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Changeset 11422 for NEMO/branches/2019/fix_vvl_ticket1791/doc/latex/NEMO/subfiles/chap_model_basics.tex – NEMO

Ignore:
Timestamp:
2019-08-08T15:40:47+02:00 (5 years ago)
Author:
jchanut
Message:

#1791, merge with trunk

Location:
NEMO/branches/2019/fix_vvl_ticket1791/doc
Files:
4 edited

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    r10544 r11422  
    3232\begin{enumerate} 
    3333\item 
    34   \textit{spherical earth approximation}: the geopotential surfaces are assumed to be spheres so that 
    35   gravity (local vertical) is parallel to the earth's radius 
     34  \textit{spherical Earth approximation}: the geopotential surfaces are assumed to be oblate spheriods 
     35  that follow the Earth's bulge; these spheroids are approximated by spheres with 
     36  gravity locally vertical (parallel to the Earth's radius) and independent of latitude  
     37  \citep[][section 2]{white.hoskins.ea_QJRMS05}.    
    3638\item 
    3739  \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius 
    … …  
    6365    \nabla \cdot \vect U = 0 
    6466  \end{equation} 
     67 \item  
     68  \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected.  
     69  These terms may be non-negligible where the Brunt-Vaisala frequency $N$ is small, either in the deep ocean or 
     70  in the sub-mesoscale motions of the mixed layer, or near the equator \citep[][section 1]{white.hoskins.ea_QJRMS05}.  
     71  They can be consistently included as part of the ocean dynamics \citep[][section 3(d)]{white.hoskins.ea_QJRMS05} and are  
     72  retained in the MIT ocean model.      
    6573\end{enumerate} 
    6674 
    6775Because the gravitational force is so dominant in the equations of large-scale motions, 
    68 it is useful to choose an orthogonal set of unit vectors $(i,j,k)$ linked to the earth such that 
     76it is useful to choose an orthogonal set of unit vectors $(i,j,k)$ linked to the Earth such that 
    6977$k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$, 
    7078\ie tangent to the geopotential surfaces. 
    … …  
    107115an air-sea or ice-sea interface at its top. 
    108116These boundaries can be defined by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,k,t)$, 
    109 where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface. 
    110 Both $H$ and $\eta$ are usually referenced to a given surface, $z = 0$, chosen as a mean sea surface 
     117where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface  
     118(discretisation can introduce additional artificial ``side-wall'' boundaries).  
     119Both $H$ and $\eta$ are referenced to a surface of constant geopotential (\ie a mean sea surface height) on which $z = 0$.  
    111120(\autoref{fig:ocean_bc}). 
    112121Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with 
    … …  
    120129\begin{figure}[!ht] 
    121130  \begin{center} 
    122     \includegraphics[]{Fig_I_ocean_bc} 
     131    \includegraphics[width=\textwidth]{Fig_I_ocean_bc} 
    123132    \caption{ 
    124133      \protect\label{fig:ocean_bc} 
    … …  
    210219The flow is barotropic and the surface moves up and down with gravity as the restoring force. 
    211220The phase speed of such waves is high (some hundreds of metres per second) so that 
    212 the time step would have to be very short if they were present in the model. 
     221the time step has to be very short when they are present in the model. 
    213222The latter strategy filters out these waves since the rigid lid approximation implies $\eta = 0$, 
    214223\ie the sea surface is the surface $z = 0$. 
    … …  
    217226The rigid-lid hypothesis is an obsolescent feature in modern OGCMs. 
    218227It has been available until the release 3.1 of \NEMO, and it has been removed in release 3.2 and followings. 
    219 Only the free surface formulation is now described in the this document (see the next sub-section). 
     228Only the free surface formulation is now described in this document (see the next sub-section). 
    220229 
    221230% ------------------------------------------------------------------------------------------------------------- 
    … …  
    237246Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as 
    238247a class of solution of the primitive equations. 
    239 These waves are barotropic because of hydrostatic assumption, and their phase speed is quite high. 
     248These waves are barotropic (\ie nearly independent of depth) and their phase speed is quite high. 
    240249Their time scale is short with respect to the other processes described by the primitive equations. 
    241250 
    … …  
    258267If further, an approximative conservation of heat and salt contents is sufficient for the problem solved, 
    259268then it is sufficient to solve a linearized version of \autoref{eq:PE_ssh}, 
    260 which still allows to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}. 
     269which still allows to take into account freshwater fluxes applied at the ocean surface \citep{roullet.madec_JGR00}. 
    261270Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost. 
    262271 
    263272The filtering of EGWs in models with a free surface is usually a matter of discretisation of 
    264273the temporal derivatives, 
    265 using a split-explicit method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92} or 
    266 the implicit scheme \citep{Dukowicz1994} or 
    267 the addition of a filtering force in the momentum equation \citep{Roullet_Madec_JGR00}. 
    268 With the present release, \NEMO offers the choice between 
     274using a split-explicit method \citep{killworth.webb.ea_JPO91, zhang.endoh_JGR92} or 
     275the implicit scheme \citep{dukowicz.smith_JGR94} or 
     276the addition of a filtering force in the momentum equation \citep{roullet.madec_JGR00}. 
     277With the present release, \NEMO  offers the choice between 
    269278an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or 
    270 a split-explicit scheme strongly inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05} 
     279a split-explicit scheme strongly inspired the one proposed by \citet{shchepetkin.mcwilliams_OM05} 
    271280(see \autoref{subsec:DYN_spg_ts}). 
    272281 
    … …  
    292301cannot be easily treated in a global model without filtering. 
    293302A solution consists of introducing an appropriate coordinate transformation that 
    294 shifts the singular point onto land \citep{Madec_Imbard_CD96, Murray_JCP96}. 
     303shifts the singular point onto land \citep{madec.imbard_CD96, murray_JCP96}. 
    295304As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems. 
    296305An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism. 
    … …  
    298307Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation), 
    299308with preservation of the local vertical. Here we give the simplified equations for this particular case. 
    300 The general case is detailed by \citet{Eiseman1980} in their survey of the conservation laws of fluid dynamics. 
     309The general case is detailed by \citet{eiseman.stone_SR80} in their survey of the conservation laws of fluid dynamics. 
    301310 
    302311Let $(i,j,k)$ be a set of orthogonal curvilinear coordinates on 
    … …  
    323332\begin{figure}[!tb] 
    324333  \begin{center} 
    325     \includegraphics[]{Fig_I_earth_referential} 
     334    \includegraphics[width=\textwidth]{Fig_I_earth_referential} 
    326335    \caption{ 
    327336      \protect\label{fig:referential} 
    … …  
    338347the vertical scale factor is a single function of $k$ as $k$ is parallel to $z$. 
    339348The scalar and vector operators that appear in the primitive equations 
    340 (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can be written in the tensorial form, 
     349(\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can then be written in the tensorial form, 
    341350invariant in any orthogonal horizontal curvilinear coordinate system transformation: 
    342351\begin{subequations} 
    … …  
    384393\end{gather} 
    385394 
    386 Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that 
     395Using again the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that 
    387396$e_3$  is a function of the single variable $k$, 
    388397$NLT$ the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows: 
    … …  
    456465  &      &= &\nabla \cdot (\vect U \, u) - (\nabla \cdot \vect U) \ u 
    457466            + \frac{1}{e_1 e_2} \lt( -v^2 \pd[e_2]{i} + u v \, \pd[e_1]{j} \rt) \\ 
    458   \intertext{as $\nabla \cdot {\vect U} \; = 0$ (incompressibility) it comes:} 
     467  \intertext{as $\nabla \cdot {\vect U} \; = 0$ (incompressibility) it becomes:} 
    459468  &      &= &\, \nabla \cdot (\vect U \, u) + \frac{1}{e_1 e_2} \lt( v \; \pd[e_2]{i} - u \; \pd[e_1]{j} \rt) (-v) 
    460469\end{alignat*} 
    … …  
    516525    % \label{eq:PE_dyn_flux_v} 
    517526    \pd[v]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u \\ 
    518                 + \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\ 
     527                - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\ 
    519528                - \frac{1}{e_3} \pd[(w \, v)]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) 
    520529                + D_v^{\vect U} + F_v^{\vect U} 
    … …  
    526535    p_s = \rho \,g \, \eta 
    527536  \] 
    528   with $\eta$ is solution of \autoref{eq:PE_ssh}. 
     537  and $\eta$ is the solution of \autoref{eq:PE_ssh}. 
    529538 
    530539  The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: 
    … …  
    536545  \] 
    537546  where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}. 
    538 \item \textit{tracer equations}: 
    539   \[ 
    540     %\label{eq:S} 
    541     \pd[T]{t} = - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 T \, u)]{i} + \pd[(e_1 T \, v)]{j} \rt] 
     547 
     548\item  
     549  \textbf{tracer equations}: 
     550  \begin{equation} 
     551  \begin{split} 
     552    \pd[T]{t} = & - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 T \, u)]{i} + \pd[(e_1 T \, v)]{j} \rt] 
    542553                - \frac{1}{e_3} \pd[(T \, w)]{k} + D^T + F^T \\ 
    543     %\label{eq:T} 
    544     \pd[S]{t} = - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 S \, u)]{i} + \pd[(e_1 S \, v)]{j} \rt] 
    545                 - \frac{1}{e_3} \pd[(S \, w)]{k} + D^S + F^S 
    546     %\label{eq:rho} 
    547     \rho = \rho \big( T,S,z(k) \big) 
    548   \] 
     554    \pd[S]{t} = & - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 S \, u)]{i} + \pd[(e_1 S \, v)]{j} \rt] 
     555                - \frac{1}{e_3} \pd[(S \, w)]{k} + D^S + F^S \\ 
     556    \rho = & \rho \big( T,S,z(k) \big) 
     557  \end{split} 
     558  \end{equation} 
    549559\end{itemize} 
    550560 
    … …  
    575585follows the isopycnal surfaces, \eg an isopycnic coordinate. 
    576586 
    577 In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in 
     587In order to satisfy two or more constraints one can even be tempted to mixed these coordinate systems, as in 
    578588HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at 
    579 the ocean bottom) \citep{Chassignet_al_JPO03} or 
     589the ocean bottom) \citep{chassignet.smith.ea_JPO03} or 
    580590OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere) 
    581 \citep{Madec_al_JPO96} among others. 
     591\citep{madec.delecluse.ea_JPO96} among others. 
    582592 
    583593In fact one is totally free to choose any space and time vertical coordinate by 
    … …  
    592602the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through 
    593603\autoref{eq:PE_s}. 
    594 This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact 
     604This so-called \textit{generalised vertical coordinate} \citep{kasahara_MWR74} is in fact 
    595605an Arbitrary Lagrangian--Eulerian (ALE) coordinate. 
    596 Indeed, choosing an expression for $s$ is an arbitrary choice that determines 
     606Indeed, one has a great deal of freedom in the choice of expression for $s$. The choice determines 
    597607which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and 
    598608which part will be used to move them (Lagrangian part). 
    599 The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, 
     609The coordinate is also sometime referenced as an adaptive coordinate \citep{hofmeister.burchard.ea_OM10}, 
    600610since the coordinate system is adapted in the course of the simulation. 
    601611Its most often used implementation is via an ALE algorithm, 
    602612in which a pure lagrangian step is followed by regridding and remapping steps, 
    603 the later step implicitly embedding the vertical advection 
    604 \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. 
    605 Here we follow the \citep{Kasahara_MWR74} strategy: 
    606 a regridding step (an update of the vertical coordinate) followed by an eulerian step with 
     613the latter step implicitly embedding the vertical advection 
     614\citep{hirt.amsden.ea_JCP74, chassignet.smith.ea_JPO03, white.adcroft.ea_JCP09}. 
     615Here we follow the \citep{kasahara_MWR74} strategy: 
     616a regridding step (an update of the vertical coordinate) followed by an Eulerian step with 
    607617an explicit computation of vertical advection relative to the moving s-surfaces. 
    608618 
    609619%\gmcomment{ 
    610620%A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient... 
    611 the generalized vertical coordinates used in ocean modelling are not orthogonal, 
     621The generalized vertical coordinates used in ocean modelling are not orthogonal, 
    612622which contrasts with many other applications in mathematical physics. 
    613623Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter. 
    … …  
    615625The horizontal velocity in ocean models measures motions in the horizontal plane, 
    616626perpendicular to the local gravitational field. 
    617 That is, horizontal velocity is mathematically the same regardless the vertical coordinate, be it geopotential, 
     627That is, horizontal velocity is mathematically the same regardless of the vertical coordinate, be it geopotential, 
    618628isopycnal, pressure, or terrain following. 
    619629The key motivation for maintaining the same horizontal velocity component is that 
    … …  
    660670\[ 
    661671  % \label{eq:PE_sco_w} 
    662   \omega = w - e_3 \, \pd[s]{t} - \sigma_1 \, u - \sigma_2 \, v 
     672  \omega = w -  \, \lt. \pd[z]{t} \rt|_s - \sigma_1 \, u - \sigma_2 \, v 
    663673\] 
    664674 
    … …  
    671681  % \label{eq:PE_sco_u_vector} 
    672682    \pd[u]{t} = + (\zeta + f) \, v - \frac{1}{2 \, e_1} \pd[]{i} (u^2 + v^2) - \frac{1}{e_3} \omega \pd[u]{k} \\ 
    673                 - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) + g \frac{\rho}{\rho_o} \sigma_1 
     683                - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 
    674684                + D_u^{\vect U} + F_u^{\vect U} 
    675685  \end{multline*} 
    … …  
    677687  % \label{eq:PE_sco_v_vector} 
    678688    \pd[v]{t} = - (\zeta + f) \, u - \frac{1}{2 \, e_2} \pd[]{j}(u^2 + v^2) - \frac{1}{e_3} \omega \pd[v]{k} \\ 
    679                 - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) + g \frac{\rho}{\rho_o} \sigma_2 
     689                - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_2 
    680690                + D_v^{\vect U} + F_v^{\vect U} 
    681691  \end{multline*} 
    … …  
    687697                                       - \frac{1}{e_3} \pd[(\omega \, u)]{k} 
    688698                                       - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) 
    689                                        + g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U} 
     699                                       - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U} 
    690700  \end{multline*} 
    691701  \begin{multline*} 
    … …  
    695705                                       - \frac{1}{e_3} \pd[(\omega \, v)]{k} 
    696706                                       - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) 
    697                                        + g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U} 
     707                                       - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U} 
    698708  \end{multline*} 
    699709  where the relative vorticity, $\zeta$, the surface pressure gradient, 
    … …  
    738748\begin{figure}[!b] 
    739749  \begin{center} 
    740     \includegraphics[]{Fig_z_zstar} 
     750    \includegraphics[width=\textwidth]{Fig_z_zstar} 
    741751    \caption{ 
    742752      \protect\label{fig:z_zstar} 
    … …  
    744754      (b) $z$-coordinate in non-linear free surface case ; 
    745755      (c) re-scaled height coordinate 
    746       (become popular as the \zstar-coordinate \citep{Adcroft_Campin_OM04}). 
     756      (become popular as the \zstar-coordinate \citep{adcroft.campin_OM04}). 
    747757    } 
    748758  \end{center} 
    … …  
    750760%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    751761 
    752 In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 
    753 These coordinates systems is presented in a report \citep{Levier2007} available on the \NEMO web site. 
     762In this case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 
     763These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO web site. 
    754764 
    755765The \zstar coordinate approach is an unapproximated, non-linear free surface implementation which allows one to 
    756 deal with large amplitude free-surface variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. 
     766deal with large amplitude free-surface variations relative to the vertical resolution \citep{adcroft.campin_OM04}. 
    757767In the \zstar formulation, 
    758768the variation of the column thickness due to sea-surface undulations is not concentrated in the surface level, 
    … …  
    766776The position (\zstar) and vertical discretization (\zstar) are expressed as: 
    767777\[ 
    768   % \label{eq:z-star} 
     778  % \label{eq:PE_z-star} 
    769779  H + \zstar = (H + z)  / r \quad \text{and}  \quad \delta \zstar 
    770780              = \delta z / r \quad \text{with} \quad r 
    771               = \frac{H + \eta}{H} 
     781              = \frac{H + \eta}{H} . 
     782\] 
     783Simple re-organisation of the above expressions gives 
     784\[ 
     785  % \label{eq:PE_zstar_2} 
     786  \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) .  
    772787\] 
    773788Since the vertical displacement of the free surface is incorporated in the vertical coordinate \zstar, 
    … …  
    776791Also the divergence of the flow field is no longer zero as shown by the continuity equation: 
    777792\[ 
    778   \pd[r]{t} = \nabla_{\zstar} \cdot \lt( r \; \vect U_h \rt) (r \; w *) = 0 
     793  \pd[r]{t} = \nabla_{\zstar} \cdot \lt( r \; \vect U_h \rt) + \pd[r \; w^*]{\zstar} = 0 . 
    779794\] 
    780  
    781 % from MOM4p1 documentation 
    782 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate  
    783 \[ 
    784   % \label{eq:PE_} 
    785   \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) 
    786 \] 
    787  
    788 This coordinate is closely related to the "eta" coordinate used in many atmospheric models 
     795This \zstar coordinate is closely related to the "eta" coordinate used in many atmospheric models 
    789796(see Black (1994) for a review of eta coordinate atmospheric models). 
    790797It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, 
    … …  
    798805These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 
    799806terrain following sigma models discussed in \autoref{subsec:PE_sco}. 
    800 Additionally, since \zstar when $\eta = 0$, 
     807Additionally, since $\zstar = z$ when $\eta = 0$, 
    801808no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography. 
    802809This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of 
    … …  
    804811depending on the sophistication of the pressure gradient solver. 
    805812The quasi -horizontal nature of the coordinate surfaces also facilitates the implementation of 
    806 neutral physics parameterizations in \zstar models using the same techniques as in $z$-models 
    807 (see Chapters 13-16 of \cite{Griffies_Bk04}) for a discussion of neutral physics in $z$-models, 
     813neutral physics parameterizations in \zstar  models using the same techniques as in $z$-models 
     814(see Chapters 13-16 of \cite{griffies_bk04}) for a discussion of neutral physics in $z$-models, 
    808815as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). 
    809816 
    810 The range over which \zstar varies is time independent $-H \leq \zstar \leq 0$. 
    811 Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. 
     817The range over which \zstar  varies is time independent $-H \leq \zstar \leq 0$. 
     818Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > -H$. 
    812819This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. 
    813820 
    814 Because \zstar has a time independent range, all grid cells have static increments ds, 
    815 and the sum of the ver tical increments yields the time independent ocean depth. %k ds = H. 
     821Because \zstar  has a time independent range, all grid cells have static increments ds, 
     822and the sum of the vertical increments yields the time independent ocean depth. %k ds = H. 
    816823The \zstar coordinate is therefore invisible to undulations of the free surface, 
    817824since it moves along with the free surface. 
    818 This proper ty means that no spurious vertical transport is induced across surfaces of constant \zstar by 
     825This property means that no spurious vertical transport is induced across surfaces of constant \zstar by 
    819826the motion of external gravity waves. 
    820 Such spurious transpor t can be a problem in z-models, especially those with tidal forcing. 
    821 Quite generally, the time independent range for the \zstar coordinate is a very convenient property that 
    822 allows for a nearly arbitrary ver tical resolution even in the presence of large amplitude fluctuations of 
     827Such spurious transport can be a problem in z-models, especially those with tidal forcing. 
     828Quite generally, the time independent range for the \zstar  coordinate is a very convenient property that 
     829allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of 
    823830the surface height, again so long as $\eta > -H$. 
    824831%end MOM doc %%% 
    … …  
    849856The response to such a velocity field often leads to numerical dispersion effects. 
    850857One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of 
    851 a full step one \cite{Pacanowski_Gnanadesikan_MWR98}. 
     858a full step one \cite{pacanowski.gnanadesikan_MWR98}. 
    852859Another solution is to introduce a terrain-following coordinate system (hereafter $s$-coordinate). 
    853860 
    … …  
    870877\begin{equation} 
    871878  \label{eq:PE_p_sco} 
    872   \nabla p |_z = \nabla p |_s - \pd[p]{s} \nabla z |_s 
     879  \nabla p |_z = \nabla p |_s - \frac{1}{e_3} \pd[p]{s} \nabla z |_s 
    873880\end{equation} 
    874881 
    875882The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface and 
    876 introduces a truncation error that is not present in a $z$-model. 
     883leads to a truncation error that is not present in a $z$-model. 
    877884In the special case of a $\sigma$-coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), 
    878 \citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude of this truncation error. 
     885\citet{haney_JPO91} and \citet{beckmann.haidvogel_JPO93} have given estimates of the magnitude of this truncation error. 
    879886It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state, 
    880887and the finite difference scheme. 
    … …  
    884891The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive. 
    885892This problem can be at least partially overcome by mixing $s$-coordinate and 
    886 step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec_al_JPO96}. 
     893step-like representation of bottom topography \citep{gerdes_JGR93*a,gerdes_JGR93*b,madec.delecluse.ea_JPO96}. 
    887894However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for 
    888895a realistic bottom topography: 
    889 a envelope topography is defined in $s$-coordinate on which a full or 
     896an envelope topography is defined in $s$-coordinate on which a full or 
    890897partial step bottom topography is then applied in order to adjust the model depth to the observed one 
    891898(see \autoref{sec:DOM_zgr}. 
    … …  
    904911In contrast, the ocean will stay at rest in a $z$-model. 
    905912As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below 
    906 the strongly stratified portion of the water column (\ie the main thermocline) \citep{Madec_al_JPO96}. 
     913the strongly stratified portion of the water column (\ie the main thermocline) \citep{madec.delecluse.ea_JPO96}. 
    907914An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces 
    908915(see \autoref{subsec:PE_ldf}). 
    … …  
    910917strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}). 
    911918 
    912 The $s$-coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} differ mainly in two aspects from 
     919The $s$-coordinates introduced here \citep{lott.madec.ea_OM90,madec.delecluse.ea_JPO96} differ mainly in two aspects from 
    913920similar models: 
    914921it allows a representation of bottom topography with mixed full or partial step-like/terrain following topography; 
    … …  
    921928\label{subsec:PE_zco_tilde} 
    922929 
    923 The \ztilde -coordinate has been developed by \citet{Leclair_Madec_OM11}. 
    924 It is available in \NEMO since the version 3.4. 
     930The \ztilde -coordinate has been developed by \citet{leclair.madec_OM11}. 
     931It is available in \NEMO since the version 3.4 and is more robust in version 4.0 than previously.  
    925932Nevertheless, it is currently not robust enough to be used in all possible configurations. 
    926933Its use is therefore not recommended. 
    … …  
    934941\label{sec:PE_zdf_ldf} 
    935942 
    936 The primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than 
     943The hydrostatic primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than 
    937944a few kilometres in the horizontal, a few meters in the vertical and a few minutes. 
    938945They are usually solved at larger scales: the specified grid spacing and time step of the numerical model. 
    … …  
    984991All the vertical physics is embedded in the specification of the eddy coefficients. 
    985992They can be assumed to be either constant, or function of the local fluid properties 
    986 (\eg Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), 
     993(\eg Richardson number, Brunt-Vais\"{a}l\"{a} frequency, distance from the boundary ...), 
    987994or computed from a turbulent closure model. 
    988995The choices available in \NEMO are discussed in \autoref{chap:ZDF}). 
    … …  
    10051012The resulting lateral diffusive and dissipative operators are of second order. 
    10061013Observations show that lateral mixing induced by mesoscale turbulence tends to be along isopycnal surfaces 
    1007 (or more precisely neutral surfaces \cite{McDougall1987}) rather than across them. 
     1014(or more precisely neutral surfaces \cite{mcdougall_JPO87}) rather than across them. 
    10081015As the slope of neutral surfaces is small in the ocean, a common approximation is to assume that 
    10091016the `lateral' direction is the horizontal, \ie the lateral mixing is performed along geopotential surfaces. 
    … …  
    10161023both horizontal and isoneutral operators have no effect on mean (\ie large scale) potential energy whereas 
    10171024potential energy is a main source of turbulence (through baroclinic instabilities). 
    1018 \citet{Gent1990} have proposed a parameterisation of mesoscale eddy-induced turbulence which 
     1025\citet{gent.mcwilliams_JPO90} proposed a parameterisation of mesoscale eddy-induced turbulence which 
    10191026associates an eddy-induced velocity to the isoneutral diffusion. 
    10201027Its mean effect is to reduce the mean potential energy of the ocean. 
    … …  
    10331040Another approach is becoming more and more popular: 
    10341041instead of specifying explicitly a sub-grid scale term in the momentum and tracer time evolution equations, 
    1035 one uses a advective scheme which is diffusive enough to maintain the model stability. 
     1042one uses an advective scheme which is diffusive enough to maintain the model stability. 
    10361043It must be emphasised that then, all the sub-grid scale physics is included in the formulation of 
    10371044the advection scheme. 
    10381045 
    10391046All these parameterisations of subgrid scale physics have advantages and drawbacks. 
    1040 There are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are: 
     1047They are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are: 
    10411048Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces, 
    1042 \citet{Gent1990} parameterisation, and various slightly diffusive advection schemes. 
     1049\citet{gent.mcwilliams_JPO90} parameterisation, and various slightly diffusive advection schemes. 
    10431050For momentum, the main ones are: Laplacian and bilaplacian operators acting along geopotential surfaces, 
    10441051and UBS advection schemes when flux form is chosen for the momentum advection. 
    … …  
    10621069the rotation between geopotential and $s$-surfaces, 
    10631070while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces. 
    1064 Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}. 
     1071Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{cox_OM87}. 
    10651072First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and 
    10661073dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity. 
    … …  
    10871094\subsubsection{Eddy induced velocity} 
    10881095 
    1089 When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used, 
     1096When the \textit{eddy induced velocity} parametrisation (eiv) \citep{gent.mcwilliams_JPO90} is used, 
    10901097an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 
    10911098\[ 
    … …  
    11411148                         - \nabla_h \times \big( A^{lm} \, \zeta \; \vect k \big) \\ 
    11421149                      &= \lt(   \frac{1}{e_1}     \pd[ \lt( A^{lm}    \chi      \rt) ]{i} \rt. 
    1143                               - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} 
     1150                              - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} ,  
    11441151                                \frac{1}{e_2}     \pd[ \lt( A^{lm}    \chi      \rt) ]{j} 
    11451152                         \lt. + \frac{1}{e_1 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{i} \rt) 
    … …  
    11621169\ie on a $f$- or $\beta$-plane, not on the sphere. 
    11631170It is also a very good approximation in vicinity of the Equator in 
    1164 a geographical coordinate system \citep{Lengaigne_al_JGR03}. 
    1165  
    1166 \subsubsection{lateral bilaplacian momentum diffusive operator} 
     1171a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}. 
     1172 
     1173\subsubsection{Lateral bilaplacian momentum diffusive operator} 
    11671174 
    11681175As for tracers, the bilaplacian order momentum diffusive operator is a re-entering Laplacian operator with 
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