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1% ================================================================
2% Chapter 1 Ñ Model Basics
3% ================================================================
4
5\chapter{Model basics}
6\label{PE}
7\minitoc
8
9\newpage
10$\ $\newline    % force a new ligne
11
12% ================================================================
13% Primitive Equations
14% ================================================================
15\section{Primitive Equations}
16\label{PE_PE}
17
18% -------------------------------------------------------------------------------------------------------------
19%        Vector Invariant Formulation
20% -------------------------------------------------------------------------------------------------------------
21
22\subsection{Vector Invariant Formulation}
23\label{PE_Vector}
24
25
26The ocean is a fluid that can be described to a good approximation by the primitive
27equations, $i.e.$ the Navier-Stokes equations along with a nonlinear equation of
28state which couples the two active tracers (temperature and salinity) to the fluid
29velocity, plus the following additional assumptions made from scale considerations:
30
31\textit{(1) spherical earth approximation: }the geopotential surfaces are assumed to
32be spheres so that gravity (local vertical) is parallel to the earth's radius
33
34\textit{(2) thin-shell approximation: }the ocean depth is neglected compared to the earth's radius
35
36\textit{(3) turbulent closure hypothesis: }the turbulent fluxes (which represent the effect
37of small scale processes on the large-scale) are expressed in terms of large-scale features
38
39\textit{(4) Boussinesq hypothesis:} density variations are neglected except in their
40contribution to the buoyancy force
41
42\textit{(5) Hydrostatic hypothesis: }the vertical momentum equation is reduced to a
43balance between the vertical pressure gradient and the buoyancy force (this removes
44convective processes from the initial Navier-Stokes equations and so convective processes
45must be parameterized instead)
46
47\textit{(6) Incompressibility hypothesis: }the three dimensional divergence of the velocity
48vector is assumed to be zero.
49
50Because the gravitational force is so dominant in the equations of large-scale motions,
51it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked
52to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two
53vectors orthogonal to \textbf{k}, $i.e.$ tangent to the geopotential surfaces. Let us define
54the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$ 
55(the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane),
56$T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density.
57The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k})
58vector system provides the following six equations (namely the momentum balance, the
59hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation
60equations and an equation of state):
61\begin{subequations} \label{Eq_PE}
62  \begin{equation}     \label{Eq_PE_dyn}
63\frac{\partial {\rm {\bf U}}_h }{\partial t}=
64-\left[    {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}}
65            +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)}    \right]_h
66 -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h
67-\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}}
68  \end{equation}
69  \begin{equation}     \label{Eq_PE_hydrostatic}
70\frac{\partial p }{\partial z} = - \rho \ g
71  \end{equation}
72  \begin{equation}     \label{Eq_PE_continuity}
73\nabla \cdot {\bf U}=  0
74  \end{equation}
75\begin{equation} \label{Eq_PE_tra_T}
76\frac{\partial T}{\partial t} = - \nabla \cdot  \left( T \ \rm{\bf U} \right) + D^T + F^T
77  \end{equation}
78  \begin{equation}     \label{Eq_PE_tra_S}
79\frac{\partial S}{\partial t} = - \nabla \cdot  \left( S \ \rm{\bf U} \right) + D^S + F^S
80  \end{equation}
81  \begin{equation}     \label{Eq_PE_eos}
82\rho = \rho \left( T,S,p \right)
83  \end{equation}
84\end{subequations}
85where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions,
86$t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by
87the equation of state (\ref{Eq_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure,
88$f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's
89angular velocity vector), and $g$ is the gravitational acceleration.
90${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterisations of small-scale
91physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ 
92and $F^S$ surface forcing terms. Their nature and formulation are discussed in
93\S\ref{PE_zdf_ldf} and page \S\ref{PE_boundary_condition}.
94
95.
96
97% -------------------------------------------------------------------------------------------------------------
98% Boundary condition
99% -------------------------------------------------------------------------------------------------------------
100\subsection{Boundary Conditions}
101\label{PE_boundary_condition}
102
103An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea
104or ice-sea interface at its top. These boundaries can be defined by two surfaces, $z=-H(i,j)$ 
105and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height
106of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$,
107chosen as a mean sea surface (Fig.~\ref{Fig_ocean_bc}). Through these two boundaries,
108the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth,
109the continental margins, the sea ice and the atmosphere. However, some of these fluxes are
110so weak that even on climatic time scales of thousands of years they can be neglected.
111In the following, we briefly review the fluxes exchanged at the interfaces between the ocean
112and the other components of the earth system.
113
114%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
115\begin{figure}[!ht]   \begin{center}
116\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_I_ocean_bc.pdf}
117\caption{    \label{Fig_ocean_bc} 
118The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,t)$, where $H$ 
119is the depth of the sea floor and $\eta$ the height of the sea surface.
120Both $H$ and $\eta$ are referenced to $z=0$.}
121\end{center}   \end{figure}
122%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
123
124
125\begin{description}
126\item[Land - ocean interface:] the major flux between continental margins and the ocean is
127a mass exchange of fresh water through river runoff. Such an exchange modifies the sea
128surface salinity especially in the vicinity of major river mouths. It can be neglected for short
129range integrations but has to be taken into account for long term integrations as it influences
130the characteristics of water masses formed (especially at high latitudes). It is required in order
131to close the water cycle of the climate system. It is usually specified as a fresh water flux at
132the air-sea interface in the vicinity of river mouths.
133\item[Solid earth - ocean interface:] heat and salt fluxes through the sea floor are small,
134except in special areas of little extent. They are usually neglected in the model
135\footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling
136($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world
137ocean (see \ref{TRA_bbc}).}.
138The boundary condition is thus set to no flux of heat and salt across solid boundaries.
139For momentum, the situation is different. There is no flow across solid boundaries,
140$i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words,
141the bottom velocity is parallel to solid boundaries). This kinematic boundary condition
142can be expressed as:
143\begin{equation} \label{Eq_PE_w_bbc}
144w = -{\rm {\bf U}}_h \cdot  \nabla _h \left( H \right)
145\end{equation}
146In addition, the ocean exchanges momentum with the earth through frictional processes.
147Such momentum transfer occurs at small scales in a boundary layer. It must be parameterized
148in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification
149depends on the nature of the physical parameterisation used for ${\rm {\bf D}}^{\rm {\bf U}}$ 
150in \eqref{Eq_PE_dyn}. It is discussed in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9.
151\item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux
152of fresh water PE  (the precipitation minus evaporation budget) leads to:
153\begin{equation} \label{Eq_PE_w_sbc}
154w = \frac{\partial \eta }{\partial t} 
155    + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot  \nabla _h \left( \eta \right)
156    + P-E
157\end{equation}
158The dynamic boundary condition, neglecting the surface tension (which removes capillary
159waves from the system) leads to the continuity of pressure across the interface $z=\eta$.
160The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat.
161\item[Sea ice - ocean interface:] the ocean and sea ice exchange heat, salt, fresh water
162and momentum. The sea surface temperature is constrained to be at the freezing point
163at the interface. Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the
164ocean ($\sim34 \,psu$). The cycle of freezing/melting is associated with fresh water and
165salt fluxes that cannot be neglected.
166\end{description}
167
168
169%\newpage
170%$\ $\newline    % force a new ligne
171
172% ================================================================
173% The Horizontal Pressure Gradient
174% ================================================================
175\section{The Horizontal Pressure Gradient }
176\label{PE_hor_pg}
177
178% -------------------------------------------------------------------------------------------------------------
179% Pressure Formulation
180% -------------------------------------------------------------------------------------------------------------
181\subsection{Pressure Formulation}
182\label{PE_p_formulation}
183
184The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a
185reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that:
186$p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ref{Eq_PE_hydrostatic}),
187assuming that pressure in decibars can be approximated by depth in meters in (\ref{Eq_PE_eos}).
188The hydrostatic pressure is then given by:
189\begin{equation} \label{Eq_PE_pressure}
190p_h \left( {i,j,z,t} \right)
191 = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma } 
192\end{equation}
193 Two strategies can be considered for the surface pressure term: $(a)$ introduce of a
194 new variable $\eta$, the free-surface elevation, for which a prognostic equation can be
195 established and solved; $(b)$ assume that the ocean surface is a rigid lid, on which the
196 pressure (or its horizontal gradient) can be diagnosed. When the former strategy is used,
197 one solution of the free-surface elevation consists of the excitation of external gravity waves.
198 The flow is barotropic and the surface moves up and down with gravity as the restoring force.
199 The phase speed of such waves is high (some hundreds of metres per second) so that
200 the time step would have to be very short if they were present in the model. The latter
201 strategy filters out these waves since the rigid lid approximation implies $\eta=0$, $i.e.$ 
202 the sea surface is the surface $z=0$. This well known approximation increases the surface
203 wave speed to infinity and modifies certain other longwave dynamics ($e.g.$ barotropic
204 Rossby or planetary waves). The rigid-lid hypothesis is an obsolescent feature in modern
205 OGCMs. It has been available until the release 3.1 of  \NEMO, and it has been removed
206 in release 3.2 and followings. Only the free surface formulation is now described in the
207 this document (see the next sub-section).
208
209% -------------------------------------------------------------------------------------------------------------
210% Free Surface Formulation
211% -------------------------------------------------------------------------------------------------------------
212\subsection{Free Surface Formulation}
213\label{PE_free_surface}
214
215In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced
216which describes the shape of the air-sea interface. This variable is solution of a
217prognostic equation which is established by forming the vertical average of the kinematic
218surface condition (\ref{Eq_PE_w_bbc}):
219\begin{equation} \label{Eq_PE_ssh}
220\frac{\partial \eta }{\partial t}=-D+P-E
221   \quad \text{where} \
222D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]
223\end{equation}
224and using (\ref{Eq_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.
225
226Allowing the air-sea interface to move introduces the external gravity waves (EGWs)
227as a class of solution of the primitive equations. These waves are barotropic because
228of hydrostatic assumption, and their phase speed is quite high. Their time scale is
229short with respect to the other processes described by the primitive equations.
230
231Two choices can be made regarding the implementation of the free surface in the model,
232depending on the physical processes of interest.
233
234$\bullet$ If one is interested in EGWs, in particular the tides and their interaction
235with the baroclinic structure of the ocean (internal waves) possibly in shallow seas,
236then a non linear free surface is the most appropriate. This means that no
237approximation is made in (\ref{Eq_PE_ssh}) and that the variation of the ocean
238volume is fully taken into account. Note that in order to study the fast time scales
239associated with EGWs it is necessary to minimize time filtering effects (use an
240explicit time scheme with very small time step, or a split-explicit scheme with
241reasonably small time step, see \S\ref{DYN_spg_exp} or \S\ref{DYN_spg_ts}.
242
243$\bullet$ If one is not interested in EGW but rather sees them as high frequency
244noise, it is possible to apply an explicit filter to slow down the fastest waves while
245not altering the slow barotropic Rossby waves. If further, an approximative conservation
246of heat and salt contents is sufficient for the problem solved, then it is
247sufficient to solve a linearized version of (\ref{Eq_PE_ssh}), which still allows
248to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}.
249Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost.
250
251The filtering of EGWs in models with a free surface is usually a matter of discretisation
252of the temporal derivatives, using a split-explicit method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92} 
253or the implicit scheme \citep{Dukowicz1994} or the addition of a filtering force in the momentum equation
254\citep{Roullet_Madec_JGR00}. With the present release, \NEMO offers the choice between
255an explicit free surface (see \S\ref{DYN_spg_exp}) or a split-explicit scheme strongly
256inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05} (see \S\ref{DYN_spg_ts}).
257
258%\newpage
259%$\ $\newline    % force a new ligne
260
261% ================================================================
262% Curvilinear z-coordinate System
263% ================================================================
264\section{Curvilinear \textit{z-}coordinate System}
265\label{PE_zco}
266
267
268% -------------------------------------------------------------------------------------------------------------
269% Tensorial Formalism
270% -------------------------------------------------------------------------------------------------------------
271\subsection{Tensorial Formalism}
272\label{PE_tensorial}
273
274In many ocean circulation problems, the flow field has regions of enhanced dynamics
275($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts).
276The representation of such dynamical processes can be improved by specifically increasing
277the model resolution in these regions. As well, it may be convenient to use a lateral
278boundary-following coordinate system to better represent coastal dynamics. Moreover,
279the common geographical coordinate system has a singular point at the North Pole that
280cannot be easily treated in a global model without filtering. A solution consists of introducing
281an appropriate coordinate transformation that shifts the singular point onto land
282\citep{Madec_Imbard_CD96, Murray_JCP96}. As a consequence, it is important to solve the primitive
283equations in various curvilinear coordinate systems. An efficient way of introducing an
284appropriate coordinate transform can be found when using a tensorial formalism.
285This formalism is suited to any multidimensional curvilinear coordinate system. Ocean
286modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth
287approximation), with preservation of the local vertical. Here we give the simplified equations
288for this particular case. The general case is detailed by \citet{Eiseman1980} in their survey
289of the conservation laws of fluid dynamics.
290
291Let (\textit{i},\textit{j},\textit{k}) be a set of orthogonal curvilinear coordinates on the sphere
292associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k})
293linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are
294two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (Fig.\ref{Fig_referential}).
295Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined
296by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of
297the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea
298level (Fig.\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is
299given by $e_1$, $e_2$ and $e_3$, the three scale factors:
300\begin{equation} \label{Eq_scale_factors}
301\begin{aligned}
302 e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda 
303}{\partial i}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi 
304}{\partial i}} \right)^2} \right]^{1/2} \\ 
305 e_2 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda 
306}{\partial j}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi 
307}{\partial j}} \right)^2} \right]^{1/2} \\ 
308 e_3 &=\left( {\frac{\partial z}{\partial k}} \right) \\ 
309 \end{aligned}
310 \end{equation}
311
312%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
313\begin{figure}[!tb]   \begin{center}
314\includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_I_earth_referential.pdf}
315\caption{   \label{Fig_referential} 
316the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear
317coordinate system (\textbf{i},\textbf{j},\textbf{k}). }
318\end{center}   \end{figure}
319%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
320
321Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by
322$a$ in (\ref{Eq_scale_factors}) (thin-shell approximation). The resulting horizontal scale
323factors $e_1$, $e_2$  are independent of $k$ while the vertical scale factor is a single
324function of $k$ as \textbf{k} is parallel to \textbf{z}. The scalar and vector operators that
325appear in the primitive equations (Eqs. \eqref{Eq_PE_dyn} to \eqref{Eq_PE_eos}) can
326be written in the tensorial form, invariant in any orthogonal horizontal curvilinear coordinate
327system transformation:
328\begin{subequations} \label{Eq_PE_discrete_operators}
329\begin{equation} \label{Eq_PE_grad}
330\nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf 
331i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3
332}\frac{\partial q}{\partial k}\;{\rm {\bf k}}    \\
333\end{equation}
334\begin{equation} \label{Eq_PE_div}
335\nabla \cdot {\rm {\bf A}} 
336= \frac{1}{e_1 \; e_2} \left[
337  \frac{\partial \left(e_2 \; a_1\right)}{\partial i }
338+\frac{\partial \left(e_1 \; a_2\right)}{\partial j }       \right]
339+ \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k }   \right]
340\end{equation}
341\begin{equation} \label{Eq_PE_curl}
342   \begin{split}
343\nabla \times \vect{A} =
344    \left[ {\frac{1}{e_2 }\frac{\partial a_3}{\partial j}
345            -\frac{1}{e_3 }\frac{\partial a_2 }{\partial k}} \right] \; \vect{i}
346&+\left[ {\frac{1}{e_3 }\frac{\partial a_1 }{\partial k}
347           -\frac{1}{e_1 }\frac{\partial a_3 }{\partial i}} \right] \; \vect{j}     \\
348&+\frac{1}{e_1 e_2 } \left[ {\frac{\partial \left( {e_2 a_2 } \right)}{\partial i}
349                                       -\frac{\partial \left( {e_1 a_1 } \right)}{\partial j}} \right] \; \vect{k} 
350   \end{split}
351\end{equation}
352\begin{equation} \label{Eq_PE_lap}
353\Delta q = \nabla \cdot \left\nabla q \right)
354\end{equation}
355\begin{equation} \label{Eq_PE_lap_vector}
356\Delta {\rm {\bf A}} =
357  \nabla \left( \nabla \cdot {\rm {\bf A}} \right)
358- \nabla \times \left\nabla \times {\rm {\bf A}} \right)
359\end{equation}
360\end{subequations}
361where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinate system.
362
363% -------------------------------------------------------------------------------------------------------------
364% Continuous Model Equations
365% -------------------------------------------------------------------------------------------------------------
366\subsection{Continuous Model Equations}
367\label{PE_zco_Eq}
368
369In order to express the Primitive Equations in tensorial formalism, it is necessary to compute
370the horizontal component of the non-linear and viscous terms of the equation using
371\eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}.
372Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate
373system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity
374field $\chi$, by:
375\begin{equation} \label{Eq_PE_curl_Uh}
376\zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} 
377\right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}} 
378\right]
379\end{equation}
380\begin{equation} \label{Eq_PE_div_Uh}
381\chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u} 
382\right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}} 
383\right]
384\end{equation}
385
386Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ 
387and that $e_3$  is a function of the single variable $k$, the nonlinear term of
388\eqref{Eq_PE_dyn} can be transformed as follows:
389\begin{flalign*}
390&\left[ {\left( { \nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
391+\frac{1}{2}   \nabla \left( {{\rm {\bf U}}^2} \right)}   \right]_h        &
392\end{flalign*}
393\begin{flalign*}
394&\qquad=\left( {{\begin{array}{*{20}c}
395 {\left[    {   \frac{1}{e_3} \frac{\partial u  }{\partial k}
396         -\frac{1}{e_1} \frac{\partial w  }{\partial i} } \right] w - \zeta \; v }     \\
397      {\zeta \; u - \left[ {   \frac{1}{e_2} \frac{\partial w}{\partial j}
398                     -\frac{1}{e_3} \frac{\partial v}{\partial k} } \right] \ w}  \\
399       \end{array} }} \right)       
400+\frac{1}{2}   \left( {{\begin{array}{*{20}c}
401       { \frac{1}{e_1}  \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial i}}  \hfill    \\
402       { \frac{1}{e_2}  \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial j}}  \hfill    \\
403       \end{array} }} \right)       &
404\end{flalign*}
405\begin{flalign*}
406& \qquad =\left( {{  \begin{array}{*{20}c}
407 {-\zeta \; v} \hfill \\
408 { \zeta \; u} \hfill \\
409         \end{array} }} \right)
410+\frac{1}{2}\left( {{   \begin{array}{*{20}c}
411 {\frac{1}{e_1 }\frac{\partial \left( {u^2+v^2} \right)}{\partial i}} \hfill  \\
412 {\frac{1}{e_2 }\frac{\partial \left( {u^2+v^2} \right)}{\partial j}} \hfill  \\
413                  \end{array} }} \right)       
414+\frac{1}{e_3 }\left( {{      \begin{array}{*{20}c}
415 { w \; \frac{\partial u}{\partial k}}    \\
416 { w \; \frac{\partial v}{\partial k}}    \\
417                     \end{array} }} \right
418-\left( {{  \begin{array}{*{20}c}
419 {\frac{w}{e_1}\frac{\partial w}{\partial i}
420 -\frac{1}{2e_1}\frac{\partial w^2}{\partial i}} \hfill \\
421 {\frac{w}{e_2}\frac{\partial w}{\partial j}
422  -\frac{1}{2e_2}\frac{\partial w^2}{\partial j}} \hfill \\
423         \end{array} }} \right)        &
424\end{flalign*}
425
426The last term of the right hand side is obviously zero, and thus the nonlinear term of
427\eqref{Eq_PE_dyn} is written in the $(i,j,k)$ coordinate system:
428\begin{equation} \label{Eq_PE_vector_form}
429\left[ {\left( {  \nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
430+\frac{1}{2}   \nabla \left( {{\rm {\bf U}}^2} \right)}   \right]_h
431=\zeta 
432\;{\rm {\bf k}}\times {\rm {\bf U}}_h +\frac{1}{2}\nabla _h \left( {{\rm 
433{\bf U}}_h^2 } \right)+\frac{1}{e_3 }w\frac{\partial {\rm {\bf U}}_h
434}{\partial k}     
435\end{equation}
436
437This is the so-called \textit{vector invariant form} of the momentum advection term.
438For some purposes, it can be advantageous to write this term in the so-called flux form,
439$i.e.$ to write it as the divergence of fluxes. For example, the first component of
440\eqref{Eq_PE_vector_form} (the $i$-component) is transformed as follows:
441\begin{flalign*}
442&{ \begin{array}{*{20}l}
443\left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U}
444          +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i   % \\
445%\\
446     = - \zeta \;v
447     + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
448     + \frac{1}{e_3}w \ \frac{\partial u}{\partial k}          \\
449\\
450\qquad =\frac{1}{e_1 \; e_2} \left(    -v\frac{\partial \left( {e_2 \,v} \right)}{\partial i}
451                     +v\frac{\partial \left( {e_1 \,u} \right)}{\partial j}    \right)
452+\frac{1}{e_1 e_2 }\left(  +e_2 \; u\frac{\partial u}{\partial i}
453                     +e_2 \; v\frac{\partial v}{\partial i}              \right)
454+\frac{1}{e_3}       \left(   w\;\frac{\partial u}{\partial k}       \right)   \\
455\end{array} }        &
456\end{flalign*}
457\begin{flalign*}
458&{ \begin{array}{*{20}l}
459\qquad =\frac{1}{e_1 \; e_2}  \left\{ 
460 -\left(        v^\frac{\partial e_2                                }{\partial i} 
461      +e_2 \,v    \frac{\partial v                                   }{\partial i}     \right)
462+\left(           \frac{\partial \left( {e_1 \,u\,v}  \right)}{\partial j}
463      -e_1 \,u    \frac{\partial v                                   }{\partial j}  \right\right.
464\\  \left\qquad \qquad \quad
465+\left(           \frac{\partial \left( {e_2 u\,u}     \right)}{\partial i}
466      -u       \frac{\partial \left( {e_2 u}         \right)}{\partial i}  \right)
467+e_2 v            \frac{\partial v                                    }{\partial i}
468                  \right\} 
469+\frac{1}{e_3} \left(
470               \frac{\partial \left( {w\,u} \right)         }{\partial k}
471       -u         \frac{\partial w                    }{\partial k}  \right) \\
472\end{array} }     &
473\end{flalign*}
474\begin{flalign*}
475&{ \begin{array}{*{20}l}
476\qquad =\frac{1}{e_1 \; e_2}  \left(
477               \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
478      +        \frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j}  \right)
479+\frac{1}{e_3 }      \frac{\partial \left( {w\,u       } \right)}{\partial k}
480\\  \qquad \qquad \quad
481+\frac{1}{e_1 e_2 }     \left(
482      -u \left(   \frac{\partial \left( {e_1 v   } \right)}{\partial j}
483               -v\,\frac{\partial e_1 }{\partial j}             \right)
484      -u       \frac{\partial \left( {e_2 u   } \right)}{\partial i}
485                  \right)
486 -\frac{1}{e_3 }     \frac{\partial w}{\partial k} u
487 +\frac{1}{e_1 e_2 }\left(    -v^2\frac{\partial e_2   }{\partial i}     \right)
488\end{array} }     &
489\end{flalign*}
490\begin{flalign*}
491&{ \begin{array}{*{20}l}
492\qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u} \right)
493-   \left( \nabla \cdot {\rm {\bf U}} \right) \ u
494+\frac{1}{e_1 e_2 }\left(
495      -v^2     \frac{\partial e_2 }{\partial i}
496      +uv   \,    \frac{\partial e_1 }{\partial j}    \right) \\
497\end{array} }     &
498\end{flalign*}
499as $\nabla \cdot {\rm {\bf U}}\;=0$ (incompressibility) it comes:
500\begin{flalign*}
501&{ \begin{array}{*{20}l}
502\qquad = \nabla \cdot \left{{\rm {\bf U}}\,u}      \right)
503\frac{1}{e_1 e_2 }   \left( v \; \frac{\partial e_2}{\partial i}
504                         -u \; \frac{\partial e_1}{\partial j}    \right\left( -v \right)
505\end{array} }     &
506\end{flalign*}
507
508The flux form of the momentum advection term is therefore given by:
509\begin{multline} \label{Eq_PE_flux_form}
510      \left[
511  \left(    {\nabla \times {\rm {\bf U}}}    \right) \times {\rm {\bf U}}
512+\frac{1}{2}   \nabla \left{{\rm {\bf U}}^2}    \right)
513      \right]_h
514\\
515= \nabla \cdot    \left( {{\begin{array}{*{20}c}   {\rm {\bf U}} \, u   \hfill \\
516                                    {\rm {\bf U}} \, v   \hfill \\
517                  \end{array} }}   
518            \right)
519+\frac{1}{e_1 e_2 }     \left(
520       v\frac{\partial e_2}{\partial i}
521      -u\frac{\partial e_1}{\partial j} 
522                  \right) {\rm {\bf k}} \times {\rm {\bf U}}_h
523\end{multline}
524
525The flux form has two terms, the first one is expressed as the divergence of momentum
526fluxes (hence the flux form name given to this formulation) and the second one is due to
527the curvilinear nature of the coordinate system used. The latter is called the \emph{metric} 
528term and can be viewed as a modification of the Coriolis parameter:
529\begin{equation} \label{Eq_PE_cor+metric}
530f \to f + \frac{1}{e_1\;e_2}  \left(  v \frac{\partial e_2}{\partial i}
531                        -u \frac{\partial e_1}{\partial j}  \right)
532\end{equation}
533
534Note that in the case of geographical coordinate, $i.e.$ when $(i,j) \to (\lambda ,\varphi )$ 
535and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, we recover the commonly used modification of
536the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
537
538
539$\ $\newline    % force a new ligne
540
541To sum up, the curvilinear $z$-coordinate equations solved by the ocean model can be
542written in the following tensorial formalism:
543
544\vspace{+10pt}
545$\bullet$ \textbf{Vector invariant form of the momentum equations} :
546
547\begin{subequations} \label{Eq_PE_dyn_vect}
548\begin{equation} \label{Eq_PE_dyn_vect_u} \begin{split}
549\frac{\partial u}{\partial t} 
550= +   \left( {\zeta +f} \right)\,v                                   
551   -   \frac{1}{2\,e_1}           \frac{\partial}{\partial i} \left(  u^2+v^2   \right)
552   -   \frac{1}{e_3    }  w     \frac{\partial u}{\partial k}      &      \\
553   -   \frac{1}{e_1    }            \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho _o}    \right)   
554   &+   D_u^{\vect{U}}  +   F_u^{\vect{U}}      \\
555\\
556\frac{\partial v}{\partial t} =
557       -   \left( {\zeta +f} \right)\,u   
558       -   \frac{1}{2\,e_2 }        \frac{\partial }{\partial j}\left(  u^2+v^\right)   
559       -   \frac{1}{e_3     }   w  \frac{\partial v}{\partial k}     &      \\
560       -   \frac{1}{e_2     }        \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)   
561    &+  D_v^{\vect{U}}  +   F_v^{\vect{U}}
562\end{split} \end{equation}
563\end{subequations}
564
565
566\vspace{+10pt}
567$\bullet$ \textbf{flux form of the momentum equations} :
568\begin{subequations} \label{Eq_PE_dyn_flux}
569\begin{multline} \label{Eq_PE_dyn_flux_u}
570\frac{\partial u}{\partial t}=
571+   \left( { f + \frac{1}{e_1 \; e_2}
572               \left(    v \frac{\partial e_2}{\partial i}
573                  -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, v    \\
574- \frac{1}{e_1 \; e_2}  \left(
575               \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
576      +        \frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j}  \right)
577                 - \frac{1}{e_3 }\frac{\partial \left( {         w\,u} \right)}{\partial k}    \\
578-   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho _o}   \right)
579+   D_u^{\vect{U}} +   F_u^{\vect{U}}
580\end{multline}
581\begin{multline} \label{Eq_PE_dyn_flux_v}
582\frac{\partial v}{\partial t}=
583-   \left( { f + \frac{1}{e_1 \; e_2}
584               \left(    v \frac{\partial e_2}{\partial i}
585                  -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, u   \\
586 \frac{1}{e_1 \; e_2}   \left(
587               \frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i}
588      +        \frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j}  \right)
589                 - \frac{1}{e_3 } \frac{\partial \left( {        w\,v} \right)}{\partial k}    \\
590-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}    \right)
591+  D_v^{\vect{U}} +  F_v^{\vect{U}} 
592\end{multline}
593\end{subequations}
594where $\zeta$, the relative vorticity, is given by \eqref{Eq_PE_curl_Uh} and $p_s $,
595the surface pressure, is given by:
596\begin{equation} \label{Eq_PE_spg}
597p_s =  \rho \,g \,\eta 
598\end{equation}
599with $\eta$ is solution of \eqref{Eq_PE_ssh}
600
601The vertical velocity and the hydrostatic pressure are diagnosed from the following equations:
602\begin{equation} \label{Eq_w_diag}
603\frac{\partial w}{\partial k}=-\chi \;e_3
604\end{equation}
605\begin{equation} \label{Eq_hp_diag}
606\frac{\partial p_h }{\partial k}=-\rho \;g\;e_3
607\end{equation}
608where the divergence of the horizontal velocity, $\chi$ is given by \eqref{Eq_PE_div_Uh}.
609
610\vspace{+10pt}
611$\bullet$ \textit{tracer equations} :
612\begin{equation} \label{Eq_S}
613\frac{\partial T}{\partial t} =
614-\frac{1}{e_1 e_2 }\left[ {      \frac{\partial \left( {e_2 T\,u} \right)}{\partial i}
615                  +\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right]
616-\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T
617\end{equation}
618\begin{equation} \label{Eq_T}
619\frac{\partial S}{\partial t} =
620-\frac{1}{e_1 e_2 }\left[    {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i}
621                  +\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right]
622-\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S
623\end{equation}
624\begin{equation} \label{Eq_rho}
625\rho =\rho \left( {T,S,z(k)} \right)
626\end{equation}
627
628The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale
629parameterisation used. It will be defined in \S\ref{PE_zdf}. The nature and formulation of
630${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed
631in Chapter~\ref{SBC}.
632
633
634\newpage 
635$\ $\newline    % force a new ligne
636% ================================================================
637% Curvilinear generalised vertical coordinate System
638% ================================================================
639\section{Curvilinear generalised vertical coordinate System}
640\label{PE_gco}
641
642The ocean domain presents a huge diversity of situation in the vertical. First the ocean surface is a time dependent surface (moving surface). Second the ocean floor depends on the geographical position, varying from more than 6,000 meters in abyssal trenches to zero at the coast. Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing.
643Therefore, in order to represent the ocean with respect to the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height $e.g.$ an $z$*-coordinate; for the second point, a space variation to fit the change of bottom topography $e.g.$ a terrain-following or $\sigma$-coordinate; and for the third point, one will be tempted to use a space and time dependent coordinate that follows the isopycnal surfaces, $e.g.$ an isopycnic coordinate.
644
645In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at the ocean bottom) \citep{Chassignet_al_JPO03}  or OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere) \citep{Madec_al_JPO96} among others.
646
647In fact one is totally free to choose any space and time vertical coordinate by introducing an arbitrary vertical coordinate :
648\begin{equation} \label{Eq_s}
649s=s(i,j,k,t)
650\end{equation}
651with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \eqref{Eq_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \eqref{Eq_s}.
652This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact an Arbitrary Lagrangian--Eulerian (ALE) coordinate. Indeed, choosing an expression for $s$ is an arbitrary choice that determines which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and which part will be used to move them (Lagrangian part).
653The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces.
654
655%\gmcomment{
656
657%A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient...
658
659the generalized vertical coordinates used in ocean modelling are not orthogonal,
660which contrasts with many other applications in mathematical physics.
661Hence, it is useful to keep in mind the following properties that may seem
662odd on initial encounter.
663
664The horizontal velocity in ocean models measures motions in the horizontal plane,
665perpendicular to the local gravitational field. That is, horizontal velocity is mathematically
666the same regardless the vertical coordinate, be it geopotential, isopycnal, pressure,
667or terrain following. The key motivation for maintaining the same horizontal velocity
668component is that the hydrostatic and geostrophic balances are dominant in the large-scale ocean.
669Use of an alternative quasi-horizontal velocity, for example one oriented parallel
670to the generalized surface, would lead to unacceptable numerical errors.
671Correspondingly, the vertical direction is anti-parallel to the gravitational force in all
672of the coordinate systems. We do not choose the alternative of a quasi-vertical
673direction oriented normal to the surface of a constant generalized vertical coordinate.
674
675It is the method used to measure transport across the generalized vertical coordinate
676surfaces which differs between the vertical coordinate choices. That is, computation
677of the dia-surface velocity component represents the fundamental distinction between
678the various coordinates. In some models, such as geopotential, pressure, and
679terrain following, this transport is typically diagnosed from volume or mass conservation.
680In other models, such as isopycnal layered models, this transport is prescribed based
681on assumptions about the physical processes producing a flux across the layer interfaces.
682
683
684In this section we first establish the PE in the generalised vertical $s$-coordinate,
685then we discuss the particular cases available in \NEMO, namely $z$, $z$*, $s$, and $\tilde z$
686%}
687
688% -------------------------------------------------------------------------------------------------------------
689% The s-coordinate Formulation
690% -------------------------------------------------------------------------------------------------------------
691\subsection{The \textit{s-}coordinate Formulation}
692
693Starting from the set of equations established in \S\ref{PE_zco} for the special case $k=z$ 
694and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes
695$z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and
696$s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed
697equations is given in Appendix~\ref{Apdx_A}. Let us define the vertical scale factor by
698$e_3=\partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the
699(\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by :
700\begin{equation} \label{Eq_PE_sco_slope}
701\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
702\quad \text{, and } \quad 
703\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
704\end{equation}
705We also introduce  $\omega $, a dia-surface velocity component, defined as the velocity
706relative to the moving $s$-surfaces and normal to them:
707\begin{equation} \label{Eq_PE_sco_w}
708\omega  = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v    \\
709\end{equation}
710
711The equations solved by the ocean model \eqref{Eq_PE} in $s-$coordinate can be written as follows (see Appendix~\ref{Apdx_A_momentum}):
712
713 \vspace{0.5cm}
714$\bullet$ Vector invariant form of the momentum equation :
715\begin{multline} \label{Eq_PE_sco_u}
716\frac{\partial  u   }{\partial t}=
717   +   \left( {\zeta +f} \right)\,v                                   
718   -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)
719   -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
720   -   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)   
721   +  g\frac{\rho }{\rho _o}\sigma _1
722   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
723\end{multline}
724\begin{multline} \label{Eq_PE_sco_v}
725\frac{\partial v }{\partial t}=
726   -   \left( {\zeta +f} \right)\,u   
727   -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^\right)       
728   -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
729   -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)
730    +  g\frac{\rho }{\rho _o }\sigma _2   
731   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
732\end{multline}
733
734 \vspace{0.5cm}
735$\bullet$ Vector invariant form of the momentum equation :
736\begin{multline} \label{Eq_PE_sco_u}
737\frac{1}{e_3} \frac{\partial \left(  e_3\,\right) }{\partial t}=
738   +   \left( { f + \frac{1}{e_1 \; e_2 }
739               \left(    v \frac{\partial e_2}{\partial i}
740                  -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, v    \\
741   - \frac{1}{e_1 \; e_2 \; e_3 }   \left(
742               \frac{\partial \left( {e_2 \, e_3 \, u\,u} \right)}{\partial i}
743      +        \frac{\partial \left( {e_1 \, e_3 \, v\,u} \right)}{\partial j}   \right)
744   - \frac{1}{e_3 }\frac{\partial \left( { \omega\,u} \right)}{\partial k}    \\
745   - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)   
746   +  g\frac{\rho }{\rho _o}\sigma _1
747   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
748\end{multline}
749\begin{multline} \label{Eq_PE_sco_v}
750\frac{1}{e_3} \frac{\partial \left(  e_3\,\right) }{\partial t}=
751   -   \left( { f + \frac{1}{e_1 \; e_2}
752               \left(    v \frac{\partial e_2}{\partial i}
753                  -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, u   \\
754   - \frac{1}{e_1 \; e_2 \; e_3 }   \left(
755               \frac{\partial \left( {e_2 \; e_\,u\,v} \right)}{\partial i}
756      +        \frac{\partial \left( {e_1 \; e_\,v\,v} \right)}{\partial j}   \right)
757                 - \frac{1}{e_3 } \frac{\partial \left( { \omega\,v} \right)}{\partial k}    \\
758   -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)
759    +  g\frac{\rho }{\rho _o }\sigma _2   
760   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
761\end{multline}
762
763where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic
764pressure have the same expressions as in $z$-coordinates although they do not represent
765exactly the same quantities. $\omega$ is provided by the continuity equation
766(see Appendix~\ref{Apdx_A}):
767\begin{equation} \label{Eq_PE_sco_continuity}
768\frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0   
769\qquad \text{with }\;\; 
770\chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u} 
771\right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial 
772j}} \right]
773\end{equation}
774
775 \vspace{0.5cm}
776$\bullet$ tracer equations:
777\begin{multline} \label{Eq_PE_sco_t}
778\frac{1}{e_3} \frac{\partial \left(  e_3\,\right) }{\partial t}=
779-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i}
780                                           +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right]   \\
781-\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k}   + D^T + F^S   \qquad
782\end{multline}
783
784\begin{multline} \label{Eq_PE_sco_s}
785\frac{1}{e_3} \frac{\partial \left(  e_3\,\right) }{\partial t}=
786-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i}
787                                           +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right]    \\
788-\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k}     + D^S + F^S   \qquad
789\end{multline}
790
791The equation of state has the same expression as in $z$-coordinate, and similar expressions
792are used for mixing and forcing terms.
793
794\gmcomment{
795\colorbox{yellow}{ to be updated $= = >$}
796Add a few works on z and zps and s and underlies the differences between all of them
797\colorbox{yellow}{ $< = =$ end update}  }
798
799
800
801% -------------------------------------------------------------------------------------------------------------
802% Curvilinear z*-coordinate System
803% -------------------------------------------------------------------------------------------------------------
804\subsection{Curvilinear \textit{z*}--coordinate System}
805\label{PE_zco_star}
806
807%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
808\begin{figure}[!b]    \begin{center}
809\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zstar.pdf}
810\caption{   \label{Fig_z_zstar} 
811(a) $z$-coordinate in linear free-surface case ;
812(b) $z-$coordinate in non-linear free surface case ;
813(c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate
814\citep{Adcroft_Campin_OM04} ).}
815\end{center}   \end{figure}
816%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
817
818
819In that case, the free surface equation is nonlinear, and the variations of volume are fully
820taken into account. These coordinates systems is presented in a report \citep{Levier2007} 
821available on the \NEMO web site.
822
823%\gmcomment{
824The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation
825which allows one to deal with large amplitude free-surface
826variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. In
827the  \textit{z*} formulation, the variation of the column thickness due to sea-surface
828undulations is not concentrated in the surface level, as in the $z$-coordinate formulation,
829but is equally distributed over the full water column. Thus vertical
830levels naturally follow sea-surface variations, with a linear attenuation with
831depth, as illustrated by figure fig.1c . Note that with a flat bottom, such as in
832fig.1c, the bottom-following  $z$ coordinate and  \textit{z*} are equivalent.
833The definition and modified oceanic equations for the rescaled vertical coordinate
834 \textit{z*}, including the treatment of fresh-water flux at the surface, are
835detailed in Adcroft and Campin (2004). The major points are summarized
836here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as:
837\begin{equation} \label{Eq_z-star}
838H +  \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H}
839\end{equation} 
840Since the vertical displacement of the free surface is incorporated in the vertical
841coordinate  \textit{z*}, the upper and lower boundaries are at fixed  \textit{z*} position, 
842$\textit{z*} = 0$ and  $\textit{z*} = -H$ respectively. Also the divergence of the flow field
843is no longer zero as shown by the continuity equation:
844\begin{equation*} 
845\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right)
846      \left( r \; w\textit{*} \right) = 0
847\end{equation*} 
848%}
849
850
851% from MOM4p1 documentation
852
853To overcome problems with vanishing surface and/or bottom cells, we consider the
854zstar coordinate
855\begin{equation} \label{PE_}
856   z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
857\end{equation}
858
859This coordinate is closely related to the "eta" coordinate used in many atmospheric
860models (see Black (1994) for a review of eta coordinate atmospheric models). It
861was originally used in ocean models by Stacey et al. (1995) for studies of tides
862next to shelves, and it has been recently promoted by Adcroft and Campin (2004)
863for global climate modelling.
864
865The 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
866undulations of the bottom topography versus undulations in the surface height, it
867is 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}.
868Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an
869unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in
870the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure
871gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using
872the same techniques as in $z$-models (see Chapters 13-16 of \cite{Griffies_Bk04}) for a
873discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp} 
874in this document for treatment in \NEMO).
875
876The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all
877cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This
878is a minor constraint relative to that encountered on the surface height when using
879$s = z$ or $s = z - \eta$.
880
881Because $z^\star$ has a time independent range, all grid cells have static increments
882ds, and the sum of the ver tical increments yields the time independent ocean
883depth %·k ds = H.
884The $z^\star$ coordinate is therefore invisible to undulations of the
885free surface, since it moves along with the free surface. This proper ty means that
886no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the
887motion of external gravity waves. Such spurious transpor t can be a problem in
888z-models, especially those with tidal forcing. Quite generally, the time independent
889range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly
890arbitrary ver tical resolution even in the presence of large amplitude fluctuations of
891the surface height, again so long as $\eta > -H$.
892
893%end MOM doc %%%
894
895
896
897\newpage 
898% -------------------------------------------------------------------------------------------------------------
899% Terrain following  coordinate System
900% -------------------------------------------------------------------------------------------------------------
901\subsection{Curvilinear Terrain-following \textit{s}--coordinate}
902\label{PE_sco}
903
904% -------------------------------------------------------------------------------------------------------------
905% Introduction
906% -------------------------------------------------------------------------------------------------------------
907\subsubsection{Introduction}
908
909Several important aspects of the ocean circulation are influenced by bottom topography.
910Of course, the most important is that bottom topography determines deep ocean sub-basins,
911barriers, sills and channels that strongly constrain the path of water masses, but more subtle
912effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary
913one along continental slopes. Topographic Rossby waves can be excited and can interact
914with the mean current. In the $z-$coordinate system presented in the previous section
915(\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is
916discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom
917and to large localized depth gradients associated with large localized vertical velocities.
918The response to such a velocity field often leads to numerical dispersion effects.
919One solution to strongly reduce this error is to use a partial step representation of bottom
920topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}.
921Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate)
922
923The $s$-coordinate avoids the discretisation error in the depth field since the layers of
924computation are gradually adjusted with depth to the ocean bottom. Relatively small
925topographic features as well as  gentle, large-scale slopes of the sea floor in the deep
926ocean, which would be ignored in typical $z$-model applications with the largest grid
927spacing at greatest depths, can easily be represented (with relatively low vertical resolution).
928A terrain-following model (hereafter $s-$model) also facilitates the modelling of the
929boundary layer flows over a large depth range, which in the framework of the $z$-model
930would require high vertical resolution over the whole depth range. Moreover, with a
931$s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface
932as the only boundaries of the domain (nomore lateral boundary condition to specify).
933Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a
934homogeneous ocean, it has strong limitations as soon as stratification is introduced.
935The main two problems come from the truncation error in the horizontal pressure
936gradient and a possibly increased diapycnal diffusion. The horizontal pressure force
937in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}),
938
939\begin{equation} \label{Eq_PE_p_sco}
940\left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial 
941p}{\partial s}\left. {\nabla z} \right|_s
942\end{equation}
943
944The second term in \eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface
945and introduces a truncation error that is not present in a $z$-model. In the special case
946of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$),
947\citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude
948of this truncation error. It depends on topographic slope, stratification, horizontal and
949vertical resolution, the equation of state, and the finite difference scheme. This error
950limits the possible topographic slopes that a model can handle at a given horizontal
951and vertical resolution. This is a severe restriction for large-scale applications using
952realistic bottom topography. The large-scale slopes require high horizontal resolution,
953and the computational cost becomes prohibitive. This problem can be at least partially
954overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec_al_JPO96}. However, the definition of the model
955domain vertical coordinate becomes then a non-trivial thing for a realistic bottom
956topography: a envelope topography is defined in $s$-coordinate on which a full or
957partial step bottom topography is then applied in order to adjust the model depth to
958the observed one (see \S\ref{DOM_zgr}.
959
960For numerical reasons a minimum of diffusion is required along the coordinate surfaces
961of any finite difference model. It causes spurious diapycnal mixing when coordinate
962surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as
963well as for a $s$-model. However, density varies more strongly on $s-$surfaces than
964on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal
965diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a
966$z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal
967circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation.
968For example, imagine an isolated bump of topography in an ocean at rest with a horizontally
969uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral
970surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast,
971the 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
972($i.e.$ the main thermocline) \citep{Madec_al_JPO96}. An alternate solution consists of rotating
973the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \S\ref{PE_ldf}.
974Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large,
975strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}).
976
977The $s-$coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} differ mainly in two
978aspects from similar models:  it allows  a representation of bottom topography with mixed
979full or partial step-like/terrain following topography ; It also offers a completely general
980transformation, $s=s(i,j,z)$ for the vertical coordinate.
981
982
983\newpage 
984% -------------------------------------------------------------------------------------------------------------
985% Curvilinear z-tilde coordinate System
986% -------------------------------------------------------------------------------------------------------------
987\subsection{Curvilinear $\tilde{z}$--coordinate}
988\label{PE_zco_tilde}
989
990The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM10s}.
991It is available in \NEMO since the version 3.4. Nevertheless, it is currently not robust enough
992to be used in all possible configurations. Its use is therefore not recommended.
993We
994
995\newpage 
996% ================================================================
997% Subgrid Scale Physics
998% ================================================================
999\section{Subgrid Scale Physics}
1000\label{PE_zdf_ldf}
1001
1002The primitive equations describe the behaviour of a geophysical fluid at
1003space and time scales larger than a few kilometres in the horizontal, a few
1004meters in the vertical and a few minutes. They are usually solved at larger
1005scales: the specified grid spacing and time step of the numerical model. The
1006effects of smaller scale motions (coming from the advective terms in the
1007Navier-Stokes equations) must be represented entirely in terms of
1008large-scale patterns to close the equations. These effects appear in the
1009equations as the divergence of turbulent fluxes ($i.e.$ fluxes associated with
1010the mean correlation of small scale perturbations). Assuming a turbulent
1011closure hypothesis is equivalent to choose a formulation for these fluxes.
1012It is usually called the subgrid scale physics. It must be emphasized that
1013this is the weakest part of the primitive equations, but also one of the
1014most important for long-term simulations as small scale processes \textit{in fine} 
1015balance the surface input of kinetic energy and heat.
1016
1017The control exerted by gravity on the flow induces a strong anisotropy
1018between the lateral and vertical motions. Therefore subgrid-scale physics 
1019\textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$  in \eqref{Eq_PE_dyn},
1020\eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part 
1021\textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part 
1022\textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms
1023and their underlying physics are briefly discussed in the next two subsections.
1024
1025% -------------------------------------------------------------------------------------------------------------
1026% Vertical Subgrid Scale Physics
1027% -------------------------------------------------------------------------------------------------------------
1028\subsection{Vertical Subgrid Scale Physics}
1029\label{PE_zdf}
1030
1031The model resolution is always larger than the scale at which the major
1032sources of vertical turbulence occur (shear instability, internal wave
1033breaking...). Turbulent motions are thus never explicitly solved, even
1034partially, but always parameterized. The vertical turbulent fluxes are
1035assumed to depend linearly on the gradients of large-scale quantities (for
1036example, the turbulent heat flux is given by $\overline{T'w'}=-A^{vT} \partial_z \overline T$,
1037where $A^{vT}$ is an eddy coefficient). This formulation is
1038analogous to that of molecular diffusion and dissipation. This is quite
1039clearly a necessary compromise: considering only the molecular viscosity
1040acting on large scale severely underestimates the role of turbulent
1041diffusion and dissipation, while an accurate consideration of the details of
1042turbulent motions is simply impractical. The resulting vertical momentum and
1043tracer diffusive operators are of second order:
1044\begin{equation} \label{Eq_PE_zdf}
1045   \begin{split}
1046{\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\         
1047D^{vT}                         &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ ,
1048\quad
1049D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right)
1050   \end{split}
1051\end{equation}
1052where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients,
1053respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat
1054and salt must be specified (see Chap.~\ref{SBC} and \ref{ZDF} and \S\ref{TRA_bbl}).
1055All the vertical physics is embedded in the specification of the eddy coefficients.
1056They can be assumed to be either constant, or function of the local fluid properties
1057($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a
1058turbulent closure model. The choices available in \NEMO are discussed in \S\ref{ZDF}).
1059
1060% -------------------------------------------------------------------------------------------------------------
1061% Lateral Diffusive and Viscous Operators Formulation
1062% -------------------------------------------------------------------------------------------------------------
1063\subsection{Formulation of the Lateral Diffusive and Viscous Operators}
1064\label{PE_ldf}
1065
1066Lateral turbulence can be roughly divided into a mesoscale turbulence
1067associated with eddies (which can be solved explicitly if the resolution is
1068sufficient since their underlying physics are included in the primitive
1069equations), and a sub mesoscale turbulence which is never explicitly solved
1070even partially, but always parameterized. The formulation of lateral eddy
1071fluxes depends on whether the mesoscale is below or above the grid-spacing
1072($i.e.$ the model is eddy-resolving or not).
1073
1074In non-eddy-resolving configurations, the closure is similar to that used
1075for the vertical physics. The lateral turbulent fluxes are assumed to depend
1076linearly on the lateral gradients of large-scale quantities. The resulting
1077lateral diffusive and dissipative operators are of second order.
1078Observations show that lateral mixing induced by mesoscale turbulence tends
1079to be along isopycnal surfaces (or more precisely neutral surfaces \cite{McDougall1987})
1080rather than across them.
1081As the slope of neutral surfaces is small in the ocean, a common
1082approximation is to assume that the `lateral' direction is the horizontal,
1083$i.e.$ the lateral mixing is performed along geopotential surfaces. This leads
1084to a geopotential second order operator for lateral subgrid scale physics.
1085This assumption can be relaxed: the eddy-induced turbulent fluxes can be
1086better approached by assuming that they depend linearly on the gradients of
1087large-scale quantities computed along neutral surfaces. In such a case,
1088the diffusive operator is an isoneutral second order operator and it has
1089components in the three space directions. However, both horizontal and
1090isoneutral operators have no effect on mean ($i.e.$ large scale) potential
1091energy whereas potential energy is a main source of turbulence (through
1092baroclinic instabilities). \citet{Gent1990} have proposed a
1093parameterisation of mesoscale eddy-induced turbulence which associates an
1094eddy-induced velocity to the isoneutral diffusion. Its mean effect is to
1095reduce the mean potential energy of the ocean. This leads to a formulation
1096of lateral subgrid-scale physics made up of an isoneutral second order
1097operator and an eddy induced advective part. In all these lateral diffusive
1098formulations, the specification of the lateral eddy coefficients remains the
1099problematic point as there is no really satisfactory formulation of these
1100coefficients as a function of large-scale features.
1101
1102In eddy-resolving configurations, a second order operator can be used, but
1103usually the more scale selective biharmonic operator is preferred as the
1104grid-spacing is usually not small enough compared to the scale of the
1105eddies. The role devoted to the subgrid-scale physics is to dissipate the
1106energy that cascades toward the grid scale and thus to ensure the stability of
1107the model while not interfering with the resolved mesoscale activity. Another approach
1108is becoming more and more popular: instead of specifying explicitly a sub-grid scale
1109term in the momentum and tracer time evolution equations, one uses a advective
1110scheme which is diffusive enough to maintain the model stability. It must be emphasised
1111that then, all the sub-grid scale physics is included in the formulation of the
1112advection scheme.
1113
1114All these parameterisations of subgrid scale physics have advantages and
1115drawbacks. There are not all available in \NEMO. In the $z$-coordinate
1116formulation, five options are offered for active tracers (temperature and
1117salinity): second order geopotential operator, second order isoneutral
1118operator, \citet{Gent1990} parameterisation, fourth order
1119geopotential operator, and various slightly diffusive advection schemes.
1120The same options are available for momentum, except
1121\citet{Gent1990} parameterisation which only involves tracers. In the
1122$s$-coordinate formulation, additional options are offered for tracers: second
1123order operator acting along $s-$surfaces, and for momentum: fourth order
1124operator acting along $s-$surfaces (see \S\ref{LDF}).
1125
1126\subsubsection{Lateral second order tracer diffusive operator}
1127
1128The lateral second order tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):
1129\begin{equation} \label{Eq_PE_iso_tensor}
1130D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
1131\mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c}
1132 1 \hfill & 0 \hfill & {-r_1 } \hfill \\
1133 0 \hfill & 1 \hfill & {-r_2 } \hfill \\
1134 {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\
1135\end{array} }} \right)
1136\end{equation}
1137where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along
1138which the diffusive operator acts and the model level ($e. g.$ $z$- or
1139$s$-surfaces). Note that the formulation \eqref{Eq_PE_iso_tensor} is exact for the
1140rotation between geopotential and $s$-surfaces, while it is only an approximation
1141for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter
1142case, two assumptions are made to simplify  \eqref{Eq_PE_iso_tensor} \citep{Cox1987}.
1143First, the horizontal contribution of the dianeutral mixing is neglected since the ratio
1144between iso and dia-neutral diffusive coefficients is known to be several orders of
1145magnitude smaller than unity. Second, the two isoneutral directions of diffusion are
1146assumed to be independent since the slopes are generally less than $10^{-2}$ in the
1147ocean (see Appendix~\ref{Apdx_B}).
1148
1149For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero. $\Re $ reduces to the identity
1150in the horizontal direction, no rotation is applied.
1151
1152For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the
1153geopotential and computational surfaces: they are equal to $\sigma _1$ and $\sigma _2$,
1154respectively (see \eqref{Eq_PE_sco_slope} ).
1155
1156For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral
1157and computational surfaces. Therefore, they are different quantities,
1158but have similar expressions in $z$- and $s$-coordinates. In $z$-coordinates:
1159\begin{equation} \label{Eq_PE_iso_slopes}
1160r_1 =\frac{e_3 }{e_1 }  \left( {\frac{\partial \rho }{\partial i}} \right)
1161                  \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \ , \quad
1162r_1 =\frac{e_3 }{e_1 }  \left( {\frac{\partial \rho }{\partial i}} \right)
1163                  \left( {\frac{\partial \rho }{\partial k}} \right)^{-1},
1164\end{equation}
1165while in $s$-coordinates $\partial/\partial k$ is replaced by
1166$\partial/\partial s$.
1167
1168\subsubsection{Eddy induced velocity}
1169 When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used,
1170an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers:
1171\begin{equation} \label{Eq_PE_iso+eiv}
1172D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right)
1173           +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right)
1174\end{equation}
1175where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent,
1176eddy-induced transport velocity. This velocity field is defined by:
1177\begin{equation} \label{Eq_PE_eiv}
1178   \begin{split}
1179 u^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\ 
1180 v^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\ 
1181 w^\ast &=  -\frac{1}{e_1 e_2 }\left[
1182                      \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right)
1183                    +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right)      \right]
1184   \end{split}
1185\end{equation}
1186where $A^{eiv}$ is the eddy induced velocity coefficient (or equivalently the isoneutral
1187thickness diffusivity coefficient), and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes
1188between isoneutral and \emph{geopotential} surfaces. Their values are
1189thus independent of the vertical coordinate, but their expression depends on the coordinate:
1190\begin{align} \label{Eq_PE_slopes_eiv}
1191\tilde{r}_n = \begin{cases}
1192   r_n                  &      \text{in $z$-coordinate}    \\
1193   r_n + \sigma_n &      \text{in \textit{z*} and $s$-coordinates} 
1194                   \end{cases}
1195\quad \text{where } n=1,2
1196\end{align}
1197
1198The normal component of the eddy induced velocity is zero at all the boundaries.
1199This can be achieved in a model by tapering either the eddy coefficient or the slopes
1200to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}).
1201
1202\subsubsection{Lateral fourth order tracer diffusive operator}
1203
1204The lateral fourth order tracer diffusive operator is defined by:
1205\begin{equation} \label{Eq_PE_bilapT}
1206D^{lT}=\Delta \left( \;\Delta T \right)
1207\qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right)
1208 \end{equation}
1209It is the second order operator given by \eqref{Eq_PE_iso_tensor} applied twice with
1210the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
1211
1212
1213\subsubsection{Lateral second order momentum diffusive operator}
1214
1215The second order momentum diffusive operator along $z$- or $s$-surfaces is found by
1216applying \eqref{Eq_PE_lap_vector} to the horizontal velocity vector (see Appendix~\ref{Apdx_B}):
1217\begin{equation} \label{Eq_PE_lapU}
1218\begin{split}
1219{\rm {\bf D}}^{l{\rm {\bf U}}} 
1220&= \quad \  \nabla _h \left( {A^{lm}\chi } \right)
1221   \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right)     \\
1222&=   \left(      \begin{aligned}
1223             \frac{1}{e_1      } \frac{\partial \left( A^{lm} \chi          \right)}{\partial i} 
1224         &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j}  \\
1225             \frac{1}{e_2      }\frac{\partial \left( {A^{lm} \chi         } \right)}{\partial j}   
1226         &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i}
1227        \end{aligned}    \right)
1228\end{split}
1229\end{equation}
1230
1231Such a formulation ensures a complete separation between the vorticity and
1232horizontal divergence fields (see Appendix~\ref{Apdx_C}).
1233Unfortunately, it is only available in \textit{iso-level} direction.
1234When a rotation is required ($i.e.$ geopotential diffusion in $s-$coordinates
1235or isoneutral diffusion in both $z$- and $s$-coordinates), the $u$ and $v-$fields
1236are considered as independent scalar fields, so that the diffusive operator is given by:
1237\begin{equation} \label{Eq_PE_lapU_iso}
1238\begin{split}
1239 D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\ 
1240 D_v^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla v} \right)
1241 \end{split}
1242 \end{equation}
1243where $\Re$ is given by  \eqref{Eq_PE_iso_tensor}. It is the same expression as
1244those used for diffusive operator on tracers. It must be emphasised that such a
1245formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or
1246$\beta-$plane, not on the sphere. It is also a very good approximation in vicinity
1247of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}.
1248
1249\subsubsection{lateral fourth order momentum diffusive operator}
1250
1251As for tracers, the fourth order momentum diffusive operator along $z$ or $s$-surfaces
1252is a re-entering second order operator \eqref{Eq_PE_lapU} or \eqref{Eq_PE_lapU_iso} 
1253with the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
1254
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