Changeset 11512 for NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex
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- 2019-09-09T12:05:20+02:00 (5 years ago)
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NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex
r11263 r11512 18 18 19 19 In 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.20 These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. 21 21 22 22 \colorbox{yellow}{ end of to be updated} … … 24 24 % from MOM4p1 documentation 25 25 26 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 26 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 27 27 \[ 28 28 % \label{eq:PE_} … … 40 40 the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. 41 41 These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 42 terrain following sigma models discussed in \autoref{subsec:PE_sco}. 42 terrain following sigma models discussed in \autoref{subsec:PE_sco}. 43 43 Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in 44 44 an unforced ocean starting from rest, regardless the bottom topography. … … 49 49 neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models 50 50 (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). 51 as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). 52 52 53 53 The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. 54 54 Hence, 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$. 55 This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. 56 56 57 57 Because $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. 58 and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H. 59 59 The $z^\star$ coordinate is therefore invisible to undulations of the free surface, 60 60 since it moves along with the free surface. … … 64 64 Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that 65 65 allows 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$. 66 the surface height, again so long as $\eta > -H$. 67 67 68 68 %%% … … 78 78 %-----------------------------------------nam_dynspg---------------------------------------------------- 79 79 80 %\nlst{nam_dynspg} 80 %\nlst{nam_dynspg} 81 81 %------------------------------------------------------------------------------------------------------------ 82 Options are defined through the \n gn{nam\_dynspg} namelist variables.82 Options are defined through the \nam{\_dynspg} namelist variables. 83 83 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). 84 84 The main distinction is between the fixed volume case (linear free surface or rigid lid) and … … 116 116 and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. 117 117 The 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).118 (\ie\ the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). 119 119 120 120 The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: … … 127 127 \end{aligned} 128 128 \right. 129 \end{equation} 129 \end{equation} 130 130 131 131 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in 132 (\autoref{eq:dynspg_exp}). 132 (\autoref{eq:dynspg_exp}). 133 133 134 134 %------------------------------------------------------------- … … 140 140 %--------------------------------------------namdom---------------------------------------------------- 141 141 142 \nlst{namdom} 142 \nlst{namdom} 143 143 %-------------------------------------------------------------------------------------------------------------- 144 144 The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}. 145 145 The general idea is to solve the free surface equation with a small time step, 146 146 while the three dimensional prognostic variables are solved with a longer time step that 147 is a multiple of \np{rdtbt} in the \n gn{namdom} namelist (Figure III.3).147 is a multiple of \np{rdtbt} in the \nam{dom} namelist (Figure III.3). 148 148 149 149 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > … … 175 175 The split-explicit formulation has a damping effect on external gravity waves, 176 176 which 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. 177 the case of an analytical barotropic Kelvin wave. 178 178 179 179 %from griffies book: ..... copy past ! … … 188 188 % \label{eq:DYN_spg_ts_eta} 189 189 \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] 190 = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 191 191 \] 192 192 \begin{multline*} … … 205 205 the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for 206 206 the duration of the barotropic time stepping over a single cycle. 207 This is also the time that sets the barotropic time steps via 207 This is also the time that sets the barotropic time steps via 208 208 \[ 209 209 % \label{eq:DYN_spg_ts_t} 210 t_n=\tau+n\Delta t 210 t_n=\tau+n\Delta t 211 211 \] 212 212 with $n$ an integer. 213 The density scaled surface pressure is evaluated via 213 The density scaled surface pressure is evaluated via 214 214 \[ 215 215 % \label{eq:DYN_spg_ts_ps} … … 220 220 \end{cases} 221 221 \] 222 To get started, we assume the following initial conditions 222 To get started, we assume the following initial conditions 223 223 \[ 224 224 % \label{eq:DYN_spg_ts_eta} … … 228 228 \end{split} 229 229 \] 230 with 230 with 231 231 \[ 232 232 % \label{eq:DYN_spg_ts_etaF} … … 240 240 \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 241 241 \] 242 with 242 with 243 243 \[ 244 244 % \label{eq:DYN_spg_ts_u} … … 246 246 \] 247 247 the time averaged vertically integrated transport. 248 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 248 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 249 249 250 250 Upon 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 $\tau + \Delta \tau$ 251 produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 252 252 \[ 253 253 % \label{eq:DYN_spg_ts_u} … … 256 256 \] 257 257 The surface height on the new baroclinic time step is then determined via 258 a baroclinic leap-frog using the following form 258 a baroclinic leap-frog using the following form 259 259 \begin{equation} 260 260 \label{eq:DYN_spg_ts_ssh} … … 264 264 The use of this "big-leap-frog" scheme for the surface height ensures compatibility between 265 265 the mass/volume budgets and the tracer budgets. 266 More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 267 266 More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 267 268 268 In general, some form of time filter is needed to maintain integrity of the surface height field due to 269 269 the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. 270 270 We have tried various forms of such filtering, 271 271 with the following method discussed in Griffies et al. (2001) chosen due to its stability and 272 reasonably good maintenance of tracer conservation properties (see ??) 272 reasonably good maintenance of tracer conservation properties (see ??) 273 273 274 274 \begin{equation} … … 276 276 \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} 277 277 \end{equation} 278 Another approach tried was 278 Another approach tried was 279 279 280 280 \[ … … 289 289 surface height time filtering (see ?? for more complete discussion). 290 290 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 291 be more conservative, and so is recommended. 292 293 %------------------------------------------------------------- 294 % Filtered formulation 295 295 %------------------------------------------------------------- 296 296 \subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})] … … 307 307 308 308 %------------------------------------------------------------- 309 % Non-linear free surface formulation 309 % Non-linear free surface formulation 310 310 %------------------------------------------------------------- 311 311 \subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})] … … 314 314 315 315 In 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 NEMOweb site.316 This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. 317 317 The three time-stepping methods (explicit, split-explicit and filtered) are the same as in 318 318 \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent.
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