Changeset 2223
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branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_DYN.tex
r2197 r2223 11 11 $\ $\newline %force an empty line 12 12 13 Using the representation described in Chap .\ref{DOM}, several semi-discrete13 Using the representation described in Chapter \ref{DOM}, several semi-discrete 14 14 space forms of the dynamical equations are available depending on the vertical 15 15 coordinate used and on the conservation properties of the vorticity term. In all 16 16 the equations presented here, the masking has been omitted for simplicity. 17 One must be aware that all the quantities are masked fields and that each time a 17 One must be aware that all the quantities are masked fields and that each time an 18 18 average or difference operator is used, the resulting field is multiplied by a mask. 19 19 … … 25 25 \end{equation*} 26 26 NXT stands for next, referring to the time-stepping. The first group of terms on 27 the rhs of the momentum equationscorresponds to the Coriolis and advection27 the rhs of the this equation corresponds to the Coriolis and advection 28 28 terms that are decomposed into a vorticity part (VOR), a kinetic energy part (KEG) 29 and, a vertical advection part (ZAD) in the vector invariant formulationor a Coriolis29 and, either a vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis 30 30 and advection part (COR+ADV) in the flux formulation. The terms following these 31 31 are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient, … … 39 39 40 40 In the present chapter we also describe the diagnostic equations used to compute 41 the horizontal divergence and curl of the velocities (\emph{divcur} module) as well42 asthe vertical velocity (\emph{wzvmod} module).41 the horizontal divergence, curl of the velocities (\emph{divcur} module) and 42 the vertical velocity (\emph{wzvmod} module). 43 43 44 44 The different options available to the user are managed by namelist variables. 45 For equation term \textit{ttt}, the logical namelist variables are \textit{ln\_dynttt\_xxx},45 For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx}, 46 46 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 47 47 If a CPP key is used for this term its name is \textbf{key\_ttt}. The corresponding … … 49 49 usually computed in the \textit{dyn\_ttt\_xxx} subroutine. 50 50 51 The user has the option of extracting each tendency term of both the rhs of the52 3D momentum equation (\key{trddyn} defined) for output, as described in53 Chap.\ref{MISC}. Furthermore, the tendency terms associated tothe 2D54 barotropic vorticity balance ( \key{trdvor} defined) can be derived on-linefrom the51 The user has the option of extracting and outputting each tendency term from the 52 3D momentum equations (\key{trddyn} defined), as described in 53 Chap.\ref{MISC}. Furthermore, the tendency terms associated with the 2D 54 barotropic vorticity balance (when \key{trdvor} is defined) can be derived from the 55 55 3D terms. 56 56 %%% … … 86 86 \end{equation} 87 87 88 Note that in the $z$-coordinate with full step ( \key{zco} is defined),89 $e_{3u}$=$e_{3v}$=$e_{3f}$ so that the ycancel in \eqref{Eq_divcur_div}.90 91 Note also that whereas the vorticity havethe same discrete expression in $z$-92 and $s$-coordinate , its physical meaning is not identical. $\zeta$ is a pseudo88 Note that in the $z$-coordinate with full step (when \key{zco} is defined), 89 $e_{3u}$=$e_{3v}$=$e_{3f}$ so that these metric terms cancel in \eqref{Eq_divcur_div}. 90 91 Note also that although the vorticity has the same discrete expression in $z$- 92 and $s$-coordinates, its physical meaning is not identical. $\zeta$ is a pseudo 93 93 vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along 94 geopotential surfaces, but are no more necessary defined at the same depth).94 geopotential surfaces, but are not necessarily defined at the same depth). 95 95 96 96 The vorticity and divergence at the \textit{before} step are used in the computation … … 121 121 where \textit{emp} is the surface freshwater budget (evaporation minus precipitation), 122 122 expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,000~Kg/m$^3$ 123 is the volumic massof pure water. If river runoff is expressed as a surface freshwater123 is the density of pure water. If river runoff is expressed as a surface freshwater 124 124 flux (see \S\ref{SBC}) then \textit{emp} can be written as the evaporation minus 125 125 precipitation, minus the river runoff. The sea-surface height is evaluated 126 using exactly the same time stepping as the tracer equation \eqref{Eq_tra_nxt}:126 using exactly the same time stepping scheme as the tracer equation \eqref{Eq_tra_nxt}: 127 127 a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing 128 128 in \eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity). 129 This is of paramount importance. Substituing $T$ by$1$ in the tracer equation and summing129 This is of paramount importance. Replacing $T$ by the number $1$ in the tracer equation and summing 130 130 over the water column must lead to the sea surface height equation otherwise tracer content 131 couldnot be conserved \ref{Griffies_al_MWR01, LeclairMadec2009}.131 will not be conserved \ref{Griffies_al_MWR01, LeclairMadec2009}. 132 132 133 133 The vertical velocity is computed by an upward integration of the horizontal 134 divergence from the bottom, takeninto account the change of the thickness of the levels :134 divergence starting at the bottom, taking into account the change of the thickness of the levels : 135 135 136 136 \begin{equation} \label{Eq_wzv} 137 137 \left\{ \begin{aligned} 138 138 &\left. w \right|_{3/2} \quad= 0 \\ 139 &\left. w \right|_{k+1/2} = \left. w \right|_{k +1/2} + e_{3t}\; \left. \chi \right|_k139 &\left. w \right|_{k+1/2} = \left. w \right|_{k-1/2} + e_{3t}\; \left. \chi \right|_k 140 140 - \frac{ e_{3t}^{t+1} - e_{3t}^{t-1} } {2 \rdt} 141 141 \end{aligned} \right. 142 142 \end{equation} 143 144 In case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$, 145 as changes in the the divergence of the barotropic transport is absorbed in the change 146 of the levels thickness.re-oriented downward co 147 In case of linear free surface, the time derivative in \eqref{Eq_wzv} cancel out. 143 \sgacomment{should e3t involve k in this equation?} 144 145 In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$, 146 as changes in the divergence of the barotropic transport are absorbed into the change 147 of the level thicknesses, re-orientated downward. 148 In the case of a linear free surface, the time derivative in \eqref{Eq_wzv} disappears. 148 149 The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity 149 150 is thus equal to the divergence of the barotropic transport ($i.e.$ the first term in the … … 151 152 152 153 Note also that whereas the vertical velocity has the same discrete 153 expression in $z$- and $s$-coordinate , its physical meaning is not the same:154 expression in $z$- and $s$-coordinates, its physical meaning is not the same: 154 155 in the second case, $w$ is the velocity normal to the $s$-surfaces. 155 Note also that the $k$-axis is re-orient ed downward in the \textsc{fortran} code compare156 Note also that the $k$-axis is re-orientated downwards in the \textsc{fortran} code compared 156 157 to the indexing used in the semi-discrete equations such as \eqref{Eq_wzv} 157 158 (see \S\ref{DOM_Num_Index_vertical}). … … 168 169 169 170 The vector invariant form of the momentum equations is the one most 170 often used in applications of \NEMO ocean model. The flux form option171 (see next section) has been introducedsince version $2$.171 often used in applications of the \NEMO ocean model. The flux form option 172 (see next section) has been present since version $2$. 172 173 Coriolis and momentum advection terms are evaluated using a leapfrog 173 174 scheme, $i.e.$ the velocity appearing in these expressions is centred in … … 194 195 The vorticity terms are given below for the general case, but note that in the full step 195 196 $z$-coordinate (\key{zco} is defined), $e_{3u}$=$e_{3v}$=$e_{3f}$ so that the vertical scale 196 factors disappear. The yare all computed in dedicated routines that can be found in197 factors disappear. The vorticity terms are all computed in dedicated routines that can be found in 197 198 the \mdl{dynvor} module. 198 199 … … 241 242 \label{DYN_vor_mix} 242 243 243 The mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the244 For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the 244 245 two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens}) 245 tothe relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied246 for the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied 246 247 to the planetary vorticity term. 247 248 \begin{equation} \label{Eq_dynvor_mix} … … 267 268 averages of the velocity allows for the presence of grid point oscillation structures 268 269 that will be invisible to the operator. These structures are \textit{computational modes} 269 that will beat least partly damped by the momentum diffusive operator ($i.e.$ the 270 subgrid-scale advection), but not by the resolved advection term. These two schemes 271 therefore do not contribute to dump grid point noise in the horizontal velocity field, 272 which results in more noise in vertical velocity field, an undesired feature. This is a well-known 273 characteristics of $C$-grid discretization where $u$ and $v$ are located at different grid point, 274 a price to pay to avoid a double averaging on the pressure gradient term as in $B$-grid. 275 To circumvent this, Adcroft (ADD REF HERE) 276 we have proposed to introduce a second velocity ... blahblah.... 277 278 Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves.... 279 280 A very nice solution to that problem was proposed by \citet{Arakawa_Hsu_MWR90}. The idea is 270 that will be at least partly damped by the momentum diffusion operator ($i.e.$ the 271 subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes 272 therefore do not contribute to any grid point noise in the horizontal velocity field. 273 Such noise would result in more noise in the vertical velocity field, an undesirable feature. This is a well-known 274 characteristic of $C$-grid discretization where $u$ and $v$ are located at different grid points, 275 a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid. 276 \gmcomment{ To circumvent this, Adcroft (ADD REF HERE) 277 278 Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....} 279 280 A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}. The idea is 281 281 to get rid of the double averaging by considering triad combinations of vorticity. 282 282 It is noteworthy that this solution is conceptually quite similar to the one proposed by 283 \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusi veoperator.283 \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator. 284 284 285 285 The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified 286 286 for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme. 287 Let first providesthe discrete expression of the potential vorticity, $q$, defined at an $f$-point:287 First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: 288 288 \begin{equation} \label{Eq_pot_vor} 289 289 q = \frac{\zeta +f} {e_{3f} } … … 305 305 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 306 306 307 Note that a key point in \eqref{Eq_een_e3f} is that the averaging in \textbf{i}- and307 Note that a key point in \eqref{Eq_een_e3f} is that the averaging in the \textbf{i}- and 308 308 \textbf{j}- directions uses the masked vertical scale factor but is always divided by 309 $4$, not by the sum of the mask at $T$-point. This preserves the continuity of309 $4$, not by the sum of the masks at the four $T$-points. This preserves the continuity of 310 310 $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and 311 extends by continuity the value of $e_{3f}$ in the land areas. This feature is essential for312 $z$-coordinate with partial step.313 314 315 Then, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at$T$-point as316 the following triad combinations of the neighbouring potential vorticities defined at f-point 311 extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for 312 the $z$-coordinate with partial steps. 313 314 315 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as 316 the following triad combinations of the neighbouring potential vorticities defined at f-points 317 317 (Fig.~\ref{Fig_DYN_een_triad}): 318 318 \begin{equation} \label{Q_triads} … … 320 320 = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) 321 321 \end{equation} 322 where the indices $i_p$ and $k_p$ take the following value: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.323 324 The vorticity terms are represented as:322 where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$. 323 324 Finally, the vorticity terms are represented as: 325 325 \begin{equation} \label{Eq_dynvor_een} 326 326 \left\{ { … … 334 334 \end{equation} 335 335 336 This EEN scheme in fact combines the conservation properties of ENS and ENE schemes.336 This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes. 337 337 It conserves both total energy and potential enstrophy in the limit of horizontally 338 338 nondivergent flow ($i.e.$ $\chi$=$0$) (see Appendix~\ref{Apdx_C_vor_zad}). 339 Applied to a realistic ocean configuration, it has been shown that it s larger mantis339 Applied to a realistic ocean configuration, it has been shown that it 340 340 leads to a significant reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}. 341 Furthermore, used in combination with partial steprepresentation of bottom topography,341 Furthermore, used in combination with a partial steps representation of bottom topography, 342 342 it improves the interaction between current and topography, leading to a larger 343 343 topostrophy of the flow \citep{Barnier_al_OD06, Penduff_al_OS07}. … … 350 350 \label{DYN_keg} 351 351 352 As demonst arted in Appendix~\ref{Apdx_C}, there is a single discrete formulation352 As demonstrated in Appendix~\ref{Apdx_C}, there is a single discrete formulation 353 353 of the kinetic energy gradient term that, together with the formulation chosen for 354 354 the vertical advection (see below), conserves the total kinetic energy: … … 492 492 permitted. But the amplitudes of the false extrema are significantly reduced over 493 493 those in the centred second order method. As the scheme already includes 494 a diffusi ve component, it can be used without explicit lateral diffusion on moment494 a diffusion component, it can be used without explicit lateral diffusion on momentum 495 495 ($i.e.$ \np{ln\_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so. 496 496 … … 503 503 For stability reasons, the first term in (\ref{Eq_dynadv_ubs}), which corresponds 504 504 to a second order centred scheme, is evaluated using the \textit{now} velocity 505 (centred in time), while the second term, which is the diffusi vepart of the scheme,505 (centred in time), while the second term, which is the diffusion part of the scheme, 506 506 is evaluated using the \textit{before} velocity (forward in time). This is discussed 507 507 by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. 508 508 509 Note that the UBS and Q uadratic Upstream Interpolation for Convective Kinematics510 (QUICK) schemes only differ by one coefficient. Substituting $1/6$ with$1/8$ in509 Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) 510 schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in 511 511 (\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. 512 512 This option is not available through a namelist parameter, since the $1/6$ coefficient 513 is hard coded. Nevertheless it is quite easy to make the substitution in 513 is hard coded. Nevertheless it is quite easy to make the substitution in the 514 514 \mdl{dynadv\_ubs} module and obtain a QUICK scheme. 515 515 … … 610 610 \label{DYN_hpg_sco} 611 611 612 Pressure gradient formulations in $s$-coordinate have been the subject of a vast613 literature($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}).612 Pressure gradient formulations in an $s$-coordinate have been the subject of a vast 613 number of papers ($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}). 614 614 A number of different pressure gradient options are coded, but they are not yet fully 615 615 documented or tested. … … 656 656 above is the \textit{now} density, computed from the \textit{now} temperature and 657 657 salinity. In some specific cases (usually high resolution simulations over an ocean 658 domain which includes weakly stratified regions) the physical phenomen umthat658 domain which includes weakly stratified regions) the physical phenomenon that 659 659 controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for 660 660 doubling the stability limit associated with IGWs can be used \citep{Brown_Campana_MWR78, 661 661 Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an 662 662 average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ ($i.e.$ 663 \textit{before}, \textit{now} and \textit{after} time-steps), rather than at central663 \textit{before}, \textit{now} and \textit{after} time-steps), rather than at the central 664 664 time level $t$ only, as in the standard leapfrog scheme. 665 665 … … 723 723 724 724 The form of the surface pressure gradient term depends on how the user wants to handle 725 the fast external gravity waves that are solution of the analytical equation (\S\ref{PE_hor_pg}).726 Three formulation are available, all controlled by a CPP key (ln\_dynspg\_xxx):727 a explicit formulation which requireda small time step ;725 the fast external gravity waves that are a solution of the analytical equation (\S\ref{PE_hor_pg}). 726 Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx): 727 an explicit formulation which requires a small time step ; 728 728 a filtered free surface formulation which allows a larger time step by adding a filtering 729 term in the momentum equation ;730 and a plit-explicit free surface formulation, described below, which also allows a larger time step.729 term into the momentum equation ; 730 and a split-explicit free surface formulation, described below, which also allows a larger time step. 731 731 732 732 The extra term introduced in the filtered method is calculated 733 implicitly, so that a solver is used to compute it and thatthe update of the $next$733 implicitly, so that a solver is used to compute it. As a consequence the update of the $next$ 734 734 velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 735 735 … … 743 743 744 744 In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step 745 is chosen to be small enough to describe the external gravity waves (typically a few tens of seconds).745 is chosen to be small enough to resolve the external gravity waves (typically a few tens of seconds). 746 746 The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time), 747 747 is thus simply given by : … … 755 755 Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined), the surface pressure 756 756 gradient is already included in the momentum tendency through the level thickness variation 757 when computingthe hydrostatic pressure gradient. Thus, nothing is done in the \mdl{dynspg\_exp} module.757 allowed in the computation of the hydrostatic pressure gradient. Thus, nothing is done in the \mdl{dynspg\_exp} module. 758 758 759 759 %-------------------------------------------------------------------------------------------------------------- … … 763 763 \label{DYN_spg_ts} 764 764 765 In the split-explicit free surface formulation, also called time-splitting formulation 766 (\key{dynspg\_ts} defined) 767 768 769 The split-explicit free surface formulation used in \NEMO follows the one 765 The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined), 766 also called the time-splitting formulation, follows the one 770 767 proposed by \citet{Griffies_Bk04}. The general idea is to solve the free surface 771 768 equation and the associated barotropic velocity equations with a smaller time … … 773 770 variables (Fig.\ref {Fig_DYN_dynspg_ts}). 774 771 The size of the small time step, $\Delta_e$ (the external mode or barotropic time step) 775 is provided through \np{nn\_baro} namelist parameter as:772 is provided through the \np{nn\_baro} namelist parameter as: 776 773 $\Delta_e = \Delta / nn\_baro$. 777 774 … … 804 801 805 802 The split-explicit formulation has a damping effect on external gravity waves, 806 which is weaker damping than for the filtered free surface but still significantas803 which is weaker damping than that for the filtered free surface but still significant, as 807 804 shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. 808 805 … … 917 914 918 915 Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth 919 is time-dependent and so is the matrix to be inverted. It is computed once for all the 920 ocean time steps. 916 is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps. 921 917 922 918 % ================================================================ … … 930 926 %------------------------------------------------------------------------------------------------------------- 931 927 932 The options available for lateral diffusion are for the choice oflaplacian928 The options available for lateral diffusion are to use either laplacian 933 929 (rotated or not) or biharmonic operators. The coefficients may be constant 934 930 or spatially variable; the description of the coefficients is found in the chapter … … 973 969 \label{DYN_ldf_iso} 974 970 975 A rotation of the lateral momentum diffusi veoperator is needed in several cases:976 for iso-neutral diffusion in $z$-coordinate (\np{ln\_dynldf\_iso}=true) and for971 A rotation of the lateral momentum diffusion operator is needed in several cases: 972 for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}=true) and for 977 973 either iso-neutral (\np{ln\_dynldf\_iso}=true) or geopotential 978 (\np{ln\_dynldf\_hor}=true) diffusion in $s$-coordinate. In the partial step979 case, coordinates are horizontal except edat the deepest level and no980 rotation is performed when \np{ln\_dynldf\_hor}=true. The diffusi veoperator974 (\np{ln\_dynldf\_hor}=true) diffusion in the $s$-coordinate. In the partial step 975 case, coordinates are horizontal except at the deepest level and no 976 rotation is performed when \np{ln\_dynldf\_hor}=true. The diffusion operator 981 977 is defined simply as the divergence of down gradient momentum fluxes on each 982 978 momentum component. It must be emphasized that this formulation ignores … … 1031 1027 \end{equation} 1032 1028 where $r_1$ and $r_2$ are the slopes between the surface along which the 1033 diffusi veoperator acts and the surface of computation ($z$- or $s$-surfaces).1029 diffusion operator acts and the surface of computation ($z$- or $s$-surfaces). 1034 1030 The way these slopes are evaluated is given in the lateral physics chapter 1035 1031 (Chap.\ref{LDF}). … … 1086 1082 depends on the vertical physics used (see \S\ref{ZDF}). 1087 1083 1088 The surface boundary condition on momentum is given bythe stress exerted by1084 The surface boundary condition on momentum is the stress exerted by 1089 1085 the wind. At the surface, the momentum fluxes are prescribed as the boundary 1090 1086 condition on the vertical turbulent momentum fluxes, … … 1114 1110 enter the dynamical equations. 1115 1111 1116 One is the effect of atmospheric pressure on the ocean dynamics (to be 1117 introduced later). 1118 1119 Another forcing term is the tidal potential, which will be introduced in the 1120 reference version soon. 1112 One is the effect of atmospheric pressure on the ocean dynamics. 1113 Another forcing term is the tidal potential. 1114 Both of which will be introduced into the reference version soon. 1121 1115 1122 1116 % ================================================================ … … 1133 1127 The general framework for dynamics time stepping is a leap-frog scheme, 1134 1128 $i.e.$ a three level centred time scheme associated with an Asselin time filter 1135 (cf. Chap.\ref{STP}). The scheme is applied to the velocity except when using1136 the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in variable1137 volume levelcase (\key{vvl} defined), where it has to be applied to the thickness1129 (cf. Chap.\ref{STP}). The scheme is applied to the velocity, except when using 1130 the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in the variable 1131 volume case (\key{vvl} defined), where it has to be applied to the thickness 1138 1132 weighted velocity (see \S\ref{Apdx_A_momentum}) 1139 1133 1140 $\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}=true ; not \key{vvl}):1134 $\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}=true ; \key{vvl} not defined): 1141 1135 \begin{equation} \label{Eq_dynnxt_vec} 1142 1136 \left\{ \begin{aligned} … … 1157 1151 denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is 1158 1152 initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = $10^{-3}$. 1159 In both cases, the modified Asselin filter is not applied since aperfect conservation1160 is not an issue for momentum equation.1161 1162 Note that with the filtered free surface, the update of the \textit{ next} velocities1163 is done in the \mdl{dynsp\_flt} module, and only the swap of arrays1153 In both cases, the modified Asselin filter is not applied since perfect conservation 1154 is not an issue for the momentum equations. 1155 1156 Note that with the filtered free surface, the update of the \textit{after} velocities 1157 is done in the \mdl{dynsp\_flt} module, and only array swapping 1164 1158 and Asselin filtering is done in \mdl{dynnxt}. 1165 1159
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