Changeset 11558 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex
r11543 r11558 23 23 These three aspects of the lateral diffusion are set through namelist parameters 24 24 (see the \nam{tra\_ldf} and \nam{dyn\_ldf} below). 25 Note that this chapter describes the standard implementation of iso-neutral tracer mixing. 25 Note that this chapter describes the standard implementation of iso-neutral tracer mixing. 26 26 Griffies's implementation, which is used if \np{ln\_traldf\_triad}\forcode{=.true.}, 27 27 is described in \autoref{apdx:TRIADS} … … 29 29 %-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- 30 30 31 \nlst{namtra_ldf}32 33 \nlst{namdyn_ldf}34 31 %-------------------------------------------------------------------------------------------------------------- 35 32 … … 45 42 {No lateral mixing (\protect\np{ln\_traldf\_OFF}, \protect\np{ln\_dynldf\_OFF})} 46 43 47 It is possible to run without explicit lateral diffusion on tracers (\protect\np{ln\_traldf\_OFF}\forcode{=.true.}) and/or 48 momentum (\protect\np{ln\_dynldf\_OFF}\forcode{=.true.}). The latter option is even recommended if using the 44 It is possible to run without explicit lateral diffusion on tracers (\protect\np{ln\_traldf\_OFF}\forcode{=.true.}) and/or 45 momentum (\protect\np{ln\_dynldf\_OFF}\forcode{=.true.}). The latter option is even recommended if using the 49 46 UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{=.true.}, 50 47 see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes. … … 52 49 \subsection[Laplacian mixing (\forcode{ln_traldf_lap}, \forcode{ln_dynldf_lap})] 53 50 {Laplacian mixing (\protect\np{ln\_traldf\_lap}, \protect\np{ln\_dynldf\_lap})} 54 Setting \protect\np{ln\_traldf\_lap}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_lap}\forcode{=.true.} enables 55 a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine 51 Setting \protect\np{ln\_traldf\_lap}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_lap}\forcode{=.true.} enables 52 a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine 56 53 Laplacian and Bilaplacian operators for the same variable. 57 54 58 55 \subsection[Bilaplacian mixing (\forcode{ln_traldf_blp}, \forcode{ln_dynldf_blp})] 59 56 {Bilaplacian mixing (\protect\np{ln\_traldf\_blp}, \protect\np{ln\_dynldf\_blp})} 60 Setting \protect\np{ln\_traldf\_blp}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_blp}\forcode{=.true.} enables 61 a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice. 57 Setting \protect\np{ln\_traldf\_blp}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_blp}\forcode{=.true.} enables 58 a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice. 62 59 We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. 63 60 … … 84 81 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), 85 82 while for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and 86 $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 83 $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 87 84 88 85 %gm% add here afigure of the slope in i-direction … … 94 91 Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when 95 92 the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform, 96 \ie\ a linear function of $z_T$, the depth of a $T$-point. 93 \ie\ a linear function of $z_T$, the depth of a $T$-point. 97 94 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 98 95 … … 113 110 \end{equation} 114 111 115 %gm% caution I'm not sure the simplification was a good idea! 112 %gm% caution I'm not sure the simplification was a good idea! 116 113 117 114 These slopes are computed once in \rou{ldf\_slp\_init} when \np{ln\_sco}\forcode{=.true.}, 118 and either \np{ln\_traldf\_hor}\forcode{=.true.} or \np{ln\_dynldf\_hor}\forcode{=.true.}. 115 and either \np{ln\_traldf\_hor}\forcode{=.true.} or \np{ln\_dynldf\_hor}\forcode{=.true.}. 119 116 120 117 \subsection{Slopes for tracer iso-neutral mixing} … … 145 142 146 143 %gm% rewrite this as the explanation is not very clear !!! 147 %In practice, \autoref{eq:LDF_slp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \autoref{eq:LDF_slp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth. 144 %In practice, \autoref{eq:LDF_slp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \autoref{eq:LDF_slp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth. 148 145 149 146 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq:LDF_slp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters). 150 147 151 %In the $z$-coordinate, the derivative of the \autoref{eq:LDF_slp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so the $in situ$ density can be used for its evaluation. 148 %In the $z$-coordinate, the derivative of the \autoref{eq:LDF_slp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so the $in situ$ density can be used for its evaluation. 152 149 153 150 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to … … 164 161 This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, 165 162 where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87} 166 (see \autoref{subsec:TRA_bn2}). 163 (see \autoref{subsec:TRA_bn2}). 167 164 168 165 \item[$z$-coordinate with partial step: ] … … 179 176 will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. 180 177 181 %gm% 178 %gm% 182 179 Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for 183 180 the constraint on iso-neutral fluxes. … … 240 237 241 238 Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, 242 contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property. 239 contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property. 243 240 244 241 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 245 242 \begin{figure}[!ht] 246 \begin{center} 247 \includegraphics[width=\textwidth]{Fig_LDF_ZDF1} 248 \caption { 249 \protect\label{fig:LDF_ZDF1} 250 averaging procedure for isopycnal slope computation. 251 } 252 \end{center} 243 \centering 244 \includegraphics[width=\textwidth]{Fig_LDF_ZDF1} 245 \caption{Averaging procedure for isopycnal slope computation} 246 \label{fig:LDF_ZDF1} 253 247 \end{figure} 254 248 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 255 249 256 %There are three additional questions about the slope calculation. 257 %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. 258 %Second, numerical stability issues also require a bound on slopes. 250 %There are three additional questions about the slope calculation. 251 %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. 252 %Second, numerical stability issues also require a bound on slopes. 259 253 %Third, the question of boundary condition specified on slopes... 260 254 … … 263 257 264 258 265 % In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04}, 266 % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly 267 % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the 259 % In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04}, 260 % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly 261 % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the 268 262 % surface motivates this flattening of isopycnals near the surface). 269 263 … … 277 271 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 278 272 \begin{figure}[!ht] 279 \begin{center} 280 \includegraphics[width=\textwidth]{Fig_eiv_slp} 281 \caption{ 282 \protect\label{fig:LDF_eiv_slp} 283 Vertical profile of the slope used for lateral mixing in the mixed layer: 284 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 285 which has to be adjusted at the surface boundary 286 \ie\ it must tend to zero at the surface since there is no mixing across the air-sea interface: 287 wall boundary condition). 288 Nevertheless, the profile between the surface zero value and the interior iso-neutral one is unknown, 289 and especially the value at the base of the mixed layer; 290 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and 291 imposing a maximum slope of 1/100; 292 \textit{(c)} profile of slope actually used in \NEMO: a linear decrease of the slope from 293 zero at the surface to its ocean interior value computed just below the mixed layer. 294 Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}. 295 } 296 \end{center} 273 \centering 274 \includegraphics[width=\textwidth]{Fig_eiv_slp} 275 \caption[Vertical profile of the slope used for lateral mixing in the mixed layer]{ 276 Vertical profile of the slope used for lateral mixing in the mixed layer: 277 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 278 which has to be adjusted at the surface boundary 279 \ie\ it must tend to zero at the surface since there is no mixing across the air-sea interface: 280 wall boundary condition). 281 Nevertheless, 282 the profile between the surface zero value and the interior iso-neutral one is unknown, 283 and especially the value at the base of the mixed layer; 284 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and 285 imposing a maximum slope of 1/100; 286 \textit{(c)} profile of slope actually used in \NEMO: 287 a linear decrease of the slope from zero at the surface to 288 its ocean interior value computed just below the mixed layer. 289 Note the huge change in the slope at the base of the mixed layer between 290 \textit{(b)} and \textit{(c)}.} 291 \label{fig:LDF_eiv_slp} 297 292 \end{figure} 298 293 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 326 321 (see \autoref{sec:LBC_coast}). 327 322 328 323 329 324 % ================================================================ 330 325 % Lateral Mixing Coefficients … … 334 329 \label{sec:LDF_coef} 335 330 336 The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. 331 The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. 337 332 The way the mixing coefficients are set in the reference version can be described as follows: 338 333 … … 340 335 { Mixing coefficients read from file (\protect\np{nn\_aht\_ijk\_t}\forcode{=-20, -30}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=-20, -30})} 341 336 342 Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model, 337 Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model, 343 338 the laplacian viscosity operator uses $A^l$~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and 344 decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}. 345 Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05. 339 decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}. 340 Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05. 346 341 The provided fields can either be 2d (\np{nn\_aht\_ijk\_t}\forcode{=-20}, \np{nn\_ahm\_ijk\_t}\forcode{=-20}) or 3d (\np{nn\_aht\_ijk\_t}\forcode{=-30}, \np{nn\_ahm\_ijk\_t}\forcode{=-30}). They must be given at U, V points for tracers and T, F points for momentum (see \autoref{tab:LDF_files}). 347 342 348 343 %-------------------------------------------------TABLE--------------------------------------------------- 349 344 \begin{table}[htb] 350 \begin{center} 351 \begin{tabular}{|l|l|l|l|} 352 \hline 353 Namelist parameter & Input filename & dimensions & variable names \\ \hline 354 \np{nn\_ahm\_ijk\_t}\forcode{=-20} & \forcode{eddy_viscosity_2D.nc } & $(i,j)$ & \forcode{ahmt_2d, ahmf_2d} \\ \hline 355 \np{nn\_aht\_ijk\_t}\forcode{=-20} & \forcode{eddy_diffusivity_2D.nc } & $(i,j)$ & \forcode{ahtu_2d, ahtv_2d} \\ \hline 356 \np{nn\_ahm\_ijk\_t}\forcode{=-30} & \forcode{eddy_viscosity_3D.nc } & $(i,j,k)$ & \forcode{ahmt_3d, ahmf_3d} \\ \hline 357 \np{nn\_aht\_ijk\_t}\forcode{=-30} & \forcode{eddy_diffusivity_3D.nc } & $(i,j,k)$ & \forcode{ahtu_3d, ahtv_3d} \\ \hline 358 \end{tabular} 359 \caption{ 360 \protect\label{tab:LDF_files} 361 Description of expected input files if mixing coefficients are read from NetCDF files. 362 } 363 \end{center} 345 \centering 346 \begin{tabular}{|l|l|l|l|} 347 \hline 348 Namelist parameter & Input filename & dimensions & variable names \\ \hline 349 \np{nn\_ahm\_ijk\_t}\forcode{=-20} & \forcode{eddy_viscosity_2D.nc } & $(i,j)$ & \forcode{ahmt_2d, ahmf_2d} \\ \hline 350 \np{nn\_aht\_ijk\_t}\forcode{=-20} & \forcode{eddy_diffusivity_2D.nc } & $(i,j)$ & \forcode{ahtu_2d, ahtv_2d} \\ \hline 351 \np{nn\_ahm\_ijk\_t}\forcode{=-30} & \forcode{eddy_viscosity_3D.nc } & $(i,j,k)$ & \forcode{ahmt_3d, ahmf_3d} \\ \hline 352 \np{nn\_aht\_ijk\_t}\forcode{=-30} & \forcode{eddy_diffusivity_3D.nc } & $(i,j,k)$ & \forcode{ahtu_3d, ahtv_3d} \\ \hline 353 \end{tabular} 354 \caption{Description of expected input files if mixing coefficients are read from NetCDF files} 355 \label{tab:LDF_files} 364 356 \end{table} 365 357 %-------------------------------------------------------------------------------------------------------------- … … 421 413 The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above, 422 414 \ie\ a hyperbolic tangent variation with depth associated with a grid size dependence of 423 the magnitude of the coefficient. 415 the magnitude of the coefficient. 424 416 425 417 \subsection[Velocity dependent mixing coefficients (\forcode{nn_aht_ijk_t=31}, \forcode{nn_ahm_ijk_t=31})] … … 433 425 \begin{aligned} 434 426 & \frac{1}{12} \lvert U \rvert e & \text{for laplacian operator } \\ 435 & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } 427 & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } 436 428 \end{aligned} 437 429 \right. … … 441 433 {Deformation rate dependent viscosities (\protect\np{nn\_ahm\_ijk\_t}\forcode{=32})} 442 434 443 This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a 435 This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a 444 436 characteristic time scale $T_{smag}$ the deformation rate and for the lengthscale $L_{smag}$ the maximum wavenumber possible on the horizontal grid, e.g.: 445 437 … … 459 451 \begin{aligned} 460 452 & C^2 T_{smag}^{-1} L_{smag}^2 & \text{for laplacian operator } \\ 461 & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4 & \text{for bilaplacian operator } 453 & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4 & \text{for bilaplacian operator } 462 454 \end{aligned} 463 455 \right. … … 469 461 \begin{aligned} 470 462 & C_{min} \frac{1}{2} \lvert U \rvert e < A_{smag} < C_{max} \frac{e^2}{ 8\rdt} & \text{for laplacian operator } \\ 471 & C_{min} \frac{1}{12} \lvert U \rvert e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt} & \text{for bilaplacian operator } 463 & C_{min} \frac{1}{12} \lvert U \rvert e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt} & \text{for bilaplacian operator } 472 464 \end{aligned} 473 465 \end{equation} … … 482 474 divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). 483 475 Although the eddy coefficient could be set to different values in these two terms, 484 this option is not currently available. 476 this option is not currently available. 485 477 486 478 (2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of … … 498 490 %--------------------------------------------namtra_eiv--------------------------------------------------- 499 491 500 \nlst{namtra_eiv} 492 \begin{listing} 493 \nlst{namtra_eiv} 494 \caption{\texttt{namtra\_eiv}} 495 \label{lst:namtra_eiv} 496 \end{listing} 501 497 502 498 %-------------------------------------------------------------------------------------------------------------- … … 530 526 and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates. 531 527 532 If isopycnal mixing is used in the standard way, \ie\ \np{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by: 528 If isopycnal mixing is used in the standard way, \ie\ \np{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by: 533 529 \begin{equation} 534 530 \label{eq:LDF_eiv} … … 539 535 \end{split} 540 536 \end{equation} 541 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn\_aei\_ijk\_t} \nam{tra\_eiv} namelist parameter. 537 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn\_aei\_ijk\_t} \nam{tra\_eiv} namelist parameter. 542 538 The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and 543 539 added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed. … … 547 543 previous releases of OPA \citep{madec.delecluse.ea_NPM98}. 548 544 This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of 549 paramount importance. 545 paramount importance. 550 546 551 547 At the surface, lateral and bottom boundaries, the eddy induced velocity, 552 and thus the advective eddy fluxes of heat and salt, are set to zero. 553 The value of the eddy induced mixing coefficient and its space variation is controlled in a similar way as for lateral mixing coefficient described in the preceding subsection (\np{nn\_aei\_ijk\_t}, \np{rn\_Ue}, \np{rn\_Le} namelist parameters). 548 and thus the advective eddy fluxes of heat and salt, are set to zero. 549 The value of the eddy induced mixing coefficient and its space variation is controlled in a similar way as for lateral mixing coefficient described in the preceding subsection (\np{nn\_aei\_ijk\_t}, \np{rn\_Ue}, \np{rn\_Le} namelist parameters). 554 550 \colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added} 555 551 … … 566 562 %--------------------------------------------namtra_eiv--------------------------------------------------- 567 563 568 \nlst{namtra_mle} 564 \begin{listing} 565 \nlst{namtra_mle} 566 \caption{\texttt{namtra\_mle}} 567 \label{lst:namtra_mle} 568 \end{listing} 569 569 570 570 %--------------------------------------------------------------------------------------------------------------
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