{ 't :=
+  ""
+  address empty$ t empty$ and
+    'skip$
+    {
+      t empty$
+        { address "address" bibinfo.check *
+        }
+        { t *
+          address empty$
+            'skip$
+            { ", " * address "address" bibinfo.check * }
+          if$
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.publisher.address}
+{ publisher "publisher" bibinfo.warn format.org.or.pub
+}
+
+FUNCTION {format.organization.address}
+{ organization "organization" bibinfo.check format.org.or.pub
+}
+
+STRINGS {oldname}
+
+FUNCTION {name.or.dash}
+{ 's :=
+   oldname empty$
+     { s 'oldname := s }
+     { s oldname =
+         { "---{}---{}---" }
+         { s 'oldname := s }
+       if$
+     }
+   if$
+}
+
+FUNCTION {article}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  new.sentence
+  crossref missing$
+    {
+      journal
+      "journal" bibinfo.check
+      emphasize
+      "journal" output.check
+      format.vol.num.pages output
+  format.doi output
+    }
+    { format.article.crossref output.nonnull
+      format.pages output
+    }
+  if$
+  format.note output
+  format.eprint output
+  fin.entry
+}
+FUNCTION {book}
+{ output.bibitem
+  author empty$
+    { format.editors "author and editor" output.check
+      editor format.key output
+      add.blank
+      name.or.dash
+    }
+    { format.authors output.nonnull
+      name.or.dash
+      crossref missing$
+        { "author and editor" editor either.or.check }
+        'skip$
+      if$
+    }
+  if$
+  format.date "year" output.check
+  date.block
+  format.btitle "title" output.check
+  crossref missing$
+    { format.bvolume output
+  new.sentence
+      format.edition output
+      format.number.series output
+      format.publisher.address output
+    }
+    {
+  new.sentence
+      format.book.crossref output.nonnull
+    }
+  if$
+  format.book.pages output
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+FUNCTION {booklet}
+{ output.bibitem
+  format.authors output
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  new.sentence
+  howpublished "howpublished" bibinfo.check output
+  address "address" bibinfo.check output
+  format.book.pages output
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+  author empty$
+    { format.editors "author and editor" output.check
+      editor format.key output
+      name.or.dash
+    }
+    { format.authors output.nonnull
+      name.or.dash
+      crossref missing$
+        { "author and editor" editor either.or.check }
+        'skip$
+      if$
+    }
+  if$
+  format.date "year" output.check
+  date.block
+  format.btitle "title" output.check
+  crossref missing$
+    {
+      format.bvolume output
+      format.chapter.pages "chapter and pages" output.check
+  new.sentence
+      format.edition output
+      format.number.series output
+      format.publisher.address output
+    }
+    {
+      format.chapter.pages "chapter and pages" output.check
+  new.sentence
+      format.book.crossref output.nonnull
+    }
+  if$
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  new.sentence
+  crossref missing$
+    { format.in.ed.booktitle "booktitle" output.check
+      format.publisher.address output      
+      format.bvolume output
+      format.number.series output
+      format.edition output
+      format.chapter.pages output
+    }
+    { format.incoll.inproc.crossref output.nonnull
+      format.chapter.pages output
+    }
+  if$
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+FUNCTION {inproceedings}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  new.sentence
+  crossref missing$
+    { format.in.ed.booktitle "booktitle" output.check
+      publisher empty$
+        { format.organization.address output }
+        { organization "organization" bibinfo.check output
+          format.publisher.address output
+        }
+      if$
+      format.bvolume output
+      format.pages output
+      format.number.series output
+      howpublished "howpublished" bibinfo.check output
+    }
+    { format.incoll.inproc.crossref output.nonnull
+      format.pages output
+    }
+  if$
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+FUNCTION {conference} { inproceedings }
+FUNCTION {manual}
+{ output.bibitem
+  format.authors output
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.btitle "title" output.check
+  new.sentence
+  organization "organization" bibinfo.check output
+  address "address" bibinfo.check output
+  format.edition output
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.title
+  "title" output.check
+  new.sentence
+  bbl.mthesis format.thesis.type output.nonnull
+  department "department" bibinfo.warn output
+  school "school" bibinfo.warn output
+  address "address" bibinfo.check output
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+  format.authors output
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.title output
+  new.sentence
+  howpublished "howpublished" bibinfo.check output
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+FUNCTION {phdthesis}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.title
+  "title" output.check
+  new.sentence
+  bbl.phdthesis format.thesis.type output.nonnull
+  school "school" bibinfo.warn output
+  address "address" bibinfo.check output
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+  format.editors output
+  editor format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.btitle "title" output.check
+  format.bvolume output
+  format.number.series output
+  publisher empty$
+    { format.organization.address output }
+    { organization "organization" bibinfo.check output
+      format.publisher.address output
+    }
+  if$
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.title
+  "title" output.check
+  new.sentence
+  format.tr.number output.nonnull
+  institution "institution" bibinfo.warn output
+  address "address" bibinfo.check output
+  format.doi output
+  format.note output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  name.or.dash
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  format.doi output
+  format.note "note" output.check
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {default.type} { misc }
+READ
+FUNCTION {sortify}
+{ purify$
+  "l" change.case$
+}
+INTEGERS { len }
+FUNCTION {chop.word}
+{ 's :=
+  'len :=
+  s #1 len substring$ =
+    { s len #1 + global.max$ substring$ }
+    's
+  if$
+}
+FUNCTION {format.lab.names}
+{ 's :=
+  "" 't :=
+  s #1 "{vv~}{ll}" format.name$
+  s num.names$ duplicate$
+  #2 >
+    { pop$
+      " " * bbl.etal *
+    }
+    { #2 <
+        'skip$
+        { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+            {
+              " " * bbl.etal *
+            }
+            { bbl.and space.word * s #2 "{vv~}{ll}" format.name$
+              * }
+          if$
+        }
+      if$
+    }
+  if$
+}
+
+FUNCTION {author.key.label}
+{ author empty$
+    { key empty$
+        { cite$ #1 #3 substring$ }
+        'key
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {author.editor.key.label}
+{ author empty$
+    { editor empty$
+        { key empty$
+            { cite$ #1 #3 substring$ }
+            'key
+          if$
+        }
+        { editor format.lab.names }
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {editor.key.label}
+{ editor empty$
+    { key empty$
+        { cite$ #1 #3 substring$ }
+        'key
+      if$
+    }
+    { editor format.lab.names }
+  if$
+}
+
+FUNCTION {calc.short.authors}
+{ type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.key.label
+    { type$ "proceedings" =
+        'editor.key.label
+        'author.key.label
+      if$
+    }
+  if$
+  'short.list :=
+}
+
+FUNCTION {calc.label}
+{ calc.short.authors
+  short.list
+  "("
+  *
+  year duplicate$ empty$
+     { pop$ "????" }
+     { purify$ #-1 #4 substring$ }
+  if$
+  *
+  'label :=
+}
+
+FUNCTION {sort.format.names}
+{ 's :=
+  #1 'nameptr :=
+  ""
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr
+      "{vv{ } }{ll{ }}{  f{ }}{  jj{ }}"
+      format.name$ 't :=
+      nameptr #1 >
+        {
+          "   "  *
+          namesleft #1 = t "others" = and
+            { "zzzzz" * }
+            { t sortify * }
+          if$
+        }
+        { t sortify * }
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+}
+
+FUNCTION {sort.format.title}
+{ 't :=
+  "A " #2
+    "An " #3
+      "The " #4 t chop.word
+    chop.word
+  chop.word
+  sortify
+  #1 global.max$ substring$
+}
+FUNCTION {author.sort}
+{ author empty$
+    { key empty$
+        { "to sort, need author or key in " cite$ * warning$
+          ""
+        }
+        { key sortify }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+FUNCTION {author.editor.sort}
+{ author empty$
+    { editor empty$
+        { key empty$
+            { "to sort, need author, editor, or key in " cite$ * warning$
+              ""
+            }
+            { key sortify }
+          if$
+        }
+        { editor sort.format.names }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+FUNCTION {editor.sort}
+{ editor empty$
+    { key empty$
+        { "to sort, need editor or key in " cite$ * warning$
+          ""
+        }
+        { key sortify }
+      if$
+    }
+    { editor sort.format.names }
+  if$
+}
+FUNCTION {presort}
+{ calc.label
+  label sortify
+  "    "
+  *
+  type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.sort
+    { type$ "proceedings" =
+        'editor.sort
+        'author.sort
+      if$
+    }
+  if$
+  #1 entry.max$ substring$
+  'sort.label :=
+  sort.label
+  *
+  "    "
+  *
+  title field.or.null
+  sort.format.title
+  *
+  #1 entry.max$ substring$
+  'sort.key$ :=
+}
+
+ITERATE {presort}
+SORT
+STRINGS { last.label next.extra }
+INTEGERS { last.extra.num number.label }
+FUNCTION {initialize.extra.label.stuff}
+{ #0 int.to.chr$ 'last.label :=
+  "" 'next.extra :=
+  #0 'last.extra.num :=
+  #0 'number.label :=
+}
+FUNCTION {forward.pass}
+{ last.label label =
+    { last.extra.num #1 + 'last.extra.num :=
+      last.extra.num int.to.chr$ 'extra.label :=
+    }
+    { "a" chr.to.int$ 'last.extra.num :=
+      "" 'extra.label :=
+      label 'last.label :=
+    }
+  if$
+  number.label #1 + 'number.label :=
+}
+FUNCTION {reverse.pass}
+{ next.extra "b" =
+    { "a" 'extra.label := }
+    'skip$
+  if$
+  extra.label 'next.extra :=
+  extra.label
+  duplicate$ empty$
+    'skip$
+    { "{\natexlab{" swap$ * "}}" * }
+  if$
+  'extra.label :=
+  label extra.label * 'label :=
+}
+EXECUTE {initialize.extra.label.stuff}
+ITERATE {forward.pass}
+REVERSE {reverse.pass}
+FUNCTION {bib.sort.order}
+{ sort.label
+  "    "
+  *
+  year field.or.null sortify
+  *
+  "    "
+  *
+  title field.or.null
+  sort.format.title
+  *
+  #1 entry.max$ substring$
+  'sort.key$ :=
+}
+ITERATE {bib.sort.order}
+SORT
+FUNCTION {begin.bib}
+{ preamble$ empty$
+    'skip$
+    { preamble$ write$ newline$ }
+  if$
+  "\begin{thebibliography}{" number.label int.to.str$ * "}" *
+  write$ newline$
+  "\providecommand{\natexlab}[1]{#1}"
+  write$ newline$
+  "\providecommand{\url}[1]{\texttt{#1}}"
+  write$ newline$
+  "\providecommand{\urlprefix}{URL }"
+  write$ newline$
+  "\expandafter\ifx\csname urlstyle\endcsname\relax"
+  write$ newline$
+  "  \providecommand{\doi}[1]{doi:\discretionary{}{}{}#1}\else"
+  write$ newline$
+  "  \providecommand{\doi}{doi:\discretionary{}{}{}\begingroup \urlstyle{rm}\Url}\fi"
+  write$ newline$
+  "\providecommand{\eprint}[2][]{\url{#2}}"
+  write$ newline$
+}
+EXECUTE {begin.bib}
+EXECUTE {init.state.consts}
+ITERATE {call.type$}
+FUNCTION {end.bib}
+{ newline$
+  "\end{thebibliography}" write$ newline$
+}
+EXECUTE {end.bib}
+%% End of customized bst file
+%%
+%% End of file `ametsoc7.bst'.
Index: trunk/DOC/TexFiles/clean.sh
===================================================================
--- trunk/DOC/TexFiles/clean.sh	(revision 994)
+++ trunk/DOC/TexFiles/clean.sh	(revision 994)
@@ -0,0 +1,2 @@
+rm -f $( ls -1 NEMO_book.* | egrep -v "(tex|pdf)" ) *mpgraph*
+rm -f Chapters/*.aux Chapters/*.log
Index: trunk/DOC/TexFiles/namlist_opa_New
===================================================================
--- trunk/DOC/TexFiles/namlist_opa_New	(revision 994)
+++ trunk/DOC/TexFiles/namlist_opa_New	(revision 994)
@@ -0,0 +1,513 @@
+!!>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+!! OPA namelists :   1 - run manager      (namrun, nam_ctl, nam_mpp)
+!!                   2 - Domain           (nam_zgr, nam_zgr_sco, namdom)
+!!                   3 - Surface boundary (namtau, namflx, namqsr, namfwb, namalb, namcpl)
+!!                   4 - lateral boundary (namlbc, namobc, namcla)
+!!                   5 - bottom  boundary (nambfr, nambbc, nambbl)
+!!                   6 - Tracer           (nam_traadv, nam_traldf, namtdp)
+!!                   7 - dynamics         (nam_dynvor, nam_dynhpg, namcfg, nam_dynldf)
+!!                   8 - Verical physics  (namzdf, namnpc, namgap, namznl, namspr, namflo)
+!!                   9 - diagnostics      (namtrd, namgap, namznl, namptr, namspr, namflo)
+!!                   9 - miscellaneous    (solver)
+!!>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+!  CAUTION: some scripts does not support CAPITALs for logical definition 
+!  *******       use .true. or .false. and NOT .TRUE. or .FALSE.
+
+!!======================================================================
+!!                   ***  Run management namelists  ***
+!!======================================================================
+!!   namrun       parameters of the run
+!!   nam_ctl      Control prints & Benchmark
+!!   nam_mpp      Massively Parallel Processing
+!!======================================================================
+
+!-----------------------------------------------------------------------
+&namrun   !   parameters of the run
+!-----------------------------------------------------------------------
+   no         =       0    !  job number
+   cexper     =  "ORCA2"   !  experience name for vairmer format
+   ln_rstart  = .false.    !  boolean term for restart (true or false)
+   nrstdt     =       0    !  restart control = 0 restart, do not control nit000 in the restart file.
+                           !                  = 1 restart, control nit000 in the restart file. Do not
+                           !                      use the date in the restart file (use ndate0 in namelist)
+                           !                  = 2 restart, control nit000 in the restart file, use the date
+                           !                      in the restart file. ndate0 in the namelist is ignored.
+   nit000     =       1    !  number of the first time step
+   nitend     =    5475    !  number of the first time step
+   ndate0     =  010101    !  initial calendar date aammjj
+   nleapy     =       0    !  Leap year calendar (0/1)
+   ninist     =       0    !  initial state output flag (0/1)
+   nstock     =    5475    !  number of the first time step
+   nwrite     =    5475    !  frequency of OUTPUT file
+   nrunoff    =       2    !  = 0 no, 1 runoff, 2 runoff+river mouth ups adv
+/
+!-----------------------------------------------------------------------
+&nam_ctl      !   Control prints & Benchmark
+!-----------------------------------------------------------------------
+   ln_ctl =  .false.       !  trends control print (expensive!)
+   nprint =       0        !  level of print (0 no print)
+   nictls =       0        !  start i indice to make the control SUM (very usefull to compare mono-
+   nictle =       0        !  end   i indice to make the control SUM (-versus multi processor runs)
+   njctls =       0        !  start j indice to make the control SUM (very usefull to compare mono-
+   njctle =       0        !  end   j indice to make the control SUM (-versus multi processor runs)
+   isplt  =       1        !  number of processors following i
+   jsplt  =       1        !  number of processors following j
+   nbench =       0        !  Bench parameter (0/1): CAUTION it must be zero except for bench
+                           !     for which we don't care about physical meaning of the results
+   nbit_cmp =     0        !  bit comparison mode parameter (0/1): enables bit comparison between 
+                           !             single and multiple processor runs
+/
+!-----------------------------------------------------------------------
+&nam_mpp      !   Massively Parallel Processing
+!-----------------------------------------------------------------------
+   c_mpi_send =  'S'       !  mpi send/recieve type   ='S', 'B', or 'I' for standard send,
+                           !  buffer blocking send or immediate non-blocking sends, resp.
+/
+
+!!======================================================================
+!!                      ***  Domain namelists  ***
+!!======================================================================
+!!   nam_zgr       vertical coordinate
+!!   nam_zgr_sco   s-coordinate or hybrid z-s-coordinate
+!!   namdom        space and time domain (bathymetry, mesh, timestep)
+!!======================================================================
+!-----------------------------------------------------------------------
+&nam_zgr       !   vertical coordinate
+!-----------------------------------------------------------------------
+   ln_zco   =  .false.     !  z-coordinate - full    steps   (T/F)      ("key_zco" may also be defined)
+   ln_zps   =  .true.      !  z-coordinate - partial steps   (T/F)
+   ln_sco   =  .false.     !  s- or hybrid z-s-coordinate    (T/F)
+/
+!-----------------------------------------------------------------------
+&nam_zgr_sco   !   s-coordinate or hybrid z-s-coordinate
+!-----------------------------------------------------------------------
+   sbot_min =  300.        !  minimum depth of s-bottom surface (>0) (m)
+   sbot_max = 5250.        !  maximum depth of s-bottom surface (= ocean depth) (>0) (m)
+   theta    =    6.0       !  surface control parameter (0<=theta<=20)
+   thetb    =    0.75      !  bottom control parameter  (0<=thetb<= 1)
+   r_max    =    0.15      !  maximum cut-off r-value allowed (0<r_max<1)
+/
+!-----------------------------------------------------------------------
+&namdom   !   space and time domain (bathymetry, mesh, timestep)
+!-----------------------------------------------------------------------
+   ntopo     =     1       !  = 0/1 ,compute/read the bathymetry file (mbathy, nb of T-ocean levels)
+   e3zps_min =     5.      !  the thickness of the partial step is set larger than the minimum
+   e3zps_rat =     0.1     !  of e3zps_min and e3zps_rat * e3t   (N.B. 0<e3zps_rat<1)
+   nmsh      =     0       !  =1 create a mesh file (coordinates, scale factors, masks)
+   nacc      =     0       !  the acceleration of convergence method   = 0, no acceleration, rdt = rdttra
+                           !                                           = 1, acceleration used, rdt < rdttra(k)
+   atfp      =     0.1     !  asselin time filter parameter
+   rdt       =  5760.      !  time step for the dynamics (and tracer if nacc=0)
+   rdtmin    =  5760.      !  minimum time step on tracers
+   rdtmax    =  5760.      !  maximum time step on tracers
+   rdth      =   800.      !  depth variation of tracer time step
+   rdtbt     =    90.      !  barotropic time step (for the time splitting algorithm)
+   nfice     =     5       !  frequency of ice model call
+   nfbulk    =     5       !  frequency of bulk formulea call (not used if ice used)
+   nclosea   =     0       !  = 0 no closed sea in the model domain
+                           !  = 1 closed sea (Black Sea, Caspian Sea, Great US Lakes...) 
+/
+
+!!======================================================================
+!!            ***  Surface Boundary Condition namelists  ***
+!!======================================================================
+!!   namtau      surface wind stress
+!!   namflx      surface fluxes
+!!   namqsr      penetrative solar radiation
+!!   namfwb      freshwater budget correction
+!!   namalb      albedo parameters
+!!   namcpl      coupled ocean/atmosphere model (#ifdef "key_coupled")
+!!======================================================================
+!-----------------------------------------------------------------------
+&namtau   !   surface wind stress
+!-----------------------------------------------------------------------
+   ntau000 =      0        !  gently increase the stress over the first ntau_rst time-steps
+   tau0x   =      0.e0     !  uniform value used as default surface heat flux
+   tau0y   =      0.e0     !  uniform value used as default solar radiation flux
+/
+!-----------------------------------------------------------------------
+&namflx   !   surface fluxes
+!-----------------------------------------------------------------------
+   q0      =      0.e0     !  uniform value used as default surface heat flux
+   qsr0    =      0.e0     !  uniform value used as default solar radiation flux
+   emp0    =      0.e0     !  uniform value used as default surface freswater budget (E-P)
+   dqdt0   =      -40.     !  feedback coefficient for SST damping (W/m2/K)
+   deds0   =        0.     !  feedback coefficient for SSS damping (mm/day)
+/
+!-----------------------------------------------------------------------
+&namqsr   !   penetrative solar radiation
+!-----------------------------------------------------------------------
+   ln_traqsr = .true.      !  penetrative solar radiation (T) or not (F)     (Default=T)
+   rabs     =   0.58       !  fraction of qsr associated with xsi1
+   xsi1     =   0.35       !  first depth of extinction
+   xsi2     =   23.0       !  second depth of extinction
+/
+!-----------------------------------------------------------------------
+&namfwb   !   freshwater budget correction
+!-----------------------------------------------------------------------
+   ln_fwb    = .true.      !  flag for freshwater budget correction (0 annual mean)
+/
+!-----------------------------------------------------------------------
+&namalb   !   albedo parameters
+!-----------------------------------------------------------------------
+   cgren    =      0.06    !  correction of the snow or ice albedo to take into account
+   albice   =      0.5     !  albedo of melting ice in the arctic and antarctic
+   alphd    =      0.80    !  coefficients for linear interpolation used to
+   alphc    =      0.65    !  compute albedo between two extremes values 
+   alphdi   =      0.72    !  (Pyane, 1972)
+/
+!-----------------------------------------------------------------------
+&namcpl   !   coupled ocean/atmosphere model                            ("key_coupled")
+!-----------------------------------------------------------------------
+/
+
+!!======================================================================
+!!               ***  Lateral boundary condition  ***
+!!======================================================================
+!!   namlbc        lateral momentum boundary condition
+!!   namcro        cross land advection
+!!   namobc        open boundaries parameters                           ("key_obc")
+!!======================================================================
+
+!-----------------------------------------------------------------------
+&namlbc   !   lateral momentum boundary condition
+!-----------------------------------------------------------------------
+   shlat  =      2.        !      shlat = 0 : free slip
+                           !  0 < shlat < 2 : partial slip
+                           !      shlat = 2 : no slip
+                           !  2 < shlat     : strong slip
+/
+!-----------------------------------------------------------------------
+&namcro   !   cross land advection
+!-----------------------------------------------------------------------
+!  n_cla   advection between 2 ocean pts separates by land 
+   n_cla   = 1
+/
+!-----------------------------------------------------------------------
+&namobc   !   open boundaries parameters                            ("key_obc")
+!-----------------------------------------------------------------------
+    nobc_dta    =    0     !  = 0 the obc data are equal to the initial state
+                           !  = 1 the obc data are read in 'obc   .dta' files
+    rdpein      =    1.    !  ???
+    rdpwin      =    1.    !  ???
+    rdpnin      =   30.    !  ???
+    rdpsin      =    1.    !  ???
+    rdpeob      = 1500.    !  time relaxation (days) for the east  open boundary
+    rdpwob      =   15.    !    "        "           for the west  open boundary
+    rdpnob      =  150.    !    "        "           for the north open boundary
+    rdpsob      =   15.    !    "        "           for the south open boundary
+    zbsic1      =  140.e+6 !  barotropic stream function on first  isolated coastline
+    zbsic2      =    1.e+6 !    "                   "    on second isolated coastline
+    zbsic3      =    0.    !    "                   "    on thrid  isolated coastline
+    ln_obc_clim = .true.   !  climatological obc data files (default T)
+    ln_vol_cst  = .true.   !  total volume conserved
+/
+
+!!======================================================================
+!!                 ***  Bottom boundary condition  ***
+!!======================================================================
+!!   nambfr        bottom friction
+!!   nambbc        bottom temperature boundary condition                ("key_trabbc")
+!!======================================================================
+
+!-----------------------------------------------------------------------
+&nambfr   !   bottom friction
+!-----------------------------------------------------------------------
+   nbotfr =       1        !  type of bottom friction :   = 0 : no slip        ,  = 2 : nonlinear friction
+                           !                              = 1 : linear friction,  = 3 : free slip
+   bfri1  =   4.e-4        !  bottom drag coefficient (linear case)
+   bfri2  =   1.e-3        !  bottom drag coefficient (non linear case)
+   bfeb2  =  2.5e-3        !  bottom turbulent kinetic energy background  (m^2/s^2)
+/
+!-----------------------------------------------------------------------
+&nambbc   !   bottom temperature boundary condition
+!-----------------------------------------------------------------------
+   ngeo_flux =  2             !  = 0 no geothermal heat flux
+                              !  = 1 constant geothermal heat flux
+                              !  = 2 variable geothermal heat flux (read in geothermal_heating.nc in mW/m2) 
+   ngeo_flux_const = 86.4e-3  !  Constant value of geothermal heat flux (W/m2)
+/
+
+!!======================================================================
+!!                        Tracer (T & S ) namelists
+!!======================================================================
+!!   nameos        equation of state
+!!   nam_traadv    advection scheme
+!!   nam_traldf    lateral diffusion scheme
+!!   nambbl        bottom boundary layer scheme                         ("key_trabbl_dif","key_trabbl_adv")
+!!   namdmp        tracer newtonian damping                             ("key_tradmp")
+!!======================================================================
+
+!-----------------------------------------------------------------------
+&nameos   !   ocean physical parameters
+!-----------------------------------------------------------------------
+   neos   =      0         !  type of equation of state and Brunt-Vaisala frequency
+                           !     = 0, UNESCO (formulation of Jackett and McDougall (1994) and of McDougall (1987) )
+                           !     = 1, linear: rho(T)   = rau0 * ( 1.028 - ralpha * T )
+                           !     = 2, linear: rho(T,S) = rau0 * ( rbeta * S - ralpha * T )
+   ralpha =  2.e-4         !  thermal expension coefficient (neos= 1 or 2)
+   rbeta  =  0.001         !  saline  expension coefficient (neos= 2)
+/
+!-----------------------------------------------------------------------
+&nam_traadv   !   advection scheme for tracer 
+!-----------------------------------------------------------------------
+   ln_traadv_cen2   =  .true.    !  2nd order centered scheme    (default T)
+   ln_traadv_tvd    =  .false.   !  TVD scheme                   (default F)
+   ln_traadv_muscl  =  .false.   !  MUSCL scheme                 (default F)
+   ln_traadv_muscl2 =  .false.   !  MUSCL2 scheme                (default F)
+   ln_traadv_ubs    =  .false.   !  UBS scheme                   (default F)
+/
+!-----------------------------------------------------------------------
+&nam_traldf   !   lateral diffusion scheme for tracer 
+!-----------------------------------------------------------------------
+!                                !  Type of the operator : 
+   ln_traldf_lap    =  .true.    !     laplacian operator          (default T)
+   ln_traldf_bilap  =  .false.   !     bilaplacian operator        (default F)
+                                 !  Direction of action  :
+   ln_traldf_level  =  .false.   !     iso-level                   (default F)
+   ln_traldf_hor    =  .false.   !     horizontal (geopotential)   (default F)   (require "key_ldfslp" when ln_sco=T)
+   ln_traldf_iso    =  .true.    !     iso-neutral                 (default T)   (require "key_ldfslp")
+!                                !  Coefficient
+   aht0    =  2000.              !     horizontal eddy diffusivity for tracers (m2/s)
+   ahtb0   =     0.              !     background eddy diffusivity for isopycnal diffusion (m2/s)
+   aeiv0   =  2000.              !     eddy induced velocity coefficient (m2/s)  (require "key_traldf_eiv")
+/
+!-----------------------------------------------------------------------
+&nam_trabbl   !   bottom boundary layer scheme                          ("key_trabbl_dif","key_trabbl_adv")
+!-----------------------------------------------------------------------
+!  "key_trabbl	                  ! diffusive bbl 
+!  "key_trabbl_adv               ! advective bbl 
+   atrbbl = 10000.               !  lateral tracer coeff. for bottom boundary layer scheme (m2/s)
+/
+!-----------------------------------------------------------------------
+&namdmp   tracer newtonian damping ('key_tradmp')
+!-----------------------------------------------------------------------
+   ndmp   =   -1                 !  type of damping in temperature and salinity 
+                                 !     ='latitude', damping poleward of 'ndmp' degrees and function 
+                                 !                  of the distance-to-coast. Red and Med Seas as ndmp=-1
+                                 !     =-1 damping only in Med and Red Seas
+   ndmpf  =    1                 !  =1 create a damping.coeff NetCDF file (the 3D damping array)
+   nmldmp =    1                 !  type of damping in the mixed layer
+                                 !     =0 damping throughout the water column
+                                 !     =1 no damping in the mixed layer defined by avt >5cm2/s )
+                                 !     =2 no damping in the mixed layer defined rho<rho(surf)+.01 )
+   sdmp   =  50.                 !  surface time scale for internal damping (days)
+   bdmp   = 360.                 !  bottom  time scale for internal damping (days)
+   hdmp   = 800.                 !  depth of transition between sdmp and bdmp (meters)
+/
+
+!!======================================================================
+!!                      ***  Dynamics namelists  ***
+!!======================================================================
+!!   nam_dynvor    advection scheme
+!!   nam_dynhpg    hydrostatic pressure gradient
+!!   namflg        hydrostatic pressure gradient time stepping
+!!   nam_dynldf    lateral diffusion scheme
+!!======================================================================
+
+!-----------------------------------------------------------------------
+&nam_dynvor   !   option of physics/algorithm 
+!-----------------------------------------------------------------------
+   ln_dynvor_ene = .false.    !  enstrophy conserving scheme (default T)
+   ln_dynvor_ens = .true.     !  energy conserving scheme    (default F)
+   ln_dynvor_mix = .false.    !  mixed scheme                (default F)
+   ln_dynvor_een = .false.    !  energy & enstrophy scheme   (default F)
+/
+!-----------------------------------------------------------------------
+&nam_dynhpg   !   Hydrostatic pressure gradient option
+!-----------------------------------------------------------------------
+   ln_hpg_zco = .false.       !  z-coordinate - full steps                   
+   ln_hpg_zps = .true.        !  z-coordinate - partial steps (interpolation)
+   ln_hpg_sco = .false.       !  s-coordinate (standard jacobian formulation)
+   ln_hpg_hel = .false.       !  s-coordinate (helsinki modification)
+   ln_hpg_wdj = .false.       !  s-coordinate (weighted density jacobian)
+   ln_hpg_djc = .false.       !  s-coordinate (Density Jacobian with Cubic polynomial)
+   ln_hpg_rot = .false.       !  s-coordinate (ROTated axes scheme)
+   gamm       = 0.e0          !  weighting coefficient (wdj scheme)
+/
+!-----------------------------------------------------------------------
+&namflg   !   algorithm flags (algorithm not control by CPP keys)
+!-----------------------------------------------------------------------
+   ln_dynhpg_imp   =  .false. !  hydrostatic pressure gradient: semi-implicit time scheme  (T)
+                              !                                 centered      time scheme  (F)
+   nn_dynhpg_rst   =  0       !  add dynhpg implicit variables in restart ot not (1/0)
+/
+!-----------------------------------------------------------------------
+!nam_dynspg   !   surface pressure gradient   (CPP key only)
+!-----------------------------------------------------------------------
+!                                !  explicit free surface               ("key_dynspg_exp")
+!                                !  filtered free surface               ("key_dynspg_flt")
+!                                !  split-explicit free surface         ("key_dynspg_ts")
+!                                !  rigid-lid                           ("key_dynspg_rl")
+
+!-----------------------------------------------------------------------
+&nam_dynldf   !   lateral diffusion on momentum
+!-----------------------------------------------------------------------
+                                 !  Type of the operator : 
+   ln_dynldf_lap    =  .true.    !     laplacian operator          (default T)
+   ln_dynldf_bilap  =  .false.   !     bilaplacian operator        (default F)
+                                 !  Direction of action  : 
+   ln_dynldf_level  =  .false.   !     iso-level                   (default F)
+   ln_dynldf_hor    =  .true.    !     horizontal (geopotential)   (default F)   (require "key_ldfslp" in s-coord.)
+   ln_dynldf_iso    =  .false.   !     iso-neutral                 (default T)   (require "key_ldfslp")
+                                 !  Coefficient
+   ahm0    = 40000.              !     horizontal eddy viscosity for the dynamics (m2/s)
+   ahmb0   =     0.              !     background eddy viscosity for isopycnal diffusion (m2/s)
+/
+
+!!======================================================================
+!!             Tracers & Dynamics vertical physics namelists
+!!======================================================================
+!!       namzdf        vertical physics
+!!       namnpc        non penetrative convection                       
+!!       nambbl        bottom boundary layer scheme                     ("key_trabbl_dif", "key_trabbl_adv")
+!!       namric        richardson number dependent vertical mixing      ("key_zdfric"      )
+!!       namtke        TKE dependent vertical mixing                    ("key_zdftke"      )
+!!       namkpp        KPP dependent vertical mixing                    ("key_zdfkpp"      )
+!!       namddm        double diffusive mixing parameterization         ("key_zdfddm"      )
+!!======================================================================
+
+!-----------------------------------------------------------------------
+&namzdf   !   vertical physics
+!-----------------------------------------------------------------------
+   ln_zdfnpc = .false.        !  Non-Penetrative Convection          (default T)
+   avm0      = 1.2e-4         !  vertical eddy viscosity   for the dynamic (m2/s)   (background Kz if not "key_zdfcst")
+   avt0      = 1.2e-5         !  vertical eddy diffusivity for tracers     (m2/s)   (background Kz if not "key_zdfcst")
+   ln_zdfevd = .true.         !  enhanced vertical diffusion         (default T)
+   avevd     =   100.         !  vertical coefficient for enhanced diffusion scheme (m2/s)
+   n_evdm    =     0          !  enhanced mixing apply on tracer (=0) or on tracer and momentum (=1)
+   ln_zdfexp =  .false.       !  =T/F  split explicit / implicit      (Default=F)
+   n_zdfexp  =     3          !  number of sub-timestep for ln_zdfexp=T
+/
+!-----------------------------------------------------------------------
+&namnpc   !   non penetrative convection
+!-----------------------------------------------------------------------
+   nnpc1  =      1            !  non penetrative convective scheme computation frequency
+   nnpc2  =    365            !  non penetrative convective scheme print frequency
+/
+!-----------------------------------------------------------------------
+&namric   richardson number dependent vertical diffusion                ("key_zdfric" )
+!-----------------------------------------------------------------------
+   avmri = 100.e-4            !  maximum value of the vertical viscosity
+   alp   =      5.            !  coefficient of the parameterization
+   nric  =       2            !  coefficient of the parameterization
+/
+!-----------------------------------------------------------------------
+&namtke   !   turbulent eddy kinetic dependent vertical diffusion       ("key_zdftke" )
+!-----------------------------------------------------------------------
+   ln_rstke = .false.         !  flag to restart with tke from a run without tke (default F)
+   ediff =       0.1          !  coef. to compute vertical eddy coef. (avt=ediff*mxl*sqrt(e) )
+   ediss =       0.7          !  coef. of the Kolmogoroff dissipation
+   ebb   =      3.75          !  coef. of the surface input of tke
+   efave =        1.          !  coef. to applied to the tke diffusion ( avtke=efave*avm )
+   emin  =     1.e-6          !  minimum value of tke (m^2/s^2)
+   emin0 =     1.e-4          !  surface minimum value of tke (m^2/s^2)
+   nitke =        50          !  number of restart iterative loops
+!! ri_c  =         0.22222222 !  critic richardson number (default = 2/9)
+   nmxl  =         2          !  mixing length    = 0 bounded by the distance to surface and bottom
+                              !                   = 1 bounded by the local vertical scale factor
+                              !                   = 2 first vertical derivative of mixing length bounded by 1
+   npdl  =         1          !  prandtl number   = 0 no vertical prandtl number (avt=avm)
+                              !                   = 1 prandtl number function of richarson number (avt=pdl*avm)
+                              !                   = 2 same as = 1 but a shapiro filter is applied on pdl
+   nave  =         1          !  horizontal averaged (=1) or not (=0) on avt and amv   (default =1)
+   navb  =         0          !  constant (=0) or profile (=1) background on avt
+/
+!-----------------------------------------------------------------------
+&namkpp   !   K-Profile Parameterization dependent vertical mixing      ("key_zdfkpp", and optionnally:
+!                                                                        "key_kppcustom" or "key_kpplktb")
+!-----------------------------------------------------------------------
+   ln_kpprimix  = .true.      !  shear instability mixing  (default T)
+   difmiw       =  1.0e-04    !  constant internal wave viscosity (m2/s)
+   difsiw       =  0.1e-04    !  constant internal wave diffusivity (m2/s)
+   Riinfty      =  0.8        !  local Richardson Number limit for shear instability
+   difri        =  0.0050     !  maximum shear mixing at Rig = 0    (m2/s)
+   bvsqcon      = -0.01e-07   !  Brunt-Vaisala squared (1/s**2) for maximum convection
+   difcon       =  1.         !  maximum mixing in interior convection (m2/s)
+   navb         =  0          !  horizontal averaged (=1) or not (=0) on avt and amv   (default =1)
+   nave         =  1          !  constant (=0) or profile (=1) background on avt
+/
+!-----------------------------------------------------------------------
+&namddm   !   double diffusive mixing parameterization                  ("key_zdfddm")
+!-----------------------------------------------------------------------
+      avts  = 1.e-4           !  maximum avs (vertical mixing on salinity)
+      hsbfr = 1.6             !  heat/salt buoyancy flux ratio
+/
+
+!!======================================================================
+!!                  ***  Miscelaneous namelists  ***
+!!======================================================================
+!!   namsol      elliptic solver / island / free surface 
+!!======================================================================
+
+!-----------------------------------------------------------------------
+&namsol   !   elliptic solver / island / free surface 
+!-----------------------------------------------------------------------
+   nsolv     =      1         !  type of elliptic solver  =1 preconditioned conjugate gradient (pcg)
+                              !                           =2 successive-over-relaxation (sor)
+                              !                           =3 FETI (fet)                               ("key_feti")
+                              !                           =4 sor with extra outer halo
+   nsol_arp  =      0         !  absolute/relative (0/1) precision convergence test
+   nmin      =    300         !  minimum of iterations for the SOR solver
+   nmax      =    800         !  maximum of iterations for the SOR solver
+   nmod      =     10         !  frequency of test for the SOR solver
+   eps       =  1.E-6         !  absolute precision of the solver
+   resmax    = 1.E-10         !  absolute precision for the SOR solver
+   sor       =   1.92         !  optimal coefficient for SOR solver
+   epsisl    = 1.e-10         !  absolute precision on stream function solver
+   nmisl     =   4000         !  maximum pcg iterations for island                                    ("key_islands")
+   rnu       =     1.         !  strength of the additional force used in filtered free surface
+/
+
+!!======================================================================
+!!                       ***  Diagnostics namelists  ***
+!!======================================================================
+!!   namtrd       dynamics and/or tracer trends                         ("key_trddyn","key_trdtra","key_trdmld")
+!!   namgap       level mean model-data gap                             ("key_diagap")
+!!   namptr       Poleward Transport Diagnostics                        ("key_diaspr")
+!!   namspr       surface pressure diagnosed in rigid-lid               ("key_diaspr")
+!!   namflo       float parameters                                      ("key_float")
+!!======================================================================
+
+!-----------------------------------------------------------------------
+&namtrd    !   diagnostics on dynamics and/or tracer trends             ("key_trddyn" and/or "key_trdtra")
+!                          or mixed-layer trends                        ('key_trdmld')
+!                          or barotropic vorticity                      ("key_trdvor")
+!-----------------------------------------------------------------------
+   ntrd  = 365                   !  time step frequency dynamics and tracers trends
+   nctls =   0                   !  control surface type in mixed-layer trends (0,1 or n<jpk)
+   ln_trdmld_restart = .false.   !  restart for ML diagnostics
+   ucf   =  1.                   !  unit conversion factor (=1 -> /seconds | =86400. -> /day)
+   ln_trdmld_instant = .false.   !  flag to diagnose trends of instantantaneous or mean ML T/S
+/
+!-----------------------------------------------------------------------
+&namgap    !   level mean model-data gap                                ('key_diagap')
+!-----------------------------------------------------------------------
+   ngap =  15                    !  time-step frequency of model-data gap computation
+   nprg =  10                    !  time-step frequency of gap print in model output
+/
+!-----------------------------------------------------------------------
+&namptr   !   Poleward Transport Diagnostic
+!-----------------------------------------------------------------------
+   ln_diaptr = .false.           !  logical flag for Poleward transport computation
+   ln_subbas = .false.           !  logical flag for Atlantic/Pacific/Indian basins computation
+                                 !  need input basins mask file named "subbasins.nc"
+   nf_ptr    =  15               !  Frequency of computation
+/
+!-----------------------------------------------------------------------
+&namspr  !   surface pressure diagnostic
+!-----------------------------------------------------------------------
+   nmaxp   =   1000              !  maximum of iterations for the solver
+   epsp    =  1.e-3              !  absolute precision of the solver
+   niterp  =    400              !  number of iteration done by the solver
+/
+!-----------------------------------------------------------------------
+&namflo    !   float parameters                                         ("key_float")
+!-----------------------------------------------------------------------
+    ln_rstflo = .false.          !  boolean term for float restart (true or false)
+    nwritefl  =      75          !  frequency of float output file 
+    nstockfl  =    5475          !  frequency of float restart file 
+    ln_argo   = .false.          !  Argo type floats (stay at the surface each 10 days)
+    ln_flork4 = .false.          !  = T trajectories computed with a 4th order Runge-Kutta
+                                 !  = F  (default)   computed with Blanke' scheme
+/
