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branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_LDF.tex
r1954 r2282 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 … … 18 21 and for tracers only, eddy induced advection on tracers). These three aspects 19 22 of the lateral diffusion are set through namelist parameters and CPP keys 20 (see the nam\_traldf and nam\_dynldfbelow).23 (see the \textit{nam\_traldf} and \textit{nam\_dynldf} below). 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 … … 29 32 % Lateral Mixing Coefficients 30 33 % ================================================================ 31 \section {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn)} } 34 \section [Lateral Mixing Coefficient (\textit{ldftra}, \textit{ldfdyn})] 35 {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn}) } 32 36 \label{LDF_coef} 33 37 … … 38 42 momentum. Six CPP keys control the space variation of eddy coefficients: 39 43 three for momentum and three for tracer. The three choices allow: 40 a space variation in the three space directions, in the horizontal plane, 41 or in the vertical only. The default option is a constant value over the whole 42 ocean on both momentum and tracers. 43 44 a space variation in the three space directions (\key{traldf\_c3d}, \key{dynldf\_c3d}), 45 in the horizontal plane (\key{traldf\_c2d}, \key{dynldf\_c2d}), 46 or in the vertical only (\key{traldf\_c1d}, \key{dynldf\_c1d}). 47 The default option is a constant value over the whole ocean on both momentum and tracers. 48 44 49 The number of additional arrays that have to be defined and the gridpoint 45 50 position at which they are defined depend on both the space variation chosen … … 60 65 When none of the \textbf{key\_ldfdyn\_...} and \textbf{key\_ldftra\_...} keys are 61 66 defined, a constant value is used over the whole ocean for momentum and 62 tracers, which is specified through the \np{ ahm0} and \np{aht0} namelist67 tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist 63 68 parameters. 64 69 … … 67 72 Indeed in all the other types of vertical coordinate, the depth is a 3D function 68 73 of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent 69 mixing coefficients will require 3D arrays , $i.e.$ \key{ldftra\_c3d} and \key{ldftra\_c3d}.70 In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced71 in which the surface value is \np{aht0} (\np{ahm0}), the bottom value is 1/4 of72 the surface value, and the transition takes place around z=300~m with a width73 of 300~m($i.e.$ both the depth and the width of the inflection point are set to 300~m).74 mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation 75 of the lateral mixing coefficient is introduced in which the surface value is 76 \np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value, 77 and the transition takes place around z=300~m with a width of 300~m 78 ($i.e.$ both the depth and the width of the inflection point are set to 300~m). 74 79 This profile is hard coded in file \hf{ldftra\_c1d}, but can be easily modified by users. 75 80 … … 81 86 \begin{aligned} 82 87 & \frac{\max(e_1,e_2)}{e_{max}} A_o^l & \text{for laplacian operator } \\ 83 & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l & \text{for bilaplacian operator }88 & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l & \text{for bilaplacian operator } 84 89 \end{aligned} \right. 85 \quad \text{comments}86 90 \end{equation} 87 91 where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked 88 ocean domain, and $A_o^l$ is the \np{ ahm0} (momentum) or \np{aht0} (tracer)92 ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) 89 93 namelist parameter. This variation is intended to reflect the lesser need for subgrid 90 94 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{Willebrand2001}. 92 %%% 93 \gmcomment { not only that! stability reasons: with non uniform grid size, it is common 94 to face a blow up of the model due to to large diffusive coefficient compare to the smallest 95 grid size... this is especially true for bilaplacian (to be added in the text!) } 95 the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}. 96 Note that such a grid scale dependance of mixing coefficients significantly increase 97 the range of stability of model configurations presenting large changes in grid pacing 98 such as global ocean models. Indeed, in such a case, a constant mixing coefficient 99 can lead to a blow up of the model due to large coefficient compare to the smallest 100 grid size (see \S\ref{STP_forward_imp}), especially when using a bilaplacian operator. 96 101 97 102 Other formulations can be introduced by the user for a given configuration. 98 103 For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian 99 viscosity operator uses \np{ ahm0}~=~$4.10^4 m^2/s$poleward of 20$^{\circ}$100 north and south and decreases linearly to \np{ aht0}~=~$2.10^3 m^2/s$101 at the equator \citep{Madec 1996, Delecluse_Madec_Bk00}. This modification104 viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ 105 north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s 106 at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification 102 107 can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. 103 108 Similar modified horizontal variations can be found with the Antarctic or Arctic … … 121 126 and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined. 122 127 128 $\ $\newline % force a new ligne 129 123 130 A space variation in the eddy coefficient appeals several remarks: 124 131 … … 135 142 (3) for isopycnal diffusion on momentum or tracers, an additional purely 136 143 horizontal background diffusion with uniform coefficient can be added by 137 setting a non zero value of \np{ ahmb0} or \np{ahtb0}, a background horizontal144 setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal 138 145 eddy viscosity or diffusivity coefficient (namelist parameters whose default 139 146 values are $0$). However, the technique used to compute the isopycnal … … 150 157 (6) it is possible to use both the laplacian and biharmonic operators concurrently. 151 158 152 (7) for testing purposes it is possible to run without lateral diffusion on momentum. 153 159 (7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap} = 160 \np{ln\_dynldf\_bilap} = false). This is recommended when using the UBS advection 161 scheme on momentum (\np{ln\_dynadv\_ubs} = true, see \ref{DYN_adv_ubs}) 162 and can be useful for testing purposes. 154 163 155 164 % ================================================================ … … 162 171 %%% 163 172 \gmcomment{ we should emphasize here that the implementation is a rather old one. 164 Better work can be achieved by using \citet{Griffies 1998, Griffies2004} iso-neutral scheme. }173 Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. } 165 174 166 175 A direction for lateral mixing has to be defined when the desired operator does … … 172 181 quantity to be diffused. For a tracer, this leads to the following four slopes : 173 182 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while 174 for momentum the slopes are $r_{1 T}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for175 $u$ and $r_{1f}$, $r_{1vw}$, $r_{2 T}$, $r_{2vw}$ for $v$.183 for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for 184 $u$ and $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 176 185 177 186 %gm% add here afigure of the slope in i-direction … … 190 199 \begin{aligned} 191 200 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} 192 \;\delta_{i+1/2}[z_ T]193 &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_ T]201 \;\delta_{i+1/2}[z_t] 202 &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] 194 203 \\ 195 204 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 196 \;\delta_{j+1/2} [z_ T]197 &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_ T]198 \\ 199 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_ T]}}^{\,i,\,k+1/2}205 \;\delta_{j+1/2} [z_t] 206 &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] 207 \\ 208 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2} 200 209 &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}] 201 210 \\ 202 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_ T]}}^{\,j,\,k+1/2}211 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2} 203 212 &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] 204 213 \\ … … 268 277 there is no specific treatment for iso-neutral mixing in the $s$-coordinate. 269 278 In other words, iso-neutral mixing will only be accurately represented with a 270 linear equation of state (\np{n eos}=1 or 2). In the case of a "true" equation279 linear equation of state (\np{nn\_eos}=1 or 2). In the case of a "true" equation 271 280 of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} 272 281 will include a pressure dependent part, leading to the wrong evaluation of … … 276 285 Note: The solution for $s$-coordinate passes trough the use of different 277 286 (and better) expression for the constraint on iso-neutral fluxes. Following 278 \citet{Griffies 2004}, instead of specifying directly that there is a zero neutral287 \citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral 279 288 diffusive flux of locally referenced potential density, we stay in the $T$-$S$ 280 289 plane and consider the balance between the neutral direction diffusive fluxes … … 328 337 a minimum background horizontal diffusion for numerical stability reasons. 329 338 To overcome this problem, several techniques have been proposed in which 330 the numerical schemes of the ocean model are modified \citep{Weaver 1997,331 Griffies 1998}. Here, another strategy has been chosen \citep{Lazar1997}:339 the numerical schemes of the ocean model are modified \citep{Weaver_Eby_JPO97, 340 Griffies_al_JPO98}. Here, another strategy has been chosen \citep{Lazar_PhD97}: 332 341 a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents 333 342 the development of grid point noise generated by the iso-neutral diffusion … … 341 350 342 351 Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, 343 contrary to the \citet{Griffies 1998} operator which has that property.352 contrary to the \citet{Griffies_al_JPO98} operator which has that property. 344 353 345 354 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 359 368 360 369 361 In addition and also for numerical stability reasons \citep{Cox1987, Griffies 2004},370 In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04}, 362 371 the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly 363 372 to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the 364 373 surface motivates this flattening of isopycnals near the surface). 365 374 366 For numerical stability reasons \citep{Cox1987, Griffies 2004}, the slopes must also375 For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also 367 376 be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear 368 377 fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter … … 402 411 \begin{equation} \label{Eq_ldfslp_dyn} 403 412 \begin{aligned} 404 &r_{1 T}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\405 &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2 T}\ &= \overline{r_{2v}}^{\,j} \\413 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ 414 &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2t}\ &= \overline{r_{2v}}^{\,j} \\ 406 415 &r_{1uw} = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ & r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\ 407 416 &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&& r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\ … … 437 446 \end{equation} 438 447 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set 439 through \np{ aeiv}, a \textit{nam\_traldf} namelist parameter.448 through \np{rn\_aeiv}, a \textit{nam\_traldf} namelist parameter. 440 449 The three components of the eddy induced velocity are computed and add 441 450 to the eulerian velocity in \mdl{traadv\_eiv}. 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