Changeset 11596 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_time_domain.tex
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- 2019-09-25T19:06:37+02:00 (5 years ago)
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_time_domain.tex
r11584 r11596 3 3 \begin{document} 4 4 5 % ================================================================6 % Chapter 2 ——— Time Domain (step.F90)7 % ================================================================8 5 \chapter{Time Domain} 9 6 \label{chap:TD} … … 16 13 would help ==> to be added} 17 14 %%%% 18 19 \newpage20 15 21 16 Having defined the continuous equations in \autoref{chap:MB}, we need now to choose a time discretization, … … 25 20 the consequences for the order in which the equations are solved. 26 21 27 % ================================================================28 % Time Discretisation29 % ================================================================30 22 \section{Time stepping environment} 31 23 \label{sec:TD_environment} … … 55 47 The time stepping itself is performed once at each time step where implicit vertical diffusion is computed, \ie\ in the \mdl{trazdf} and \mdl{dynzdf} modules. 56 48 57 % -------------------------------------------------------------------------------------------------------------58 % Non-Diffusive Part---Leapfrog Scheme59 % -------------------------------------------------------------------------------------------------------------60 49 \section{Non-diffusive part --- Leapfrog scheme} 61 50 \label{sec:TD_leap_frog} … … 98 87 filter parameter and the viscosity and diffusion coefficients. 99 88 100 % -------------------------------------------------------------------------------------------------------------101 % Diffusive Part---Forward or Backward Scheme102 % -------------------------------------------------------------------------------------------------------------103 89 \section{Diffusive part --- Forward or backward scheme} 104 90 \label{sec:TD_forward_imp} … … 165 151 (see for example \citet{richtmyer.morton_bk67}). 166 152 167 % -------------------------------------------------------------------------------------------------------------168 % Surface Pressure gradient169 % -------------------------------------------------------------------------------------------------------------170 153 \section{Surface pressure gradient} 171 154 \label{sec:TD_spg_ts} … … 184 167 185 168 %\gmcomment{ 186 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>187 169 \begin{figure}[!t] 188 170 \centering … … 197 179 \label{fig:TD_TimeStep_flowchart} 198 180 \end{figure} 199 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>200 181 %} 201 182 202 % -------------------------------------------------------------------------------------------------------------203 % The Modified Leapfrog -- Asselin Filter scheme204 % -------------------------------------------------------------------------------------------------------------205 183 \section{Modified Leapfrog -- Asselin filter scheme} 206 184 \label{sec:TD_mLF} … … 245 223 even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term. 246 224 247 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>248 225 \begin{figure}[!t] 249 226 \centering … … 261 238 \label{fig:TD_MLF_forcing} 262 239 \end{figure} 263 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 264 265 % ------------------------------------------------------------------------------------------------------------- 266 % Start/Restart strategy 267 % ------------------------------------------------------------------------------------------------------------- 240 268 241 \section{Start/Restart strategy} 269 242 \label{sec:TD_rst} … … 315 288 %------------------------------------------------------------------------------------------------------------- 316 289 % Time Domain 317 % -------------------------------------------------------------------------------------------------------------318 \subsection{Time domain}319 \label{subsec:TD_time}320 %--------------------------------------------namrun-------------------------------------------321 322 %--------------------------------------------------------------------------------------------------------------323 324 Options are defined through the \nam{dom}{dom} namelist variables.325 \colorbox{yellow}{add here a few word on nit000 and nitend}326 327 \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj}328 329 add a description of daymod, and the model calandar (leap-year and co)330 331 } %% end add332 333 334 335 %%336 \gmcomment{ % add implicit in vvl case and Crant-Nicholson scheme337 338 Implicit time stepping in case of variable volume thickness.339 340 Tracer case (NB for momentum in vector invariant form take care!)341 342 \begin{flalign*}343 &\frac{\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1}}{2\rdt}344 \equiv \text{RHS}+ \delta_k \lt[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k + 1/2} \lt[ {T^{t+1}} \rt]}345 \rt] \\346 &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1}347 \equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta_k \lt[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k + 1/2} \lt[ {T^{t+1}} \rt]}348 \rt] \\349 &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1}350 \equiv 2\rdt \ \text{RHS}351 + 2\rdt \ \lt\{ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} [ T_{k +1}^{t+1} - T_k ^{t+1} ]352 - \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} [ T_k ^{t+1} - T_{k -1}^{t+1} ] \rt\} \\353 &\\354 &\lt( e_{3t}\,T \rt)_k^{t+1}355 - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} T_{k +1}^{t+1}356 + {2\rdt} \ \lt\{ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2}357 + \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \rt\} T_{k }^{t+1}358 - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} T_{k -1}^{t+1} \\359 &\equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS} \\360 %361 \end{flalign*}362 \begin{flalign*}363 \allowdisplaybreaks364 \intertext{ Tracer case }365 %366 & \qquad \qquad \quad - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2}367 \qquad \qquad \qquad \qquad T_{k +1}^{t+1} \\368 &+ {2\rdt} \ \biggl\{ (e_{3t})_{k }^{t+1} \bigg. + \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2}369 + \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \bigg. \biggr\} \ \ \ T_{k }^{t+1} &&\\370 & \qquad \qquad \qquad \qquad \qquad \quad \ \ - {2\rdt} \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \quad \ \ T_{k -1}^{t+1}371 \ \equiv \ \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS} \\372 %373 \end{flalign*}374 \begin{flalign*}375 \allowdisplaybreaks376 \intertext{ Tracer content case }377 %378 & - {2\rdt} \ & \frac{(A_w^{vt})_{k + 1/2}} {(e_{3w})_{k + 1/2}^{t+1}\;(e_{3t})_{k +1}^{t+1}} && \ \lt( e_{3t}\,T \rt)_{k +1}^{t+1} &\\379 & + {2\rdt} \ \lt[ 1 \rt.+ & \frac{(A_w^{vt})_{k + 1/2}} {(e_{3w})_{k + 1/2}^{t+1}\;(e_{3t})_k^{t+1}}380 + & \frac{(A_w^{vt})_{k - 1/2}} {(e_{3w})_{k - 1/2}^{t+1}\;(e_{3t})_k^{t+1}} \lt. \rt] & \lt( e_{3t}\,T \rt)_{k }^{t+1} &\\381 & - {2\rdt} \ & \frac{(A_w^{vt})_{k - 1/2}} {(e_{3w})_{k - 1/2}^{t+1}\;(e_{3t})_{k -1}^{t+1}} &\ \lt( e_{3t}\,T \rt)_{k -1}^{t+1}382 \equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS} &383 \end{flalign*}384 385 %%386 }387 388 \onlyinsubfile{\input{../../global/epilogue}}389 390 \end{document}
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