Changeset 9392 for branches/2017/dev_merge_2017/DOC/tex_sub/chap_DYN.tex
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branches/2017/dev_merge_2017/DOC/tex_sub/chap_DYN.tex
r9389 r9392 200 200 % enstrophy conserving scheme 201 201 %------------------------------------------------------------- 202 \subsubsection{Enstrophy conserving scheme (\protect\ np{ln\_dynvor\_ens}=true)}202 \subsubsection{Enstrophy conserving scheme (\protect\forcode{ln_dynvor_ens = .true.})} 203 203 \label{DYN_vor_ens} 204 204 … … 221 221 % energy conserving scheme 222 222 %------------------------------------------------------------- 223 \subsubsection{Energy conserving scheme (\protect\ np{ln\_dynvor\_ene}=true)}223 \subsubsection{Energy conserving scheme (\protect\forcode{ln_dynvor_ene = .true.})} 224 224 \label{DYN_vor_ene} 225 225 … … 238 238 % mix energy/enstrophy conserving scheme 239 239 %------------------------------------------------------------- 240 \subsubsection{Mixed energy/enstrophy conserving scheme (\protect\ np{ln\_dynvor\_mix}=true) }240 \subsubsection{Mixed energy/enstrophy conserving scheme (\protect\forcode{ln_dynvor_mix = .true.}) } 241 241 \label{DYN_vor_mix} 242 242 … … 261 261 % energy and enstrophy conserving scheme 262 262 %------------------------------------------------------------- 263 \subsubsection{Energy and enstrophy conserving scheme (\protect\ np{ln\_dynvor\_een}=true) }263 \subsubsection{Energy and enstrophy conserving scheme (\protect\forcode{ln_dynvor_een = .true.}) } 264 264 \label{DYN_vor_een} 265 265 … … 305 305 A key point in \eqref{Eq_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 306 306 It uses the sum of masked t-point vertical scale factor divided either 307 by the sum of the four t-point masks (\np{nn \_een\_e3f}~=~1),308 or just by $4$ (\np{nn \_een\_e3f}~=~true).307 by the sum of the four t-point masks (\np{nn_een_e3f}~=~1), 308 or just by $4$ (\np{nn_een_e3f}~=~true). 309 309 The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ 310 310 tends to zero and extends by continuity the value of $e_{3f}$ into the land areas. … … 377 377 \end{aligned} \right. 378 378 \end{equation} 379 When \np{ln \_dynzad\_zts}~=~\textit{true}, a split-explicit time stepping with 5 sub-timesteps is used379 When \np{ln_dynzad_zts}~=~\textit{true}, a split-explicit time stepping with 5 sub-timesteps is used 380 380 on the vertical advection term. 381 381 This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. 382 382 Note that in this case, a similar split-explicit time stepping should be used on 383 383 vertical advection of tracer to ensure a better stability, 384 an option which is only available with a TVD scheme (see \np{ln \_traadv\_tvd\_zts} in \S\ref{TRA_adv_tvd}).384 an option which is only available with a TVD scheme (see \np{ln_traadv_tvd_zts} in \S\ref{TRA_adv_tvd}). 385 385 386 386 … … 451 451 difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS. 452 452 The latter is described in \citet{Shchepetkin_McWilliams_OM05}. The schemes are 453 selected using the namelist logicals \np{ln \_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.453 selected using the namelist logicals \np{ln_dynadv_cen2} and \np{ln_dynadv_ubs}. 454 454 In flux form, the schemes differ by the choice of a space and time interpolation to 455 455 define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells, … … 460 460 % 2nd order centred scheme 461 461 %------------------------------------------------------------- 462 \subsubsection{$2^{nd}$ order centred scheme (cen2) (\protect\ np{ln\_dynadv\_cen2}=true)}462 \subsubsection{$2^{nd}$ order centred scheme (cen2) (\protect\forcode{ln_dynadv_cen2 = .true.})} 463 463 \label{DYN_adv_cen2} 464 464 … … 481 481 % UBS scheme 482 482 %------------------------------------------------------------- 483 \subsubsection{Upstream Biased Scheme (UBS) (\protect\ np{ln\_dynadv\_ubs}=true)}483 \subsubsection{Upstream Biased Scheme (UBS) (\protect\forcode{ln_dynadv_ubs = .true.})} 484 484 \label{DYN_adv_ubs} 485 485 … … 501 501 those in the centred second order method. As the scheme already includes 502 502 a diffusion component, it can be used without explicit lateral diffusion on momentum 503 ($i.e.$ \np{ln \_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so.503 ($i.e.$ \np{ln_dynldf_lap}=\forcode{ln_dynldf_bilap = .false.}), and it is recommended to do so. 504 504 505 505 The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ … … 554 554 % z-coordinate with full step 555 555 %-------------------------------------------------------------------------------------------------------------- 556 \subsection [$z$-coordinate with full step (\protect\np{ln \_dynhpg\_zco}) ]557 {$z$-coordinate with full step (\protect\ np{ln\_dynhpg\_zco}=true)}556 \subsection [$z$-coordinate with full step (\protect\np{ln_dynhpg_zco}) ] 557 {$z$-coordinate with full step (\protect\forcode{ln_dynhpg_zco = .true.})} 558 558 \label{DYN_hpg_zco} 559 559 … … 595 595 % z-coordinate with partial step 596 596 %-------------------------------------------------------------------------------------------------------------- 597 \subsection [$z$-coordinate with partial step (\protect\np{ln \_dynhpg\_zps})]598 {$z$-coordinate with partial step (\protect\ np{ln\_dynhpg\_zps}=true)}597 \subsection [$z$-coordinate with partial step (\protect\np{ln_dynhpg_zps})] 598 {$z$-coordinate with partial step (\protect\forcode{ln_dynhpg_zps = .true.})} 599 599 \label{DYN_hpg_zps} 600 600 … … 624 624 cubic polynomial method is currently disabled whilst known bugs are under investigation. 625 625 626 $\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\ np{ln\_dynhpg\_sco}=true)626 $\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\forcode{ln_dynhpg_sco = .true.}) 627 627 \begin{equation} \label{Eq_dynhpg_sco} 628 628 \left\{ \begin{aligned} … … 639 639 ($e_{3w}$). 640 640 641 $\bullet$ Traditional coding with adaptation for ice shelf cavities (\ np{ln\_dynhpg\_isf}=true).642 This scheme need the activation of ice shelf cavities (\ np{ln\_isfcav}=true).643 644 $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\ np{ln\_dynhpg\_prj}=true)641 $\bullet$ Traditional coding with adaptation for ice shelf cavities (\forcode{ln_dynhpg_isf = .true.}). 642 This scheme need the activation of ice shelf cavities (\forcode{ln_isfcav = .true.}). 643 644 $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\forcode{ln_dynhpg_prj = .true.}) 645 645 646 646 $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} 647 (\ np{ln\_dynhpg\_djc}=true) (currently disabled; under development)647 (\forcode{ln_dynhpg_djc = .true.}) (currently disabled; under development) 648 648 649 649 Note that expression \eqref{Eq_dynhpg_sco} is commonly used when the variable volume formulation is 650 650 activated (\key{vvl}) because in that case, even with a flat bottom, the coordinate surfaces are not 651 651 horizontal but follow the free surface \citep{Levier2007}. The pressure jacobian scheme 652 (\ np{ln\_dynhpg\_prj}=true) is available as an improved option to \np{ln\_dynhpg\_sco}=truewhen652 (\forcode{ln_dynhpg_prj = .true.}) is available as an improved option to \forcode{ln_dynhpg_sco = .true.} when 653 653 \key{vvl} is active. The pressure Jacobian scheme uses a constrained cubic spline to reconstruct 654 654 the density profile across the water column. This method maintains the monotonicity between the … … 660 660 \label{DYN_hpg_isf} 661 661 Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and 662 the pressure gradient due to the ocean load. If cavity opened (\np{ln \_isfcav}~=~true) these 2 terms can be663 calculated by setting \np{ln \_dynhpg\_isf}~=~true. No other scheme are working with the ice shelf.\\662 the pressure gradient due to the ocean load. If cavity opened (\np{ln_isfcav}~=~true) these 2 terms can be 663 calculated by setting \np{ln_dynhpg_isf}~=~true. No other scheme are working with the ice shelf.\\ 664 664 665 665 $\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium. … … 673 673 % Time-scheme 674 674 %-------------------------------------------------------------------------------------------------------------- 675 \subsection [Time-scheme (\protect\np{ln \_dynhpg\_imp}) ]676 {Time-scheme (\protect\np{ln \_dynhpg\_imp}= true/false)}675 \subsection [Time-scheme (\protect\np{ln_dynhpg_imp}) ] 676 {Time-scheme (\protect\np{ln_dynhpg_imp}= true/false)} 677 677 \label{DYN_hpg_imp} 678 678 … … 689 689 time level $t$ only, as in the standard leapfrog scheme. 690 690 691 $\bullet$ leapfrog scheme (\ np{ln\_dynhpg\_imp}=true):691 $\bullet$ leapfrog scheme (\forcode{ln_dynhpg_imp = .true.}): 692 692 693 693 \begin{equation} \label{Eq_dynhpg_lf} … … 696 696 \end{equation} 697 697 698 $\bullet$ semi-implicit scheme (\ np{ln\_dynhpg\_imp}=true):698 $\bullet$ semi-implicit scheme (\forcode{ln_dynhpg_imp = .true.}): 699 699 \begin{equation} \label{Eq_dynhpg_imp} 700 700 \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; … … 713 713 the stability limits associated with advection or diffusion. 714 714 715 In practice, the semi-implicit scheme is used when \ np{ln\_dynhpg\_imp}=true.715 In practice, the semi-implicit scheme is used when \forcode{ln_dynhpg_imp = .true.}. 716 716 In this case, we choose to apply the time filter to temperature and salinity used in 717 717 the equation of state, instead of applying it to the hydrostatic pressure or to the … … 727 727 Note that in the semi-implicit case, it is necessary to save the filtered density, an 728 728 extra three-dimensional field, in the restart file to restart the model with exact 729 reproducibility. This option is controlled by \np{nn \_dynhpg\_rst}, a namelist parameter.729 reproducibility. This option is controlled by \np{nn_dynhpg_rst}, a namelist parameter. 730 730 731 731 % ================================================================ … … 806 806 variables (Fig.~\ref{Fig_DYN_dynspg_ts}). 807 807 The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) 808 is provided through the \np{nn \_baro} namelist parameter as:809 $\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\ np{ln\_bt\_nn\_auto}=true)808 is provided through the \np{nn_baro} namelist parameter as: 809 $\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\forcode{ln_bt_nn_auto = .true.}) 810 810 considering that the stability of the barotropic system is essentially controled by external waves propagation. 811 811 Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry. 812 Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn \_bt\_cmax}.812 Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn_bt_cmax}. 813 813 814 814 %%% … … 839 839 The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged 840 840 transports to advect tracers. 841 a) Forward time integration: \protect\ np{ln\_bt\_fw}=true, \protect\np{ln\_bt\_av}=true.842 b) Centred time integration: \protect\ np{ln\_bt\_fw}=false, \protect\np{ln\_bt\_av}=true.843 c) Forward time integration with no time filtering (POM-like scheme): \protect\ np{ln\_bt\_fw}=true, \protect\np{ln\_bt\_av}=false. }841 a) Forward time integration: \protect\forcode{ln_bt_fw = .true.}, \protect\forcode{ln_bt_av = .true.}. 842 b) Centred time integration: \protect\forcode{ln_bt_fw = .false.}, \protect\forcode{ln_bt_av = .true.}. 843 c) Forward time integration with no time filtering (POM-like scheme): \protect\forcode{ln_bt_fw = .true.}, \protect\forcode{ln_bt_av = .false.}. } 844 844 \end{center} \end{figure} 845 845 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > 846 846 847 In the default case (\ np{ln\_bt\_fw}=true), the external mode is integrated847 In the default case (\forcode{ln_bt_fw = .true.}), the external mode is integrated 848 848 between \textit{now} and \textit{after} baroclinic time-steps (Fig.~\ref{Fig_DYN_dynspg_ts}a). To avoid aliasing of fast barotropic motions into three dimensional equations, time filtering is eventually applied on barotropic 849 quantities (\ np{ln\_bt\_av}=true). In that case, the integration is extended slightly beyond \textit{after} time step to provide time filtered quantities.849 quantities (\forcode{ln_bt_av = .true.}). In that case, the integration is extended slightly beyond \textit{after} time step to provide time filtered quantities. 850 850 These are used for the subsequent initialization of the barotropic mode in the following baroclinic step. 851 851 Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme, 852 852 asselin filtering is not applied to barotropic quantities. \\ 853 853 Alternatively, one can choose to integrate barotropic equations starting 854 from \textit{before} time step (\ np{ln\_bt\_fw}=false). Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary), the baroclinic to barotropic forcing term given at \textit{now} time step854 from \textit{before} time step (\forcode{ln_bt_fw = .false.}). Although more computationaly expensive ( \np{nn_baro} additional iterations are indeed necessary), the baroclinic to barotropic forcing term given at \textit{now} time step 855 855 become centred in the middle of the integration window. It can easily be shown that this property 856 856 removes part of splitting errors between modes, which increases the overall numerical robustness. … … 868 868 %%% 869 869 870 One can eventually choose to feedback instantaneous values by not using any time filter (\ np{ln\_bt\_av}=false).870 One can eventually choose to feedback instantaneous values by not using any time filter (\forcode{ln_bt_av = .false.}). 871 871 In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new 872 872 sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost) … … 1036 1036 1037 1037 % ================================================================ 1038 \subsection [Iso-level laplacian operator (\protect\np{ln \_dynldf\_lap}) ]1039 {Iso-level laplacian operator (\protect\ np{ln\_dynldf\_lap}=true)}1038 \subsection [Iso-level laplacian operator (\protect\np{ln_dynldf_lap}) ] 1039 {Iso-level laplacian operator (\protect\forcode{ln_dynldf_lap = .true.})} 1040 1040 \label{DYN_ldf_lap} 1041 1041 … … 1060 1060 % Rotated laplacian operator 1061 1061 %-------------------------------------------------------------------------------------------------------------- 1062 \subsection [Rotated laplacian operator (\protect\np{ln \_dynldf\_iso}) ]1063 {Rotated laplacian operator (\protect\ np{ln\_dynldf\_iso}=true)}1062 \subsection [Rotated laplacian operator (\protect\np{ln_dynldf_iso}) ] 1063 {Rotated laplacian operator (\protect\forcode{ln_dynldf_iso = .true.})} 1064 1064 \label{DYN_ldf_iso} 1065 1065 1066 1066 A rotation of the lateral momentum diffusion operator is needed in several cases: 1067 for iso-neutral diffusion in the $z$-coordinate (\ np{ln\_dynldf\_iso}=true) and for1068 either iso-neutral (\ np{ln\_dynldf\_iso}=true) or geopotential1069 (\ np{ln\_dynldf\_hor}=true) diffusion in the $s$-coordinate. In the partial step1067 for iso-neutral diffusion in the $z$-coordinate (\forcode{ln_dynldf_iso = .true.}) and for 1068 either iso-neutral (\forcode{ln_dynldf_iso = .true.}) or geopotential 1069 (\forcode{ln_dynldf_hor = .true.}) diffusion in the $s$-coordinate. In the partial step 1070 1070 case, coordinates are horizontal except at the deepest level and no 1071 rotation is performed when \ np{ln\_dynldf\_hor}=true. The diffusion operator1071 rotation is performed when \forcode{ln_dynldf_hor = .true.}. The diffusion operator 1072 1072 is defined simply as the divergence of down gradient momentum fluxes on each 1073 1073 momentum component. It must be emphasized that this formulation ignores … … 1129 1129 % Iso-level bilaplacian operator 1130 1130 %-------------------------------------------------------------------------------------------------------------- 1131 \subsection [Iso-level bilaplacian operator (\protect\np{ln \_dynldf\_bilap})]1132 {Iso-level bilaplacian operator (\protect\ np{ln\_dynldf\_bilap}=true)}1131 \subsection [Iso-level bilaplacian operator (\protect\np{ln_dynldf_bilap})] 1132 {Iso-level bilaplacian operator (\protect\forcode{ln_dynldf_bilap = .true.})} 1133 1133 \label{DYN_ldf_bilap} 1134 1134 … … 1157 1157 would be too restrictive a constraint on the time step. Two time stepping schemes 1158 1158 can be used for the vertical diffusion term : $(a)$ a forward time differencing 1159 scheme (\ np{ln\_zdfexp}=true) using a time splitting technique1160 (\np{nn \_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme1161 (\ np{ln\_zdfexp}=false) (see \S\ref{STP}). Note that namelist variables1162 \np{ln \_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.1159 scheme (\forcode{ln_zdfexp = .true.}) using a time splitting technique 1160 (\np{nn_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme 1161 (\forcode{ln_zdfexp = .false.}) (see \S\ref{STP}). Note that namelist variables 1162 \np{ln_zdfexp} and \np{nn_zdfexp} apply to both tracers and dynamics. 1163 1163 1164 1164 The formulation of the vertical subgrid scale physics is the same whatever … … 1206 1206 may enter the dynamical equations by affecting the surface pressure gradient. 1207 1207 1208 (1) When \np{ln \_apr\_dyn}~=~true (see \S\ref{SBC_apr}), the atmospheric pressure is taken1208 (1) When \np{ln_apr_dyn}~=~true (see \S\ref{SBC_apr}), the atmospheric pressure is taken 1209 1209 into account when computing the surface pressure gradient. 1210 1210 1211 (2) When \np{ln \_tide\_pot}~=~true and \np{ln\_tide}~=~true (see \S\ref{SBC_tide}),1211 (2) When \np{ln_tide_pot}~=~true and \np{ln_tide}~=~true (see \S\ref{SBC_tide}), 1212 1212 the tidal potential is taken into account when computing the surface pressure gradient. 1213 1213 1214 (3) When \np{nn \_ice\_embd}~=~2 and LIM or CICE is used ($i.e.$ when the sea-ice is embedded in the ocean),1214 (3) When \np{nn_ice_embd}~=~2 and LIM or CICE is used ($i.e.$ when the sea-ice is embedded in the ocean), 1215 1215 the snow-ice mass is taken into account when computing the surface pressure gradient. 1216 1216 … … 1238 1238 weighted velocity (see \S\ref{Apdx_A_momentum}) 1239 1239 1240 $\bullet$ vector invariant form or linear free surface (\ np{ln\_dynhpg\_vec}=true; \key{vvl} not defined):1240 $\bullet$ vector invariant form or linear free surface (\forcode{ln_dynhpg_vec = .true.} ; \key{vvl} not defined): 1241 1241 \begin{equation} \label{Eq_dynnxt_vec} 1242 1242 \left\{ \begin{aligned} … … 1246 1246 \end{equation} 1247 1247 1248 $\bullet$ flux form and nonlinear free surface (\ np{ln\_dynhpg\_vec}=false; \key{vvl} defined):1248 $\bullet$ flux form and nonlinear free surface (\forcode{ln_dynhpg_vec = .false.} ; \key{vvl} defined): 1249 1249 \begin{equation} \label{Eq_dynnxt_flux} 1250 1250 \left\{ \begin{aligned} … … 1256 1256 where RHS is the right hand side of the momentum equation, the subscript $f$ 1257 1257 denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is 1258 initialized as \np{nn \_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = $10^{-3}$.1258 initialized as \np{nn_atfp} (namelist parameter). Its default value is \np{nn_atfp} = $10^{-3}$. 1259 1259 In both cases, the modified Asselin filter is not applied since perfect conservation 1260 1260 is not an issue for the momentum equations.
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