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branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_DYN.tex
r2282 r2285 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 th e this equation corresponds to the Coriolis and advection28 terms that are decomposed into a vorticity part (VOR), a kinetic energy part (KEG)29 and , eithera vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis27 the rhs of this equation corresponds to the Coriolis and advection 28 terms that are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) 29 and 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, … … 86 86 \end{equation} 87 87 88 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 $z$- 88 Note that although the vorticity has the same discrete expression in $z$- 92 89 and $s$-coordinates, its physical meaning is not identical. $\zeta$ is a pseudo 93 90 vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along … … 113 110 \begin{aligned} 114 111 \frac{\partial \eta }{\partial t} 115 &\equiv \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left (\delta _i \left[ {e_{2u}\,e_{3u}\;u} \right]116 +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right] \right )}112 &\equiv \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{ \delta _i \left[ {e_{2u}\,e_{3u}\;u} \right] 113 +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right] \right\} } 117 114 - \frac{\textit{emp}}{\rho _w } \\ 118 115 &\equiv \sum\limits_k {\chi \ e_{3t}} - \frac{\textit{emp}}{\rho _w } … … 120 117 \end{equation} 121 118 where \textit{emp} is the surface freshwater budget (evaporation minus precipitation), 122 expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,000~Kg/m$^3$ 123 is the density of pure water. If river runoff is expressed as a surface freshwater 124 flux (see \S\ref{SBC}) then \textit{emp} can be written as the evaporation minus 125 precipitation, minus the river runoff. The sea-surface height is evaluated 126 using exactly the same time stepping scheme as the tracer equation \eqref{Eq_tra_nxt}: 119 expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,035~Kg/m$^3$ 120 is the reference density of sea water (Boussinesq approximation). If river runoff is 121 expressed as a surface freshwater flux (see \S\ref{SBC}) then \textit{emp} can be 122 written as the evaporation minus precipitation, minus the river runoff. 123 The sea-surface height is evaluated using exactly the same time stepping scheme 124 as the tracer equation \eqref{Eq_tra_nxt}: 127 125 a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing 128 126 in \eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity). … … 133 131 The vertical velocity is computed by an upward integration of the horizontal 134 132 divergence starting at the bottom, taking into account the change of the thickness of the levels : 135 136 133 \begin{equation} \label{Eq_wzv} 137 134 \left\{ \begin{aligned} 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|_k140 - \frac{ e_{3t}^{t+1} - e_{3t}^{t-1} } {2 \rdt}135 &\left. w \right|_{k_b-1/2} \quad= 0 \qquad \text{where } k_b \text{ is the level just above the sea floor } \\ 136 &\left. w \right|_{k+1/2} = \left. w \right|_{k-1/2} + \left. e_{3t} \right|_{k}\; \left. \chi \right|_k 137 - \frac{1} {2 \rdt} \left( \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right) 141 138 \end{aligned} \right. 142 139 \end{equation} 143 \sgacomment{should e3t involve k in this equation?}144 140 145 141 In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$, 146 142 as changes in the divergence of the barotropic transport are absorbed into the change 147 143 of the level thicknesses, re-orientated downward. 144 \gmcomment{not sure of this... to be modified with the change in emp setting} 148 145 In the case of a linear free surface, the time derivative in \eqref{Eq_wzv} disappears. 149 146 The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity … … 193 190 term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent 194 191 flow and horizontal kinetic energy (ENE scheme) (see Appendix~\ref{Apdx_C_vor_zad}). 195 The vorticity terms are given below for the general case, but note that in the full step 196 $z$-coordinate (\key{zco} is defined), $e_{3u}$=$e_{3v}$=$e_{3f}$ so that the vertical scale 197 factors disappear. The vorticity terms are all computed in dedicated routines that can be found in 192 The vorticity terms are all computed in dedicated routines that can be found in 198 193 the \mdl{dynvor} module. 199 194 … … 270 265 that will be at least partly damped by the momentum diffusion operator ($i.e.$ the 271 266 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. 267 therefore do not contribute to dump any grid point noise in the horizontal velocity field. 268 Such noise would result in more noise in the vertical velocity field, an undesirable feature. 269 This is a well-known characteristic of $C$-grid discretization where $u$ and $v$ are located 270 at different grid points, a price worth paying to avoid a double averaging in the pressure 271 gradient term as in the $B$-grid. 276 272 \gmcomment{ To circumvent this, Adcroft (ADD REF HERE) 277 278 273 Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....} 279 274 280 A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}. The idea is281 to get rid of the double averaging by considering triad combinations of vorticity.275 A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}. 276 The idea is to get rid of the double averaging by considering triad combinations of vorticity. 282 277 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 diffusion operator .278 \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see App.\ref{Apdx_C}). 284 279 285 280 The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified … … 311 306 extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for 312 307 the $z$-coordinate with partial steps. 313 314 308 315 309 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as … … 337 331 It conserves both total energy and potential enstrophy in the limit of horizontally 338 332 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 340 leads to a significantreduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}.333 Applied to a realistic ocean configuration, it has been shown that it leads to a significant 334 reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}. 341 335 Furthermore, used in combination with a partial steps representation of bottom topography, 342 336 it improves the interaction between current and topography, leading to a larger … … 959 953 and curl of the vorticity) preserves symmetry and ensures a complete 960 954 separation between the vorticity and divergence parts of the momentum diffusion. 961 Note that in the full step $z$-coordinate (\key{zco} is defined), $e_{3u} =e_{3v} =e_{3f}$962 so that they cancel in the rotational part of \eqref{Eq_dynldf_lap}.963 955 964 956 %-------------------------------------------------------------------------------------------------------------- … … 1114 1106 Both of which will be introduced into the reference version soon. 1115 1107 1108 \gmcomment{atmospheric pressure is there!!!! include its description } 1109 1116 1110 % ================================================================ 1117 1111 % Time evolution term
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