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branches/2015/nemo_v3_6_STABLE/DOC/TexFiles/Chapters/Annex_C.tex
r3294 r6275 410 410 \end{aligned} } \right. 411 411 \end{equation} 412 where the indices $i_p$ and $ k_p$ take the following value:412 where the indices $i_p$ and $j_p$ take the following value: 413 413 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 414 414 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: … … 1103 1103 The discrete formulation of the horizontal diffusion of momentum ensures the 1104 1104 conservation of potential vorticity and the horizontal divergence, and the 1105 dissipation of the square of these quantities ( i.e.enstrophy and the1105 dissipation of the square of these quantities ($i.e.$ enstrophy and the 1106 1106 variance of the horizontal divergence) as well as the dissipation of the 1107 1107 horizontal kinetic energy. In particular, when the eddy coefficients are … … 1127 1127 &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 1128 1128 \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) 1129 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv = 01130 \end{flalign*}1129 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ 1130 %\end{flalign*} 1131 1131 %%%%%%%%%% recheck here.... (gm) 1132 \begin{flalign*}1133 = \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times1134 \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv &&&\\1135 \end{flalign*}1136 \begin{flalign*}1132 %\begin{flalign*} 1133 =& \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 1134 \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ 1135 %\end{flalign*} 1136 %\begin{flalign*} 1137 1137 \equiv& \sum\limits_{i,j} 1138 1138 \left\{ 1139 \delta_{i+1/2} 1140 \left[ 1141 \frac {e_{2v}} {e_{1v}\,e_{3v}} \delta_i 1142 \left[ A_f^{\,lm} e_{3f} \zeta \right] 1143 \right] 1144 + \delta_{j+1/2} 1145 \left[ 1146 \frac {e_{1u}} {e_{2u}\,e_{3u}} \delta_j 1147 \left[ A_f^{\,lm} e_{3f} \zeta \right] 1148 \right] 1149 \right\} 1150 && \\ 1139 \delta_{i+1/2} \left[ \frac {e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] \right] 1140 + \delta_{j+1/2} \left[ \frac {e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] \right] 1141 \right\} \\ 1151 1142 % 1152 1143 \intertext{Using \eqref{DOM_di_adj}, it follows:} … … 1154 1145 \equiv& \sum\limits_{i,j,k} 1155 1146 -\,\left\{ 1156 \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i 1157 \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_i \left[ 1\right] 1158 + \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j 1159 \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_j \left[ 1\right] 1147 \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_i \left[ 1\right] 1148 + \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_j \left[ 1\right] 1160 1149 \right\} \quad \equiv 0 1161 &&\\1150 \\ 1162 1151 \end{flalign*} 1163 1152 … … 1167 1156 \subsection{Dissipation of Horizontal Kinetic Energy} 1168 1157 \label{Apdx_C.3.2} 1169 1170 1158 1171 1159 The lateral momentum diffusion term dissipates the horizontal kinetic energy: … … 1221 1209 \label{Apdx_C.3.3} 1222 1210 1223 1224 1211 The lateral momentum diffusion term dissipates the enstrophy when the eddy 1225 1212 coefficients are horizontally uniform: … … 1228 1215 \left[ \nabla_h \left( A^{\,lm}\;\chi \right) 1229 1216 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \right]\;dv &&&\\ 1230 & = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times1217 &\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 1231 1218 \left[ \nabla_h \times \left( \zeta \; \textbf{k} \right) \right]\;dv &&&\\ 1232 &\ equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f}1219 &\quad \equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} 1233 1220 \left\{ \delta_{i+1/2} \left[ \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right] 1234 1221 + \delta_{j+1/2} \left[ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right] \right\} &&&\\ … … 1236 1223 \intertext{Using \eqref{DOM_di_adj}, it follows:} 1237 1224 % 1238 &\ equiv - A^{\,lm} \sum\limits_{i,j,k}1225 &\quad \equiv - A^{\,lm} \sum\limits_{i,j,k} 1239 1226 \left\{ \left( \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right)^2 b_v 1240 + \left( \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right)^2 b_u \right\} &&&\\ 1241 & \leq \;0 &&&\\ 1227 + \left( \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right)^2 b_u \right\} \quad \leq \;0 &&&\\ 1242 1228 \end{flalign*} 1243 1229 … … 1250 1236 When the horizontal divergence of the horizontal diffusion of momentum 1251 1237 (discrete sense) is taken, the term associated with the vertical curl of the 1252 vorticity is zero locally, due to (!!! II.1.8 !!!!!). The resulting term conserves the1253 $\chi$ and dissipates $\chi^2$ when the eddy coefficients are1254 horizontally uniform.1238 vorticity is zero locally, due to \eqref{Eq_DOM_div_curl}. 1239 The resulting term conserves the $\chi$ and dissipates $\chi^2$ 1240 when the eddy coefficients are horizontally uniform. 1255 1241 \begin{flalign*} 1256 1242 & \int\limits_D \nabla_h \cdot 1257 1243 \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) 1258 1244 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \Bigr] dv 1259 = \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv &&&\\1245 = \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv \\ 1260 1246 % 1261 1247 &\equiv \sum\limits_{i,j,k} 1262 1248 \left\{ \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] 1263 + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} &&&\\1249 + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} \\ 1264 1250 % 1265 1251 \intertext{Using \eqref{DOM_di_adj}, it follows:} … … 1267 1253 &\equiv \sum\limits_{i,j,k} 1268 1254 - \left\{ \frac{e_{2u}\,e_{3u}} {e_{1u}} A_u^{\,lm} \delta_{i+1/2} \left[ \chi \right] \delta_{i+1/2} \left[ 1 \right] 1269 + \frac{e_{1v}\,e_{3v}} 1270 \q quad \equiv 0 &&&\\1255 + \frac{e_{1v}\,e_{3v}} {e_{2v}} A_v^{\,lm} \delta_{j+1/2} \left[ \chi \right] \delta_{j+1/2} \left[ 1 \right] \right\} 1256 \quad \equiv 0 \\ 1271 1257 \end{flalign*} 1272 1258 … … 1281 1267 \left[ \nabla_h \left( A^{\,lm}\;\chi \right) 1282 1268 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right]\; dv 1283 = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv &&&\\1269 = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv \\ 1284 1270 % 1285 1271 &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1t}\,e_{2t}\,e_{3t}} \chi … … 1287 1273 \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] 1288 1274 + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] 1289 \right\} \; e_{1t}\,e_{2t}\,e_{3t} &&&\\1275 \right\} \; e_{1t}\,e_{2t}\,e_{3t} \\ 1290 1276 % 1291 1277 \intertext{Using \eqref{DOM_di_adj}, it turns out to be:} … … 1293 1279 &\equiv - A^{\,lm} \sum\limits_{i,j,k} 1294 1280 \left\{ \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 b_u 1295 + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\} \; &&&\\ 1296 % 1297 &\leq 0 &&&\\ 1281 + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\} 1282 \quad \leq 0 \\ 1298 1283 \end{flalign*} 1299 1284 … … 1303 1288 \section{Conservation Properties on Vertical Momentum Physics} 1304 1289 \label{Apdx_C_4} 1305 1306 1290 1307 1291 As for the lateral momentum physics, the continuous form of the vertical diffusion … … 1319 1303 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \quad &\leq 0 \\ 1320 1304 \end{align*} 1305 1321 1306 The first property is obvious. The second results from: 1322 1323 1307 \begin{flalign*} 1324 1308 \int\limits_D … … 1359 1343 e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 && \\ 1360 1344 \end{flalign*} 1345 1361 1346 If the vertical diffusion coefficient is uniform over the whole domain, the 1362 1347 enstrophy is dissipated, $i.e.$ … … 1366 1351 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ 1367 1352 \end{flalign*} 1353 1368 1354 This property is only satisfied in $z$-coordinates: 1369 1370 1355 \begin{flalign*} 1371 1356 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times … … 1477 1462 1478 1463 The numerical schemes used for tracer subgridscale physics are written such 1479 that the heat and salt contents are conserved (equations in flux form, second 1480 order centered finite differences). Since a flux form is used to compute the 1481 temperature and salinity, the quadratic form of these quantities (i.e. their variance) 1482 globally tends to diminish. As for the advection term, there is generally no strict 1483 conservation of mass, even if in practice the mass is conserved to a very high 1484 accuracy. 1464 that the heat and salt contents are conserved (equations in flux form). 1465 Since a flux form is used to compute the temperature and salinity, 1466 the quadratic form of these quantities ($i.e.$ their variance) globally tends to diminish. 1467 As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. 1485 1468 1486 1469 % -------------------------------------------------------------------------------------------------------------
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