Changeset 6625 for branches/UKMO/dev_r5518_v3.4_asm_nemovar_community/DOC/TexFiles/Chapters/Annex_C.tex
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branches/UKMO/dev_r5518_v3.4_asm_nemovar_community/DOC/TexFiles/Chapters/Annex_C.tex
r6617 r6625 410 410 \end{aligned} } \right. 411 411 \end{equation} 412 where the indices $i_p$ and $ j_p$ take the following value:412 where the indices $i_p$ and $k_p$ 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 \\1130 %\end{flalign*}1129 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv = 0 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} \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\} \\ 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 && \\ 1142 1151 % 1143 1152 \intertext{Using \eqref{DOM_di_adj}, it follows:} … … 1145 1154 \equiv& \sum\limits_{i,j,k} 1146 1155 -\,\left\{ 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] 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] 1149 1160 \right\} \quad \equiv 0 1150 \\1161 && \\ 1151 1162 \end{flalign*} 1152 1163 … … 1156 1167 \subsection{Dissipation of Horizontal Kinetic Energy} 1157 1168 \label{Apdx_C.3.2} 1169 1158 1170 1159 1171 The lateral momentum diffusion term dissipates the horizontal kinetic energy: … … 1209 1221 \label{Apdx_C.3.3} 1210 1222 1223 1211 1224 The lateral momentum diffusion term dissipates the enstrophy when the eddy 1212 1225 coefficients are horizontally uniform: … … 1215 1228 \left[ \nabla_h \left( A^{\,lm}\;\chi \right) 1216 1229 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \right]\;dv &&&\\ 1217 & \quad= A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times1230 &= A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 1218 1231 \left[ \nabla_h \times \left( \zeta \; \textbf{k} \right) \right]\;dv &&&\\ 1219 &\ quad \equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f}1232 &\equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} 1220 1233 \left\{ \delta_{i+1/2} \left[ \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right] 1221 1234 + \delta_{j+1/2} \left[ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right] \right\} &&&\\ … … 1223 1236 \intertext{Using \eqref{DOM_di_adj}, it follows:} 1224 1237 % 1225 &\ quad \equiv - A^{\,lm} \sum\limits_{i,j,k}1238 &\equiv - A^{\,lm} \sum\limits_{i,j,k} 1226 1239 \left\{ \left( \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right)^2 b_v 1227 + \left( \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right)^2 b_u \right\} \quad \leq \;0 &&&\\ 1240 + \left( \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right)^2 b_u \right\} &&&\\ 1241 & \leq \;0 &&&\\ 1228 1242 \end{flalign*} 1229 1243 … … 1236 1250 When the horizontal divergence of the horizontal diffusion of momentum 1237 1251 (discrete sense) is taken, the term associated with the vertical curl of the 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 arehorizontally uniform.1252 vorticity is zero locally, due to (!!! II.1.8 !!!!!). The resulting term conserves the 1253 $\chi$ and dissipates $\chi^2$ when the eddy coefficients are 1254 horizontally uniform. 1241 1255 \begin{flalign*} 1242 1256 & \int\limits_D \nabla_h \cdot 1243 1257 \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) 1244 1258 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \Bigr] dv 1245 = \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv \\1259 = \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv &&&\\ 1246 1260 % 1247 1261 &\equiv \sum\limits_{i,j,k} 1248 1262 \left\{ \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \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\} \\1263 + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} &&&\\ 1250 1264 % 1251 1265 \intertext{Using \eqref{DOM_di_adj}, it follows:} … … 1253 1267 &\equiv \sum\limits_{i,j,k} 1254 1268 - \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] 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 \q uad \equiv 0\\1269 + \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\} 1270 \qquad \equiv 0 &&& \\ 1257 1271 \end{flalign*} 1258 1272 … … 1267 1281 \left[ \nabla_h \left( A^{\,lm}\;\chi \right) 1268 1282 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right]\; dv 1269 = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv \\1283 = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv &&&\\ 1270 1284 % 1271 1285 &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1t}\,e_{2t}\,e_{3t}} \chi … … 1273 1287 \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] 1274 1288 + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] 1275 \right\} \; e_{1t}\,e_{2t}\,e_{3t} \\1289 \right\} \; e_{1t}\,e_{2t}\,e_{3t} &&&\\ 1276 1290 % 1277 1291 \intertext{Using \eqref{DOM_di_adj}, it turns out to be:} … … 1279 1293 &\equiv - A^{\,lm} \sum\limits_{i,j,k} 1280 1294 \left\{ \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 b_u 1281 + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\} 1282 \quad \leq 0 \\ 1295 + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\} \; &&&\\ 1296 % 1297 &\leq 0 &&&\\ 1283 1298 \end{flalign*} 1284 1299 … … 1288 1303 \section{Conservation Properties on Vertical Momentum Physics} 1289 1304 \label{Apdx_C_4} 1305 1290 1306 1291 1307 As for the lateral momentum physics, the continuous form of the vertical diffusion … … 1303 1319 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \quad &\leq 0 \\ 1304 1320 \end{align*} 1305 1306 1321 The first property is obvious. The second results from: 1322 1307 1323 \begin{flalign*} 1308 1324 \int\limits_D … … 1343 1359 e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 && \\ 1344 1360 \end{flalign*} 1345 1346 1361 If the vertical diffusion coefficient is uniform over the whole domain, the 1347 1362 enstrophy is dissipated, $i.e.$ … … 1351 1366 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ 1352 1367 \end{flalign*} 1353 1354 1368 This property is only satisfied in $z$-coordinates: 1369 1355 1370 \begin{flalign*} 1356 1371 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times … … 1462 1477 1463 1478 The numerical schemes used for tracer subgridscale physics are written such 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. 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. 1468 1485 1469 1486 % -------------------------------------------------------------------------------------------------------------
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