[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
---|
| 2 | |
---|
[6997] | 3 | \begin{document} |
---|
[11598] | 4 | |
---|
[10442] | 5 | \chapter{ essai \zstar \sstar} |
---|
[11598] | 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 | |
---|
[11597] | 26 | %% ================================================================================================= |
---|
[10442] | 27 | \section{Curvilinear \zstar- or \sstar coordinate system} |
---|
[707] | 28 | |
---|
| 29 | \colorbox{yellow}{ to be updated } |
---|
| 30 | |
---|
[10354] | 31 | In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. |
---|
[11435] | 32 | These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. |
---|
[707] | 33 | |
---|
| 34 | \colorbox{yellow}{ end of to be updated} |
---|
| 35 | |
---|
| 36 | % from MOM4p1 documentation |
---|
| 37 | |
---|
[11435] | 38 | To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate |
---|
[10414] | 39 | \[ |
---|
[11544] | 40 | % \label{eq:MBZ_PE_} |
---|
[10414] | 41 | z^\star = H \left( \frac{z-\eta}{H+\eta} \right) |
---|
| 42 | \] |
---|
[707] | 43 | |
---|
[10354] | 44 | This coordinate is closely related to the "eta" coordinate used in many atmospheric models |
---|
| 45 | (see Black (1994) for a review of eta coordinate atmospheric models). |
---|
| 46 | It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, |
---|
| 47 | and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling. |
---|
[707] | 48 | |
---|
[10354] | 49 | The surfaces of constant $z^\star$ are quasi-horizontal. |
---|
| 50 | Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. |
---|
| 51 | In general, when noting the large differences between undulations of the bottom topography versus undulations in |
---|
| 52 | the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. |
---|
| 53 | These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to |
---|
[11543] | 54 | terrain following sigma models discussed in \autoref{subsec:MB_sco}. |
---|
[10354] | 55 | Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in |
---|
| 56 | an unforced ocean starting from rest, regardless the bottom topography. |
---|
| 57 | This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of |
---|
| 58 | nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, |
---|
| 59 | depending on the sophistication of the pressure gradient solver. |
---|
| 60 | The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of |
---|
| 61 | neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models |
---|
| 62 | (see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models, |
---|
[11435] | 63 | as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). |
---|
[707] | 64 | |
---|
[10354] | 65 | The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. |
---|
| 66 | Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. |
---|
[11435] | 67 | This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. |
---|
[707] | 68 | |
---|
[10354] | 69 | Because $z^\star$ has a time independent range, all grid cells have static increments ds, |
---|
[11435] | 70 | and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H. |
---|
[10354] | 71 | The $z^\star$ coordinate is therefore invisible to undulations of the free surface, |
---|
| 72 | since it moves along with the free surface. |
---|
| 73 | This proper ty means that no spurious ver tical transpor t is induced across surfaces of |
---|
| 74 | constant $z^\star$ by the motion of external gravity waves. |
---|
| 75 | Such spurious transpor t can be a problem in z-models, especially those with tidal forcing. |
---|
| 76 | Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that |
---|
| 77 | allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of |
---|
[11435] | 78 | the surface height, again so long as $\eta > -H$. |
---|
[707] | 79 | |
---|
| 80 | % essai update time splitting... |
---|
| 81 | |
---|
[11597] | 82 | %% ================================================================================================= |
---|
[11571] | 83 | \section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})]{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} |
---|
[11544] | 84 | \label{sec:MBZ_dyn_hpg_spg} |
---|
[10146] | 85 | |
---|
[11435] | 86 | %\nlst{nam_dynspg} |
---|
[11577] | 87 | Options are defined through the \nam{_dynspg}{\_dynspg} namelist variables. |
---|
[11543] | 88 | The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). |
---|
[10354] | 89 | The main distinction is between the fixed volume case (linear free surface or rigid lid) and |
---|
| 90 | the variable volume case (nonlinear free surface, \key{vvl} is active). |
---|
[11543] | 91 | In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), |
---|
[10354] | 92 | the vertical scale factors $e_{3}$ are fixed in time, |
---|
[11543] | 93 | while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent. |
---|
[10354] | 94 | With both linear and nonlinear free surface, external gravity waves are allowed in the equations, |
---|
| 95 | which imposes a very small time step when an explicit time stepping is used. |
---|
| 96 | Two methods are proposed to allow a longer time step for the three-dimensional equations: |
---|
[11543] | 97 | the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}), |
---|
[10354] | 98 | and the split-explicit free surface described below. |
---|
| 99 | The extra term introduced in the filtered method is calculated implicitly, |
---|
| 100 | so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
---|
[707] | 101 | |
---|
| 102 | % Explicit |
---|
[11597] | 103 | %% ================================================================================================= |
---|
[11571] | 104 | \subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})]{Explicit (\protect\key{dynspg\_exp})} |
---|
[11544] | 105 | \label{subsec:MBZ_dyn_spg_exp} |
---|
[707] | 106 | |
---|
[10354] | 107 | In the explicit free surface formulation, the model time step is chosen small enough to |
---|
| 108 | describe the external gravity waves (typically a few ten seconds). |
---|
| 109 | The sea surface height is given by: |
---|
[10414] | 110 | \begin{equation} |
---|
[11544] | 111 | \label{eq:MBZ_dynspg_ssh} |
---|
[10414] | 112 | \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T} |
---|
| 113 | e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u} |
---|
| 114 | \right]+\delta_j \left[ {e_{1v} e_{3v} v} \right]} \right)} |
---|
[707] | 115 | \end{equation} |
---|
| 116 | |
---|
[10354] | 117 | where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs |
---|
| 118 | (if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$, |
---|
[10406] | 119 | and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. |
---|
[10354] | 120 | The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, |
---|
[11543] | 121 | (\ie\ the velocity appearing in (\autoref{eq:DYN_spg_ssh}) is centred in time (\textit{now} velocity). |
---|
[707] | 122 | |
---|
[10354] | 123 | The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: |
---|
[10414] | 124 | \begin{equation} |
---|
[11544] | 125 | \label{eq:MBZ_dynspg_exp} |
---|
[10414] | 126 | \left\{ |
---|
| 127 | \begin{aligned} |
---|
| 128 | - \frac{1} {e_{1u}} \; \delta_{i+1/2} \left[ \,\eta\, \right] \\ \\ |
---|
| 129 | - \frac{1} {e_{2v}} \; \delta_{j+1/2} \left[ \,\eta\, \right] |
---|
| 130 | \end{aligned} |
---|
| 131 | \right. |
---|
[11435] | 132 | \end{equation} |
---|
[707] | 133 | |
---|
[10406] | 134 | Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in |
---|
[11543] | 135 | (\autoref{eq:DYN_spg_exp}). |
---|
[707] | 136 | |
---|
| 137 | % Split-explicit time-stepping |
---|
[11597] | 138 | %% ================================================================================================= |
---|
[11571] | 139 | \subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})]{Split-explicit time-stepping (\protect\key{dynspg\_ts})} |
---|
[11544] | 140 | \label{subsec:MBZ_dyn_spg_ts} |
---|
[10146] | 141 | |
---|
[11123] | 142 | The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}. |
---|
[10354] | 143 | The general idea is to solve the free surface equation with a small time step, |
---|
| 144 | while the three dimensional prognostic variables are solved with a longer time step that |
---|
[11578] | 145 | is a multiple of \np{rdtbt}{rdtbt} in the \nam{dom}{dom} namelist (Figure III.3). |
---|
[707] | 146 | |
---|
[10414] | 147 | \begin{figure}[!t] |
---|
[11558] | 148 | \centering |
---|
[11693] | 149 | %\includegraphics[width=0.66\textwidth]{MBZ_DYN_dynspg_ts} |
---|
[11558] | 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} |
---|
[707] | 169 | \end{figure} |
---|
| 170 | |
---|
[10354] | 171 | The split-explicit formulation has a damping effect on external gravity waves, |
---|
[11123] | 172 | which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in |
---|
[11435] | 173 | the case of an analytical barotropic Kelvin wave. |
---|
[707] | 174 | |
---|
| 175 | %from griffies book: ..... copy past ! |
---|
| 176 | |
---|
| 177 | \textbf{title: Time stepping the barotropic system } |
---|
| 178 | |
---|
[10354] | 179 | Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. |
---|
| 180 | Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using |
---|
| 181 | the small barotropic time step $\Delta t$. |
---|
| 182 | We have |
---|
[10414] | 183 | \[ |
---|
[11544] | 184 | % \label{eq:MBZ_dyn_spg_ts_eta} |
---|
[10414] | 185 | \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) |
---|
[11435] | 186 | = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] |
---|
[10414] | 187 | \] |
---|
| 188 | \begin{multline*} |
---|
[11544] | 189 | % \label{eq:MBZ_dyn_spg_ts_u} |
---|
[10414] | 190 | \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ |
---|
| 191 | = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) |
---|
| 192 | - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] |
---|
| 193 | \end{multline*} |
---|
[707] | 194 | \ |
---|
| 195 | |
---|
[10354] | 196 | In these equations, araised (b) denotes values of surface height and |
---|
| 197 | vertically integrated velocity updated with the barotropic time steps. |
---|
| 198 | The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which |
---|
| 199 | the vertically integrated forcing $\textbf{M}(\tau)$ |
---|
| 200 | (note that this forcing includes the surface freshwater forcing), the tracer fields, |
---|
| 201 | the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for |
---|
| 202 | the duration of the barotropic time stepping over a single cycle. |
---|
[11435] | 203 | This is also the time that sets the barotropic time steps via |
---|
[10414] | 204 | \[ |
---|
[11544] | 205 | % \label{eq:MBZ_dyn_spg_ts_t} |
---|
[11435] | 206 | t_n=\tau+n\Delta t |
---|
[10414] | 207 | \] |
---|
[10354] | 208 | with $n$ an integer. |
---|
[11435] | 209 | The density scaled surface pressure is evaluated via |
---|
[10414] | 210 | \[ |
---|
[11544] | 211 | % \label{eq:MBZ_dyn_spg_ts_ps} |
---|
[10414] | 212 | p_s^{(b)}(\tau,t_{n}) = |
---|
| 213 | \begin{cases} |
---|
| 214 | g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ |
---|
| 215 | g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} |
---|
| 216 | \end{cases} |
---|
| 217 | \] |
---|
[11435] | 218 | To get started, we assume the following initial conditions |
---|
[10414] | 219 | \[ |
---|
[11544] | 220 | % \label{eq:MBZ_dyn_spg_ts_eta} |
---|
[10414] | 221 | \begin{split} |
---|
| 222 | \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ |
---|
| 223 | \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
---|
| 224 | \end{split} |
---|
| 225 | \] |
---|
[11435] | 226 | with |
---|
[10414] | 227 | \[ |
---|
[11544] | 228 | % \label{eq:MBZ_dyn_spg_ts_etaF} |
---|
[10414] | 229 | \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) |
---|
| 230 | \] |
---|
[10354] | 231 | the time averaged surface height taken from the previous barotropic cycle. |
---|
| 232 | Likewise, |
---|
[10414] | 233 | \[ |
---|
[11544] | 234 | % \label{eq:MBZ_dyn_spg_ts_u} |
---|
[10414] | 235 | \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ |
---|
| 236 | \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
---|
| 237 | \] |
---|
[11435] | 238 | with |
---|
[10414] | 239 | \[ |
---|
[11544] | 240 | % \label{eq:MBZ_dyn_spg_ts_u} |
---|
[10414] | 241 | \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) |
---|
| 242 | \] |
---|
[10354] | 243 | the time averaged vertically integrated transport. |
---|
[11435] | 244 | Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. |
---|
[707] | 245 | |
---|
[10354] | 246 | Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to |
---|
[11435] | 247 | produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ |
---|
[10414] | 248 | \[ |
---|
[11544] | 249 | % \label{eq:MBZ_dyn_spg_ts_u} |
---|
[10414] | 250 | \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} |
---|
| 251 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) |
---|
| 252 | \] |
---|
[10354] | 253 | The surface height on the new baroclinic time step is then determined via |
---|
[11435] | 254 | a baroclinic leap-frog using the following form |
---|
[10414] | 255 | \begin{equation} |
---|
[11544] | 256 | \label{eq:MBZ_dyn_spg_ts_ssh} |
---|
[10414] | 257 | \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] |
---|
[707] | 258 | \end{equation} |
---|
| 259 | |
---|
[10354] | 260 | The use of this "big-leap-frog" scheme for the surface height ensures compatibility between |
---|
| 261 | the mass/volume budgets and the tracer budgets. |
---|
[11435] | 262 | More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). |
---|
| 263 | |
---|
[10354] | 264 | In general, some form of time filter is needed to maintain integrity of the surface height field due to |
---|
[11544] | 265 | the leap-frog splitting mode in equation \autoref{eq:MBZ_dyn_spg_ts_ssh}. |
---|
[10354] | 266 | We have tried various forms of such filtering, |
---|
| 267 | with the following method discussed in Griffies et al. (2001) chosen due to its stability and |
---|
[11435] | 268 | reasonably good maintenance of tracer conservation properties (see ??) |
---|
[707] | 269 | |
---|
[10414] | 270 | \begin{equation} |
---|
[11544] | 271 | \label{eq:MBZ_dyn_spg_ts_sshf} |
---|
[10414] | 272 | \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} |
---|
[707] | 273 | \end{equation} |
---|
[11435] | 274 | Another approach tried was |
---|
[707] | 275 | |
---|
[10414] | 276 | \[ |
---|
[11544] | 277 | % \label{eq:MBZ_dyn_spg_ts_sshf2} |
---|
[10414] | 278 | \eta^{F}(\tau-\Delta) = \eta(\tau) |
---|
| 279 | + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) |
---|
| 280 | + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right] |
---|
| 281 | \] |
---|
[707] | 282 | |
---|
[10354] | 283 | which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$. |
---|
| 284 | This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and |
---|
| 285 | surface height time filtering (see ?? for more complete discussion). |
---|
[11544] | 286 | However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to |
---|
[11435] | 287 | be more conservative, and so is recommended. |
---|
[707] | 288 | |
---|
[11435] | 289 | % Filtered formulation |
---|
[11597] | 290 | %% ================================================================================================= |
---|
[11571] | 291 | \subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})]{Filtered formulation (\protect\key{dynspg\_flt})} |
---|
[11544] | 292 | \label{subsec:MBZ_dyn_spg_flt} |
---|
[707] | 293 | |
---|
[11123] | 294 | The filtered formulation follows the \citet{Roullet2000?} implementation. |
---|
[10354] | 295 | The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. |
---|
| 296 | The elliptic solvers available in the code are documented in \autoref{chap:MISC}. |
---|
[11578] | 297 | The amplitude of the extra term is given by the namelist variable \np{rnu}{rnu}. |
---|
[11123] | 298 | The default value is 1, as recommended by \citet{Roullet2000?} |
---|
[707] | 299 | |
---|
[11582] | 300 | \colorbox{red}{\np[=1]{rnu}{rnu} to be suppressed from namelist !} |
---|
[707] | 301 | |
---|
[11435] | 302 | % Non-linear free surface formulation |
---|
[11597] | 303 | %% ================================================================================================= |
---|
[11571] | 304 | \subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})]{Non-linear free surface formulation (\protect\key{vvl})} |
---|
[11544] | 305 | \label{subsec:MBZ_dyn_spg_vvl} |
---|
[707] | 306 | |
---|
[10354] | 307 | In the non-linear free surface formulation, the variations of volume are fully taken into account. |
---|
[11435] | 308 | This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. |
---|
[10354] | 309 | The three time-stepping methods (explicit, split-explicit and filtered) are the same as in |
---|
[11543] | 310 | \autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent. |
---|
[10354] | 311 | In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. |
---|
[707] | 312 | |
---|
[11693] | 313 | \subinc{\input{../../global/epilogue}} |
---|
[707] | 314 | |
---|
[6997] | 315 | \end{document} |
---|