Changeset 12063 for NEMO/branches/2019/dev_ASINTER-01-05_merged/doc/latex/NEMO/subfiles/chap_conservation.tex
- Timestamp:
- 2019-12-05T11:46:38+01:00 (4 years ago)
- Location:
- NEMO/branches/2019/dev_ASINTER-01-05_merged/doc
- Files:
-
- 5 edited
Legend:
- Unmodified
- Added
- Removed
-
NEMO/branches/2019/dev_ASINTER-01-05_merged/doc
-
Property
svn:externals
set to
^/utils/badges badges
^/utils/logos logos
-
Property
svn:externals
set to
-
NEMO/branches/2019/dev_ASINTER-01-05_merged/doc/latex
- Property svn:ignore deleted
-
NEMO/branches/2019/dev_ASINTER-01-05_merged/doc/latex/NEMO
-
Property
svn:externals
set to
^/utils/figures/NEMO figures
-
Property
svn:externals
set to
-
NEMO/branches/2019/dev_ASINTER-01-05_merged/doc/latex/NEMO/subfiles
- Property svn:ignore
-
old new 1 *.aux 2 *.bbl 3 *.blg 4 *.dvi 5 *.fdb* 6 *.fls 7 *.idx 1 *.ind 8 2 *.ilg 9 *.ind10 *.log11 *.maf12 *.mtc*13 *.out14 *.pdf15 *.toc16 _minted-*
-
- Property svn:ignore
-
NEMO/branches/2019/dev_ASINTER-01-05_merged/doc/latex/NEMO/subfiles/chap_conservation.tex
r11263 r12063 3 3 \begin{document} 4 4 5 % ================================================================6 % Invariant of the Equations7 % ================================================================8 5 \chapter{Invariants of the Primitive Equations} 9 \label{chap:Invariant} 10 \minitoc 6 \label{chap:CONS} 7 8 \thispagestyle{plain} 9 10 \chaptertoc 11 12 \paragraph{Changes record} ~\\ 13 14 {\footnotesize 15 \begin{tabularx}{\textwidth}{l||X|X} 16 Release & Author(s) & Modifications \\ 17 \hline 18 {\em 4.0} & {\em ...} & {\em ...} \\ 19 {\em 3.6} & {\em ...} & {\em ...} \\ 20 {\em 3.4} & {\em ...} & {\em ...} \\ 21 {\em <=3.4} & {\em ...} & {\em ...} 22 \end{tabularx} 23 } 24 25 \clearpage 11 26 12 27 The continuous equations of motion have many analytic properties. … … 35 50 The alternative is to use diffusive schemes such as upstream or flux corrected schemes. 36 51 This last option was rejected because we prefer a complete handling of the model diffusion, 37 \ie of the model physics rather than letting the advective scheme produces its own implicit diffusion without52 \ie\ of the model physics rather than letting the advective scheme produces its own implicit diffusion without 38 53 controlling the space and time structure of this implicit diffusion. 39 54 Note that in some very specific cases as passive tracer studies, the positivity of the advective scheme is required. … … 41 56 \citep{Marti1992?, Levy1996?, Levy1998?}. 42 57 43 % ------------------------------------------------------------------------------------------------------------- 44 % Conservation Properties on Ocean Dynamics 45 % ------------------------------------------------------------------------------------------------------------- 58 %% ================================================================================================= 46 59 \section{Conservation properties on ocean dynamics} 47 \label{sec: Invariant_dyn}60 \label{sec:CONS_Invariant_dyn} 48 61 49 62 The non linear term of the momentum equations has been split into a vorticity term, … … 63 76 The continuous formulation of the vorticity term satisfies following integral constraints: 64 77 \[ 65 % \label{eq: vor_vorticity}78 % \label{eq:CONS_vor_vorticity} 66 79 \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma 67 80 \;{\mathrm {\mathbf k}}\times {\textbf {U}}_h } \right)\;dv} =0 … … 69 82 70 83 \[ 71 % \label{eq: vor_enstrophy}84 % \label{eq:CONS_vor_enstrophy} 72 85 if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot 73 86 \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 89 77 90 \[ 78 % \label{eq: vor_energy}91 % \label{eq:CONS_vor_energy} 79 92 \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 80 93 \] … … 88 101 Using the symmetry or anti-symmetry properties of the operators (Eqs II.1.10 and 11), 89 102 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). 103 while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). 91 104 Note that the enstrophy conserving scheme on total vorticity has been chosen as the standard discrete form of 92 105 the vorticity term. … … 102 115 the horizontal gradient of horizontal kinetic energy: 103 116 104 \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 117 \begin{equation} \label{eq:CONS_keg_zad} 118 \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 106 119 {\textbf{U}}_h }{\partial k}\;dv} 107 120 \end{equation} 108 121 109 122 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 of123 the mean and difference operators, \autoref{eq:CONS_keg_zad} is demonstrated in the Appendix C. 124 The main point here is that satisfying \autoref{eq:CONS_keg_zad} links the choice of the discrete forms of 112 125 the vertical advection and of the horizontal gradient of horizontal kinetic energy. 113 126 Choosing one imposes the other. … … 122 135 This properties is satisfied locally with the choice of discretization we have made (property (II.1.9)~). 123 136 In addition, when the equation of state is linear 124 (\ie when an advective-diffusive equation for density can be derived from those of temperature and salinity)137 (\ie\ when an advective-diffusive equation for density can be derived from those of temperature and salinity) 125 138 the change of horizontal kinetic energy due to the work of pressure forces is balanced by the change of 126 139 potential energy due to buoyancy forces: 127 140 128 141 \[ 129 % \label{eq: hpg_pe}142 % \label{eq:CONS_hpg_pe} 130 143 \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} 131 144 \] … … 133 146 Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of 134 147 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 148 The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of 136 149 $T$-points and of the term added to the pressure gradient in $s-$coordinates: the depth of a $T$-point, $z_T$, 137 150 is defined as the sum the vertical scale factors at $w$-points starting from the surface. … … 145 158 Nevertheless, the $\psi$-equation is solved numerically by an iterative solver (see {\S}~III.5), 146 159 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 160 In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of 148 161 surface pressure forces is exactly zero: 149 162 \[ 150 % \label{eq: spg}163 % \label{eq:CONS_spg} 151 164 \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 152 165 \] … … 157 170 otherwise there is no guarantee that the surface pressure force work vanishes. 158 171 159 % ------------------------------------------------------------------------------------------------------------- 160 % Conservation Properties on Ocean Thermodynamics 161 % ------------------------------------------------------------------------------------------------------------- 172 %% ================================================================================================= 162 173 \section{Conservation properties on ocean thermodynamics} 163 \label{sec: Invariant_tra}174 \label{sec:CONS_Invariant_tra} 164 175 165 176 In continuous formulation, the advective terms of the tracer equations conserve the tracer content and 166 177 the quadratic form of the tracer, \ie 167 178 \[ 168 % \label{eq: tra_tra2}179 % \label{eq:CONS_tra_tra2} 169 180 \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 170 181 \;\text{and} … … 176 187 Note that in both continuous and discrete formulations, there is generally no strict conservation of mass, 177 188 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. 179 180 % ------------------------------------------------------------------------------------------------------------- 181 % Conservation Properties on Momentum Physics 182 % ------------------------------------------------------------------------------------------------------------- 189 In practice, the mass is conserved with a very good accuracy. 190 191 %% ================================================================================================= 183 192 \subsection{Conservation properties on momentum physics} 184 \label{subsec: Invariant_dyn_physics}193 \label{subsec:CONS_Invariant_dyn_physics} 185 194 186 195 \textbf{* lateral momentum diffusion term} … … 188 197 The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~: 189 198 \[ 190 % \label{eq: dynldf_dyn}199 % \label{eq:CONS_dynldf_dyn} 191 200 \int\limits_D {\frac{1}{e_3 }{\mathrm {\mathbf k}}\cdot \nabla \times \left[ {\nabla 192 201 _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta … … 195 204 196 205 \[ 197 % \label{eq: dynldf_div}206 % \label{eq:CONS_dynldf_div} 198 207 \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 199 208 \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} … … 202 211 203 212 \[ 204 % \label{eq: dynldf_curl}213 % \label{eq:CONS_dynldf_curl} 205 214 \int_D {{\mathrm {\mathbf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 206 215 \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} … … 209 218 210 219 \[ 211 % \label{eq: dynldf_curl2}220 % \label{eq:CONS_dynldf_curl2} 212 221 \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\mathrm {\mathbf k}}\cdot 213 222 \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h … … 217 226 218 227 \[ 219 % \label{eq: dynldf_div2}228 % \label{eq:CONS_dynldf_div2} 220 229 \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ 221 230 {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( 222 231 {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} \right]\;dv} \leqslant 0 223 232 \] 224 225 233 226 234 (II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum conserve both the potential vorticity and … … 250 258 251 259 \[ 252 % \label{eq: dynzdf_dyn}260 % \label{eq:CONS_dynzdf_dyn} 253 261 \begin{aligned} 254 262 & \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}} \\ … … 258 266 conservation of vorticity, dissipation of enstrophy 259 267 \[ 260 % \label{eq: dynzdf_vor}268 % \label{eq:CONS_dynzdf_vor} 261 269 \begin{aligned} 262 270 & \int_D {\frac{1}{e_3 }{\mathrm {\mathbf k}}\cdot \nabla \times \left( {\frac{1}{e_3 … … 270 278 conservation of horizontal divergence, dissipation of square of the horizontal divergence 271 279 \[ 272 % \label{eq: dynzdf_div}280 % \label{eq:CONS_dynzdf_div} 273 281 \begin{aligned} 274 282 &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial … … 283 291 In discrete form, all these properties are satisfied in $z$-coordinate (see Appendix C). 284 292 In $s$-coordinates, only first order properties can be demonstrated, 285 \ie the vertical momentum physics conserve momentum, potential vorticity, and horizontal divergence. 286 287 % ------------------------------------------------------------------------------------------------------------- 288 % Conservation Properties on Tracer Physics 289 % ------------------------------------------------------------------------------------------------------------- 293 \ie\ the vertical momentum physics conserve momentum, potential vorticity, and horizontal divergence. 294 295 %% ================================================================================================= 290 296 \subsection{Conservation properties on tracer physics} 291 \label{subsec: Invariant_tra_physics}297 \label{subsec:CONS_Invariant_tra_physics} 292 298 293 299 The numerical schemes used for tracer subgridscale physics are written in such a way that 294 300 the heat and salt contents are conserved (equations in flux form, second order centred finite differences). 295 301 As a form flux is used to compute the temperature and salinity, 296 the quadratic form of these quantities (\ie their variance) globally tends to diminish.302 the quadratic form of these quantities (\ie\ their variance) globally tends to diminish. 297 303 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 304 in practice, the mass is conserved with a very good accuracy. 305 306 \textbf{* lateral physics: }conservation of tracer, dissipation of tracer 301 307 variance, i.e. 302 308 303 309 \[ 304 % \label{eq: traldf_t_t2}310 % \label{eq:CONS_traldf_t_t2} 305 311 \begin{aligned} 306 312 &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ … … 312 318 313 319 \[ 314 % \label{eq: trazdf_t_t2}320 % \label{eq:CONS_trazdf_t_t2} 315 321 \begin{aligned} 316 322 & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv = 0 \\ … … 328 334 It has not been implemented. 329 335 330 \biblio 331 332 \pindex 336 \subinc{\input{../../global/epilogue}} 333 337 334 338 \end{document}
Note: See TracChangeset
for help on using the changeset viewer.