Changeset 3294 for trunk/DOC/TexFiles/Chapters/Chap_Model_Basics.tex
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- 2012-01-28T17:44:18+01:00 (12 years ago)
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r2376 r3294 1094 1094 % Lateral Diffusive and Viscous Operators Formulation 1095 1095 % ------------------------------------------------------------------------------------------------------------- 1096 \subsection{ Lateral Diffusive and Viscous Operators Formulation}1096 \subsection{Formulation of the Lateral Diffusive and Viscous Operators} 1097 1097 \label{PE_ldf} 1098 1098 … … 1134 1134 1135 1135 In eddy-resolving configurations, a second order operator can be used, but 1136 usually a more scale selective one (biharmonic operator)is preferred as the1136 usually the more scale selective biharmonic operator is preferred as the 1137 1137 grid-spacing is usually not small enough compared to the scale of the 1138 1138 eddies. The role devoted to the subgrid-scale physics is to dissipate the 1139 energy that cascades toward the grid scale and thus ensuresthe stability of1140 the model while not interfering with the solved mesoscale activity. Another approach1139 energy that cascades toward the grid scale and thus to ensure the stability of 1140 the model while not interfering with the resolved mesoscale activity. Another approach 1141 1141 is becoming more and more popular: instead of specifying explicitly a sub-grid scale 1142 1142 term in the momentum and tracer time evolution equations, one uses a advective 1143 1143 scheme which is diffusive enough to maintain the model stability. It must be emphasised 1144 that then, all the sub-grid scale physics is in this case includein the formulation of the1144 that then, all the sub-grid scale physics is included in the formulation of the 1145 1145 advection scheme. 1146 1146 1147 All these parameterisations of subgrid scale physics presentadvantages and1147 All these parameterisations of subgrid scale physics have advantages and 1148 1148 drawbacks. There are not all available in \NEMO. In the $z$-coordinate 1149 1149 formulation, five options are offered for active tracers (temperature and … … 1157 1157 operator acting along $s-$surfaces (see \S\ref{LDF}). 1158 1158 1159 \subsubsection{ lateral second order tracer diffusive operator}1159 \subsubsection{Lateral second order tracer diffusive operator} 1160 1160 1161 1161 The lateral second order tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}): … … 1186 1186 1187 1187 For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral 1188 and computational surfaces. Therefore, they have a same expression in $z$- and $s$-coordinates: 1188 and computational surfaces. Therefore, they are different quantities, 1189 but have similar expressions in $z$- and $s$-coordinates. In $z$-coordinates: 1189 1190 \begin{equation} \label{Eq_PE_iso_slopes} 1190 1191 r_1 =\frac{e_3 }{e_1 } \left( {\frac{\partial \rho }{\partial i}} \right) 1191 1192 \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \ , \quad 1192 1193 r_1 =\frac{e_3 }{e_1 } \left( {\frac{\partial \rho }{\partial i}} \right) 1193 \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 1194 \end{equation} 1195 1196 When the \textit{Eddy Induced Velocity} parametrisation (eiv) \citep{Gent1990} is used, 1194 \left( {\frac{\partial \rho }{\partial k}} \right)^{-1}, 1195 \end{equation} 1196 while in $s$-coordinates $\partial/\partial k$ is replaced by 1197 $\partial/\partial s$. 1198 1199 \subsubsection{Eddy induced velocity} 1200 When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used, 1197 1201 an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 1198 1202 \begin{equation} \label{Eq_PE_iso+eiv} … … 1213 1217 where $A^{eiv}$ is the eddy induced velocity coefficient (or equivalently the isoneutral 1214 1218 thickness diffusivity coefficient), and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes 1215 between isoneutral and \emph{geopotential} surfaces and thus depends on the coordinate1216 considered:1219 between isoneutral and \emph{geopotential} surfaces. Their values are 1220 thus independent of the vertical coordinate, but their expression depends on the coordinate: 1217 1221 \begin{align} \label{Eq_PE_slopes_eiv} 1218 1222 \tilde{r}_n = \begin{cases} … … 1227 1231 to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}). 1228 1232 1229 \subsubsection{ lateral fourth order tracer diffusive operator}1233 \subsubsection{Lateral fourth order tracer diffusive operator} 1230 1234 1231 1235 The lateral fourth order tracer diffusive operator is defined by: … … 1239 1243 1240 1244 1241 \subsubsection{ lateral second order momentum diffusive operator}1245 \subsubsection{Lateral second order momentum diffusive operator} 1242 1246 1243 1247 The second order momentum diffusive operator along $z$- or $s$-surfaces is found by
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