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Changeset 11692 for NEMO/branches/2019/dev_r11514_HPC-02_single-core-extrahalo/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex – NEMO

Ignore:
Timestamp:
2019-10-12T16:08:18+02:00 (5 years ago)
Author:
francesca
Message:

Update branch to integrate the development starting from the current v4.01 ready trunk

Location:
NEMO/branches/2019/dev_r11514_HPC-02_single-core-extrahalo/doc
Files:
5 edited

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  • NEMO/branches/2019/dev_r11514_HPC-02_single-core-extrahalo/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

    r11435 r11692  
    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} 
    12  
    13 % ------------------------------------------------------------------------------------------------------------- 
    14 % ???? 
    15 % ------------------------------------------------------------------------------------------------------------- 
    1628 
    1729\colorbox{yellow}{ to be updated } 
    … …  
    2638To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 
    2739\[ 
    28   % \label{eq:PE_} 
     40  % \label{eq:MBZ_PE_} 
    2941  z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 
    3042\] 
    … …  
    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. 
    … …  
    6678the surface height, again so long as $\eta > -H$. 
    6779 
    68 %%% 
    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---------------------------------------------------- 
     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} 
    7985 
    8086%\nlst{nam_dynspg} 
    81 %------------------------------------------------------------------------------------------------------------ 
    82 Options are defined through the \nam{\_dynspg} namelist variables. 
    83 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). 
     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} 
    … …  
    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 \nam{dom} 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, 
    … …  
    186182We have 
    187183\[ 
    188   % \label{eq:DYN_spg_ts_eta} 
     184  % \label{eq:MBZ_dyn_spg_ts_eta} 
    189185  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 
    190186  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
    191187\] 
    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}) 
    … …  
    207203This is also the time that sets the barotropic time steps via 
    208204\[ 
    209   % \label{eq:DYN_spg_ts_t} 
     205  % \label{eq:MBZ_dyn_spg_ts_t} 
    210206  t_n=\tau+n\Delta t 
    211207\] 
    … …  
    213209The density scaled surface pressure is evaluated via 
    214210\[ 
    215   % \label{eq:DYN_spg_ts_ps} 
     211  % \label{eq:MBZ_dyn_spg_ts_ps} 
    216212  p_s^{(b)}(\tau,t_{n}) = 
    217213  \begin{cases} 
    … …  
    222218To get started, we assume the following initial conditions 
    223219\[ 
    224   % \label{eq:DYN_spg_ts_eta} 
     220  % \label{eq:MBZ_dyn_spg_ts_eta} 
    225221  \begin{split} 
    226222    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ 
    … …  
    230226with 
    231227\[ 
    232   % \label{eq:DYN_spg_ts_etaF} 
     228  % \label{eq:MBZ_dyn_spg_ts_etaF} 
    233229  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 
    234230\] 
    … …  
    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} 
    … …  
    242238with 
    243239\[ 
    244   % \label{eq:DYN_spg_ts_u} 
     240  % \label{eq:MBZ_dyn_spg_ts_u} 
    245241  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) 
    246242\] 
    … …  
    251247produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 
    252248\[ 
    253   % \label{eq:DYN_spg_ts_u} 
     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}) 
    … …  
    258254a 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} 
    … …  
    267263 
    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 
    … …  
    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} 
    … …  
    279275 
    280276\[ 
    281   % \label{eq:DYN_spg_ts_sshf2} 
     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 
     286However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to 
    291287be more conservative, and so is recommended. 
    292288 
    293 %------------------------------------------------------------- 
    294289% 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} 
     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 %------------------------------------------------------------- 
     300\colorbox{red}{\np[=1]{rnu}{rnu} to be suppressed from namelist !} 
     301 
    309302% 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} 
     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. 
    316308This 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\onlyinsubfile{\input{../../global/epilogue}} 
    324314 
    325315\end{document} 
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