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NEMO/trunk/doc/latex/NEMO/subfiles/chap_conservation.tex
r11435 r11544 7 7 % ================================================================ 8 8 \chapter{Invariants of the Primitive Equations} 9 \label{chap:Invariant} 9 \label{chap:CONS} 10 10 11 \chaptertoc 11 12 … … 45 46 % ------------------------------------------------------------------------------------------------------------- 46 47 \section{Conservation properties on ocean dynamics} 47 \label{sec: Invariant_dyn}48 \label{sec:CONS_Invariant_dyn} 48 49 49 50 The non linear term of the momentum equations has been split into a vorticity term, … … 63 64 The continuous formulation of the vorticity term satisfies following integral constraints: 64 65 \[ 65 % \label{eq: vor_vorticity}66 % \label{eq:CONS_vor_vorticity} 66 67 \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma 67 68 \;{\mathrm {\mathbf k}}\times {\textbf {U}}_h } \right)\;dv} =0 … … 69 70 70 71 \[ 71 % \label{eq: vor_enstrophy}72 % \label{eq:CONS_vor_enstrophy} 72 73 if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot 73 74 \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv} … … 76 77 77 78 \[ 78 % \label{eq: vor_energy}79 % \label{eq:CONS_vor_energy} 79 80 \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 80 81 \] … … 88 89 Using the symmetry or anti-symmetry properties of the operators (Eqs II.1.10 and 11), 89 90 it can be shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), 90 while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). 91 while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). 91 92 Note that the enstrophy conserving scheme on total vorticity has been chosen as the standard discrete form of 92 93 the vorticity term. … … 102 103 the horizontal gradient of horizontal kinetic energy: 103 104 104 \begin{equation} \label{eq: keg_zad}105 \int_D {{\textbf{U}}_h \cdot \nabla _h \left( {1/2\;{\textbf{U}}_h ^2} \right)\;dv} =-\int_D {{\textbf{U}}_h \cdot \frac{w}{e_3 }\;\frac{\partial 105 \begin{equation} \label{eq:CONS_keg_zad} 106 \int_D {{\textbf{U}}_h \cdot \nabla _h \left( {1/2\;{\textbf{U}}_h ^2} \right)\;dv} =-\int_D {{\textbf{U}}_h \cdot \frac{w}{e_3 }\;\frac{\partial 106 107 {\textbf{U}}_h }{\partial k}\;dv} 107 108 \end{equation} 108 109 109 110 Using the discrete form given in {\S}II.2-a and the symmetry or anti-symmetry properties of 110 the mean and difference operators, \autoref{eq: keg_zad} is demonstrated in the Appendix C.111 The main point here is that satisfying \autoref{eq: keg_zad} links the choice of the discrete forms of111 the mean and difference operators, \autoref{eq:CONS_keg_zad} is demonstrated in the Appendix C. 112 The main point here is that satisfying \autoref{eq:CONS_keg_zad} links the choice of the discrete forms of 112 113 the vertical advection and of the horizontal gradient of horizontal kinetic energy. 113 114 Choosing one imposes the other. … … 127 128 128 129 \[ 129 % \label{eq: hpg_pe}130 % \label{eq:CONS_hpg_pe} 130 131 \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} 131 132 \] … … 133 134 Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of 134 135 the mean and difference operators, (II.4.3) is demonstrated in the Appendix C. 135 The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of 136 The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of 136 137 $T$-points and of the term added to the pressure gradient in $s-$coordinates: the depth of a $T$-point, $z_T$, 137 138 is defined as the sum the vertical scale factors at $w$-points starting from the surface. … … 145 146 Nevertheless, the $\psi$-equation is solved numerically by an iterative solver (see {\S}~III.5), 146 147 thus the property is only satisfied with the accuracy required on the solver. 147 In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of 148 In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of 148 149 surface pressure forces is exactly zero: 149 150 \[ 150 % \label{eq: spg}151 % \label{eq:CONS_spg} 151 152 \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 152 153 \] … … 161 162 % ------------------------------------------------------------------------------------------------------------- 162 163 \section{Conservation properties on ocean thermodynamics} 163 \label{sec: Invariant_tra}164 \label{sec:CONS_Invariant_tra} 164 165 165 166 In continuous formulation, the advective terms of the tracer equations conserve the tracer content and 166 167 the quadratic form of the tracer, \ie 167 168 \[ 168 % \label{eq: tra_tra2}169 % \label{eq:CONS_tra_tra2} 169 170 \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 170 171 \;\text{and} … … 176 177 Note that in both continuous and discrete formulations, there is generally no strict conservation of mass, 177 178 since the equation of state is non linear with respect to $T$ and $S$. 178 In practice, the mass is conserved with a very good accuracy. 179 In practice, the mass is conserved with a very good accuracy. 179 180 180 181 % ------------------------------------------------------------------------------------------------------------- … … 182 183 % ------------------------------------------------------------------------------------------------------------- 183 184 \subsection{Conservation properties on momentum physics} 184 \label{subsec: Invariant_dyn_physics}185 \label{subsec:CONS_Invariant_dyn_physics} 185 186 186 187 \textbf{* lateral momentum diffusion term} … … 188 189 The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~: 189 190 \[ 190 % \label{eq: dynldf_dyn}191 % \label{eq:CONS_dynldf_dyn} 191 192 \int\limits_D {\frac{1}{e_3 }{\mathrm {\mathbf k}}\cdot \nabla \times \left[ {\nabla 192 193 _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta … … 195 196 196 197 \[ 197 % \label{eq: dynldf_div}198 % \label{eq:CONS_dynldf_div} 198 199 \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 199 200 \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} … … 202 203 203 204 \[ 204 % \label{eq: dynldf_curl}205 % \label{eq:CONS_dynldf_curl} 205 206 \int_D {{\mathrm {\mathbf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 206 207 \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} … … 209 210 210 211 \[ 211 % \label{eq: dynldf_curl2}212 % \label{eq:CONS_dynldf_curl2} 212 213 \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\mathrm {\mathbf k}}\cdot 213 214 \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h … … 217 218 218 219 \[ 219 % \label{eq: dynldf_div2}220 % \label{eq:CONS_dynldf_div2} 220 221 \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ 221 222 {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( … … 250 251 251 252 \[ 252 % \label{eq: dynzdf_dyn}253 % \label{eq:CONS_dynzdf_dyn} 253 254 \begin{aligned} 254 255 & \int_D {\frac{1}{e_3 }} \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\ … … 258 259 conservation of vorticity, dissipation of enstrophy 259 260 \[ 260 % \label{eq: dynzdf_vor}261 % \label{eq:CONS_dynzdf_vor} 261 262 \begin{aligned} 262 263 & \int_D {\frac{1}{e_3 }{\mathrm {\mathbf k}}\cdot \nabla \times \left( {\frac{1}{e_3 … … 270 271 conservation of horizontal divergence, dissipation of square of the horizontal divergence 271 272 \[ 272 % \label{eq: dynzdf_div}273 % \label{eq:CONS_dynzdf_div} 273 274 \begin{aligned} 274 275 &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial … … 289 290 % ------------------------------------------------------------------------------------------------------------- 290 291 \subsection{Conservation properties on tracer physics} 291 \label{subsec: Invariant_tra_physics}292 \label{subsec:CONS_Invariant_tra_physics} 292 293 293 294 The numerical schemes used for tracer subgridscale physics are written in such a way that … … 296 297 the quadratic form of these quantities (\ie\ their variance) globally tends to diminish. 297 298 As for the advective term, there is generally no strict conservation of mass even if, 298 in practice, the mass is conserved with a very good accuracy. 299 300 \textbf{* lateral physics: }conservation of tracer, dissipation of tracer 299 in practice, the mass is conserved with a very good accuracy. 300 301 \textbf{* lateral physics: }conservation of tracer, dissipation of tracer 301 302 variance, i.e. 302 303 303 304 \[ 304 % \label{eq: traldf_t_t2}305 % \label{eq:CONS_traldf_t_t2} 305 306 \begin{aligned} 306 307 &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ … … 312 313 313 314 \[ 314 % \label{eq: trazdf_t_t2}315 % \label{eq:CONS_trazdf_t_t2} 315 316 \begin{aligned} 316 317 & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv = 0 \\
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