Changeset 11596 for NEMO/trunk/doc/latex/NEMO/subfiles/apdx_invariants.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/apdx_invariants.tex
r11584 r11596 2 2 3 3 \begin{document} 4 % ================================================================5 % Chapter Ñ Appendix C : Discrete Invariants of the Equations6 % ================================================================7 4 \chapter{Discrete Invariants of the Equations} 8 5 \label{apdx:INVARIANTS} … … 15 12 %\gmcomment{ 16 13 17 \newpage18 19 % ================================================================20 % Introduction / Notations21 % ================================================================22 14 \section{Introduction / Notations} 23 15 \label{sec:INVARIANTS_0} … … 93 85 \end{flalign} 94 86 95 % ================================================================96 % Continuous Total energy Conservation97 % ================================================================98 87 \section{Continuous conservation} 99 88 \label{sec:INVARIANTS_1} … … 322 311 % 323 312 324 % ================================================================325 % Discrete Total energy Conservation : vector invariant form326 % ================================================================327 313 \section{Discrete total energy conservation: vector invariant form} 328 314 \label{sec:INVARIANTS_2} 329 315 330 % -------------------------------------------------------------------------------------------------------------331 % Total energy conservation332 % -------------------------------------------------------------------------------------------------------------333 316 \subsection{Total energy conservation} 334 317 \label{subsec:INVARIANTS_KE+PE_vect} … … 354 337 leads to the discrete equivalent of the four equations \autoref{eq:INVARIANTS_E_tot_flux}. 355 338 356 % -------------------------------------------------------------------------------------------------------------357 % Vorticity term (coriolis + vorticity part of the advection)358 % -------------------------------------------------------------------------------------------------------------359 339 \subsection{Vorticity term (coriolis + vorticity part of the advection)} 360 340 \label{subsec:INVARIANTS_vor} … … 363 343 or the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$). 364 344 Two discretisation of the vorticity term (ENE and EEN) allows the conservation of the kinetic energy. 365 % -------------------------------------------------------------------------------------------------------------366 % Vorticity Term with ENE scheme367 % -------------------------------------------------------------------------------------------------------------368 345 \subsubsection{Vorticity term with ENE scheme (\protect\np[=.true.]{ln_dynvor_ene}{ln\_dynvor\_ene})} 369 346 \label{subsec:INVARIANTS_vorENE} … … 403 380 In other words, the domain averaged kinetic energy does not change due to the vorticity term. 404 381 405 % -------------------------------------------------------------------------------------------------------------406 % Vorticity Term with EEN scheme407 % -------------------------------------------------------------------------------------------------------------408 382 \subsubsection{Vorticity term with EEN scheme (\protect\np[=.true.]{ln_dynvor_een}{ln\_dynvor\_een})} 409 383 \label{subsec:INVARIANTS_vorEEN_vect} … … 475 449 \end{flalign*} 476 450 477 % -------------------------------------------------------------------------------------------------------------478 % Gradient of Kinetic Energy / Vertical Advection479 % -------------------------------------------------------------------------------------------------------------480 451 \subsubsection{Gradient of kinetic energy / Vertical advection} 481 452 \label{subsec:INVARIANTS_zad} … … 585 556 Blah blah required on the the step representation of bottom topography..... 586 557 587 588 % -------------------------------------------------------------------------------------------------------------589 % Pressure Gradient Term590 % -------------------------------------------------------------------------------------------------------------591 558 \subsection{Pressure gradient term} 592 559 \label{subsec:INVARIANTS_2.6} … … 731 698 Nevertheless, it is almost never satisfied since a linear equation of state is rarely used. 732 699 733 % ================================================================734 % Discrete Total energy Conservation : flux form735 % ================================================================736 700 \section{Discrete total energy conservation: flux form} 737 701 \label{sec:INVARIANTS_3} 738 702 739 % -------------------------------------------------------------------------------------------------------------740 % Total energy conservation741 % -------------------------------------------------------------------------------------------------------------742 703 \subsection{Total energy conservation} 743 704 \label{subsec:INVARIANTS_KE+PE_flux} … … 760 721 vector invariant or in flux form, leads to the discrete equivalent of the ???? 761 722 762 763 % -------------------------------------------------------------------------------------------------------------764 % Coriolis and advection terms: flux form765 % -------------------------------------------------------------------------------------------------------------766 723 \subsection{Coriolis and advection terms: flux form} 767 724 \label{subsec:INVARIANTS_3.2} 768 725 769 % -------------------------------------------------------------------------------------------------------------770 % Coriolis plus ``metric'' Term771 % -------------------------------------------------------------------------------------------------------------772 726 \subsubsection{Coriolis plus ``metric'' term} 773 727 \label{subsec:INVARIANTS_3.3} … … 788 742 The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:INVARIANTS_vor}). 789 743 790 % -------------------------------------------------------------------------------------------------------------791 % Flux form advection792 % -------------------------------------------------------------------------------------------------------------793 744 \subsubsection{Flux form advection} 794 745 \label{subsec:INVARIANTS_3.4} … … 869 820 The horizontal kinetic energy is not conserved, but forced to decay (\ie\ the scheme is diffusive). 870 821 871 % ================================================================872 % Discrete Enstrophy Conservation873 % ================================================================874 822 \section{Discrete enstrophy conservation} 875 823 \label{sec:INVARIANTS_4} 876 824 877 % -------------------------------------------------------------------------------------------------------------878 % Vorticity Term with ENS scheme879 % -------------------------------------------------------------------------------------------------------------880 825 \subsubsection{Vorticity term with ENS scheme (\protect\np[=.true.]{ln_dynvor_ens}{ln\_dynvor\_ens})} 881 826 \label{subsec:INVARIANTS_vorENS} … … 944 889 The later equality is obtain only when the flow is horizontally non-divergent, \ie\ $\chi$=$0$. 945 890 946 % -------------------------------------------------------------------------------------------------------------947 % Vorticity Term with EEN scheme948 % -------------------------------------------------------------------------------------------------------------949 891 \subsubsection{Vorticity Term with EEN scheme (\protect\np[=.true.]{ln_dynvor_een}{ln\_dynvor\_een})} 950 892 \label{subsec:INVARIANTS_vorEEN} … … 1017 959 \end{flalign*} 1018 960 1019 % ================================================================1020 % Conservation Properties on Tracers1021 % ================================================================1022 961 \section{Conservation properties on tracers} 1023 962 \label{sec:INVARIANTS_5} … … 1033 972 as the equation of state is non linear with respect to $T$ and $S$. 1034 973 In practice, the mass is conserved to a very high accuracy. 1035 % -------------------------------------------------------------------------------------------------------------1036 % Advection Term1037 % -------------------------------------------------------------------------------------------------------------1038 974 \subsection{Advection term} 1039 975 \label{subsec:INVARIANTS_5.1} … … 1099 1035 which is the discrete form of $ \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv }$. 1100 1036 1101 % ================================================================1102 % Conservation Properties on Lateral Momentum Physics1103 % ================================================================1104 1037 \section{Conservation properties on lateral momentum physics} 1105 1038 \label{sec:INVARIANTS_dynldf_properties} … … 1120 1053 the term associated with the horizontal gradient of the divergence is locally zero. 1121 1054 1122 % -------------------------------------------------------------------------------------------------------------1123 % Conservation of Potential Vorticity1124 % -------------------------------------------------------------------------------------------------------------1125 1055 \subsection{Conservation of potential vorticity} 1126 1056 \label{subsec:INVARIANTS_6.1} … … 1154 1084 \end{flalign*} 1155 1085 1156 % -------------------------------------------------------------------------------------------------------------1157 % Dissipation of Horizontal Kinetic Energy1158 % -------------------------------------------------------------------------------------------------------------1159 1086 \subsection{Dissipation of horizontal kinetic energy} 1160 1087 \label{subsec:INVARIANTS_6.2} … … 1206 1133 \] 1207 1134 1208 % -------------------------------------------------------------------------------------------------------------1209 % Dissipation of Enstrophy1210 % -------------------------------------------------------------------------------------------------------------1211 1135 \subsection{Dissipation of enstrophy} 1212 1136 \label{subsec:INVARIANTS_6.3} … … 1230 1154 \end{flalign*} 1231 1155 1232 % -------------------------------------------------------------------------------------------------------------1233 % Conservation of Horizontal Divergence1234 % -------------------------------------------------------------------------------------------------------------1235 1156 \subsection{Conservation of horizontal divergence} 1236 1157 \label{subsec:INVARIANTS_6.4} … … 1257 1178 \end{flalign*} 1258 1179 1259 % -------------------------------------------------------------------------------------------------------------1260 % Dissipation of Horizontal Divergence Variance1261 % -------------------------------------------------------------------------------------------------------------1262 1180 \subsection{Dissipation of horizontal divergence variance} 1263 1181 \label{subsec:INVARIANTS_6.5} … … 1283 1201 \end{flalign*} 1284 1202 1285 % ================================================================1286 % Conservation Properties on Vertical Momentum Physics1287 % ================================================================1288 1203 \section{Conservation properties on vertical momentum physics} 1289 1204 \label{sec:INVARIANTS_7} … … 1454 1369 \end{flalign*} 1455 1370 1456 % ================================================================1457 % Conservation Properties on Tracer Physics1458 % ================================================================1459 1371 \section{Conservation properties on tracer physics} 1460 1372 \label{sec:INVARIANTS_8} … … 1466 1378 As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. 1467 1379 1468 % -------------------------------------------------------------------------------------------------------------1469 % Conservation of Tracers1470 % -------------------------------------------------------------------------------------------------------------1471 1380 \subsection{Conservation of tracers} 1472 1381 \label{subsec:INVARIANTS_8.1} … … 1499 1408 In fact, this property simply results from the flux form of the operator. 1500 1409 1501 % -------------------------------------------------------------------------------------------------------------1502 % Dissipation of Tracer Variance1503 % -------------------------------------------------------------------------------------------------------------1504 1410 \subsection{Dissipation of tracer variance} 1505 1411 \label{subsec:INVARIANTS_8.2}
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