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branches/UKMO/dev_r5518_GC3p0_package/DOC/TexFiles/Chapters/Chap_DYN.tex
r5120 r6440 1 1 % ================================================================ 2 % Chapter �Ocean Dynamics (DYN)2 % Chapter ——— Ocean Dynamics (DYN) 3 3 % ================================================================ 4 4 \chapter{Ocean Dynamics (DYN)} 5 5 \label{DYN} 6 6 \minitoc 7 8 % add a figure for dynvor ens, ene latices9 7 10 8 %\vspace{2.cm} … … 165 163 %------------------------------------------------------------------------------------------------------------- 166 164 167 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). 165 The vector invariant form of the momentum equations (\np{ln\_dynhpg\_vec}~=~true) is the one most 166 often used in applications of the \NEMO ocean model. The flux form option (\np{ln\_dynhpg\_vec}~=false) 167 (see next section) has been present since version $2$. 168 Options are defined through the \ngn{namdyn\_adv} namelist variables. 169 Coriolis and momentum advection terms are evaluated using a leapfrog scheme, 170 $i.e.$ the velocity appearing in these expressions is centred in time (\textit{now} velocity). 174 171 At the lateral boundaries either free slip, no slip or partial slip boundary 175 172 conditions are applied following Chap.\ref{LBC}. … … 303 300 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 304 301 305 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. 302 A key point in \eqref{Eq_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 303 It uses the sum of masked t-point vertical scale factor divided either 304 by the sum of the four t-point masks (\np{ln\_dynvor\_een\_old}~=~false), 305 or just by $4$ (\np{ln\_dynvor\_een\_old}~=~true). 306 The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ 307 tends to zero and extends by continuity the value of $e_{3f}$ into the land areas. 308 This case introduces a sub-grid-scale topography at f-points (with a systematic reduction of $e_{3f}$ 309 when a model level intercept the bathymetry) that tends to reinforce the topostrophy of the flow 310 ($i.e.$ the tendency of the flow to follow the isobaths) \citep{Penduff_al_OS07}. 311 311 312 312 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as … … 374 374 \end{aligned} \right. 375 375 \end{equation} 376 When \np{ln\_dynzad\_zts}~=~\textit{true}, a split-explicit time stepping with 5 sub-timesteps is used 377 on the vertical advection term. 378 This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. 379 Note that in this case, a similar split-explicit time stepping should be used on 380 vertical advection of tracer to ensure a better stability, 381 an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \S\ref{TRA_adv_tvd}). 382 376 383 377 384 % ================================================================ … … 491 498 those in the centred second order method. As the scheme already includes 492 499 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. 500 ($i.e.$ setting both \np{ln\_dynldf\_lap} and \np{ln\_dynldf\_bilap} to \textit{false}), 501 and it is recommended to do so. 494 502 495 503 The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ … … 629 637 ($e_{3w}$). 630 638 631 $\bullet$ 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).633 634 639 $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}=true) 635 640 … … 646 651 pressure Jacobian method is used to solve the horizontal pressure gradient. This method can provide 647 652 a more accurate calculation of the horizontal pressure gradient than the standard scheme. 653 654 \subsection{Ice shelf cavity} 655 \label{DYN_hpg_isf} 656 Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and 657 the pressure gradient due to the ocean load. If cavities are present (\np{ln\_isfcav}~=~true) these two terms can be 658 calculated by setting \np{ln\_dynhpg\_isf}~=~true. No other scheme is working with ice shelves.\\ 659 660 $\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in isostatic equilibrium. 661 The top pressure is computed integrating a reference density profile (prescribed as density of a water at 34.4 662 PSU and -1.9$\degres C$) from the sea surface to the ice shelf base, which corresponds to the load of the water 663 column in which the ice shelf is floatting. This top pressure is constant over time. A detailed description of 664 this method is described in \citet{Losch2008}.\\ 665 666 $\bullet$ The ocean load is computed using the expression \eqref{Eq_dynhpg_sco} described in \ref{DYN_hpg_sco}. 667 A treatment of the top and bottom partial cells similar to the one described in \ref{DYN_hpg_zps} is done 668 to reduce the residual circulation generated by the top partial cell. 648 669 649 670 %-------------------------------------------------------------------------------------------------------------- … … 718 739 $\ $\newline %force an empty line 719 740 720 %%%721 741 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 %%% 742 The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). 743 The main distinction is between the fixed volume case (linear free surface) and the variable volume case 744 (nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\S\ref{PE_free_surface}) 745 the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case 746 (\S\ref{PE_free_surface}). 747 With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 748 which imposes a very small time step when an explicit time stepping is used. 749 Two methods are proposed to allow a longer time step for the three-dimensional equations: 750 the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), 751 and the split-explicit free surface described below. 752 The extra term introduced in the filtered method is calculated implicitly, 753 so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 725 754 726 755 … … 736 765 implicitly, so that a solver is used to compute it. As a consequence the update of the $next$ 737 766 velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 738 739 767 740 768 … … 779 807 $\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}=true) 780 808 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. 809 Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry. 810 Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn\_bt\_cmax}. 782 811 783 812 %%% … … 802 831 Schematic of the split-explicit time stepping scheme for the external 803 832 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 $nn\_baro=5$.833 a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$. 805 834 Internal mode time steps (which are also the model time steps) are denoted 806 835 by $t-\rdt$, $t$ and $t+\rdt$. Variables with $k$ superscript refer to instantaneous barotropic variables, … … 808 837 The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged 809 838 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. }839 a) Forward time integration: \np{ln\_bt\_fw}=true, \np{ln\_bt\_av}=true. 840 b) Centred time integration: \np{ln\_bt\_fw}=false, \np{ln\_bt\_av}=true. 841 c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}=true, \np{ln\_bt\_av}=false. } 813 842 \end{center} \end{figure} 814 843 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > … … 816 845 In the default case (\np{ln\_bt\_fw}=true), the external mode is integrated 817 846 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.847 quantities (\np{ln\_bt\_av}=true). In that case, the integration is extended slightly beyond \textit{after} time step to provide time filtered quantities. 819 848 These are used for the subsequent initialization of the barotropic mode in the following baroclinic step. 820 849 Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme, … … 837 866 %%% 838 867 839 One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_av e}=false).868 One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_av}=false). 840 869 In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new 841 870 sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost) … … 1158 1187 1159 1188 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 } 1189 introduced as boundary conditions on the vertical mixing, three other forcings 1190 may enter the dynamical equations by affecting the surface pressure gradient. 1191 1192 (1) When \np{ln\_apr\_dyn}~=~true (see \S\ref{SBC_apr}), the atmospheric pressure is taken 1193 into account when computing the surface pressure gradient. 1194 1195 (2) When \np{ln\_tide\_pot}~=~true and \key{tide} is defined (see \S\ref{SBC_tide}), 1196 the tidal potential is taken into account when computing the surface pressure gradient. 1197 1198 (3) When \np{nn\_ice\_embd}~=~2 and LIM or CICE is used ($i.e.$ when the sea-ice is embedded in the ocean), 1199 the snow-ice mass is taken into account when computing the surface pressure gradient. 1200 1201 1202 \gmcomment{ missing : the lateral boundary condition !!! another external forcing 1203 } 1168 1204 1169 1205 % ================================================================
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