Changeset 9414 for branches/2017/dev_merge_2017/DOC/tex_sub/annex_C.tex
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branches/2017/dev_merge_2017/DOC/tex_sub/annex_C.tex
r9408 r9414 122 122 123 123 advection term (vector invariant form): 124 \begin{equation} \label{eq:E_tot_vect_vor }124 \begin{equation} \label{eq:E_tot_vect_vor_1} 125 125 \int\limits_D \zeta \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 126 126 \end{equation} 127 127 % 128 \begin{equation} \label{eq:E_tot_vect_adv }128 \begin{equation} \label{eq:E_tot_vect_adv_1} 129 129 \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv 130 130 + \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv … … 151 151 152 152 pressure gradient: 153 \begin{equation} \label{eq:E_tot_pg }153 \begin{equation} \label{eq:E_tot_pg_1} 154 154 - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 155 155 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 172 172 Vector invariant form: 173 173 \begin{subequations} \label{eq:E_tot_vect} 174 \begin{equation} \label{eq:E_tot_vect_vor }174 \begin{equation} \label{eq:E_tot_vect_vor_2} 175 175 \int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 \\ 176 176 \end{equation} 177 \begin{equation} \label{eq:E_tot_vect_adv }177 \begin{equation} \label{eq:E_tot_vect_adv_2} 178 178 \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv 179 179 + \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 180 180 - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\ 181 181 \end{equation} 182 \begin{equation} \label{eq:E_tot_pg }182 \begin{equation} \label{eq:E_tot_pg_2} 183 183 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 184 184 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 189 189 Flux form: 190 190 \begin{subequations} \label{eq:E_tot_flux} 191 \begin{equation} \label{eq:E_tot_flux_metric }191 \begin{equation} \label{eq:E_tot_flux_metric_2} 192 192 \int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 \\ 193 193 \end{equation} 194 \begin{equation} \label{eq:E_tot_flux_adv }194 \begin{equation} \label{eq:E_tot_flux_adv_2} 195 195 \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv 196 196 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\ 197 197 \end{equation} 198 \begin{equation} \label{eq:E_tot_pg }198 \begin{equation} \label{eq:E_tot_pg_3} 199 199 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 200 200 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 207 207 208 208 209 \autoref{eq:E_tot_pg } is the balance between the conversion KE to PE and PE to KE.210 Indeed the left hand side of \autoref{eq:E_tot_pg } can be transformed as follows:209 \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 210 Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 211 211 \begin{flalign*} 212 212 \partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right) … … 223 223 the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}). 224 224 225 The left hand side of \autoref{eq:E_tot_pg } can be transformed as follows:225 The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 226 226 \begin{flalign*} 227 227 - \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv … … 325 325 % ================================================================ 326 326 \section{Discrete total energy conservation: vector invariant form} 327 \label{sec:C. 1}327 \label{sec:C.2} 328 328 329 329 % ------------------------------------------------------------------------------------------------------------- … … 331 331 % ------------------------------------------------------------------------------------------------------------- 332 332 \subsection{Total energy conservation} 333 \label{subsec:C_KE+PE }333 \label{subsec:C_KE+PE_vect} 334 334 335 335 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: … … 401 401 % ------------------------------------------------------------------------------------------------------------- 402 402 \subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 403 \label{subsec:C_vorEEN }403 \label{subsec:C_vorEEN_vect} 404 404 405 405 With the EEN scheme, the vorticity terms are represented as: 406 \begin{equation} \ label{eq:dynvor_een}406 \begin{equation} \tag{\ref{eq:dynvor_een}} 407 407 \left\{ { \begin{aligned} 408 408 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} … … 415 415 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 416 416 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 417 \begin{equation} \ label{eq:Q_triads}417 \begin{equation} \tag{\ref{eq:Q_triads}} 418 418 _i^j \mathbb{Q}^{i_p}_{j_p} 419 419 = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) … … 580 580 % ------------------------------------------------------------------------------------------------------------- 581 581 \subsection{Pressure gradient term} 582 \label{subsec:C. 1.4}582 \label{subsec:C.2.6} 583 583 584 584 \gmcomment{ … … 733 733 % ================================================================ 734 734 \section{Discrete total energy conservation: flux form} 735 \label{sec:C. 1}735 \label{sec:C.3} 736 736 737 737 % ------------------------------------------------------------------------------------------------------------- … … 739 739 % ------------------------------------------------------------------------------------------------------------- 740 740 \subsection{Total energy conservation} 741 \label{subsec:C_KE+PE }741 \label{subsec:C_KE+PE_flux} 742 742 743 743 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: … … 763 763 % ------------------------------------------------------------------------------------------------------------- 764 764 \subsection{Coriolis and advection terms: flux form} 765 \label{subsec:C. 1.3}765 \label{subsec:C.3.2} 766 766 767 767 % ------------------------------------------------------------------------------------------------------------- … … 769 769 % ------------------------------------------------------------------------------------------------------------- 770 770 \subsubsection{Coriolis plus ``metric'' term} 771 \label{subsec:C. 1.3.1}771 \label{subsec:C.3.3} 772 772 773 773 In flux from the vorticity term reduces to a Coriolis term in which the Coriolis … … 789 789 % ------------------------------------------------------------------------------------------------------------- 790 790 \subsubsection{Flux form advection} 791 \label{subsec:C. 1.3.2}791 \label{subsec:C.3.4} 792 792 793 793 The flux form operator of the momentum advection is evaluated using a … … 877 877 % ================================================================ 878 878 \section{Discrete enstrophy conservation} 879 \label{sec:C. 1}879 \label{sec:C.4} 880 880 881 881 … … 887 887 888 888 In the ENS scheme, the vorticity term is descretized as follows: 889 \begin{equation} \ label{eq:dynvor_ens}889 \begin{equation} \tag{\ref{eq:dynvor_ens}} 890 890 \left\{ \begin{aligned} 891 891 +\frac{1}{e_{1u}} & \overline{q}^{\,i} & {\overline{ \overline{\left( e_{1v}\,e_{3v}\; v \right) } } }^{\,i, j+1/2} \\ … … 947 947 948 948 With the EEN scheme, the vorticity terms are represented as: 949 \begin{equation} \ label{eq:dynvor_een}949 \begin{equation} \tag{\ref{eq:dynvor_een}} 950 950 \left\{ { \begin{aligned} 951 951 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} … … 958 958 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 959 959 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 960 \begin{equation} \ label{eq:Q_triads}960 \begin{equation} \tag{\ref{eq:Q_triads}} 961 961 _i^j \mathbb{Q}^{i_p}_{j_p} 962 962 = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) … … 1017 1017 % ================================================================ 1018 1018 \section{Conservation properties on tracers} 1019 \label{sec:C. 2}1019 \label{sec:C.5} 1020 1020 1021 1021 … … 1033 1033 % ------------------------------------------------------------------------------------------------------------- 1034 1034 \subsection{Advection term} 1035 \label{subsec:C. 2.1}1035 \label{subsec:C.5.1} 1036 1036 1037 1037 conservation of a tracer, $T$: … … 1123 1123 % ------------------------------------------------------------------------------------------------------------- 1124 1124 \subsection{Conservation of potential vorticity} 1125 \label{subsec:C. 3.1}1125 \label{subsec:C.6.1} 1126 1126 1127 1127 The lateral momentum diffusion term conserves the potential vorticity : … … 1157 1157 % ------------------------------------------------------------------------------------------------------------- 1158 1158 \subsection{Dissipation of horizontal kinetic energy} 1159 \label{subsec:C. 3.2}1159 \label{subsec:C.6.2} 1160 1160 1161 1161 The lateral momentum diffusion term dissipates the horizontal kinetic energy: … … 1209 1209 % ------------------------------------------------------------------------------------------------------------- 1210 1210 \subsection{Dissipation of enstrophy} 1211 \label{subsec:C. 3.3}1211 \label{subsec:C.6.3} 1212 1212 1213 1213 The lateral momentum diffusion term dissipates the enstrophy when the eddy … … 1234 1234 % ------------------------------------------------------------------------------------------------------------- 1235 1235 \subsection{Conservation of horizontal divergence} 1236 \label{subsec:C. 3.4}1236 \label{subsec:C.6.4} 1237 1237 1238 1238 When the horizontal divergence of the horizontal diffusion of momentum … … 1263 1263 % ------------------------------------------------------------------------------------------------------------- 1264 1264 \subsection{Dissipation of horizontal divergence variance} 1265 \label{subsec:C. 3.5}1265 \label{subsec:C.6.5} 1266 1266 1267 1267 \begin{flalign*} … … 1289 1289 % ================================================================ 1290 1290 \section{Conservation properties on vertical momentum physics} 1291 \label{sec:C _4}1291 \label{sec:C.7} 1292 1292 1293 1293 As for the lateral momentum physics, the continuous form of the vertical diffusion … … 1461 1461 % ================================================================ 1462 1462 \section{Conservation properties on tracer physics} 1463 \label{sec:C. 5}1463 \label{sec:C.8} 1464 1464 1465 1465 The numerical schemes used for tracer subgridscale physics are written such … … 1473 1473 % ------------------------------------------------------------------------------------------------------------- 1474 1474 \subsection{Conservation of tracers} 1475 \label{subsec:C. 5.1}1475 \label{subsec:C.8.1} 1476 1476 1477 1477 constraint of conservation of tracers: … … 1507 1507 % ------------------------------------------------------------------------------------------------------------- 1508 1508 \subsection{Dissipation of tracer variance} 1509 \label{subsec:C. 5.2}1509 \label{subsec:C.8.2} 1510 1510 1511 1511 constraint on the dissipation of tracer variance:
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