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branches/2015/dev_r5003_MERCATOR6_CRS/DOC/TexFiles/Chapters/Chap_DYN.tex
r5602 r7260 1 % ================================================================ 2 % Chapter � Ocean Dynamics (DYN) 1 \documentclass[NEMO_book]{subfiles} 2 \begin{document} 3 % ================================================================ 4 % Chapter ——— Ocean Dynamics (DYN) 3 5 % ================================================================ 4 6 \chapter{Ocean Dynamics (DYN)} 5 7 \label{DYN} 6 8 \minitoc 7 8 % add a figure for dynvor ens, ene latices9 9 10 10 %\vspace{2.cm} … … 165 165 %------------------------------------------------------------------------------------------------------------- 166 166 167 The vector invariant form of the momentum equations is the one most 168 often used in applications of the \NEMO ocean model. The flux form option 169 (see next section) has been present since version $2$. Options are defined 170 through the \ngn{namdyn\_adv} namelist variables 171 Coriolis and momentum advection terms are evaluated using a leapfrog 172 scheme, $i.e.$ the velocity appearing in these expressions is centred in 173 time (\textit{now} velocity). 167 The vector invariant form of the momentum equations (\np{ln\_dynhpg\_vec}~=~true) is the one most 168 often used in applications of the \NEMO ocean model. The flux form option (\np{ln\_dynhpg\_vec}~=false) 169 (see next section) has been present since version $2$. 170 Options are defined through the \ngn{namdyn\_adv} namelist variables. 171 Coriolis and momentum advection terms are evaluated using a leapfrog scheme, 172 $i.e.$ the velocity appearing in these expressions is centred in time (\textit{now} velocity). 174 173 At the lateral boundaries either free slip, no slip or partial slip boundary 175 174 conditions are applied following Chap.\ref{LBC}. … … 296 295 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 297 296 \begin{figure}[!ht] \begin{center} 298 \includegraphics[width=0.70\textwidth]{ ./TexFiles/Figures/Fig_DYN_een_triad.pdf}297 \includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad} 299 298 \caption{ \label{Fig_DYN_een_triad} 300 299 Triads used in the energy and enstrophy conserving scheme (een) for … … 303 302 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 304 303 305 Note that a key point in \eqref{Eq_een_e3f} is that the averaging in the \textbf{i}- and 306 \textbf{j}- directions uses the masked vertical scale factor but is always divided by 307 $4$, not by the sum of the masks at the four $T$-points. This preserves the continuity of 308 $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and 309 extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for 310 the $z$-coordinate with partial steps. 304 A key point in \eqref{Eq_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 305 It uses the sum of masked t-point vertical scale factor divided either 306 by the sum of the four t-point masks (\np{ln\_dynvor\_een\_old}~=~false), 307 or just by $4$ (\np{ln\_dynvor\_een\_old}~=~true). 308 The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ 309 tends to zero and extends by continuity the value of $e_{3f}$ into the land areas. 310 This case introduces a sub-grid-scale topography at f-points (with a systematic reduction of $e_{3f}$ 311 when a model level intercept the bathymetry) that tends to reinforce the topostrophy of the flow 312 ($i.e.$ the tendency of the flow to follow the isobaths) \citep{Penduff_al_OS07}. 311 313 312 314 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as … … 374 376 \end{aligned} \right. 375 377 \end{equation} 378 When \np{ln\_dynzad\_zts}~=~\textit{true}, a split-explicit time stepping with 5 sub-timesteps is used 379 on the vertical advection term. 380 This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. 381 Note that in this case, a similar split-explicit time stepping should be used on 382 vertical advection of tracer to ensure a better stability, 383 an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \S\ref{TRA_adv_tvd}). 384 376 385 377 386 % ================================================================ … … 491 500 those in the centred second order method. As the scheme already includes 492 501 a diffusion component, it can be used without explicit lateral diffusion on momentum 493 ($i.e.$ \np{ln\_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so. 502 ($i.e.$ setting both \np{ln\_dynldf\_lap} and \np{ln\_dynldf\_bilap} to \textit{false}), 503 and it is recommended to do so. 494 504 495 505 The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ … … 629 639 ($e_{3w}$). 630 640 631 $\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}=true).632 This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}=true).633 634 641 $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}=true) 635 642 … … 646 653 pressure Jacobian method is used to solve the horizontal pressure gradient. This method can provide 647 654 a more accurate calculation of the horizontal pressure gradient than the standard scheme. 655 656 \subsection{Ice shelf cavity} 657 \label{DYN_hpg_isf} 658 Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and 659 the pressure gradient due to the ocean load. If cavities are present (\np{ln\_isfcav}~=~true) these two terms can be 660 calculated by setting \np{ln\_dynhpg\_isf}~=~true. No other scheme is working with ice shelves.\\ 661 662 $\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in isostatic equilibrium. 663 The top pressure is computed integrating a reference density profile (prescribed as density of a water at 34.4 664 PSU and -1.9$^{\circ}C$) from the sea surface to the ice shelf base, which corresponds to the load of the water 665 column in which the ice shelf is floatting. This top pressure is constant over time. A detailed description of 666 this method is described in \citet{Losch2008}.\\ 667 668 $\bullet$ The ocean load is computed using the expression \eqref{Eq_dynhpg_sco} described in \ref{DYN_hpg_sco}. 669 A treatment of the top and bottom partial cells similar to the one described in \ref{DYN_hpg_zps} is done 670 to reduce the residual circulation generated by the top partial cell. 648 671 649 672 %-------------------------------------------------------------------------------------------------------------- … … 718 741 $\ $\newline %force an empty line 719 742 720 %%%721 743 Options are defined through the \ngn{namdyn\_spg} namelist variables. 722 The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface) and the variable volume case (nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\S\ref{PE_free_surface}) the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case (\S\ref{PE_free_surface}). With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 723 724 %%% 744 The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). 745 The main distinction is between the fixed volume case (linear free surface) and the variable volume case 746 (nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\S\ref{PE_free_surface}) 747 the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case 748 (\S\ref{PE_free_surface}). 749 With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 750 which imposes a very small time step when an explicit time stepping is used. 751 Two methods are proposed to allow a longer time step for the three-dimensional equations: 752 the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), 753 and the split-explicit free surface described below. 754 The extra term introduced in the filtered method is calculated implicitly, 755 so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 725 756 726 757 … … 736 767 implicitly, so that a solver is used to compute it. As a consequence the update of the $next$ 737 768 velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 738 739 769 740 770 … … 779 809 $\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}=true) 780 810 considering that the stability of the barotropic system is essentially controled by external waves propagation. 781 Maximum allowed Courant number is in that case time independent, and easily computed online from the input bathymetry. 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}. 782 813 783 814 %%% … … 798 829 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > 799 830 \begin{figure}[!t] \begin{center} 800 \includegraphics[width=0.7\textwidth]{ ./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf}831 \includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts} 801 832 \caption{ \label{Fig_DYN_dynspg_ts} 802 833 Schematic of the split-explicit time stepping scheme for the external 803 834 and internal modes. Time increases to the right. In this particular exemple, 804 a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_f ilt=1$) and $nn\_baro=5$.835 a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$. 805 836 Internal mode time steps (which are also the model time steps) are denoted 806 837 by $t-\rdt$, $t$ and $t+\rdt$. Variables with $k$ superscript refer to instantaneous barotropic variables, … … 808 839 The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged 809 840 transports to advect tracers. 810 a) Forward time integration: \np{ln\_bt\_fw}=true, \np{ln\_bt\_av e}=true.811 b) Centred time integration: \np{ln\_bt\_fw}=false, \np{ln\_bt\_av e}=true.812 c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}=true, \np{ln\_bt\_av e}=false. }841 a) Forward time integration: \np{ln\_bt\_fw}=true, \np{ln\_bt\_av}=true. 842 b) Centred time integration: \np{ln\_bt\_fw}=false, \np{ln\_bt\_av}=true. 843 c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}=true, \np{ln\_bt\_av}=false. } 813 844 \end{center} \end{figure} 814 845 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > … … 816 847 In the default case (\np{ln\_bt\_fw}=true), the external mode is integrated 817 848 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 818 quantities (\np{ln\_bt\_av e}=true). In that case, the integration is extended slightly beyond \textit{after} time step to provide time filtered quantities.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. 819 850 These are used for the subsequent initialization of the barotropic mode in the following baroclinic step. 820 851 Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme, … … 837 868 %%% 838 869 839 One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_av e}=false).870 One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_av}=false). 840 871 In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new 841 872 sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost) … … 1158 1189 1159 1190 Besides the surface and bottom stresses (see the above section) which are 1160 introduced as boundary conditions on the vertical mixing, two other forcings 1161 enter the dynamical equations. 1162 1163 One is the effect of atmospheric pressure on the ocean dynamics. 1164 Another forcing term is the tidal potential. 1165 Both of which will be introduced into the reference version soon. 1166 1167 \gmcomment{atmospheric pressure is there!!!! include its description } 1191 introduced as boundary conditions on the vertical mixing, three other forcings 1192 may enter the dynamical equations by affecting the surface pressure gradient. 1193 1194 (1) When \np{ln\_apr\_dyn}~=~true (see \S\ref{SBC_apr}), the atmospheric pressure is taken 1195 into account when computing the surface pressure gradient. 1196 1197 (2) When \np{ln\_tide\_pot}~=~true and \key{tide} is defined (see \S\ref{SBC_tide}), 1198 the tidal potential is taken into account when computing the surface pressure gradient. 1199 1200 (3) When \np{nn\_ice\_embd}~=~2 and LIM or CICE is used ($i.e.$ when the sea-ice is embedded in the ocean), 1201 the snow-ice mass is taken into account when computing the surface pressure gradient. 1202 1203 1204 \gmcomment{ missing : the lateral boundary condition !!! another external forcing 1205 } 1168 1206 1169 1207 % ================================================================ … … 1296 1334 1297 1335 % ================================================================ 1336 \end{document}
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