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branches/2016/dev_INGV_UKMO_2016/DOC/TexFiles/Chapters/Chap_DYN.tex
r5120 r7351 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} … … 296 296 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 297 297 \begin{figure}[!ht] \begin{center} 298 \includegraphics[width=0.70\textwidth]{ ./TexFiles/Figures/Fig_DYN_een_triad.pdf}298 \includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad} 299 299 \caption{ \label{Fig_DYN_een_triad} 300 300 Triads used in the energy and enstrophy conserving scheme (een) for … … 303 303 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 304 304 305 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. 305 A key point in \eqref{Eq_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 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). 309 The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ 310 tends to zero and extends by continuity the value of $e_{3f}$ into the land areas. 311 This case introduces a sub-grid-scale topography at f-points (with a systematic reduction of $e_{3f}$ 312 when a model level intercept the bathymetry) that tends to reinforce the topostrophy of the flow 313 ($i.e.$ the tendency of the flow to follow the isobaths) \citep{Penduff_al_OS07}. 311 314 312 315 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as … … 374 377 \end{aligned} \right. 375 378 \end{equation} 379 When \np{ln\_dynzad\_zts}~=~\textit{true}, a split-explicit time stepping with 5 sub-timesteps is used 380 on the vertical advection term. 381 This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. 382 Note that in this case, a similar split-explicit time stepping should be used on 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}). 385 376 386 377 387 % ================================================================ … … 647 657 a more accurate calculation of the horizontal pressure gradient than the standard scheme. 648 658 659 \subsection{Ice shelf cavity} 660 \label{DYN_hpg_isf} 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 be 663 calculated by setting \np{ln\_dynhpg\_isf}~=~true. No other scheme are working with the ice shelf.\\ 664 665 $\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium. 666 The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile 667 (prescribed as density of a water at 34.4 PSU and -1.9\degC) and corresponds to the water replaced by the ice shelf. 668 This top pressure is constant over time. A detailed description of this method is described in \citet{Losch2008}.\\ 669 670 $\bullet$ The ocean load is computed using the expression \eqref{Eq_dynhpg_sco} described in \ref{DYN_hpg_sco}. 671 649 672 %-------------------------------------------------------------------------------------------------------------- 650 673 % Time-scheme … … 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\_ave}=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) … … 989 1020 At the lateral boundaries either free slip, no slip or partial slip boundary 990 1021 conditions are applied according to the user's choice (see Chap.\ref{LBC}). 1022 1023 \gmcomment{ 1024 Hyperviscous operators are frequently used in the simulation of turbulent flows to control 1025 the dissipation of unresolved small scale features. 1026 Their primary role is to provide strong dissipation at the smallest scale supported by the grid 1027 while minimizing the impact on the larger scale features. 1028 Hyperviscous operators are thus designed to be more scale selective than the traditional, 1029 physically motivated Laplace operator. 1030 In finite difference methods, the biharmonic operator is frequently the method of choice to achieve 1031 this scale selective dissipation since its damping time ($i.e.$ its spin down time) 1032 scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$ 1033 (so that short waves damped more rapidelly than long ones), 1034 whereas the Laplace operator damping time scales only like $\lambda^{-2}$. 1035 } 991 1036 992 1037 % ================================================================ … … 1158 1203 1159 1204 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 } 1205 introduced as boundary conditions on the vertical mixing, three other forcings 1206 may enter the dynamical equations by affecting the surface pressure gradient. 1207 1208 (1) When \np{ln\_apr\_dyn}~=~true (see \S\ref{SBC_apr}), the atmospheric pressure is taken 1209 into account when computing the surface pressure gradient. 1210 1211 (2) When \np{ln\_tide\_pot}~=~true and \key{tide} is defined (see \S\ref{SBC_tide}), 1212 the tidal potential is taken into account when computing the surface pressure gradient. 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), 1215 the snow-ice mass is taken into account when computing the surface pressure gradient. 1216 1217 1218 \gmcomment{ missing : the lateral boundary condition !!! another external forcing 1219 } 1168 1220 1169 1221 % ================================================================ … … 1213 1265 1214 1266 % ================================================================ 1215 % Neptune effect 1216 % ================================================================ 1217 \section [Neptune effect (\textit{dynnept})] 1218 {Neptune effect (\mdl{dynnept})} 1219 \label{DYN_nept} 1220 1221 The "Neptune effect" (thus named in \citep{HollowayOM86}) is a 1222 parameterisation of the potentially large effect of topographic form stress 1223 (caused by eddies) in driving the ocean circulation. Originally developed for 1224 low-resolution models, in which it was applied via a Laplacian (second-order) 1225 diffusion-like term in the momentum equation, it can also be applied in eddy 1226 permitting or resolving models, in which a more scale-selective bilaplacian 1227 (fourth-order) implementation is preferred. This mechanism has a 1228 significant effect on boundary currents (including undercurrents), and the 1229 upwelling of deep water near continental shelves. 1230 1231 The theoretical basis for the method can be found in 1232 \citep{HollowayJPO92}, including the explanation of why form stress is not 1233 necessarily a drag force, but may actually drive the flow. 1234 \citep{HollowayJPO94} demonstrate the effects of the parameterisation in 1235 the GFDL-MOM model, at a horizontal resolution of about 1.8 degrees. 1236 \citep{HollowayOM08} demonstrate the biharmonic version of the 1237 parameterisation in a global run of the POP model, with an average horizontal 1238 grid spacing of about 32km. 1239 1240 The NEMO implementation is a simplified form of that supplied by 1241 Greg Holloway, the testing of which was described in \citep{HollowayJGR09}. 1242 The major simplification is that a time invariant Neptune velocity 1243 field is assumed. This is computed only once, during start-up, and 1244 made available to the rest of the code via a module. Vertical 1245 diffusive terms are also ignored, and the model topography itself 1246 is used, rather than a separate topographic dataset as in 1247 \citep{HollowayOM08}. This implementation is only in the iso-level 1248 formulation, as is the case anyway for the bilaplacian operator. 1249 1250 The velocity field is derived from a transport stream function given by: 1251 1252 \begin{equation} \label{Eq_dynnept_sf} 1253 \psi = -fL^2H 1254 \end{equation} 1255 1256 where $L$ is a latitude-dependant length scale given by: 1257 1258 \begin{equation} \label{Eq_dynnept_ls} 1259 L = l_1 + (l_2 -l_1)\left ( {1 + \cos 2\phi \over 2 } \right ) 1260 \end{equation} 1261 1262 where $\phi$ is latitude and $l_1$ and $l_2$ are polar and equatorial length scales respectively. 1263 Neptune velocity components, $u^*$, $v^*$ are derived from the stremfunction as: 1264 1265 \begin{equation} \label{Eq_dynnept_vel} 1266 u^* = -{1\over H} {\partial \psi \over \partial y}\ \ \ ,\ \ \ v^* = {1\over H} {\partial \psi \over \partial x} 1267 \end{equation} 1268 1269 \smallskip 1270 %----------------------------------------------namdom---------------------------------------------------- 1271 \namdisplay{namdyn_nept} 1272 %-------------------------------------------------------------------------------------------------------- 1273 \smallskip 1274 1275 The Neptune effect is enabled when \np{ln\_neptsimp}=true (default=false). 1276 \np{ln\_smooth\_neptvel} controls whether a scale-selective smoothing is applied 1277 to the Neptune effect flow field (default=false) (this smoothing method is as 1278 used by Holloway). \np{rn\_tslse} and \np{rn\_tslsp} are the equatorial and 1279 polar values respectively of the length-scale parameter $L$ used in determining 1280 the Neptune stream function \eqref{Eq_dynnept_sf} and \eqref{Eq_dynnept_ls}. 1281 Values at intermediate latitudes are given by a cosine fit, mimicking the 1282 variation of the deformation radius with latitude. The default values of 12km 1283 and 3km are those given in \citep{HollowayJPO94}, appropriate for a coarse 1284 resolution model. The finer resolution study of \citep{HollowayOM08} increased 1285 the values of L by a factor of $\sqrt 2$ to 17km and 4.2km, thus doubling the 1286 stream function for a given topography. 1287 1288 The simple formulation for ($u^*$, $v^*$) can give unacceptably large velocities 1289 in shallow water, and \citep{HollowayOM08} add an offset to the depth in the 1290 denominator to control this problem. In this implementation we offer instead (at 1291 the suggestion of G. Madec) the option of ramping down the Neptune flow field to 1292 zero over a finite depth range. The switch \np{ln\_neptramp} activates this 1293 option (default=false), in which case velocities at depths greater than 1294 \np{rn\_htrmax} are unaltered, but ramp down linearly with depth to zero at a 1295 depth of \np{rn\_htrmin} (and shallower). 1296 1297 % ================================================================ 1267 \end{document}
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