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Changeset 11564 for NEMO/branches/2019/dev_r10973_AGRIF-01_jchanut_small_jpi_jpj/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex – NEMO

Ignore:
Timestamp:
2019-09-18T16:11:52+02:00 (5 years ago)
Author:
jchanut
Message:

#2199, merged with trunk

Location:
NEMO/branches/2019/dev_r10973_AGRIF-01_jchanut_small_jpi_jpj/doc
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4 edited

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    r10544 r11564  
    1818 
    1919In 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{Levier2007} available on the \NEMO web site.  
     20These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. 
    2121 
    2222\colorbox{yellow}{  end of to be updated} 
    … …  
    2424% from MOM4p1 documentation 
    2525 
    26 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate  
    27 \[ 
    28   % \label{eq:PE_} 
     26To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 
     27\[ 
     28  % \label{eq:MBZ_PE_} 
    2929  z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 
    3030\] 
    … …  
    4040the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. 
    4141These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 
    42 terrain following sigma models discussed in \autoref{subsec:PE_sco}.  
     42terrain following sigma models discussed in \autoref{subsec:MB_sco}. 
    4343Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in 
    4444an unforced ocean starting from rest, regardless the bottom topography. 
    … …  
    4949neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models 
    5050(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).  
     51as well as  \autoref{sec:LDF_slp} in this document for treatment in \NEMO). 
    5252 
    5353The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. 
    5454Hence, 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 $s = z - \eta$.  
     55This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. 
    5656 
    5757Because $z^\star$ 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.  
     58and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H. 
    5959The $z^\star$ coordinate is therefore invisible to undulations of the free surface, 
    6060since it moves along with the free surface. 
    … …  
    6464Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that 
    6565allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of 
    66 the surface height, again so long as $\eta > -H$.  
     66the surface height, again so long as $\eta > -H$. 
    6767 
    6868%%% 
    … …  
    7373% Surface Pressure Gradient and Sea Surface Height 
    7474% ================================================================ 
    75 \section{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} 
    76 \label{sec:DYN_hpg_spg} 
     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:MBZ_dyn_hpg_spg} 
    7778%-----------------------------------------nam_dynspg---------------------------------------------------- 
    7879 
    79 %\nlst{nam_dynspg}  
     80%\nlst{nam_dynspg} 
    8081%------------------------------------------------------------------------------------------------------------ 
    81 Options are defined through the \ngn{nam\_dynspg} namelist variables. 
    82 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). 
     82Options are defined through the \nam{\_dynspg} namelist variables. 
     83The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). 
    8384The main distinction is between the fixed volume case (linear free surface or rigid lid) and 
    8485the variable volume case (nonlinear free surface, \key{vvl} is active). 
    85 In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), 
     86In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), 
    8687the vertical scale factors $e_{3}$ are fixed in time, 
    87 while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent. 
     88while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent. 
    8889With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 
    8990which imposes a very small time step when an explicit time stepping is used. 
    9091Two methods are proposed to allow a longer time step for the three-dimensional equations: 
    91 the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:PE_flt}), 
     92the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}), 
    9293and the split-explicit free surface described below. 
    9394The extra term introduced in the filtered method is calculated implicitly, 
    … …  
    9798% Explicit 
    9899%------------------------------------------------------------- 
    99 \subsubsection{Explicit (\protect\key{dynspg\_exp})} 
    100 \label{subsec:DYN_spg_exp} 
     100\subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})] 
     101{Explicit (\protect\key{dynspg\_exp})} 
     102\label{subsec:MBZ_dyn_spg_exp} 
    101103 
    102104In the explicit free surface formulation, the model time step is chosen small enough to 
    … …  
    104106The sea surface height is given by: 
    105107\begin{equation} 
    106   \label{eq:dynspg_ssh} 
     108  \label{eq:MBZ_dynspg_ssh} 
    107109  \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T} 
    108110    e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u} 
    … …  
    114116and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. 
    115117The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, 
    116 (\ie the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity).  
     118(\ie\ the velocity appearing in (\autoref{eq:DYN_spg_ssh}) is centred in time (\textit{now} velocity). 
    117119 
    118120The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: 
    119121\begin{equation} 
    120   \label{eq:dynspg_exp} 
     122  \label{eq:MBZ_dynspg_exp} 
    121123  \left\{ 
    122124    \begin{aligned} 
    … …  
    125127    \end{aligned} 
    126128  \right. 
    127 \end{equation}  
     129\end{equation} 
    128130 
    129131Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in 
    130 (\autoref{eq:dynspg_exp}).  
     132(\autoref{eq:DYN_spg_exp}). 
    131133 
    132134%------------------------------------------------------------- 
    133135% Split-explicit time-stepping 
    134136%------------------------------------------------------------- 
    135 \subsubsection{Split-explicit time-stepping (\protect\key{dynspg\_ts})} 
    136 \label{subsec:DYN_spg_ts} 
     137\subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})] 
     138{Split-explicit time-stepping (\protect\key{dynspg\_ts})} 
     139\label{subsec:MBZ_dyn_spg_ts} 
    137140%--------------------------------------------namdom---------------------------------------------------- 
    138141 
    139 \nlst{namdom}  
    140142%-------------------------------------------------------------------------------------------------------------- 
    141 The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}. 
     143The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}. 
    142144The general idea is to solve the free surface equation with a small time step, 
    143145while the three dimensional prognostic variables are solved with a longer time step that 
    144 is a multiple of \np{rdtbt} in the \ngn{namdom} namelist (Figure III.3).  
     146is a multiple of \np{rdtbt} in the \nam{dom} namelist (Figure III.3). 
    145147 
    146148%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
    147149\begin{figure}[!t] 
    148   \begin{center} 
    149     \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts} 
    150     \caption{ 
    151       \protect\label{fig:DYN_dynspg_ts} 
    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 the convergence of 
    166       the time averaged vertically integrated velocity taken from baroclinic time step t. 
    167     } 
    168   \end{center} 
     150  \centering 
     151  \includegraphics[width=\textwidth]{Fig_DYN_dynspg_ts} 
     152  \caption[Schematic of the split-explicit time stepping scheme for 
     153  the barotropic and baroclinic modes, after \citet{Griffies2004?}]{ 
     154    Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 
     155    after \citet{Griffies2004?}. 
     156    Time increases to the right. 
     157    Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
     158    The curved line represents a leap-frog time step, 
     159    and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. 
     160    The vertically integrated forcing \textbf{M}(t) computed at 
     161    baroclinic time step t represents the interaction between the barotropic and baroclinic motions. 
     162    While keeping the total depth, tracer, and freshwater forcing fields fixed, 
     163    a leap-frog integration carries the surface height and vertically integrated velocity from 
     164    t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. 
     165    Time averaging the barotropic fields over the N+1 time steps (endpoints included) 
     166    centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. 
     167    A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using 
     168    the convergence of the time averaged vertically integrated velocity taken from 
     169    baroclinic time step t.} 
     170  \label{fig:MBZ_dyn_dynspg_ts} 
    169171\end{figure} 
    170172%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
    171173 
    172174The split-explicit formulation has a damping effect on external gravity waves, 
    173 which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in 
    174 the case of an analytical barotropic Kelvin wave.  
     175which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in 
     176the case of an analytical barotropic Kelvin wave. 
    175177 
    176178%from griffies book: .....   copy past ! 
    … …  
    183185We have 
    184186\[ 
    185   % \label{eq:DYN_spg_ts_eta} 
     187  % \label{eq:MBZ_dyn_spg_ts_eta} 
    186188  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 
    187   = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]  
     189  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
    188190\] 
    189191\begin{multline*} 
    190   % \label{eq:DYN_spg_ts_u} 
     192  % \label{eq:MBZ_dyn_spg_ts_u} 
    191193  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\ 
    192194  = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) 
    … …  
    202204the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for 
    203205the duration of the barotropic time stepping over a single cycle. 
    204 This is also the time that sets the barotropic time steps via  
    205 \[ 
    206   % \label{eq:DYN_spg_ts_t} 
    207   t_n=\tau+n\Delta t    
     206This is also the time that sets the barotropic time steps via 
     207\[ 
     208  % \label{eq:MBZ_dyn_spg_ts_t} 
     209  t_n=\tau+n\Delta t 
    208210\] 
    209211with $n$ an integer. 
    210 The density scaled surface pressure is evaluated via  
    211 \[ 
    212   % \label{eq:DYN_spg_ts_ps} 
     212The density scaled surface pressure is evaluated via 
     213\[ 
     214  % \label{eq:MBZ_dyn_spg_ts_ps} 
    213215  p_s^{(b)}(\tau,t_{n}) = 
    214216  \begin{cases} 
    … …  
    217219  \end{cases} 
    218220\] 
    219 To get started, we assume the following initial conditions  
    220 \[ 
    221   % \label{eq:DYN_spg_ts_eta} 
     221To get started, we assume the following initial conditions 
     222\[ 
     223  % \label{eq:MBZ_dyn_spg_ts_eta} 
    222224  \begin{split} 
    223225    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ 
    … …  
    225227  \end{split} 
    226228\] 
    227 with  
    228 \[ 
    229   % \label{eq:DYN_spg_ts_etaF} 
     229with 
     230\[ 
     231  % \label{eq:MBZ_dyn_spg_ts_etaF} 
    230232  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 
    231233\] 
    … …  
    233235Likewise, 
    234236\[ 
    235   % \label{eq:DYN_spg_ts_u} 
     237  % \label{eq:MBZ_dyn_spg_ts_u} 
    236238  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ 
    237239  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 
    238240\] 
    239 with  
    240 \[ 
    241   % \label{eq:DYN_spg_ts_u} 
     241with 
     242\[ 
     243  % \label{eq:MBZ_dyn_spg_ts_u} 
    242244  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) 
    243245\] 
    244246the time averaged vertically integrated transport. 
    245 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.  
     247Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 
    246248 
    247249Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to 
    248 produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$  
    249 \[ 
    250   % \label{eq:DYN_spg_ts_u} 
     250produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 
     251\[ 
     252  % \label{eq:MBZ_dyn_spg_ts_u} 
    251253  \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 
    252254  = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) 
    253255\] 
    254256The surface height on the new baroclinic time step is then determined via 
    255 a baroclinic leap-frog using the following form  
     257a baroclinic leap-frog using the following form 
    256258\begin{equation} 
    257   \label{eq:DYN_spg_ts_ssh} 
     259  \label{eq:MBZ_dyn_spg_ts_ssh} 
    258260  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
    259261\end{equation} 
    … …  
    261263The use of this "big-leap-frog" scheme for the surface height ensures compatibility between 
    262264the mass/volume budgets and the tracer budgets. 
    263 More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).  
    264   
     265More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 
     266 
    265267In general, some form of time filter is needed to maintain integrity of the surface height field due to 
    266 the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. 
     268the leap-frog splitting mode in equation \autoref{eq:MBZ_dyn_spg_ts_ssh}. 
    267269We have tried various forms of such filtering, 
    268270with the following method discussed in Griffies et al. (2001) chosen due to its stability and 
    269 reasonably good maintenance of tracer conservation properties (see ??)  
     271reasonably good maintenance of tracer conservation properties (see ??) 
    270272 
    271273\begin{equation} 
    272   \label{eq:DYN_spg_ts_sshf} 
     274  \label{eq:MBZ_dyn_spg_ts_sshf} 
    273275  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
    274276\end{equation} 
    275 Another approach tried was  
    276  
    277 \[ 
    278   % \label{eq:DYN_spg_ts_sshf2} 
     277Another approach tried was 
     278 
     279\[ 
     280  % \label{eq:MBZ_dyn_spg_ts_sshf2} 
    279281  \eta^{F}(\tau-\Delta) = \eta(\tau) 
    280282  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) 
    … …  
    285287This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and 
    286288surface height time filtering (see ?? for more complete discussion). 
    287 However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to 
    288 be more conservative, and so is recommended.  
    289  
    290 %------------------------------------------------------------- 
    291 % Filtered formulation  
    292 %------------------------------------------------------------- 
    293 \subsubsection{Filtered formulation (\protect\key{dynspg\_flt})} 
    294 \label{subsec:DYN_spg_flt} 
    295  
    296 The filtered formulation follows the \citet{Roullet2000} implementation. 
     289However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to 
     290be more conservative, and so is recommended. 
     291 
     292%------------------------------------------------------------- 
     293% Filtered formulation 
     294%------------------------------------------------------------- 
     295\subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})] 
     296{Filtered formulation (\protect\key{dynspg\_flt})} 
     297\label{subsec:MBZ_dyn_spg_flt} 
     298 
     299The filtered formulation follows the \citet{Roullet2000?} implementation. 
    297300The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. 
    298301The elliptic solvers available in the code are documented in \autoref{chap:MISC}. 
    299302The amplitude of the extra term is given by the namelist variable \np{rnu}. 
    300 The default value is 1, as recommended by \citet{Roullet2000} 
    301  
    302 \colorbox{red}{\np{rnu}\forcode{ = 1} to be suppressed from namelist !} 
    303  
    304 %------------------------------------------------------------- 
    305 % Non-linear free surface formulation  
    306 %------------------------------------------------------------- 
    307 \subsection{Non-linear free surface formulation (\protect\key{vvl})} 
    308 \label{subsec:DYN_spg_vvl} 
     303The default value is 1, as recommended by \citet{Roullet2000?} 
     304 
     305\colorbox{red}{\np{rnu}\forcode{=1} to be suppressed from namelist !} 
     306 
     307%------------------------------------------------------------- 
     308% Non-linear free surface formulation 
     309%------------------------------------------------------------- 
     310\subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})] 
     311{Non-linear free surface formulation (\protect\key{vvl})} 
     312\label{subsec:MBZ_dyn_spg_vvl} 
    309313 
    310314In the non-linear free surface formulation, the variations of volume are fully taken into account. 
    311 This option is presented in a report \citep{Levier2007} available on the NEMO web site. 
     315This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. 
    312316The three time-stepping methods (explicit, split-explicit and filtered) are the same as in 
    313 \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent. 
     317\autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent. 
    314318In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. 
    315319 
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