Changeset 9393 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
r9392 r9393 69 69 % Horizontal divergence and relative vorticity 70 70 %-------------------------------------------------------------------------------------------------------------- 71 \subsection [Horizontal divergence and relative vorticity (\textit{divcur})] 72 {Horizontal divergence and relative vorticity (\protect\mdl{divcur})} 71 \subsection{Horizontal divergence and relative vorticity (\protect\mdl{divcur})} 73 72 \label{DYN_divcur} 74 73 … … 102 101 % Sea Surface Height evolution 103 102 %-------------------------------------------------------------------------------------------------------------- 104 \subsection [Sea surface height evolution and vertical velocity (\textit{sshwzv})] 105 {Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})} 103 \subsection{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})} 106 104 \label{DYN_sshwzv} 107 105 … … 159 157 % Coriolis and Advection terms: vector invariant form 160 158 % ================================================================ 161 \section{Coriolis and Advection: vector invariant form}159 \section{Coriolis and advection: vector invariant form} 162 160 \label{DYN_adv_cor_vect} 163 161 %-----------------------------------------nam_dynadv---------------------------------------------------- … … 178 176 % Vorticity term 179 177 % ------------------------------------------------------------------------------------------------------------- 180 \subsection [Vorticity term (\textit{dynvor}) ] 181 {Vorticity term (\protect\mdl{dynvor})} 178 \subsection{Vorticity term (\protect\mdl{dynvor})} 182 179 \label{DYN_vor} 183 180 %------------------------------------------nam_dynvor---------------------------------------------------- … … 186 183 187 184 Options are defined through the \ngn{namdyn\_vor} namelist variables. 188 Four discretisations of the vorticity term (\ textit{ln\_dynvor\_xxx}=true) are available:185 Four discretisations of the vorticity term (\np{ln\_dynvor\_xxx}\forcode{ = .true.}) are available: 189 186 conserving potential enstrophy of horizontally non-divergent flow (ENS scheme) ; 190 187 conserving horizontal kinetic energy (ENE scheme) ; conserving potential enstrophy for … … 193 190 flow and horizontal kinetic energy (EEN scheme) (see Appendix~\ref{Apdx_C_vorEEN}). In the 194 191 case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the 195 consistency of vorticity term with analytical equations (\ textit{ln\_dynvor\_con}=true).192 consistency of vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{ = .true.}). 196 193 The vorticity terms are all computed in dedicated routines that can be found in 197 194 the \mdl{dynvor} module. … … 200 197 % enstrophy conserving scheme 201 198 %------------------------------------------------------------- 202 \subsubsection{Enstrophy conserving scheme (\protect\ forcode{ln_dynvor_ens= .true.})}199 \subsubsection{Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} 203 200 \label{DYN_vor_ens} 204 201 … … 221 218 % energy conserving scheme 222 219 %------------------------------------------------------------- 223 \subsubsection{Energy conserving scheme (\protect\ forcode{ln_dynvor_ene= .true.})}220 \subsubsection{Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} 224 221 \label{DYN_vor_ene} 225 222 … … 238 235 % mix energy/enstrophy conserving scheme 239 236 %------------------------------------------------------------- 240 \subsubsection{Mixed energy/enstrophy conserving scheme (\protect\ forcode{ln_dynvor_mix= .true.}) }237 \subsubsection{Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{ = .true.}) } 241 238 \label{DYN_vor_mix} 242 239 … … 261 258 % energy and enstrophy conserving scheme 262 259 %------------------------------------------------------------- 263 \subsubsection{Energy and enstrophy conserving scheme (\protect\ forcode{ln_dynvor_een= .true.}) }260 \subsubsection{Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.}) } 264 261 \label{DYN_vor_een} 265 262 … … 305 302 A key point in \eqref{Eq_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 306 303 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).304 by the sum of the four t-point masks (\np{nn\_een\_e3f}\forcode{ = 1}), 305 or just by $4$ (\np{nn\_een\_e3f}\forcode{ = .true.}). 309 306 The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ 310 307 tends to zero and extends by continuity the value of $e_{3f}$ into the land areas. … … 346 343 % Kinetic Energy Gradient term 347 344 %-------------------------------------------------------------------------------------------------------------- 348 \subsection [Kinetic Energy Gradient term (\textit{dynkeg})] 349 {Kinetic Energy Gradient term (\protect\mdl{dynkeg})} 345 \subsection{Kinetic energy gradient term (\protect\mdl{dynkeg})} 350 346 \label{DYN_keg} 351 347 … … 363 359 % Vertical advection term 364 360 %-------------------------------------------------------------------------------------------------------------- 365 \subsection [Vertical advection term (\textit{dynzad}) ] 366 {Vertical advection term (\protect\mdl{dynzad}) } 361 \subsection{Vertical advection term (\protect\mdl{dynzad}) } 367 362 \label{DYN_zad} 368 363 … … 377 372 \end{aligned} \right. 378 373 \end{equation} 379 When \np{ln _dynzad_zts}~=~\textit{true}, a split-explicit time stepping with 5 sub-timesteps is used374 When \np{ln\_dynzad\_zts}\forcode{ = .true.}, a split-explicit time stepping with 5 sub-timesteps is used 380 375 on the vertical advection term. 381 376 This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. 382 377 Note that in this case, a similar split-explicit time stepping should be used on 383 378 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}).379 an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \S\ref{TRA_adv_tvd}). 385 380 386 381 … … 388 383 % Coriolis and Advection : flux form 389 384 % ================================================================ 390 \section{Coriolis and Advection: flux form}385 \section{Coriolis and advection: flux form} 391 386 \label{DYN_adv_cor_flux} 392 387 %------------------------------------------nam_dynadv---------------------------------------------------- … … 405 400 % Coriolis plus curvature metric terms 406 401 %-------------------------------------------------------------------------------------------------------------- 407 \subsection [Coriolis plus curvature metric terms (\textit{dynvor}) ] 408 {Coriolis plus curvature metric terms (\protect\mdl{dynvor}) } 402 \subsection{Coriolis plus curvature metric terms (\protect\mdl{dynvor}) } 409 403 \label{DYN_cor_flux} 410 404 … … 427 421 % Flux form Advection term 428 422 %-------------------------------------------------------------------------------------------------------------- 429 \subsection [Flux form Advection term (\textit{dynadv}) ] 430 {Flux form Advection term (\protect\mdl{dynadv}) } 423 \subsection{Flux form advection term (\protect\mdl{dynadv}) } 431 424 \label{DYN_adv_flux} 432 425 … … 451 444 difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS. 452 445 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}.446 selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. 454 447 In flux form, the schemes differ by the choice of a space and time interpolation to 455 448 define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells, … … 460 453 % 2nd order centred scheme 461 454 %------------------------------------------------------------- 462 \subsubsection{ $2^{nd}$ order centred scheme (cen2) (\protect\forcode{ln_dynadv_cen2= .true.})}455 \subsubsection{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{ = .true.})} 463 456 \label{DYN_adv_cen2} 464 457 … … 481 474 % UBS scheme 482 475 %------------------------------------------------------------- 483 \subsubsection{U pstream Biased Scheme (UBS) (\protect\forcode{ln_dynadv_ubs= .true.})}476 \subsubsection{UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{ = .true.})} 484 477 \label{DYN_adv_ubs} 485 478 … … 500 493 permitted. But the amplitudes of the false extrema are significantly reduced over 501 494 those in the centred second order method. As the scheme already includes 502 a diffusion component, it can be used without explicit 503 ($i.e.$ \np{ln _dynldf_lap}=\forcode{ln_dynldf_bilap= .false.}), and it is recommended to do so.495 a diffusion component, it can be used without explicit lateral diffusion on momentum 496 ($i.e.$ \np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}), and it is recommended to do so. 504 497 505 498 The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ … … 532 525 % Hydrostatic pressure gradient term 533 526 % ================================================================ 534 \section [Hydrostatic pressure gradient (\textit{dynhpg})] 535 {Hydrostatic pressure gradient (\protect\mdl{dynhpg})} 527 \section{Hydrostatic pressure gradient (\protect\mdl{dynhpg})} 536 528 \label{DYN_hpg} 537 529 %------------------------------------------nam_dynhpg--------------------------------------------------- … … 554 546 % z-coordinate with full step 555 547 %-------------------------------------------------------------------------------------------------------------- 556 \subsection [$z$-coordinate with full step (\protect\np{ln_dynhpg_zco}) ] 557 {$z$-coordinate with full step (\protect\forcode{ln_dynhpg_zco = .true.})} 548 \subsection{Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{ = .true.})} 558 549 \label{DYN_hpg_zco} 559 550 … … 595 586 % z-coordinate with partial step 596 587 %-------------------------------------------------------------------------------------------------------------- 597 \subsection [$z$-coordinate with partial step (\protect\np{ln_dynhpg_zps})] 598 {$z$-coordinate with partial step (\protect\forcode{ln_dynhpg_zps = .true.})} 588 \subsection{Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{ = .true.})} 599 589 \label{DYN_hpg_zps} 600 590 … … 616 606 % s- and s-z-coordinates 617 607 %-------------------------------------------------------------------------------------------------------------- 618 \subsection{$ s$- and $z$-$s$-coordinates}608 \subsection{$S$- and $Z$-$S$-coordinates} 619 609 \label{DYN_hpg_sco} 620 610 … … 624 614 cubic polynomial method is currently disabled whilst known bugs are under investigation. 625 615 626 $\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\ forcode{ln_dynhpg_sco= .true.})616 $\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .true.}) 627 617 \begin{equation} \label{Eq_dynhpg_sco} 628 618 \left\{ \begin{aligned} … … 639 629 ($e_{3w}$). 640 630 641 $\bullet$ 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.})631 $\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}\forcode{ = .true.}). 632 This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}\forcode{ = .true.}). 633 634 $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) 645 635 646 636 $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} 647 (\ forcode{ln_dynhpg_djc= .true.}) (currently disabled; under development)637 (\np{ln\_dynhpg\_djc}\forcode{ = .true.}) (currently disabled; under development) 648 638 649 639 Note that expression \eqref{Eq_dynhpg_sco} is commonly used when the variable volume formulation is 650 640 activated (\key{vvl}) because in that case, even with a flat bottom, the coordinate surfaces are not 651 641 horizontal but follow the free surface \citep{Levier2007}. The pressure jacobian scheme 652 (\ forcode{ln_dynhpg_prj = .true.}) is available as an improved option to \forcode{ln_dynhpg_sco= .true.} when642 (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) is available as an improved option to \np{ln\_dynhpg\_sco}\forcode{ = .true.} when 653 643 \key{vvl} is active. The pressure Jacobian scheme uses a constrained cubic spline to reconstruct 654 644 the density profile across the water column. This method maintains the monotonicity between the … … 660 650 \label{DYN_hpg_isf} 661 651 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.\\652 the pressure gradient due to the ocean load. If cavity opened (\np{ln\_isfcav}\forcode{ = .true.}) these 2 terms can be 653 calculated by setting \np{ln\_dynhpg\_isf}\forcode{ = .true.}. No other scheme are working with the ice shelf.\\ 664 654 665 655 $\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium. … … 673 663 % Time-scheme 674 664 %-------------------------------------------------------------------------------------------------------------- 675 \subsection [Time-scheme (\protect\np{ln_dynhpg_imp}) ] 676 {Time-scheme (\protect\np{ln_dynhpg_imp}= true/false)} 665 \subsection{Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{ = .true./.false.})} 677 666 \label{DYN_hpg_imp} 678 667 … … 689 678 time level $t$ only, as in the standard leapfrog scheme. 690 679 691 $\bullet$ leapfrog scheme (\ forcode{ln_dynhpg_imp= .true.}):680 $\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}): 692 681 693 682 \begin{equation} \label{Eq_dynhpg_lf} … … 696 685 \end{equation} 697 686 698 $\bullet$ semi-implicit scheme (\ forcode{ln_dynhpg_imp= .true.}):687 $\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}): 699 688 \begin{equation} \label{Eq_dynhpg_imp} 700 689 \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; … … 713 702 the stability limits associated with advection or diffusion. 714 703 715 In practice, the semi-implicit scheme is used when \ forcode{ln_dynhpg_imp= .true.}.704 In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}\forcode{ = .true.}. 716 705 In this case, we choose to apply the time filter to temperature and salinity used in 717 706 the equation of state, instead of applying it to the hydrostatic pressure or to the … … 727 716 Note that in the semi-implicit case, it is necessary to save the filtered density, an 728 717 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.718 reproducibility. This option is controlled by \np{nn\_dynhpg\_rst}, a namelist parameter. 730 719 731 720 % ================================================================ 732 721 % Surface Pressure Gradient 733 722 % ================================================================ 734 \section [Surface pressure gradient (\textit{dynspg}) ] 735 {Surface pressure gradient (\protect\mdl{dynspg})} 723 \section{Surface pressure gradient (\protect\mdl{dynspg})} 736 724 \label{DYN_spg} 737 725 %-----------------------------------------nam_dynspg---------------------------------------------------- … … 793 781 % Split-explict free surface formulation 794 782 %-------------------------------------------------------------------------------------------------------------- 795 \subsection{Split- Explicit free surface (\protect\key{dynspg\_ts})}783 \subsection{Split-explicit free surface (\protect\key{dynspg\_ts})} 796 784 \label{DYN_spg_ts} 797 785 %------------------------------------------namsplit----------------------------------------------------------- … … 806 794 variables (Fig.~\ref{Fig_DYN_dynspg_ts}). 807 795 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 (\ forcode{ln_bt_nn_auto= .true.})796 is provided through the \np{nn\_baro} namelist parameter as: 797 $\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}\forcode{ = .true.}) 810 798 considering that the stability of the barotropic system is essentially controled by external waves propagation. 811 799 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}.800 Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn\_bt\_cmax}. 813 801 814 802 %%% … … 839 827 The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged 840 828 transports to advect tracers. 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.}. }829 a) Forward time integration: \np{ln\_bt\_fw}\forcode{ = .true.}, \np{ln\_bt\_av}\forcode{ = .true.}. 830 b) Centred time integration: \np{ln\_bt\_fw}\forcode{ = .false.}, \np{ln\_bt\_av}\forcode{ = .true.}. 831 c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}\forcode{ = .true.}, \np{ln\_bt\_av}\forcode{ = .false.}. } 844 832 \end{center} \end{figure} 845 833 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > 846 834 847 In the default case (\ forcode{ln_bt_fw= .true.}), the external mode is integrated835 In the default case (\np{ln\_bt\_fw}\forcode{ = .true.}), the external mode is integrated 848 836 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 (\ forcode{ln_bt_av= .true.}). In that case, the integration is extended slightly beyond \textit{after} time step to provide time filtered quantities.837 quantities (\np{ln\_bt\_av}\forcode{ = .true.}). In that case, the integration is extended slightly beyond \textit{after} time step to provide time filtered quantities. 850 838 These are used for the subsequent initialization of the barotropic mode in the following baroclinic step. 851 839 Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme, 852 840 asselin filtering is not applied to barotropic quantities. \\ 853 841 Alternatively, one can choose to integrate barotropic equations starting 854 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 step842 from \textit{before} time step (\np{ln\_bt\_fw}\forcode{ = .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 855 843 become centred in the middle of the integration window. It can easily be shown that this property 856 844 removes part of splitting errors between modes, which increases the overall numerical robustness. … … 868 856 %%% 869 857 870 One can eventually choose to feedback instantaneous values by not using any time filter (\ forcode{ln_bt_av= .false.}).858 One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_av}\forcode{ = .false.}). 871 859 In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new 872 860 sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost) … … 1001 989 % Lateral diffusion term 1002 990 % ================================================================ 1003 \section [Lateral diffusion term (\textit{dynldf})] 1004 {Lateral diffusion term (\protect\mdl{dynldf})} 991 \section{Lateral diffusion term and operators (\protect\mdl{dynldf})} 1005 992 \label{DYN_ldf} 1006 993 %------------------------------------------nam_dynldf---------------------------------------------------- … … 1036 1023 1037 1024 % ================================================================ 1038 \subsection [Iso-level laplacian operator (\protect\np{ln_dynldf_lap})]1039 {Iso-level laplacian operator (\protect\forcode{ln_dynldf_lap= .true.})}1025 \subsection[Iso-level laplacian (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})] 1026 {Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})} 1040 1027 \label{DYN_ldf_lap} 1041 1028 … … 1060 1047 % Rotated laplacian operator 1061 1048 %-------------------------------------------------------------------------------------------------------------- 1062 \subsection [Rotated laplacian operator (\protect\np{ln_dynldf_iso})]1063 {Rotated laplacian operator (\protect\forcode{ln_dynldf_iso= .true.})}1049 \subsection[Rotated laplacian (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})] 1050 {Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})} 1064 1051 \label{DYN_ldf_iso} 1065 1052 1066 1053 A rotation of the lateral momentum diffusion operator is needed in several cases: 1067 for iso-neutral diffusion in the $z$-coordinate (\ forcode{ln_dynldf_iso= .true.}) and for1068 either iso-neutral (\ forcode{ln_dynldf_iso= .true.}) or geopotential1069 (\ forcode{ln_dynldf_hor= .true.}) diffusion in the $s$-coordinate. In the partial step1054 for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}\forcode{ = .true.}) and for 1055 either iso-neutral (\np{ln\_dynldf\_iso}\forcode{ = .true.}) or geopotential 1056 (\np{ln\_dynldf\_hor}\forcode{ = .true.}) diffusion in the $s$-coordinate. In the partial step 1070 1057 case, coordinates are horizontal except at the deepest level and no 1071 rotation is performed when \ forcode{ln_dynldf_hor= .true.}. The diffusion operator1058 rotation is performed when \np{ln\_dynldf\_hor}\forcode{ = .true.}. The diffusion operator 1072 1059 is defined simply as the divergence of down gradient momentum fluxes on each 1073 1060 momentum component. It must be emphasized that this formulation ignores … … 1129 1116 % Iso-level bilaplacian operator 1130 1117 %-------------------------------------------------------------------------------------------------------------- 1131 \subsection [Iso-level bilaplacian operator (\protect\np{ln_dynldf_bilap})]1132 {Iso-level bilaplacian operator (\protect\forcode{ln_dynldf_bilap= .true.})}1118 \subsection[Iso-level bilaplacian (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})] 1119 {Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})} 1133 1120 \label{DYN_ldf_bilap} 1134 1121 … … 1145 1132 % Vertical diffusion term 1146 1133 % ================================================================ 1147 \section [Vertical diffusion term (\protect\mdl{dynzdf})] 1148 {Vertical diffusion term (\protect\mdl{dynzdf})} 1134 \section{Vertical diffusion term (\protect\mdl{dynzdf})} 1149 1135 \label{DYN_zdf} 1150 1136 %----------------------------------------------namzdf------------------------------------------------------ … … 1157 1143 would be too restrictive a constraint on the time step. Two time stepping schemes 1158 1144 can be used for the vertical diffusion term : $(a)$ a forward time differencing 1159 scheme (\ forcode{ln_zdfexp= .true.}) using a time splitting technique1160 (\np{nn _zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme1161 (\ forcode{ln_zdfexp= .false.}) (see \S\ref{STP}). Note that namelist variables1162 \np{ln _zdfexp} and \np{nn_zdfexp} apply to both tracers and dynamics.1145 scheme (\np{ln\_zdfexp}\forcode{ = .true.}) using a time splitting technique 1146 (\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme 1147 (\np{ln\_zdfexp}\forcode{ = .false.}) (see \S\ref{STP}). Note that namelist variables 1148 \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 1163 1149 1164 1150 The formulation of the vertical subgrid scale physics is the same whatever … … 1199 1185 % External Forcing 1200 1186 % ================================================================ 1201 \section{External Forcings}1187 \section{External forcings} 1202 1188 \label{DYN_forcing} 1203 1189 … … 1206 1192 may enter the dynamical equations by affecting the surface pressure gradient. 1207 1193 1208 (1) When \np{ln _apr_dyn}~=~true(see \S\ref{SBC_apr}), the atmospheric pressure is taken1194 (1) When \np{ln\_apr\_dyn}\forcode{ = .true.} (see \S\ref{SBC_apr}), the atmospheric pressure is taken 1209 1195 into account when computing the surface pressure gradient. 1210 1196 1211 (2) When \np{ln _tide_pot}~=~true and \np{ln_tide}~=~true(see \S\ref{SBC_tide}),1197 (2) When \np{ln\_tide\_pot}\forcode{ = .true.} and \np{ln\_tide}\forcode{ = .true.} (see \S\ref{SBC_tide}), 1212 1198 the tidal potential is taken into account when computing the surface pressure gradient. 1213 1199 1214 (3) When \np{nn _ice_embd}~=~2and LIM or CICE is used ($i.e.$ when the sea-ice is embedded in the ocean),1200 (3) When \np{nn\_ice\_embd}\forcode{ = 2} and LIM or CICE is used ($i.e.$ when the sea-ice is embedded in the ocean), 1215 1201 the snow-ice mass is taken into account when computing the surface pressure gradient. 1216 1202 … … 1222 1208 % Time evolution term 1223 1209 % ================================================================ 1224 \section [Time evolution term (\textit{dynnxt})] 1225 {Time evolution term (\protect\mdl{dynnxt})} 1210 \section{Time evolution term (\protect\mdl{dynnxt})} 1226 1211 \label{DYN_nxt} 1227 1212 … … 1238 1223 weighted velocity (see \S\ref{Apdx_A_momentum}) 1239 1224 1240 $\bullet$ vector invariant form or linear free surface (\ forcode{ln_dynhpg_vec= .true.} ; \key{vvl} not defined):1225 $\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \key{vvl} not defined): 1241 1226 \begin{equation} \label{Eq_dynnxt_vec} 1242 1227 \left\{ \begin{aligned} … … 1246 1231 \end{equation} 1247 1232 1248 $\bullet$ flux form and nonlinear free surface (\ forcode{ln_dynhpg_vec= .false.} ; \key{vvl} defined):1233 $\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \key{vvl} defined): 1249 1234 \begin{equation} \label{Eq_dynnxt_flux} 1250 1235 \left\{ \begin{aligned} … … 1256 1241 where RHS is the right hand side of the momentum equation, the subscript $f$ 1257 1242 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} = $10^{-3}$.1243 initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp}\forcode{ = 10.e-3}. 1259 1244 In both cases, the modified Asselin filter is not applied since perfect conservation 1260 1245 is not an issue for the momentum equations.
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