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branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_LDF.tex
r1224 r1831 7 7 \minitoc 8 8 9 10 \newpage 9 11 $\ $\newline % force a new ligne 12 10 13 11 14 The lateral physics terms in the momentum and tracer equations have been … … 21 24 22 25 %-----------------------------------nam_traldf - nam_dynldf-------------------------------------------- 23 \namdisplay{nam _traldf}24 \namdisplay{nam _dynldf}26 \namdisplay{namtra_ldf} 27 \namdisplay{namdyn_ldf} 25 28 %-------------------------------------------------------------------------------------------------------------- 26 29 … … 89 92 namelist parameter. This variation is intended to reflect the lesser need for subgrid 90 93 scale eddy mixing where the grid size is smaller in the domain. It was introduced in 91 the context of the DYNAMO modelling project \citep{Willebrand 2001}.94 the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}. 92 95 %%% 93 96 \gmcomment { not only that! stability reasons: with non uniform grid size, it is common … … 99 102 viscosity operator uses \np{ahm0}~=~$4.10^4 m^2/s$ poleward of 20$^{\circ}$ 100 103 north and south and decreases linearly to \np{aht0}~=~$2.10^3 m^2/s$ 101 at the equator \citep{Madec 1996, Delecluse_Madec_Bk00}. This modification104 at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification 102 105 can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. 103 106 Similar modified horizontal variations can be found with the Antarctic or Arctic … … 158 161 %%% 159 162 \gmcomment{ we should emphasize here that the implementation is a rather old one. 160 Better work can be achieved by using \citet{Griffies 1998, Griffies2004} iso-neutral scheme. }163 Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. } 161 164 162 165 A direction for lateral mixing has to be defined when the desired operator does … … 168 171 quantity to be diffused. For a tracer, this leads to the following four slopes : 169 172 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while 170 for momentum the slopes are $r_{1 T}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for171 $u$ and $r_{1f}$, $r_{1vw}$, $r_{2 T}$, $r_{2vw}$ for $v$.173 for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for 174 $u$ and $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 172 175 173 176 %gm% add here afigure of the slope in i-direction … … 186 189 \begin{aligned} 187 190 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} 188 \;\delta_{i+1/2}[z_ T]189 &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_ T]191 \;\delta_{i+1/2}[z_t] 192 &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] 190 193 \\ 191 194 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 192 \;\delta_{j+1/2} [z_ T]193 &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_ T]194 \\ 195 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_ T]}}^{\,i,\,k+1/2}195 \;\delta_{j+1/2} [z_t] 196 &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] 197 \\ 198 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2} 196 199 &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}] 197 200 \\ 198 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_ T]}}^{\,j,\,k+1/2}201 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2} 199 202 &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] 200 203 \\ … … 272 275 Note: The solution for $s$-coordinate passes trough the use of different 273 276 (and better) expression for the constraint on iso-neutral fluxes. Following 274 \citet{Griffies 2004}, instead of specifying directly that there is a zero neutral277 \citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral 275 278 diffusive flux of locally referenced potential density, we stay in the $T$-$S$ 276 279 plane and consider the balance between the neutral direction diffusive fluxes … … 324 327 a minimum background horizontal diffusion for numerical stability reasons. 325 328 To overcome this problem, several techniques have been proposed in which 326 the numerical schemes of the ocean model are modified \citep{Weaver 1997,327 Griffies 1998}. Here, another strategy has been chosen \citep{Lazar1997}:329 the numerical schemes of the ocean model are modified \citep{Weaver_Eby_JPO97, 330 Griffies_al_JPO98}. Here, another strategy has been chosen \citep{Lazar_PhD97}: 328 331 a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents 329 332 the development of grid point noise generated by the iso-neutral diffusion … … 337 340 338 341 Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, 339 contrary to the \citet{Griffies 1998} operator which has that property.342 contrary to the \citet{Griffies_al_JPO98} operator which has that property. 340 343 341 344 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 355 358 356 359 357 In addition and also for numerical stability reasons \citep{Cox1987, Griffies 2004},360 In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04}, 358 361 the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly 359 362 to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the 360 363 surface motivates this flattening of isopycnals near the surface). 361 364 362 For numerical stability reasons \citep{Cox1987, Griffies 2004}, the slopes must also365 For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also 363 366 be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear 364 367 fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter … … 398 401 \begin{equation} \label{Eq_ldfslp_dyn} 399 402 \begin{aligned} 400 &r_{1 T}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\401 &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2 T}\ &= \overline{r_{2v}}^{\,j} \\403 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ 404 &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2t}\ &= \overline{r_{2v}}^{\,j} \\ 402 405 &r_{1uw} = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ & r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\ 403 406 &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&& r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
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