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branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics_zstar.tex
r9393 r9407 27 27 To overcome problems with vanishing surface and/or bottom cells, we consider the 28 28 zstar coordinate 29 \begin{equation} \label{ PE_}29 \begin{equation} \label{eq:PE_} 30 30 z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 31 31 \end{equation} … … 39 39 The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between 40 40 undulations of the bottom topography versus undulations in the surface height, it 41 is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \ S\ref{PE_sco}.41 is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \autoref{subsec:PE_sco}. 42 42 Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an 43 43 unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in … … 45 45 gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using 46 46 the same techniques as in $z$-models (see Chapters 13-16 of Griffies (2004) for a 47 discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp}47 discussion of neutral physics in $z$-models, as well as \autoref{sec:LDF_slp} 48 48 in this document for treatment in \NEMO). 49 49 … … 76 76 % ================================================================ 77 77 \section{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} 78 \label{ DYN_hpg_spg}78 \label{sec:DYN_hpg_spg} 79 79 %-----------------------------------------nam_dynspg---------------------------------------------------- 80 80 \forfile{../namelists/nam_dynspg} 81 81 %------------------------------------------------------------------------------------------------------------ 82 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 (\ S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\S\ref{PE_free_surface}) and rigid lid (\S\ref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\S\ref{PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.83 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 84 84 85 85 %------------------------------------------------------------- … … 87 87 %------------------------------------------------------------- 88 88 \subsubsection{Explicit (\protect\key{dynspg\_exp})} 89 \label{ DYN_spg_exp}89 \label{subsec:DYN_spg_exp} 90 90 91 91 In the explicit free surface formulation, the model time step is chosen small enough to describe the external gravity waves (typically a few ten seconds). The sea surface height is given by : 92 \begin{equation} \label{ Eq_dynspg_ssh}92 \begin{equation} \label{eq:dynspg_ssh} 93 93 \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} 94 94 e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u} … … 96 96 \end{equation} 97 97 98 where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \ S\ref{SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\ref{Eq_dynspg_ssh}) is centred in time (\textit{now} velocity).98 where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). 99 99 100 100 The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by : 101 \begin{equation} \label{ Eq_dynspg_exp}101 \begin{equation} \label{eq:dynspg_exp} 102 102 \left\{ \begin{aligned} 103 103 - \frac{1} {e_{1u}} \; \delta _{i+1/2} \left[ \,\eta\, \right] \\ … … 107 107 \end{equation} 108 108 109 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\ ref{Eq_dynspg_exp}).109 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\autoref{eq:dynspg_exp}). 110 110 111 111 %------------------------------------------------------------- … … 113 113 %------------------------------------------------------------- 114 114 \subsubsection{Split-explicit time-stepping (\protect\key{dynspg\_ts})} 115 \label{ DYN_spg_ts}115 \label{subsec:DYN_spg_ts} 116 116 %--------------------------------------------namdom---------------------------------------------------- 117 117 \forfile{../namelists/namdom} … … 124 124 \begin{figure}[!t] \begin{center} 125 125 \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts} 126 \caption{ \protect\label{ Fig_DYN_dynspg_ts}126 \caption{ \protect\label{fig:DYN_dynspg_ts} 127 127 Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 128 128 after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by … … 151 151 scheme using the small barotropic time step $\Delta t$. We have 152 152 153 \begin{equation} \label{ DYN_spg_ts_eta}153 \begin{equation} \label{eq:DYN_spg_ts_eta} 154 154 \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 155 155 = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 156 156 \end{equation} 157 \begin{multline} \label{ DYN_spg_ts_u}157 \begin{multline} \label{eq:DYN_spg_ts_u} 158 158 \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ 159 159 = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) … … 165 165 and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time 166 166 that sets the barotropic time steps via 167 \begin{equation} \label{ DYN_spg_ts_t}167 \begin{equation} \label{eq:DYN_spg_ts_t} 168 168 t_n=\tau+n\Delta t 169 169 \end{equation} 170 170 with $n$ an integer. The density scaled surface pressure is evaluated via 171 \begin{equation} \label{ DYN_spg_ts_ps}171 \begin{equation} \label{eq:DYN_spg_ts_ps} 172 172 p_s^{(b)}(\tau,t_{n}) = \begin{cases} 173 173 g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ … … 176 176 \end{equation} 177 177 To get started, we assume the following initial conditions 178 \begin{equation} \label{ DYN_spg_ts_eta}178 \begin{equation} \label{eq:DYN_spg_ts_eta} 179 179 \begin{split} 180 180 \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} … … 184 184 \end{equation} 185 185 with 186 \begin{equation} \label{ DYN_spg_ts_etaF}186 \begin{equation} \label{eq:DYN_spg_ts_etaF} 187 187 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 188 188 \end{equation} 189 189 the time averaged surface height taken from the previous barotropic cycle. Likewise, 190 \begin{equation} \label{ DYN_spg_ts_u}190 \begin{equation} \label{eq:DYN_spg_ts_u} 191 191 \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ 192 192 \\ … … 194 194 \end{equation} 195 195 with 196 \begin{equation} \label{ DYN_spg_ts_u}196 \begin{equation} \label{eq:DYN_spg_ts_u} 197 197 \overline{\textbf{U}^{(b)}(\tau)} 198 198 = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) … … 201 201 202 202 Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 203 \begin{equation} \label{ DYN_spg_ts_u}203 \begin{equation} \label{eq:DYN_spg_ts_u} 204 204 \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 205 205 = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) … … 207 207 The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form 208 208 209 \begin{equation} \label{ DYN_spg_ts_ssh}209 \begin{equation} \label{eq:DYN_spg_ts_ssh} 210 210 \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 211 211 \end{equation} … … 214 214 215 215 In general, some form of time filter is needed to maintain integrity of the surface 216 height field due to the leap-frog splitting mode in equation \ ref{DYN_spg_ts_ssh}. We216 height field due to the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. We 217 217 have tried various forms of such filtering, with the following method discussed in 218 218 Griffies et al. (2001) chosen due to its stability and reasonably good maintenance of 219 tracer conservation properties (see Section??)220 221 \begin{equation} \label{ DYN_spg_ts_sshf}219 tracer conservation properties (see ??) 220 221 \begin{equation} \label{eq:DYN_spg_ts_sshf} 222 222 \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} 223 223 \end{equation} 224 224 Another approach tried was 225 225 226 \begin{equation} \label{ DYN_spg_ts_sshf2}226 \begin{equation} \label{eq:DYN_spg_ts_sshf2} 227 227 \eta^{F}(\tau-\Delta) = \eta(\tau) 228 228 + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) … … 232 232 which is useful since it isolates all the time filtering aspects into the term multiplied 233 233 by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when 234 eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.234 eliminating tracer and surface height time filtering (see ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. 235 235 236 236 … … 242 242 %------------------------------------------------------------- 243 243 \subsubsection{Filtered formulation (\protect\key{dynspg\_flt})} 244 \label{ DYN_spg_flt}244 \label{subsec:DYN_spg_flt} 245 245 246 246 The filtered formulation follows the \citet{Roullet2000} implementation. The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic solvers available in the code are 247 documented in \ S\ref{MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000}247 documented in \autoref{chap:MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000} 248 248 249 249 \colorbox{red}{\np{rnu}\forcode{ = 1} to be suppressed from namelist !} … … 253 253 %------------------------------------------------------------- 254 254 \subsection{Non-linear free surface formulation (\protect\key{vvl})} 255 \label{ DYN_spg_vvl}256 257 In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \ S\ref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.255 \label{subsec:DYN_spg_vvl} 256 257 In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. 258 258 259 259
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