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Changeset 12063 for NEMO/branches/2019/dev_ASINTER-01-05_merged/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex – NEMO

Ignore:
Timestamp:
2019-12-05T11:46:38+01:00 (4 years ago)
Author:
gsamson
Message:

dev_ASINTER-01-05_merged: update branch with dev_r11085_ASINTER-05_Brodeau_Advanced_Bulk@r12061 and trunk@r12055 + bugfix for agrif compatibility in sbcblk: sette tests with ref configs ok except ABL restartability (under investigation) (tickets #2159 and #2131)

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NEMO/branches/2019/dev_ASINTER-01-05_merged/doc
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5 edited

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  • NEMO/branches/2019/dev_ASINTER-01-05_merged/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

    r11263 r12063  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter 1 Model Basics 
    6 % ================================================================ 
    7 % ================================================================ 
    8 % Curvilinear \zstar- \sstar-coordinate System 
    9 % ================================================================ 
     4 
    105\chapter{ essai \zstar \sstar} 
     6 
     7\thispagestyle{plain} 
     8 
     9\chaptertoc 
     10 
     11\paragraph{Changes record} ~\\ 
     12 
     13{\footnotesize 
     14  \begin{tabularx}{\textwidth}{l||X|X} 
     15    Release & Author(s) & Modifications \\ 
     16    \hline 
     17    {\em   4.0} & {\em ...} & {\em ...} \\ 
     18    {\em   3.6} & {\em ...} & {\em ...} \\ 
     19    {\em   3.4} & {\em ...} & {\em ...} \\ 
     20    {\em <=3.4} & {\em ...} & {\em ...} 
     21  \end{tabularx} 
     22} 
     23 
     24\clearpage 
     25 
     26%% ================================================================================================= 
    1127\section{Curvilinear \zstar- or \sstar coordinate system} 
    1228 
    13 % ------------------------------------------------------------------------------------------------------------- 
    14 % ???? 
    15 % ------------------------------------------------------------------------------------------------------------- 
    16  
    1729\colorbox{yellow}{ to be updated } 
    1830 
    1931In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 
    20 These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO web site.  
     32These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. 
    2133 
    2234\colorbox{yellow}{  end of to be updated} 
    … …  
    2436% from MOM4p1 documentation 
    2537 
    26 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate  
    27 \[ 
    28   % \label{eq:PE_} 
     38To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 
     39\[ 
     40  % \label{eq:MBZ_PE_} 
    2941  z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 
    3042\] 
    … …  
    4052the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. 
    4153These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 
    42 terrain following sigma models discussed in \autoref{subsec:PE_sco}.  
     54terrain following sigma models discussed in \autoref{subsec:MB_sco}. 
    4355Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in 
    4456an unforced ocean starting from rest, regardless the bottom topography. 
    … …  
    4961neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models 
    5062(see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models, 
    51 as well as  \autoref{sec:LDF_slp} in this document for treatment in \NEMO).  
     63as well as  \autoref{sec:LDF_slp} in this document for treatment in \NEMO). 
    5264 
    5365The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. 
    5466Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. 
    55 This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$.  
     67This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. 
    5668 
    5769Because $z^\star$ has a time independent range, all grid cells have static increments ds, 
    58 and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H.  
     70and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H. 
    5971The $z^\star$ coordinate is therefore invisible to undulations of the free surface, 
    6072since it moves along with the free surface. 
    … …  
    6476Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that 
    6577allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of 
    66 the surface height, again so long as $\eta > -H$.  
    67  
    68 %%% 
     78the surface height, again so long as $\eta > -H$. 
     79 
    6980%  essai update time splitting... 
    70 %%% 
    71  
    72 % ================================================================ 
    73 % Surface Pressure Gradient and Sea Surface Height 
    74 % ================================================================ 
    75 \section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})] 
    76 {Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} 
    77 \label{sec:DYN_hpg_spg} 
    78 %-----------------------------------------nam_dynspg---------------------------------------------------- 
    79  
    80 %\nlst{nam_dynspg}  
    81 %------------------------------------------------------------------------------------------------------------ 
    82 Options are defined through the \ngn{nam\_dynspg} namelist variables. 
    83 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). 
     81 
     82%% ================================================================================================= 
     83\section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})]{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} 
     84\label{sec:MBZ_dyn_hpg_spg} 
     85 
     86%\nlst{nam_dynspg} 
     87Options are defined through the \nam{_dynspg}{\_dynspg} namelist variables. 
     88The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). 
    8489The main distinction is between the fixed volume case (linear free surface or rigid lid) and 
    8590the variable volume case (nonlinear free surface, \key{vvl} is active). 
    86 In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), 
     91In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), 
    8792the vertical scale factors $e_{3}$ are fixed in time, 
    88 while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent. 
     93while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent. 
    8994With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 
    9095which imposes a very small time step when an explicit time stepping is used. 
    9196Two methods are proposed to allow a longer time step for the three-dimensional equations: 
    92 the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:PE_flt?}), 
     97the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}), 
    9398and the split-explicit free surface described below. 
    9499The extra term introduced in the filtered method is calculated implicitly, 
    95100so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 
    96101 
    97 %------------------------------------------------------------- 
    98102% Explicit 
    99 %------------------------------------------------------------- 
    100 \subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})] 
    101 {Explicit (\protect\key{dynspg\_exp})} 
    102 \label{subsec:DYN_spg_exp} 
     103%% ================================================================================================= 
     104\subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})]{Explicit (\protect\key{dynspg\_exp})} 
     105\label{subsec:MBZ_dyn_spg_exp} 
    103106 
    104107In the explicit free surface formulation, the model time step is chosen small enough to 
    … …  
    106109The sea surface height is given by: 
    107110\begin{equation} 
    108   \label{eq:dynspg_ssh} 
     111  \label{eq:MBZ_dynspg_ssh} 
    109112  \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T} 
    110113    e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u} 
    … …  
    116119and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. 
    117120The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, 
    118 (\ie the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity).  
     121(\ie\ the velocity appearing in (\autoref{eq:DYN_spg_ssh}) is centred in time (\textit{now} velocity). 
    119122 
    120123The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: 
    121124\begin{equation} 
    122   \label{eq:dynspg_exp} 
     125  \label{eq:MBZ_dynspg_exp} 
    123126  \left\{ 
    124127    \begin{aligned} 
    … …  
    127130    \end{aligned} 
    128131  \right. 
    129 \end{equation}  
     132\end{equation} 
    130133 
    131134Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in 
    132 (\autoref{eq:dynspg_exp}).  
    133  
    134 %------------------------------------------------------------- 
     135(\autoref{eq:DYN_spg_exp}). 
     136 
    135137% Split-explicit time-stepping 
    136 %------------------------------------------------------------- 
    137 \subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})] 
    138 {Split-explicit time-stepping (\protect\key{dynspg\_ts})} 
    139 \label{subsec:DYN_spg_ts} 
    140 %--------------------------------------------namdom---------------------------------------------------- 
    141  
    142 \nlst{namdom}  
    143 %-------------------------------------------------------------------------------------------------------------- 
     138%% ================================================================================================= 
     139\subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})]{Split-explicit time-stepping (\protect\key{dynspg\_ts})} 
     140\label{subsec:MBZ_dyn_spg_ts} 
     141 
    144142The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}. 
    145143The general idea is to solve the free surface equation with a small time step, 
    146144while the three dimensional prognostic variables are solved with a longer time step that 
    147 is a multiple of \np{rdtbt} in the \ngn{namdom} namelist (Figure III.3).  
    148  
    149 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
     145is a multiple of \np{rdtbt}{rdtbt} in the \nam{dom}{dom} namelist (Figure III.3). 
     146 
    150147\begin{figure}[!t] 
    151   \begin{center} 
    152     \includegraphics[width=\textwidth]{Fig_DYN_dynspg_ts} 
    153     \caption{ 
    154       \protect\label{fig:DYN_dynspg_ts} 
    155       Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 
    156       after \citet{Griffies2004?}. 
    157       Time increases to the right. 
    158       Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
    159       The curved line represents a leap-frog time step, 
    160       and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. 
    161       The vertically integrated forcing \textbf{M}(t) computed at 
    162       baroclinic time step t represents the interaction between the barotropic and baroclinic motions. 
    163       While keeping the total depth, tracer, and freshwater forcing fields fixed, 
    164       a leap-frog integration carries the surface height and vertically integrated velocity from 
    165       t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. 
    166       Time averaging the barotropic fields over the N+1 time steps (endpoints included) 
    167       centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. 
    168       A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of 
    169       the time averaged vertically integrated velocity taken from baroclinic time step t. 
    170     } 
    171   \end{center} 
     148  \centering 
     149  %\includegraphics[width=0.66\textwidth]{MBZ_DYN_dynspg_ts} 
     150  \caption[Schematic of the split-explicit time stepping scheme for 
     151  the barotropic and baroclinic modes, after \citet{Griffies2004?}]{ 
     152    Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 
     153    after \citet{Griffies2004?}. 
     154    Time increases to the right. 
     155    Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
     156    The curved line represents a leap-frog time step, 
     157    and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. 
     158    The vertically integrated forcing \textbf{M}(t) computed at 
     159    baroclinic time step t represents the interaction between the barotropic and baroclinic motions. 
     160    While keeping the total depth, tracer, and freshwater forcing fields fixed, 
     161    a leap-frog integration carries the surface height and vertically integrated velocity from 
     162    t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. 
     163    Time averaging the barotropic fields over the N+1 time steps (endpoints included) 
     164    centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. 
     165    A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using 
     166    the convergence of the time averaged vertically integrated velocity taken from 
     167    baroclinic time step t.} 
     168  \label{fig:MBZ_dyn_dynspg_ts} 
    172169\end{figure} 
    173 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
    174170 
    175171The split-explicit formulation has a damping effect on external gravity waves, 
    176172which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in 
    177 the case of an analytical barotropic Kelvin wave.  
     173the case of an analytical barotropic Kelvin wave. 
    178174 
    179175%from griffies book: .....   copy past ! 
    … …  
    186182We have 
    187183\[ 
    188   % \label{eq:DYN_spg_ts_eta} 
     184  % \label{eq:MBZ_dyn_spg_ts_eta} 
    189185  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 
    190   = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]  
     186  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
    191187\] 
    192188\begin{multline*} 
    193   % \label{eq:DYN_spg_ts_u} 
     189  % \label{eq:MBZ_dyn_spg_ts_u} 
    194190  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\ 
    195191  = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) 
    … …  
    205201the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for 
    206202the duration of the barotropic time stepping over a single cycle. 
    207 This is also the time that sets the barotropic time steps via  
    208 \[ 
    209   % \label{eq:DYN_spg_ts_t} 
    210   t_n=\tau+n\Delta t    
     203This is also the time that sets the barotropic time steps via 
     204\[ 
     205  % \label{eq:MBZ_dyn_spg_ts_t} 
     206  t_n=\tau+n\Delta t 
    211207\] 
    212208with $n$ an integer. 
    213 The density scaled surface pressure is evaluated via  
    214 \[ 
    215   % \label{eq:DYN_spg_ts_ps} 
     209The density scaled surface pressure is evaluated via 
     210\[ 
     211  % \label{eq:MBZ_dyn_spg_ts_ps} 
    216212  p_s^{(b)}(\tau,t_{n}) = 
    217213  \begin{cases} 
    … …  
    220216  \end{cases} 
    221217\] 
    222 To get started, we assume the following initial conditions  
    223 \[ 
    224   % \label{eq:DYN_spg_ts_eta} 
     218To get started, we assume the following initial conditions 
     219\[ 
     220  % \label{eq:MBZ_dyn_spg_ts_eta} 
    225221  \begin{split} 
    226222    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ 
    … …  
    228224  \end{split} 
    229225\] 
    230 with  
    231 \[ 
    232   % \label{eq:DYN_spg_ts_etaF} 
     226with 
     227\[ 
     228  % \label{eq:MBZ_dyn_spg_ts_etaF} 
    233229  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 
    234230\] 
    … …  
    236232Likewise, 
    237233\[ 
    238   % \label{eq:DYN_spg_ts_u} 
     234  % \label{eq:MBZ_dyn_spg_ts_u} 
    239235  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ 
    240236  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 
    241237\] 
    242 with  
    243 \[ 
    244   % \label{eq:DYN_spg_ts_u} 
     238with 
     239\[ 
     240  % \label{eq:MBZ_dyn_spg_ts_u} 
    245241  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) 
    246242\] 
    247243the time averaged vertically integrated transport. 
    248 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.  
     244Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 
    249245 
    250246Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to 
    251 produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$  
    252 \[ 
    253   % \label{eq:DYN_spg_ts_u} 
     247produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 
     248\[ 
     249  % \label{eq:MBZ_dyn_spg_ts_u} 
    254250  \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 
    255251  = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) 
    256252\] 
    257253The surface height on the new baroclinic time step is then determined via 
    258 a baroclinic leap-frog using the following form  
     254a baroclinic leap-frog using the following form 
    259255\begin{equation} 
    260   \label{eq:DYN_spg_ts_ssh} 
     256  \label{eq:MBZ_dyn_spg_ts_ssh} 
    261257  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
    262258\end{equation} 
    … …  
    264260The use of this "big-leap-frog" scheme for the surface height ensures compatibility between 
    265261the mass/volume budgets and the tracer budgets. 
    266 More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).  
    267   
     262More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 
     263 
    268264In general, some form of time filter is needed to maintain integrity of the surface height field due to 
    269 the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. 
     265the leap-frog splitting mode in equation \autoref{eq:MBZ_dyn_spg_ts_ssh}. 
    270266We have tried various forms of such filtering, 
    271267with the following method discussed in Griffies et al. (2001) chosen due to its stability and 
    272 reasonably good maintenance of tracer conservation properties (see ??)  
     268reasonably good maintenance of tracer conservation properties (see ??) 
    273269 
    274270\begin{equation} 
    275   \label{eq:DYN_spg_ts_sshf} 
     271  \label{eq:MBZ_dyn_spg_ts_sshf} 
    276272  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
    277273\end{equation} 
    278 Another approach tried was  
    279  
    280 \[ 
    281   % \label{eq:DYN_spg_ts_sshf2} 
     274Another approach tried was 
     275 
     276\[ 
     277  % \label{eq:MBZ_dyn_spg_ts_sshf2} 
    282278  \eta^{F}(\tau-\Delta) = \eta(\tau) 
    283279  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) 
    … …  
    288284This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and 
    289285surface height time filtering (see ?? for more complete discussion). 
    290 However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to 
    291 be more conservative, and so is recommended.  
    292  
    293 %------------------------------------------------------------- 
    294 % Filtered formulation  
    295 %------------------------------------------------------------- 
    296 \subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})] 
    297 {Filtered formulation (\protect\key{dynspg\_flt})} 
    298 \label{subsec:DYN_spg_flt} 
     286However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to 
     287be more conservative, and so is recommended. 
     288 
     289% Filtered formulation 
     290%% ================================================================================================= 
     291\subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})]{Filtered formulation (\protect\key{dynspg\_flt})} 
     292\label{subsec:MBZ_dyn_spg_flt} 
    299293 
    300294The filtered formulation follows the \citet{Roullet2000?} implementation. 
    301295The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. 
    302296The elliptic solvers available in the code are documented in \autoref{chap:MISC}. 
    303 The amplitude of the extra term is given by the namelist variable \np{rnu}. 
     297The amplitude of the extra term is given by the namelist variable \np{rnu}{rnu}. 
    304298The default value is 1, as recommended by \citet{Roullet2000?} 
    305299 
    306 \colorbox{red}{\np{rnu}\forcode{ = 1} to be suppressed from namelist !} 
    307  
    308 %------------------------------------------------------------- 
    309 % Non-linear free surface formulation  
    310 %------------------------------------------------------------- 
    311 \subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})] 
    312 {Non-linear free surface formulation (\protect\key{vvl})} 
    313 \label{subsec:DYN_spg_vvl} 
     300\colorbox{red}{\np[=1]{rnu}{rnu} to be suppressed from namelist !} 
     301 
     302% Non-linear free surface formulation 
     303%% ================================================================================================= 
     304\subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})]{Non-linear free surface formulation (\protect\key{vvl})} 
     305\label{subsec:MBZ_dyn_spg_vvl} 
    314306 
    315307In the non-linear free surface formulation, the variations of volume are fully taken into account. 
    316 This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the NEMO web site. 
     308This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. 
    317309The three time-stepping methods (explicit, split-explicit and filtered) are the same as in 
    318 \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent. 
     310\autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent. 
    319311In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. 
    320312 
    321 \biblio 
    322  
    323 \pindex 
     313\subinc{\input{../../global/epilogue}} 
    324314 
    325315\end{document} 
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