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r707 r817 2 2 % Chapter Ñ Appendix C : Discrete Invariants of the Equations 3 3 % ================================================================ 4 \chapter{ Appendix C :Discrete Invariants of the Equations}4 \chapter{Discrete Invariants of the Equations} 5 5 \label{Apdx_C} 6 6 \minitoc 7 8 %%% Appendix put in gmcomment as it has not been updated for z* and s coordinate 9 I'm writting this appendix. It will be available in a forthcoming release of the documentation 10 11 \gmcomment{ 7 12 8 13 % ================================================================ … … 14 19 15 20 First, the boundary condition on the vertical velocity (no flux through the surface and the bottom) is established for the discrete set of momentum equations. Then, it is shown that the non linear terms of the momentum equation are written such that the potential enstrophy of a horizontally non divergent flow is preserved while all the other non-diffusive terms preserve the kinetic energy: the energy is also preserved in practice. In addition, an option is also offer for the vorticity term discretization which provides 16 a total kinetic energy conserving discretization for that term. Note that although these properties are established in the curvilinear $s$-coordinate system, they still hold in the curvilinear $z$-coordinate system. 21 a total kinetic energy conserving discretization for that term. 22 23 Nota Bene: these properties are established here in the rigid-lid case and for the 2nd order centered scheme. A forthcoming update will be their generalisation to the free surface case 24 and higher order scheme. 17 25 18 26 % ------------------------------------------------------------------------------------------------------------- … … 25 33 The discrete set of momentum equations used in rigid lid approximation 26 34 automatically satisfies the surface and bottom boundary conditions 27 ( $w_{surface} =w_{bottom} =~0$, no flux through the surface and the bottom).35 (no flux through the surface and the bottom: $w_{surface} =w_{bottom} =~0$). 28 36 Indeed, taking the discrete horizontal divergence of the vertical sum of the 29 37 horizontal momentum equations (Eqs. (II.2.1) and (II.2.2)~) wheighted by the 30 38 vertical scale factors, it becomes: 31 39 \begin{flalign*} 32 \frac{\partial } {\partial t} 33 \left( 34 \sum\limits_k \chi 35 \right) 36 \equiv \frac{\partial } {\partial t} 37 \left( 38 w_{surface} -w_{bottom} 39 \right) 40 &&&\\ 40 \frac{\partial } {\partial t} \left( \sum\limits_k \chi \right) 41 \equiv 42 \frac{\partial } {\partial t} \left( w_{surface} -w_{bottom} \right)&&&\\ 41 43 \end{flalign*} 42 44 \begin{flalign*} … … 91 93 % ------------------------------------------------------------------------------------------------------------- 92 94 \subsection{Coriolis and advection terms: vector invariant form} 93 \label{Apdx_C .1.2}95 \label{Apdx_C_vor_zad} 94 96 95 97 % ------------------------------------------------------------------------------------------------------------- … … 97 99 % ------------------------------------------------------------------------------------------------------------- 98 100 \subsubsection{Vorticity Term} 99 \label{Apdx_C .1.2.1}100 101 Potential vorticity is located at $f$-points and defined as: $\zeta / e_{3f}$. The standard discrete formulation of the relative vorticity term obviously conserves potential vorticity . It also conserves the potential enstrophy for a horizontally non-divergent flow (i.e. $\chi $=0) but not the total kinetic energy. Indeed, using the symmetry or skew symmetry properties of the operators (Eqs (II.1.10) and (II.1.11)), it can be shown that:101 \label{Apdx_C_vor} 102 103 Potential vorticity is located at $f$-points and defined as: $\zeta / e_{3f}$. The standard discrete formulation of the relative vorticity term obviously conserves potential vorticity (ENS scheme). It also conserves the potential enstrophy for a horizontally non-divergent flow (i.e. $\chi $=0) but not the total kinetic energy. Indeed, using the symmetry or skew symmetry properties of the operators (Eqs \eqref{DOM_mi_adj} and \eqref{DOM_di_adj}), it can be shown that: 102 104 \begin{equation} \label{Apdx_C_1.1} 103 \int_D {\zeta / e_3\,\;{\textbf{k}}\cdot \frac{1} {e_3 }\nabla \times \left( {\zeta \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0105 \int_D {\zeta / e_3\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {\zeta \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 104 106 \end{equation} 105 106 where $dv=e_1 \,e_2 \,e_3 \;di\,dj\,dk$ is the volume element. Indeed, using 107 (II.2.11), the discrete form of the right hand side of (C.1.1) can be 108 transformed as follow: 107 where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using 108 \eqref{Eq_dynvor_ens}, the discrete form of the right hand side of \eqref{Apdx_C_1.1} 109 can be transformed as follow: 109 110 \begin{flalign*} 110 \int_D \zeta / e_3\,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times111 &\int_D \zeta / e_3\,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 111 112 \left( 112 113 \zeta \; \textbf{k} \times \textbf{U}_h 113 114 \right)\; 114 115 dv 115 &&&\\ 116 \end{flalign*} 117 \begin{flalign*} 118 \equiv \sum\limits_{i,j,k} 116 &&& \displaybreak[0] \\ 117 % 118 \equiv& \sum\limits_{i,j,k} 119 119 \frac{\zeta / e_{3f}} {e_{1f}\,e_{2f}\,e_{3f}} 120 120 \biggl\{ \quad 121 121 \delta_{i+1/2} 122 &\left[122 \left[ 123 123 - \overline {\left( {\zeta / e_{3f}} \right)}^{\,i}\; 124 124 \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/ 2} 125 125 \right] 126 && 127 \biggr. \\ 128 \biggl. 126 && \\ & \qquad \qquad \qquad \;\; 129 127 - \delta_{j+1/2} 130 &\left[ \;\;\;128 \left[ \;\;\; 131 129 \overline {\left( \zeta / e_{3f} \right)}^{\,j}\; 132 130 \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 133 131 \right] 134 \biggr\} \; 135 e_{1f}\,e_{2f}\,e_{3f} 136 &&\\ 137 \end{flalign*} 138 \begin{flalign*} 139 \equiv \sum\limits_{i,j,k} 140 \biggl\{ \quad 141 \delta_i 142 &\left[ \zeta / e_{3f} \right] \; 143 \overline {\left( \zeta / e_{3f} \right)}^{\,i}\; 144 \overline{\overline {\left( e_{1u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 145 && 146 \biggr. \\ 147 \biggl. 148 + \delta_j 149 &\left[ \zeta / e_{3f} \right] \; 150 \overline {\left( \zeta / e_{3f} \right)}^{\,j} \; 151 \overline{\overline {\left( e_{2v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} \; 132 \;\;\biggr\} \; e_{1f}\,e_{2f}\,e_{3f} && \displaybreak[0] \\ 133 % 134 \equiv& \sum\limits_{i,j,k} 135 \biggl\{ \delta_i \left[ \frac{\zeta} {e_{3f}} \right] \; 136 \overline{ \left( \frac{\zeta} {e_{3f}} \right) }^{\,i}\; 137 \overline{ \overline{ \left( e_{1u}\,e_{3u}\,u \right) } }^{\,i,j+1/2} 138 + \delta_j \left[ \frac{\zeta} {e_{3f}} \right] \; 139 \overline{ \left( \frac{\zeta} {e_{3f}} \right) }^{\,j} \; 140 \overline{\overline {\left( e_{2v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} \biggr\} 141 &&&& \displaybreak[0] \\ 142 % 143 \equiv& \frac{1} {2} \sum\limits_{i,j,k} 144 \biggl\{ \delta_i \Bigl[ \left( \frac{\zeta} {e_{3f}} \right)^2 \Bigr]\; 145 \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 146 + \delta_j \Bigl[ \left( \zeta / e_{3f} \right)^2 \Bigr]\; 147 \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 152 148 \biggr\} 153 && \\ 154 \end{flalign*} 155 156 \begin{flalign*} 157 \equiv \frac{1} {2} \sum\limits_{i,j,k} 158 \biggl\{ \quad 159 \delta_i 160 &\Bigl[ 161 \left( \zeta / e_{3f} \right)^2 162 \Bigr]\; 163 \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 164 && 165 \biggr. \\ 166 \biggl. 167 + \delta_j 168 &\Bigl[ 169 \left( \zeta / e_{3f} \right)^2 170 \Bigr]\; 171 \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 172 \biggr\} 173 && \\ 174 \end{flalign*} 175 176 \begin{flalign*} 177 \equiv - \frac{1} {2} \sum\limits_{i,j,k} \left( \zeta / e_{3f} \right)^2\; 178 \biggl\{ \quad 179 \delta_{i+1/2} 180 &\left[ 181 \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 182 \right] 183 && 184 \biggr. \\ 185 \biggl. 186 + \delta_{j+1/2} 187 &\left[ 188 \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 189 \right] 190 \biggr\} 191 && \\ 192 \end{flalign*} 193 194 149 && \displaybreak[0] \\ 150 % 151 \equiv& - \frac{1} {2} \sum\limits_{i,j,k} \left( \frac{\zeta} {e_{3f}} \right)^2\; 152 \biggl\{ \delta_{i+1/2} 153 \left[ \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} \right] 154 + \delta_{j+1/2} 155 \left[ \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} \right] 156 \biggr\} && \\ 157 \end{flalign*} 195 158 Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} 196 159 \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, 197 160 and introducing the horizontal divergence $\chi $, it becomes: 198 \begin{equation*} 199 \equiv \sum\limits_{i,j,k} - \frac{1} {2} \left( \zeta / e_{3f} \right)^2 \; \overline{\overline{ e_1T\,e_2T\,e_3T\, \chi}}^{\,i+1/2,j+1/2} \equiv 0 200 \end{equation*} 161 \begin{align*} 162 \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( \frac{\zeta} {e_{3f}} \right)^2 \; \overline{\overline{ e_1T\,e_2T\,e_3T\, \chi}}^{\,i+1/2,j+1/2} \;\;\equiv 0 163 \qquad \qquad \qquad \qquad \qquad \qquad \qquad &&&&\\ 164 \end{align*} 201 165 202 166 Note that the demonstration is done here for the relative potential 203 vorticity but it still hold for the planetary ($f/e_3$ 167 vorticity but it still hold for the planetary ($f/e_3$) and the total 204 168 potential vorticity $((\zeta +f) /e_3 )$. Another formulation of 205 the two components of the vorticity term is optionally offered :169 the two components of the vorticity term is optionally offered (ENE scheme) : 206 170 \begin{equation*} 207 171 \frac{1} {e_3 }\nabla \times … … 228 192 enstrophy on the relative vorticity term and energy on the Coriolis term. 229 193 \begin{flalign*} 230 \int\limits_D \textbf{U}_h \times 231 \left( 232 \zeta \;\textbf{k} \times \textbf{U}_h 233 \right)\; 234 dv 235 &&& \\ 236 \end{flalign*} 237 238 \begin{flalign*} 239 \equiv \sum\limits_{i,j,k} 240 \biggl\{ \quad 241 & \overline {\left( \zeta / e_{3f} \right) 242 \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} \; e_{2u}\,e_{3u}\,u 243 && 244 \biggr. \\ 245 \biggl. 246 -& \overline {\left( \zeta / e_{3f} \right) 247 \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \; e_{1v}\,e_{3v}\,v \; 248 \biggr\} 249 && \\ 250 \end{flalign*} 251 252 \begin{flalign*} 253 \equiv \sum\limits_{i,j,k} 254 \Bigl( 255 \zeta / e_{3f} 256 \Bigr)\; 257 \biggl\{ \quad 258 & \overline {\left( e_{1v}\,e_{3v} \,v \right)}^{\,i+1/2}\;\; 259 \overline {\left( e_{2u}\,e_{3u} \,u \right)}^{\,j+1/2} 260 && 261 \biggr. \\ 262 \biggl. 263 -& \overline {\left( e_{2u}\,e_{3u} \,u \right)}^{\,j+1/2}\;\; 264 \overline {\left( e_{1v}\,e_{3v} \,v \right)}^{\,i+1/2}\; 265 \biggr\} \; 194 &\int\limits_D \textbf{U}_h \times \left( \zeta \;\textbf{k} \times \textbf{U}_h \right) \; dv && \\ 195 \equiv& \sum\limits_{i,j,k} \biggl\{ 196 \overline {\left( \frac{\zeta} {e_{3f}} \right) 197 \overline {\left( e_{1v}e_{3v}v \right)}^{\,i+1/2}} ^{\,j} \, e_{2u}e_{3u}u 198 - \overline {\left( \frac{\zeta} {e_{3f}} \right) 199 \overline {\left( e_{2u}e_{3u}u \right)}^{\,j+1/2}} ^{\,i} \, e_{1v}e_{3v}v \; 200 \biggr\} 201 \\ 202 \equiv& \sum\limits_{i,j,k} \frac{\zeta} {e_{3f}} 203 \biggl\{ \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2}\; 204 \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2} 205 - \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2}\; 206 \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2} 207 \biggr\} 266 208 \equiv 0 267 && \\ 268 \end{flalign*} 209 \end{flalign*} 210 269 211 270 212 % ------------------------------------------------------------------------------------------------------------- … … 272 214 % ------------------------------------------------------------------------------------------------------------- 273 215 \subsubsection{Gradient of Kinetic Energy / Vertical Advection} 274 \label{Apdx_C .1.2.2}216 \label{Apdx_C_zad} 275 217 276 218 The change of Kinetic Energy (KE) due to the vertical advection is exactly 277 219 balanced by the change of KE due to the horizontal gradient of KE~: 278 220 \begin{equation*} 279 \int_D \textbf{U}_h \cdot \nabla_h \left( 1/2\;\textbf{U}_h^2 \right)\;dv221 \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv 280 222 = - \int_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv 281 223 \end{equation*} 282 Indeed, using successively (II.1.10) and the continuity of mass (II.2.5), 283 (II.1.10), and the commutativity of operators $\overline {\,\cdot \,} $ and 284 $\delta$, and finally (II.1.11) successively in the horizontal and in the vertical 285 direction, it becomes: 286 \begin{flalign*} 287 \int_D \textbf{U}_h \cdot \nabla_h \left( 1/2\; \textbf{U}_h^2 \right)\;dv &&&\\ 288 \end{flalign*} 289 290 \begin{flalign*} 291 \equiv \frac{1} {2} \sum\limits_{i,j,k} 292 \biggl\{ \quad 293 &\frac{1} {e_{1u}} \delta_{i+1/2} 294 \left[ 295 \overline {u^2}^{\,i} 296 + \overline {v^2}^{\,j} 297 \right] 298 \;u\;e_{1u}\,e_{2u}\,e_{3u} 299 && 300 \biggr. \\ 301 \biggl. 302 & + \frac{1} {e_{2v}} \delta_{j+1/2} 303 \left[ 304 \overline {u^2}^{\,i} 305 + \overline {v^2}^{\,j} 306 \right] 307 \;v\;e_{1v}\,e_{2v}\,e_{3v} \; 224 Indeed, using successively \eqref{DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator) and the incompressibility, then again \eqref{DOM_di_adj}, then 225 the commutativity of operators $\overline {\,\cdot \,}$ and $\delta$, and finally \eqref{DOM_mi_adj} ($i.e.$ the symmetry property of the $\overline {\,\cdot \,}$ operator) applied in the horizontal and vertical direction, it becomes: 226 \begin{flalign*} 227 &\int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv &&&\\ 228 \equiv& \frac{1}{2} \sum\limits_{i,j,k} 229 \biggl\{ 230 \frac{1} {e_{1u}} \delta_{i+1/2} 231 \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] u\,e_{1u}e_{2u}e_{3u} 232 + \frac{1} {e_{2v}} \delta_{j+1/2} 233 \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] v\,e_{1v}e_{2v}e_{3v} 308 234 \biggr\} 309 && \\ 310 \end{flalign*} 311 312 \begin{flalign*} 313 \equiv &\frac{1} {2}\quad \sum\limits_{i,j,k} 314 \left( 315 \overline {u^2}^{\,i} 316 + \overline {v^2}^{\,j} 317 \right)\; 318 \delta_k 319 \left[ 320 e_{1T}\,e_{2T} \,w 321 \right] 322 && \\ 323 \end{flalign*} 324 \begin{flalign*} 325 \equiv &\frac{1} {2} - \sum\limits_{i,j,k} \delta_{k+1/2} 235 &&& \displaybreak[0] \\ 236 % 237 \equiv& \frac{1}{2} \sum\limits_{i,j,k} 238 \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)\; 239 \delta_k \left[ e_{1T}\,e_{2T} \,w \right] 240 % 241 \;\; \equiv -\frac{1}{2} \sum\limits_{i,j,k} \delta_{k+1/2} 326 242 \left[ 327 243 \overline{ u^2}^{\,i} … … 329 245 \right] \; 330 246 e_{1v}\,e_{2v}\,w 331 && \\ 332 \end{flalign*} 333 \begin{flalign*} 334 \equiv &\frac{1} {2} \quad \sum\limits_{i,j,k} 335 \left( 336 \overline {\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 337 + \overline {\delta_{k+1/2} \left[ v^2 \right]}^{\,j} 338 \right)\; 339 e_{1T}\,e_{2T} \,w 340 && \\ 341 \end{flalign*} 342 \begin{flalign*} 343 \equiv \frac{1} {2} \quad \sum\limits_{i,j,k} 344 \biggl\{ \quad 345 & \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\;2 \overline {u}^{\,k+1/2}\; \delta_{k+1/2} 346 \left[ u \right] 347 && 348 \biggr. \\ 349 \biggl. 350 & + \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2}\;2 \overline {v}^{\,k+1/2}\; \delta_{k+1/2} \left[ v \right] \; 247 &&& \displaybreak[0]\\ 248 % 249 \equiv &\frac{1} {2} \sum\limits_{i,j,k} 250 \left( \overline {\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 251 + \overline {\delta_{k+1/2} \left[ v^2 \right]}^{\,j} \right) \; e_{1T}\,e_{2T} \,w 252 && \displaybreak[0] \\ 253 % 254 \equiv &\frac{1} {2} \sum\limits_{i,j,k} 255 \biggl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\;2 256 \overline {u}^{\,k+1/2}\; \delta_{k+1/2} \left[ u \right] %&&& \\ 257 + \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2}\;2 \overline {v}^{\,k+1/2}\; \delta_{k+1/2} \left[ v \right] \; 351 258 \biggr\} 352 &&\\ 353 \end{flalign*} 354 \begin{flalign*} 355 \equiv \quad -\sum\limits_{i,j,k} 356 \biggl\{ \quad 357 &\frac{1} {b_u } \; 259 &&\displaybreak[0] \\ 260 % 261 \equiv& -\sum\limits_{i,j,k} 262 \biggl\{ 263 \quad \frac{1} {b_u } \; 358 264 \overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\,\delta_{k+1/2} 359 265 \left[ u \right] 360 266 \Bigr\} }^{\,k} \;u\;e_{1u}\,e_{2u}\,e_{3u} 361 && 362 \biggr. \\ 363 \biggl. 364 & + \frac{1} {b_v } \; 267 && \\ 268 &\qquad \quad\; + \frac{1} {b_v } \; 365 269 \overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2} \delta_{k+1/2} 366 270 \left[ v \right] … … 368 272 \biggr\} 369 273 && \\ 370 \end{flalign*} 371 \begin{flalign*} 372 \equiv -\int\limits_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv &&&\\ 373 \end{flalign*} 374 375 The main point here is that the respect of this property links the choice of the discrete formulation of vertical advection and of horizontal gradient of KE. Choosing one imposes the other. For example KE can also be defined as $1/2(\overline u^{\,i} + \overline v^{\,j})$, but this leads to the following expression for the vertical advection~: 274 \equiv& -\int\limits_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv &&&\\ 275 \end{flalign*} 276 277 The main point here is that the satisfaction of this property links the choice of the discrete formulation of vertical advection and of horizontal gradient of KE. Choosing one imposes the other. For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. This leads to the following expression for the vertical advection: 376 278 \begin{equation*} 377 279 \frac{1} {e_3 }\; w\; \frac{\partial \textbf{U}_h } {\partial k} … … 385 287 \end{array}} } \right) 386 288 \end{equation*} 387 388 This formulation requires one more horizontal mean, and thus the use of 9 velocity points instead of 3. This is the reason why it has not been retained in the model. 289 a formulation that requires a additional horizontal mean compare to the one used in NEMO. Nine velocity points have to be used instead of 3. This is the reason why it has not been choosen. 290 389 291 % ------------------------------------------------------------------------------------------------------------- 390 292 % Coriolis and advection terms: flux form … … 399 301 \label{Apdx_C.1.3.1} 400 302 401 In flux from the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the ``metric'' term. This altered Coriolis parameter is thusdiscretised at F-point. It is given by:303 In flux from the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the ``metric'' term. This altered Coriolis parameter is discretised at F-point. It is given by: 402 304 \begin{equation*} 403 305 f+\frac{1} {e_1 e_2 } … … 409 311 \end{equation*} 410 312 411 The n any of the scheme presented above for the vorticity term in the vector invariant formulation can be used, except of course the mixed scheme. However, the energy conserving scheme has exclusively been used to date.313 The ENE scheme is then applied to obtain the vorticity term in flux form. It therefore conserves the total KE. The demonstration is same as for the vorticity term in vector invariant form (\S\ref{Apdx_C_vor}). 412 314 413 315 % ------------------------------------------------------------------------------------------------------------- … … 417 319 \label{Apdx_C.1.3.2} 418 320 419 The flux form operator of the momentum advection is evaluated using a second order finite difference scheme. Because of the flux form, the discrete operator does not contribute to the global budget of linear momentum. Moreover, it conserves the horizontal kinetic energy, that is :321 The flux form operator of the momentum advection is evaluated using a centered second order finite difference scheme. Because of the flux form, the discrete operator does not contribute to the global budget of linear momentum. Because of the centered second order scheme, it conserves the horizontal kinetic energy, that is : 420 322 421 323 \begin{equation} \label{Apdx_C_I.3.10} … … 429 331 Let us demonstrate this property for the first term of the scalar product (i.e. considering just the the terms associated with the i-component of the advection): 430 332 \begin{flalign*} 431 \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right)\;dv &&&\\ 432 \end{flalign*} 433 434 \begin{flalign*} 435 \equiv \sum\limits_{i,j,k} 333 &\int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv &&&\\ 334 % 335 \equiv& \sum\limits_{i,j,k} 336 \biggl\{ \frac{1} {e_{1u}\, e_{2u}\,e_{3u}} \biggl( 337 \delta_{i+1/2} \left[ \overline {e_{2u}\,e_{3u}\,u}^{\,i} \;\overline u^{\,i} \right] 338 + \delta_j \left[ \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] 339 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad 340 + \delta_k \left[ \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;\overline u^{\,k+1/2} \right] 341 \biggr) \; \biggr\} \, e_{1u}\,e_{2u}\,e_{3u} \;u 342 &&& \displaybreak[0] \\ 343 % 344 \equiv& \sum\limits_{i,j,k} 436 345 \biggl\{ 437 \frac{1} {e_{1u}\, e_{2u}\,e_{3u}} 438 \biggl( \quad \biggr. 439 & \delta_{i+1/2} 440 \left[ 441 \overline {e_{2u}\,e_{3u}\,u}^{\,i}\;\overline u^{\,i} 442 \right] 443 && \\ 444 & + \delta_j 445 \left[ 446 \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} 447 \right] 448 && 449 \biggr.\\ 450 \biggl. 451 & + \delta_k \biggl. 452 \left[ 453 \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;\overline u^{\,k+1/2} 454 \right]\; 455 \biggr)\; 456 \biggr\} \, 457 e_{1u}\,e_{2u}\,e_{3u} \;u 458 && \\ 459 \end{flalign*} 460 461 \begin{flalign*} 462 \equiv \sum\limits_{i,j,k} 463 \biggl\{ 464 \delta_{i+1/2} 465 \left[ 466 \overline {e_{2u}\,e_{3u}\,u}^{\,i}\; \overline u^{\,i} 467 \right] 468 & + \delta_j 469 \left[ 470 \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} 471 \right] 472 && 473 \biggr.\\ 474 \biggl. 475 & + \delta_k 476 \left[ 477 \overline {e_{1w}\,e_{2w}\,w}^{\,i+12}\;\overline u^{\,k+1/2} 478 \right]\; 479 \biggr\} 480 && \\ 481 \end{flalign*} 482 483 \begin{flalign*} 484 \equiv \sum\limits_{i,j,k} 346 \delta_{i+1/2} \left[ \overline {e_{2u}\,e_{3u}\,u}^{\,i}\; \overline u^{\,i} \right] 347 + \delta_j \left[ \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] 348 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 349 + \delta_k \left[ \overline {e_{1w}\,e_{2w}\,w}^{\,i+12}\;\overline u^{\,k+1/2} \right] 350 \; \biggr\} &&& \displaybreak[0] \\ 351 % 352 \equiv& \sum\limits_{i,j,k} 485 353 \biggl\{ 486 354 \overline {e_{2u}\,e_{3u}\,u}^{\,i}\; \overline u^{\,i} \delta_i 487 355 \left[ u \right] 488 &+ \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2}356 + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2} 489 357 \left[ u \right] 490 && 491 \biggr. \\ 492 \biggl. 493 & + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\; \overline u^{\,k+1/2} \delta_{k+1/2} 494 \left[ u \right] 495 \biggr\} 496 && \\ 497 \end{flalign*} 498 499 \begin{flalign*} 500 \equiv \sum\limits_{i,j,k} 358 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 359 + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\; \overline u^{\,k+1/2} \delta_{k+1/2} \left[ u \right] \biggr\} && \displaybreak[0] \\ 360 % 361 \equiv& \sum\limits_{i,j,k} 501 362 \biggl\{ 502 363 \overline {e_{2u}\,e_{3u}\,u}^{\,i} \delta_i \left[ u^2 \right] 503 & + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2} \delta_{j+/2} \left[ u^2 \right] 504 && 505 \biggr. \\ 506 \biggl. 507 & + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2} \delta_{k+1/2} \left[ u^2 \right] 364 + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2} \delta_{j+/2} \left[ u^2 \right] 365 + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2} \delta_{k+1/2} \left[ u^2 \right] 508 366 \biggr\} 509 && \\ 510 \end{flalign*} 511 512 \begin{flalign*} 513 \equiv \sum\limits_{i,j,k} 367 && \displaybreak[0] \\ 368 % 369 \equiv& \sum\limits_{i,j,k} 514 370 \bigg\{ 515 371 e_{2u}\,e_{3u}\,u\; \delta_{i+1/2} \left[ \overline {u^2}^{\,i} \right] 516 & + e_{1u}\,e_{3u}\,v\; \delta_{j+1/2} \; \left[ \overline {u^2}^{\,i} \right] 517 && 518 \biggr.\\ 519 \biggl. 520 & + e_{1w}\,e_{2w}\,w\; \delta_{k+1/2} \left[ \overline {u^2}^{\,i} \right] 372 + e_{1u}\,e_{3u}\,v\; \delta_{j+1/2} \; \left[ \overline {u^2}^{\,i} \right] 373 + e_{1w}\,e_{2w}\,w\; \delta_{k+1/2} \left[ \overline {u^2}^{\,i} \right] 521 374 \biggr\} 522 && \\ 523 \end{flalign*} 524 525 \begin{flalign*} 526 \equiv \sum\limits_{i,j,k} 375 && \displaybreak[0] \\ 376 % 377 \equiv& \sum\limits_{i,j,k} 527 378 \overline {u^2}^{\,i} 528 379 \biggl\{ … … 530 381 + \delta_{j+1/2} \left[ e_{1u}\,e_{3u}\,v \right] 531 382 + \delta_{k+1/2} \left[ e_{1w}\,e_{2w}\,w \right] 532 \biggr\} 383 \biggr\} \;\; \equiv 0 533 384 &&& \\ 534 385 \end{flalign*} 535 386 536 \begin{flalign*} 537 \equiv 0 &&&\\ 538 \end{flalign*} 387 When the UBS scheme is used to evaluate the flux form momentum advection, the discrete operator does not contribute to the global budget of linear momentum (flux form). The horizontal kinetic energy is not conserved, but forced to decrease (the scheme is diffusive). 539 388 540 389 % ------------------------------------------------------------------------------------------------------------- … … 545 394 546 395 547 A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero. In $z -$coordinates, this properties is satisfied locally with the choice of discretization we have made (property (II.1.9)~). When the equation of state is linear (i.e. when a advective-diffusive equation for density can be derived from those of temperature and salinity) the change of KE due to the work of pressure forces is balanced by the change of potential energy due to buoyancy forces.396 A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero. In $z$-coordinate, this properties is satisfied locally on a C-grid with 2nd order finite differences (property \eqref{Eq_DOM_curl_grad}). When the equation of state is linear ($i.e.$ when an advective-diffusive equation for density can be derived from those of temperature and salinity) the change of KE due to the work of pressure forces is balanced by the change of potential energy due to buoyancy forces: 548 397 \begin{equation*} 549 398 \int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv … … 551 400 \end{equation*} 552 401 553 This property is satisfied in both $z- $and $s-$coordinates. Indeed, defining the depth of a $T$-point, $z_T$ defined as the sum of the vertical scale factors at $w$-points starting from the surface, it can be written as: 554 \begin{flalign*} 555 \int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv &&& \\ 556 \end{flalign*} 557 \begin{flalign*} 558 \equiv \sum\limits_{i,j,k} 559 \biggl\{ \; 560 & - \frac{1} {\rho_o e_{1u}} 561 \Bigl( 562 \delta_{i+1/2} \left[ p^h \right] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} 563 \left[ z_T \right] 564 \Bigr)\; 565 u\;e_{1u}\,e_{2u}\,e_{3u} 566 && 567 \biggr. \\ 568 \biggl. 569 & - \frac{1} {\rho_o e_{2v}} 570 \Bigl( 571 \delta_{j+1/2} \left[ p^h \right] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} 572 \left[ z_T \right] 573 \Bigr)\; 574 v\;e_{1v}\,e_{2v}\,e_{3v} \; 575 \biggr\} 576 && \\ 577 \end{flalign*} 578 579 Using (II.1.10), the continuity equation (II.2.5), and the hydrostatic 580 equation (II.2.4), it turns out to be: 581 \begin{multline*} 582 \equiv \frac{1} {\rho_o} \sum\limits_{i,j,k} 583 \biggl\{ 402 This property can be satisfied in discrete sense for both $z$- and $s$-coordinates. Indeed, defining the depth of a $T$-point, $z_T$ defined as the sum of the vertical scale factors at $w$-points starting from the surface, the workof pressure forces can be written as: 403 \begin{flalign*} 404 &\int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv &&& \\ 405 \equiv& \sum\limits_{i,j,k} \biggl\{ \; - \frac{1} {\rho_o e_{1u}} \Bigl( 406 \delta_{i+1/2} \left[ p^h \right] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] 407 \Bigr) \; u\;e_{1u}\,e_{2u}\,e_{3u} 408 && \\ & \qquad \qquad 409 - \frac{1} {\rho_o e_{2v}} \Bigl( 410 \delta_{j+1/2} \left[ p^h \right] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} \left[ z_T \right] 411 \Bigr) \; v\;e_{1v}\,e_{2v}\,e_{3v} \; 412 \biggr\} && \\ 413 \end{flalign*} 414 415 Using \eqref{DOM_di_adj}, $i.e.$ the skew symmetry property of the $\delta$ operator, \eqref{Eq_wzv}, the continuity equation), and \eqref{Eq_dynhpg_sco}, the hydrostatic 416 equation in $s$-coordinate, it turns out to be: 417 \begin{flalign*} 418 \equiv& \frac{1} {\rho_o} \sum\limits_{i,j,k} \biggl\{ 584 419 e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2}[ z_T] 585 + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2}[ z_T] 586 \biggr. \\ 587 \shoveright { 588 +\biggl. 589 \Bigl( 590 \delta_i[ e_{2u}\,e_{3u}\,u] + \delta_j [ e_{1v}\,e_{3v}\,v] 591 \Bigr)\;p^h 592 \biggr\} } \\ 593 \end{multline*} 594 595 \begin{multline*} 596 \equiv \frac{1} {\rho_o } \sum\limits_{i,j,k} 420 + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2}[ z_T] 421 && \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, 422 +\Bigl( \delta_i[ e_{2u}\,e_{3u}\,u] + \delta_j [ e_{1v}\,e_{3v}\,v] \Bigr)\;p^h \biggr\} &&\\ 423 % 424 \equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k} 597 425 \biggl\{ 598 426 e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2} \delta_{i+1/2} \left[ z_T \right] 599 427 + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right] 600 \biggr. \\ 601 \shoveright { 602 \biggl. 603 - \delta_k 604 \left[ e_{1w} e_{2w}\,w \right]\;p^h 605 \biggr\} } \\ 606 \end{multline*} 607 608 \begin{multline*} 609 \equiv \frac{1} {\rho_o } \sum\limits_{i,j,k} 428 &&&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, 429 - \delta_k \left[ e_{1w} e_{2w}\,w \right]\;p^h \biggr\} &&&\\ 430 % 431 \equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k} 610 432 \biggl\{ 611 433 e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] 612 + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2} \;\delta_{j+1/2} \left[ z_T \right] \biggr. \\ 613 \shoveright{ 614 \biggl. 434 + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2} \;\delta_{j+1/2} \left[ z_T \right] 435 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, 615 436 + e_{1w} e_{2w} \;w\;\delta_{k+1/2} \left[ p_h \right] 616 \biggr\} } \\ 617 \end{multline*} 618 619 \begin{multline*} 620 \equiv \frac{g} {\rho_o} \sum\limits_{i,j,k} 437 \biggr\} &&&\\ 438 % 439 \equiv& \frac{g} {\rho_o} \sum\limits_{i,j,k} 621 440 \biggl\{ 622 441 e_{2u}\,e_{3u} \;u\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] 623 + e_{1v}\,e_{3v} \;v\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2} \left[ z_T \right] \biggr. \\ 624 \shoveright{ 625 \biggl. 442 + e_{1v}\,e_{3v} \;v\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2} \left[ z_T \right] 443 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, 626 444 - e_{1w} e_{2w} \;w\;e_{3w} \overline \rho^{\,k+1/2} 627 \biggr\} }\\628 \end{ multline*}445 \biggr\} &&&\\ 446 \end{flalign*} 629 447 noting that by definition of $z_T$, $\delta_{k+1/2} \left[ z_T \right] \equiv - e_{3w} $, thus: 630 448 \begin{multline*} … … 639 457 \biggr\} } \\ 640 458 \end{multline*} 641 Using (II.1.10), it becomes~:642 \begin{flalign*} 643 \equiv - \frac{g} {\rho_o} \sum\limits_{i,j,k} z_T459 Using \eqref{DOM_di_adj}, it becomes: 460 \begin{flalign*} 461 \equiv& - \frac{g} {\rho_o} \sum\limits_{i,j,k} z_T 644 462 \biggl\{ 645 463 \delta_i \left[ e_{2u}\,e_{3u}\,u\; \overline \rho^{\,i+1/2} \right] … … 648 466 \biggr\} 649 467 &&& \\ 650 \end{flalign*} 651 \begin{flalign*} 652 \equiv -\int_D \nabla \cdot \left( \rho \, \textbf{U} \right)\;g\;z\;\;dv &&& \\ 468 % 469 \equiv& -\int_D \nabla \cdot \left( \rho \, \textbf{U} \right)\;g\;z\;\;dv &&& \\ 653 470 \end{flalign*} 654 471 655 472 Note that this property strongly constraints the discrete expression of both 656 473 the depth of $T-$points and of the term added to the pressure gradient in 657 $s-$coordinates. 474 $s$-coordinate. Nevertheless, it is almost never satisfied as a linear equation of state 475 is rarely used. 658 476 659 477 % ------------------------------------------------------------------------------------------------------------- … … 669 487 With the rigid-lid approximation, the change of KE due to the work of surface pressure forces is exactly zero. This is satisfied in discrete form, at the precision required on the elliptic solver used to solve this equation. This can be demonstrated as follows: 670 488 \begin{flalign*} 671 \int\limits_D - \frac{1} {\rho_o} \nabla_h \left( p_s \right) \cdot \textbf{U}_h \;dv &&& \\ 672 \end{flalign*} 673 674 \begin{flalign*} 675 \equiv \sum\limits_{i,j,k} 676 \biggl\{ \quad 677 & \left( 678 - M_u - \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] 679 \right)\; 680 u\;e_{1u}\,e_{2u}\,e_{3u} 681 && 682 \biggr. \\ 683 \biggl. 684 + & \left( 685 - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] 686 \right)\; 687 v\;e_{1v}\,e_{2v}\,e_{3v} \; 688 \biggr\} 689 && \\ 690 \end{flalign*} 691 692 \begin{flalign*} 693 \equiv \sum\limits_{i,j} 694 \Biggl\{ \quad 695 &\biggl( 696 - M_u - \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] 697 \biggr) 698 \biggl( 699 \sum\limits_k u\;e_{3u} 700 \biggr)\; 701 e_{1u}\,e_{2u} 702 && 703 \Biggr. \\ 704 \Biggl. 705 + & \biggl( 706 - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] 707 \biggr) 708 \biggl( 709 \sum\limits_k v\;e_{3v} 710 \biggr)\; 711 e_{1v}\,e_{2v} \; 712 \Biggr\} 489 \int\limits_D - \frac{1} {\rho_o} \nabla_h \left( p_s \right) \cdot \textbf{U}_h \;dv% &&& \\ 490 % 491 &\equiv \sum\limits_{i,j,k} \biggl\{ \; 492 \left( - M_u - \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] \right)\; 493 u\;e_{1u}\,e_{2u}\,e_{3u} 494 &&&\\& \qquad \;\;\, 495 + \left( - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] \right)\; 496 v\;e_{1v}\,e_{2v}\,e_{3v} \; \biggr\} 497 &&&\\ 498 \\ 499 % 500 &\equiv \sum\limits_{i,j} \Biggl\{ \; 501 \biggl( - M_u - \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] \biggr) 502 \biggl( \sum\limits_k u\;e_{3u} \biggr)\; e_{1u}\,e_{2u} 503 &&&\\& \qquad \;\;\, 504 + \biggl( - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] \biggr) 505 \biggl( \sum\limits_k v\;e_{3v} \biggr)\; e_{1v}\,e_{2v} \; \Biggr\} 713 506 && \\ 714 \end{flalign*} 715 using the relation between \textit{$\psi $} and the vertically sum of the velocity, it becomes~: 716 717 \begin{flalign*} 718 \equiv \sum\limits_{i,j} 719 \biggl\{ \quad 720 &\left( 507 % 508 \intertext{using the relation between \textit{$\psi $} and the vertically sum of the velocity, it becomes:} 509 % 510 &\equiv \sum\limits_{i,j} 511 \biggl\{ \; 512 \left( \;\;\, 721 513 M_u + \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] 722 514 \right)\; 723 515 e_{1u} \,\delta_j 724 516 \left[ \partial_t \psi \right] 725 && 726 \biggr. \\ 727 \biggl. 728 +& \left( 517 && \\ & \qquad \;\;\, 518 + \left( 729 519 - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] 730 520 \right)\; … … 732 522 \biggr\} 733 523 && \\ 734 \end{flalign*} 735 applying the adjoint of the $\delta$ operator, it is now: 736 737 \begin{flalign*} 738 \equiv \sum\limits_{i,j} - \partial_t \psi \; 739 \biggl\{ \quad 740 & \delta_{j+1/2} \left[ e_{1u} M_u \right] 524 % 525 \intertext{applying the adjoint of the $\delta$ operator, it is now:} 526 % 527 &\equiv \sum\limits_{i,j} - \partial_t \psi \; 528 \biggl\{ \; 529 \delta_{j+1/2} \left[ e_{1u} M_u \right] 741 530 - \delta_{i+1/2} \left[ e_{1v} M_v \right] 742 && 743 \biggr. \\ 744 \biggl. 745 +& \delta_{i+1/2} 531 && \\ & \qquad \;\;\, 532 + \delta_{i+1/2} 746 533 \left[ \frac{e_{2v}} {H_v \,e_{2v}} \delta_i \left[ \partial_t \psi \right] 747 534 \right] … … 749 536 \left[ \frac{e_{1u}} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] 750 537 \right] 751 \biggr\} 752 \equiv 0 753 && \\ 754 \end{flalign*} 755 756 The last equality is obtained using (II.2.3), the discrete barotropic streamfunction time evolution equation. By the way, this shows that (II.2.3) is the only way do compute the streamfunction, otherwise the surface pressure forces will work. Nevertheless, since the elliptic equation verified by $\psi $ is solved numerically by an iterative solver, the property is only satisfied at the precision required on the solver. 538 \biggr\} &&&\\ 539 &\equiv 0 && \\ 540 \end{flalign*} 541 542 The last equality is obtained using \eqref{Eq_dynspg_rl}, the discrete barotropic streamfunction time evolution equation. By the way, this shows that \eqref{Eq_dynspg_rl} is the only way do compute the streamfunction, otherwise the surface pressure forces will work. Nevertheless, since the elliptic equation verified by $\psi $ is solved numerically by an iterative solver, the property is only satisfied at the precision required on the solver. 757 543 758 544 % ================================================================ … … 763 549 764 550 765 The numerical schemes are written such that the heat and salt contents are conserved by the internal dynamics (equations in flux form, second order centered finite differences). As a form flux is used to compute the temperature and salinity, the quadratic form of these quantities (i.e. their variance) is globally conserved, too. There is generally no strict conservation of mass, as the equation of state is non linear with respect to $T$ and $S$. In practice, the mass is conserved with a very good accuracy. 766 551 All the numerical schemes used in NEMO are written such that the tracer content is conserved by the internal dynamics and physics (equations in flux form). For advection, only the CEN2 scheme ($i.e.$ 2nd order finite different scheme) conserves the global variance of tracer. Nevertheless the other schemes ensure that the global variance decreases ($i.e.$ they are at least slightly diffusive). For diffusion, all the schemes ensure the decrease of the total tracer variance, but the iso-neutral operator. There is generally no strict conservation of mass, as the equation of state is non linear with respect to $T$ and $S$. In practice, the mass is conserved with a very good accuracy. 767 552 % ------------------------------------------------------------------------------------------------------------- 768 553 % Advection Term … … 771 556 \label{Apdx_C.2.1} 772 557 773 Conservation of the tracer 774 775 The flux form 776 \begin{flalign*} 777 \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&& 778 \end{flalign*} 779 \begin{flalign*} 780 \equiv \sum\limits_{i,j,k} \left\{ 558 Whatever the advection scheme considered it conserves of the tracer content as all the scheme are written in flux form. Let $\tau$ be the tracer interpolated at velocity point (whatever the interpolation is). The conservation of the tracer content is obtained as follows: 559 \begin{flalign*} 560 &\int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&&\\ 561 &\equiv \sum\limits_{i,j,k} \biggl\{ 781 562 \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} 782 \left( 783 \delta_i 784 \left[ 785 e_{2u}\,e_{3u} \,\overline T^{\,i+1/2}\,u 786 \right] 787 \right. \right. 788 & + \left. \delta_j \left[ e_{1v}\,e_{3v} \,\overline T^{\,j+1/2}\,v \right] \right) && \\ 789 & + \left. \frac{1} {e_{3T}} \delta_k \left[ \overline T^{\,k+1/2}\,w \right] \ \ \right\} 790 \ \ e_{1T}\,e_{2T}\,e_{3T} && 791 \end{flalign*} 792 793 \begin{flalign*} 794 \equiv& \sum\limits_{i,j,k} \left\{ 563 \left( \delta_i \left[ e_{2u}\,e_{3u}\; u \;\tau_u \right] 564 + \delta_j \left[ e_{1v}\,e_{3v}\; v \;\tau_v \right] \right) 565 &&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 566 + \frac{1} {e_{3T}} \delta_k \left[ w\;\tau \right] \biggl\} e_{1T}\,e_{2T}\,e_{3T} &&&\\ 567 % 568 &\equiv \sum\limits_{i,j,k} \left\{ 795 569 \delta_i \left[ e_{2u}\,e_{3u} \,\overline T^{\,i+1/2}\,u \right] 796 570 + \delta_j \left[ e_{1v}\,e_{3v} \,\overline T^{\,j+1/2}\,v \right] 797 571 + \delta_k \left[ e_{1T}\,e_{2T} \,\overline T^{\,k+1/2}\,w \right] \right\} 798 && 799 \end{flalign*} 800 \begin{flalign*} 801 \equiv 0 &&& 802 \end{flalign*} 803 804 Conservation of the variance of tracer 805 \begin{flalign*} 806 \int\limits_D T\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ 807 \end{flalign*} 808 \begin{flalign*} 572 && \\ 573 &\equiv 0 &&& 574 \end{flalign*} 575 576 The conservation of the variance of tracer can be achieved only with the CEN2 scheme. It can be demonstarted as follows: 577 \begin{flalign*} 578 &\int\limits_D T\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ 809 579 \equiv& \sum\limits_{i,j,k} T\; 810 580 \left\{ … … 814 584 \right\} 815 585 && \\ 816 \end{flalign*} 817 \begin{flalign*} 818 \equiv \sum\limits_{i,j,k} 586 \equiv& \sum\limits_{i,j,k} 819 587 \left\{ 820 588 - e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u\,\delta_{i+1/2} \left[ T \right] \right. 821 - &e_{1v}\,e_{3v} \overline T^{\,j+1/2}\,v\;\delta_{j+1/2} \left[ T \right]822 &&\\ 823 - &\left. e_{1T}\,e_{2T} \overline T^{\,k+1/2}w\;\delta_{k+1/2} \left[ T \right]589 - e_{1v}\,e_{3v} \overline T^{\,j+1/2}\,v\;\delta_{j+1/2} \left[ T \right] 590 &&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \quad \; 591 - \left. e_{1T}\,e_{2T} \overline T^{\,k+1/2}w\;\delta_{k+1/2} \left[ T \right] 824 592 \right\} 825 593 &&\\ 826 \end{flalign*}827 \begin{flalign*}828 594 \equiv& -\frac{1} {2} \sum\limits_{i,j,k} 829 595 \Bigl\{ … … 833 599 \Bigr\} 834 600 && \\ 835 \end{flalign*}836 \begin{flalign*}837 601 \equiv& \frac{1} {2} \sum\limits_{i,j,k} T^2 838 602 \Bigl\{ … … 841 605 + \delta_k \left[ e_{1T}\,e_{2T}\,w \right] 842 606 \Bigr\} 843 &&\\ 844 \end{flalign*} 845 \begin{flalign*} 846 \equiv 0 &&& 607 \quad \equiv 0 &&& 847 608 \end{flalign*} 848 609 … … 866 627 867 628 These properties of the horizontal diffusive operator are a direct 868 consequence of properties (II.1.8) and (II.1.9). When the vertical curl of 869 the horizontal diffusion of momentum (discrete sense) is taken, the term 870 associated to the horizontal gradient of the divergence is zero locally. 629 consequence of properties \eqref{Eq_DOM_curl_grad} and \eqref{Eq_DOM_div_curl}. When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken, the term associated to the horizontal gradient of the divergence is zero locally. 871 630 872 631 % ------------------------------------------------------------------------------------------------------------- … … 878 637 The lateral momentum diffusion term conserves the potential vorticity : 879 638 \begin{flalign*} 880 \int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times639 &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 881 640 \Bigl[ \nabla_h 882 641 \left( A^{\,lm}\;\chi \right) 883 642 - \nabla_h \times 884 643 \left( A^{\,lm}\;\zeta \; \textbf{k} \right) 885 \Bigr]\;dv &&& \\ 886 \end{flalign*} 644 \Bigr]\;dv = 0 645 \end{flalign*} 646 %%%%%%%%%% recheck here.... (gm) 887 647 \begin{flalign*} 888 648 = \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times … … 891 651 \Bigr]\;dv &&& \\ 892 652 \end{flalign*} 893 894 653 \begin{flalign*} 895 654 \equiv& \sum\limits_{i,j} … … 907 666 \right\} 908 667 && \\ 909 \end{flalign*} 910 Using (II.1.10), it follows: 911 912 \begin{flalign*} 668 % 669 \intertext{Using \eqref{DOM_di_adj}, it follows:} 670 % 913 671 \equiv& \sum\limits_{i,j,k} 914 672 -\,\left\{ … … 928 686 929 687 930 The lateral momentum diffusion term dissipates the horizontal kinetic 931 energy: 932 \begin{flalign*} 688 The lateral momentum diffusion term dissipates the horizontal kinetic energy: 689 %\begin{flalign*} 690 \begin{equation*} 691 \begin{split} 933 692 \int_D \textbf{U}_h \cdot 934 \left[ \nabla_h 935 \left( A^{\,lm}\;\chi \right) 936 - \nabla_h \times 937 \left( A^{\,lm}\;\zeta \;\textbf{k} \right) 938 \right]\;dv &&& \\ 939 \end{flalign*} 940 \begin{flalign*} 941 \equiv \sum\limits_{i,j,k} \quad 942 &\left\{ 943 \frac{1} {e_{1u}} \delta_{i+1/2} 944 \left[ A_T^{\,lm} \chi \right] 945 - \frac{1} {e_{2u}\,e_{3u}} \delta_j 946 \left[ A_f^{\,lm} e_{3f} \zeta \right] 947 \right\}\; 948 e_{1u}\,e_{2u}\,e_{3u} \;u 949 &&\\ 950 & + \left\{ 951 \frac{1} {e_{2u}} \delta_{j+1/2} 952 \left[ A_T^{\,lm} \chi \right] 953 + \frac{1} {e_{1v}\,e_{3v}} \delta_i 954 \left[ A_f^{\,lm} e_{3f} \zeta \right] 955 \right\}\; 956 e_{1v}\,e_{2u}\,e_{3v} \;v 957 && \\ 958 \end{flalign*} 959 \begin{flalign*} 960 \equiv \sum\limits_{i,j,k} \quad 961 &\left\{ 962 e_{2u}\,e_{3u} \;u\;\delta_{i+1/2} 963 \left[ A_T^{\,lm} \chi \right] 964 - e_{1u} \;u\;\delta_j 965 \left[ A_f^{\,lm} e_{3f} \zeta \right] 966 \right\} 967 &&\\ 968 & + \left\{ 969 e_{1v}\,e_{3v} \;v\;\delta_{j+1/2} 970 \left[ A_T^{\,lm} \chi \right] 971 + e_{2v} \;v\;\delta_i 972 \left[ A_f^{\,lm} e_{3f} \zeta \right] 973 \right\} 974 &&\\ 975 \end{flalign*} 976 \begin{flalign*} 977 \equiv \sum\limits_{i,j,k} \quad 978 -& \Bigl( 979 \delta_i 980 \left[ e_{2u}\,e_{3u} \;u \right] 981 + \delta_j 982 \left[ e_{1v}\,e_{3v} \;v \right] 983 \Bigr)\; 984 A_T^{\,lm} \chi 985 && \\ 986 -& \Bigl( 987 \delta_{i+1/2} 988 \left[ e_{2v} \;v \right] 989 - \delta_{j+1/2} 990 \left[ e_{1u} \;u \right] 991 \Bigr)\; 992 A_f^{\,lm} e_{3f} \zeta 993 &&\\ 994 \end{flalign*} 995 \begin{flalign*} 996 \equiv \sum\limits_{i,j,k} 997 - A_T^{\,lm} \,\chi^2 \;e_{1T}\,e_{2T}\,e_{3T} 998 - A_f^{\,lm} \,\zeta^2 \;e_{1f}\,e_{2f}\,e_{3f} 999 \quad \leq 0 1000 &&&\\ 1001 \end{flalign*} 693 \left[ \nabla_h \right. & \left. \left( A^{\,lm}\;\chi \right) 694 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right] \; dv \\ 695 \\ %%% 696 \equiv& \sum\limits_{i,j,k} 697 \left\{ 698 \frac{1} {e_{1u}} \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] 699 - \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] 700 \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u \\ 701 &\;\; + \left\{ 702 \frac{1} {e_{2u}} \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] 703 + \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] 704 \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v \qquad \\ 705 \\ %%% 706 \equiv& \sum\limits_{i,j,k} 707 \Bigl\{ 708 e_{2u}\,e_{3u} \;u\; \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] 709 - e_{1u} \;u\; \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] 710 \Bigl\} 711 \\ 712 &\;\; + \Bigl\{ 713 e_{1v}\,e_{3v} \;v\; \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] 714 + e_{2v} \;v\; \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] 715 \Bigl\} \\ 716 \\ %%% 717 \equiv& \sum\limits_{i,j,k} 718 - \Bigl( 719 \delta_i \left[ e_{2u}\,e_{3u} \;u \right] 720 + \delta_j \left[ e_{1v}\,e_{3v} \;v \right] 721 \Bigr) \; A_T^{\,lm} \chi \\ 722 &\;\; - \Bigl( 723 \delta_{i+1/2} \left[ e_{2v} \;v \right] 724 - \delta_{j+1/2} \left[ e_{1u} \;u \right] 725 \Bigr)\; A_f^{\,lm} e_{3f} \zeta \\ 726 \\ %%% 727 \equiv& \sum\limits_{i,j,k} 728 - A_T^{\,lm} \,\chi^2 \;e_{1T}\,e_{2T}\,e_{3T} 729 - A_f ^{\,lm} \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f} 730 \quad \leq 0 \\ 731 \end{split} 732 \end{equation*} 1002 733 1003 734 % ------------------------------------------------------------------------------------------------------------- … … 1011 742 coefficients are horizontally uniform: 1012 743 \begin{flalign*} 1013 \int\limits_D \zeta \; \textbf{k} \cdot \nabla \times744 &\int\limits_D \zeta \; \textbf{k} \cdot \nabla \times 1014 745 \left[ 1015 746 \nabla_h … … 1018 749 \left( A^{\,lm}\;\zeta \; \textbf{k} \right) 1019 750 \right]\;dv &&&\\ 1020 \end{flalign*} 1021 \begin{flalign*} 1022 = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 751 &= A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 1023 752 \left[ 1024 753 \nabla_h \times 1025 754 \left( \zeta \; \textbf{k} \right) 1026 \right]\;dv &&&\\ 1027 \end{flalign*} 1028 \begin{flalign*} 1029 \equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} 755 \right]\;dv &&&\displaybreak[0]\\ 756 &\equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} 1030 757 \left\{ 1031 758 \delta_{i+1/2} … … 1041 768 \right\} 1042 769 &&&\\ 1043 \end{flalign*} 1044 Using (II.1.10), it becomes~: 1045 1046 \begin{flalign*} 1047 \equiv - A^{\,lm} \sum\limits_{i,j,k} 770 % 771 \intertext{Using \eqref{DOM_di_adj}, it follows:} 772 % 773 &\equiv - A^{\,lm} \sum\limits_{i,j,k} 1048 774 \left\{ 1049 775 \left( … … 1055 781 \left[ e_{3f} \zeta \right] 1056 782 \right)^2 e_{1u}\,e_{2u}\,e_{3u} 1057 \right\} 1058 \; \leq \;0 1059 &&&\\ 783 \right\} &&&\\ 784 & \leq \;0 &&&\\ 1060 785 \end{flalign*} 1061 786 … … 1072 797 horizontally uniform. 1073 798 \begin{flalign*} 1074 799 & \int\limits_D \nabla_h \cdot 1075 800 \Bigl[ 1076 801 \nabla_h … … 1084 809 dv 1085 810 &&&\\ 1086 \end{flalign*} 1087 \begin{flalign*} 1088 \equiv \sum\limits_{i,j,k} 811 &\equiv \sum\limits_{i,j,k} 1089 812 \left\{ 1090 813 \delta_i … … 1100 823 \right\} 1101 824 &&&\\ 1102 \end{flalign*} 1103 Using (II.1.10), it follows: 1104 1105 \begin{flalign*} 1106 \equiv \sum\limits_{i,j,k} 825 % 826 \intertext{Using \eqref{DOM_di_adj}, it follows:} 827 % 828 &\equiv \sum\limits_{i,j,k} 1107 829 - \left\{ 1108 830 \frac{e_{2u}\,e_{3u}} {e_{1u}} A_u^{\,lm} \delta_{i+1/2} … … 1126 848 1127 849 \begin{flalign*} 1128 \int\limits_D \chi \;\nabla_h \cdot 1129 \left[ 1130 \nabla_h 1131 \left( A^{\,lm}\;\chi \right) 1132 - \nabla_h \times 1133 \left( A^{\,lm}\;\zeta \;\textbf{k} \right) 1134 \right]\; 1135 dv 1136 = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h 1137 \left( \chi \right)\; 1138 dv 1139 &&&\\ 1140 \end{flalign*} 1141 1142 \begin{flalign*} 1143 \equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} \chi 850 &\int\limits_D \chi \;\nabla_h \cdot 851 \left[ \nabla_h \left( A^{\,lm}\;\chi \right) 852 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right]\; dv 853 = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv &&&\\ 854 % 855 &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} \chi 1144 856 \left\{ 1145 \delta_i 1146 \left[ 1147 \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 1148 \left[ \chi \right] 1149 \right] 1150 + \delta_j 1151 \left[ 1152 \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 1153 \left[ \chi \right] 1154 \right] 1155 \right\} \; 1156 e_{1T}\,e_{2T}\,e_{3T} 1157 &&&\\ 1158 \end{flalign*} 1159 Using (II.1.10), it turns out to be: 1160 1161 \begin{flalign*} 1162 \equiv - A^{\,lm} \sum\limits_{i,j,k} 857 \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] 858 + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] 859 \right\} \; e_{1T}\,e_{2T}\,e_{3T} &&&\\ 860 % 861 \intertext{Using \eqref{DOM_di_adj}, it turns out to be:} 862 % 863 &\equiv - A^{\,lm} \sum\limits_{i,j,k} 1163 864 \left\{ 1164 \left( 1165 \frac{1} {e_{1u}} \delta_{i+1/2} 1166 \left[ \chi \right] 1167 \right)^2 1168 e_{1u}\,e_{2u}\,e_{3u} 1169 + \left( 1170 \frac{1} {e_{2v}} \delta_{j+1/2} 1171 \left[ \chi \right] 1172 \right)^2 1173 e_{1v}\,e_{2v}\,e_{3v} 1174 \right\} \; 1175 \leq 0 1176 &&&\\ 865 \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 e_{1u}\,e_{2u}\,e_{3u} 866 + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 e_{1v}\,e_{2v}\,e_{3v} 867 \right\} \; &&&\\ 868 &\leq 0 &&&\\ 1177 869 \end{flalign*} 1178 870 … … 1185 877 1186 878 As for the lateral momentum physics, the continuous form of the vertical diffusion of momentum satisfies the several integral constraints. The first two are associated to the conservation of momentum and the dissipation of horizontal kinetic energy: 1187 \begin{ flalign*}879 \begin{align*} 1188 880 \int\limits_D 1189 881 \frac{1} {e_3 }\; \frac{\partial } {\partial k} … … 1191 883 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 1192 884 \right)\; 1193 dv \q uad= \vec{\textbf{0}}1194 &&&\\1195 \end{flalign*} 1196 and 1197 \begin{flalign*} 885 dv \qquad \quad &= \vec{\textbf{0}} 886 \\ 887 % 888 \intertext{and} 889 % 1198 890 \int\limits_D 1199 891 \textbf{U}_h \cdot … … 1202 894 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 1203 895 \right)\; 1204 dv \quad \leq 01205 &&&\\1206 \end{ flalign*}896 dv \quad &\leq 0 897 \\ 898 \end{align*} 1207 899 The first property is obvious. The second results from: 1208 900 … … 1217 909 \end{flalign*} 1218 910 \begin{flalign*} 1219 \equiv \sum\limits_{i,j,k}911 &\equiv \sum\limits_{i,j,k} 1220 912 \left( 1221 913 u\; \delta_k … … 1233 925 \right) 1234 926 &&&\\ 1235 \end{flalign*} 1236 as the horizontal scale factor do not depend on $k$, it follows: 1237 1238 \begin{flalign*} 1239 \equiv - \sum\limits_{i,j,k} 927 % 928 \intertext{as the horizontal scale factor do not depend on $k$, it follows:} 929 % 930 &\equiv - \sum\limits_{i,j,k} 1240 931 \left( 1241 932 \frac{A_u^{\,vm}} {e_{3uw}} … … 1254 945 &&&\\ 1255 946 \end{flalign*} 947 1256 948 The vorticity is also conserved. Indeed: 1257 949 \begin{flalign*} … … 1532 1224 constraint of conservation of tracers: 1533 1225 \begin{flalign*} 1534 \int\limits_D T\;\nabla \cdot \left( A\;\nabla T \right)\;dv &&&\\ 1535 \end{flalign*} 1536 \begin{flalign*} 1537 \equiv \sum\limits_{i,j,k} 1226 &\int\limits_D T\;\nabla \cdot \left( A\;\nabla T \right)\;dv &&&\\ 1227 \\ 1228 &\equiv \sum\limits_{i,j,k} 1538 1229 \biggl\{ \biggr. 1539 1230 \delta_i … … 1543 1234 \right] 1544 1235 + \delta_j 1545 &\left[1236 \left[ 1546 1237 A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 1547 1238 \left[ T \right] 1548 1239 \right] 1549 &&\\ 1550 \biggl. 1240 &&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\; 1551 1241 + \delta_k 1552 &\left[1242 \left[ 1553 1243 A_w^{\,vT} \frac{e_{1T}\,e_{2T}} {e_{3T}} \delta_{k+1/2} 1554 1244 \left[ T \right] 1555 1245 \right] 1556 \biggr\} 1246 \biggr\} \quad \equiv 0 1557 1247 &&\\ 1558 \end{flalign*}1559 \begin{flalign*}1560 \equiv 0 &&&\\1561 1248 \end{flalign*} 1562 1249 … … 1572 1259 \int\limits_D T\;\nabla \cdot \left( A\;\nabla T \right)\;dv &&&\\ 1573 1260 \end{flalign*} 1574 1575 1261 \begin{flalign*} 1576 1262 \equiv \sum\limits_{i,j,k} T … … 1596 1282 &&\\ 1597 1283 \end{flalign*} 1598 1599 1284 \begin{flalign*} 1600 1285 \equiv - \sum\limits_{i,j,k} … … 1626 1311 \end{flalign*} 1627 1312 1313 1314 %%%% end of appendix in gm comment 1315 }
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