Changeset 11336 for NEMO/trunk/doc/latex/NEMO
- Timestamp:
- 2019-07-24T12:30:54+02:00 (5 years ago)
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- NEMO/trunk/doc/latex/NEMO
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NEMO/trunk/doc/latex/NEMO/main/bibliography.bib
r11335 r11336 928 928 } 929 929 930 @article{ foxkemper.ferrari_JPO08, 931 title = "Parameterization of Mixed Layer Eddies. Part I: Theory and Diagnosis", 932 pages = "1145--1165", 933 journal = "Journal of Physical Oceanography", 934 volume = "38", 935 number = "6", 936 author = "B. Fox-Kemper and R. Ferrari and B. Hallberg", 937 year = "2008", 938 month = "jun", 939 publisher = "American Meteorological Society", 940 issn = "1520-0485", 941 doi = "10.1175/2007JPO3792.1" 942 } 943 930 944 @article{ galperin.kantha.ea_JAS88, 931 945 title = "A quasi-equilibrium turbulent energy model for geophysical … … 2639 2653 } 2640 2654 2655 @article{ smagorinsky_MW63, 2656 title = "General circulation experiments with the primitive equations: I. The basic experiment ", 2657 pages = "99--164", 2658 journal = "Monthly Weather Review", 2659 volume = "91", 2660 number = "3", 2661 author = "J. Smagorinsky", 2662 year = "1963", 2663 month = "mar", 2664 publisher = "American Meteorological Society", 2665 issn = "1520-0493", 2666 doi = "10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2" 2667 } 2668 2641 2669 @article{ song.haidvogel_JCP94, 2642 2670 title = "A semi-implicit ocean circulation model using a -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex
r11179 r11336 23 23 These three aspects of the lateral diffusion are set through namelist parameters 24 24 (see the \ngn{nam\_traldf} and \ngn{nam\_dynldf} below). 25 Note that this chapter describes the standard implementation of iso-neutral tracer mixing ,26 and Griffies's implementation, which is used if \np{traldf\_grif}\forcode{ = .true.},27 is described in Appdx\autoref{apdx:triad}28 29 %-----------------------------------nam _traldf - nam_dynldf--------------------------------------------25 Note that this chapter describes the standard implementation of iso-neutral tracer mixing. 26 Griffies's implementation, which is used if \np{ln\_traldf\_triad}\forcode{ = .true.}, 27 is described in \autoref{apdx:triad} 28 29 %-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- 30 30 31 31 \nlst{namtra_ldf} … … 34 34 %-------------------------------------------------------------------------------------------------------------- 35 35 36 % ================================================================ 37 % Lateral Mixing Operator 38 % ================================================================ 39 \section[Lateral mixing operators] 40 {Lateral mixing operators} 41 \label{sec:LDF_op} 42 We remind here the different lateral mixing operators that can be used. Further details can be found in \autoref{subsec:TRA_ldf_op} and \autoref{sec:DYN_ldf}. 43 44 \subsection[No lateral mixing (\forcode{ln_traldf_OFF}, \forcode{ln_dynldf_OFF})] 45 {No lateral mixing (\protect\np{ln\_traldf\_OFF}, \protect\np{ln\_dynldf\_OFF})} 46 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 49 UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.}, 50 see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes. 51 52 \subsection[Laplacian mixing (\forcode{ln_traldf_lap}, \forcode{ln_dynldf_lap})] 53 {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 56 Laplacian and Bilaplacian operators for the same variable. 57 58 \subsection[Bilaplacian mixing (\forcode{ln_traldf_blp}, \forcode{ln_dynldf_blp})] 59 {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. 62 We stress again that from \NEMO 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. 36 63 37 64 % ================================================================ … … 88 115 %gm% caution I'm not sure the simplification was a good idea! 89 116 90 These slopes are computed once in \rou{ldf slp\_init} when \np{ln\_sco}\forcode{ = .true.}rue,117 These slopes are computed once in \rou{ldf\_slp\_init} when \np{ln\_sco}\forcode{ = .true.}, 91 118 and either \np{ln\_traldf\_hor}\forcode{ = .true.} or \np{ln\_dynldf\_hor}\forcode{ = .true.}. 92 119 … … 145 172 \item[$s$- or hybrid $s$-$z$- coordinate: ] 146 173 in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 147 the Griffies scheme is used (\np{ traldf\_grif}\forcode{ = .true.};148 see Appdx\autoref{apdx:triad}).174 the Griffies scheme is used (\np{ln\_traldf\_triad}\forcode{ = .true.}; 175 see \autoref{apdx:triad}). 149 176 In other words, iso-neutral mixing will only be accurately represented with a linear equation of state 150 (\np{ nn\_eos}\forcode{ = 1..2}).177 (\np{ln\_seos}\forcode{ = .true.}). 151 178 In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso} 152 179 will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. … … 198 225 199 226 This implementation is a rather old one. 200 It is similar to the one proposed by Cox [1987], except for the background horizontal diffusion.201 Indeed, the Coximplementation of isopycnal diffusion in GFDL-type models requires227 It is similar to the one proposed by \citet{cox_OM87}, except for the background horizontal diffusion. 228 Indeed, the \citet{cox_OM87} implementation of isopycnal diffusion in GFDL-type models requires 202 229 a minimum background horizontal diffusion for numerical stability reasons. 203 230 To overcome this problem, several techniques have been proposed in which the numerical schemes of 204 231 the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. 205 Griffies's scheme is now available in \NEMO if \np{ traldf\_grif\_iso} is set true; see Appdx\autoref{apdx:triad}.232 Griffies's scheme is now available in \NEMO if \np{ln\_traldf\_triad}=\forcode{= .true.}; see \autoref{apdx:triad}. 206 233 Here, another strategy is presented \citep{lazar_phd97}: 207 234 a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of … … 242 269 243 270 For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by 244 $1/100$everywhere.271 the namelist scalar \np{rn\_slpmax} (usually $1/100$) everywhere. 245 272 This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to 246 273 $1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean 247 274 (the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface). 275 \colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.} 248 276 249 277 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 298 326 (see \autoref{sec:LBC_coast}). 299 327 300 301 % ================================================================302 % Lateral Mixing Operator303 % ================================================================304 \section[Lateral mixing operators (\textit{traldf.F90})]305 {Lateral mixing operators (\protect\mdl{traldf}, \protect\mdl{traldf})}306 \label{sec:LDF_op}307 308 309 328 310 329 % ================================================================ 311 330 % Lateral Mixing Coefficients 312 331 % ================================================================ 313 \section[Lateral mixing coefficient (\ textit{ldftra.F90}, \textit{ldfdyn.F90})]314 {Lateral mixing coefficient (\protect\ mdl{ldftra}, \protect\mdl{ldfdyn})}332 \section[Lateral mixing coefficient (\forcode{nn_aht_ijk_t}, \forcode{nn_ahm_ijk_t})] 333 {Lateral mixing coefficient (\protect\np{nn\_aht\_ijk\_t}, \protect\np{nn\_ahm\_ijk\_t})} 315 334 \label{sec:LDF_coef} 316 335 317 Introducing a space variation in the lateral eddy mixing coefficients changes the model core memory requirement, 318 adding up to four extra three-dimensional arrays for the geopotential or isopycnal second order operator applied to 319 momentum. 320 Six CPP keys control the space variation of eddy coefficients: three for momentum and three for tracer. 321 The three choices allow: 322 a space variation in the three space directions (\key{traldf\_c3d}, \key{dynldf\_c3d}), 323 in the horizontal plane (\key{traldf\_c2d}, \key{dynldf\_c2d}), 324 or in the vertical only (\key{traldf\_c1d}, \key{dynldf\_c1d}). 325 The default option is a constant value over the whole ocean on both momentum and tracers. 326 327 The number of additional arrays that have to be defined and the gridpoint position at which 328 they are defined depend on both the space variation chosen and the type of operator used. 329 The resulting eddy viscosity and diffusivity coefficients can be a function of more than one variable. 330 Changes in the computer code when switching from one option to another have been minimized by 331 introducing the eddy coefficients as statement functions 332 (include file \textit{ldftra\_substitute.h90} and \textit{ldfdyn\_substitute.h90}). 333 The functions are replaced by their actual meaning during the preprocessing step (CPP). 334 The specification of the space variation of the coefficient is made in \mdl{ldftra} and \mdl{ldfdyn}, 335 or more precisely in include files \textit{traldf\_cNd.h90} and \textit{dynldf\_cNd.h90}, with N=1, 2 or 3. 336 The user can modify these include files as he/she wishes. 337 The way the mixing coefficient are set in the reference version can be briefly described as follows: 338 339 \subsubsection{Constant mixing coefficients (default option)} 340 When none of the \key{dynldf\_...} and \key{traldf\_...} keys are defined, 341 a constant value is used over the whole ocean for momentum and tracers, 342 which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist parameters. 343 344 \subsubsection[Vertically varying mixing coefficients (\texttt{\textbf{key\_traldf\_c1d}} and \texttt{\textbf{key\_dynldf\_c1d}})] 345 {Vertically varying mixing coefficients (\protect\key{traldf\_c1d} and \key{dynldf\_c1d})} 346 The 1D option is only available when using the $z$-coordinate with full step. 347 Indeed in all the other types of vertical coordinate, 348 the depth is a 3D function of (\textbf{i},\textbf{j},\textbf{k}) and therefore, 349 introducing depth-dependent mixing coefficients will require 3D arrays. 350 In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced in which 351 the surface value is \np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value, 352 and the transition takes place around z=300~m with a width of 300~m 353 (\ie both the depth and the width of the inflection point are set to 300~m). 354 This profile is hard coded in file \textit{traldf\_c1d.h90}, but can be easily modified by users. 355 356 \subsubsection[Horizontally varying mixing coefficients (\texttt{\textbf{key\_traldf\_c2d}} and \texttt{\textbf{key\_dynldf\_c2d}})] 357 {Horizontally varying mixing coefficients (\protect\key{traldf\_c2d} and \protect\key{dynldf\_c2d})} 358 By default the horizontal variation of the eddy coefficient depends on the local mesh size and 336 The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. 337 The way the mixing coefficients are set in the reference version can be described as follows: 338 339 \subsection[Mixing coefficients read from file (\forcode{nn_aht_ijk_t = -20, -30}, \forcode{nn_ahm_ijk_t = -20,-30})] 340 { Mixing coefficients read from file (\protect\np{nn\_aht\_ijk\_t}\forcode{ = -20, -30}, \protect\np{nn\_ahm\_ijk\_t}\forcode{ = -20, -30})} 341 342 Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model, 343 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. 346 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}). 347 348 %-------------------------------------------------TABLE--------------------------------------------------- 349 \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} 364 \end{table} 365 %-------------------------------------------------------------------------------------------------------------- 366 367 \subsection[Constant mixing coefficients (\forcode{nn_aht_ijk_t = 0}, \forcode{nn_ahm_ijk_t = 0})] 368 { Constant mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{ = 0}, \protect\np{nn\_ahm\_ijk\_t}\forcode{ = 0})} 369 370 If constant, mixing coefficients are set thanks to a velocity and a length scales ($U_{scl}$, $L_{scl}$) such that: 371 372 \begin{equation} 373 \label{eq:constantah} 374 A_o^l = \left\{ 375 \begin{aligned} 376 & \frac{1}{2} U_{scl} L_{scl} & \text{for laplacian operator } \\ 377 & \frac{1}{12} U_{scl} L_{scl}^3 & \text{for bilaplacian operator } 378 \end{aligned} 379 \right. 380 \end{equation} 381 382 $U_{scl}$ and $L_{scl}$ are given by the namelist parameters \np{rn\_Ud}, \np{rn\_Uv}, \np{rn\_Ld} and \np{rn\_Lv}. 383 384 \subsection[Vertically varying mixing coefficients (\forcode{nn_aht_ijk_t = 10}, \forcode{nn_ahm_ijk_t = 10})] 385 {Vertically varying mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{ = 10}, \protect\np{nn\_ahm\_ijk\_t}\forcode{ = 10})} 386 387 In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which 388 the surface value is given by \autoref{eq:constantah}, the bottom value is 1/4 of the surface value, 389 and the transition takes place around z=500~m with a width of 200~m. 390 This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. 391 392 \subsection[Mesh size dependent mixing coefficients (\forcode{nn_aht_ijk_t = 20}, \forcode{nn_ahm_ijk_t = 20})] 393 {Mesh size dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{ = 20}, \protect\np{nn\_ahm\_ijk\_t}\forcode{ = 20})} 394 395 In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and 359 396 the type of operator used: 360 397 \begin{equation} … … 362 399 A_l = \left\{ 363 400 \begin{aligned} 364 & \frac{\max(e_1,e_2)}{e_{ max}} A_o^l & \text{for laplacian operator } \\365 & \frac{\max(e_1,e_2)^{3}}{e_{ max}^{3}} A_o^l & \text{for bilaplacian operator }401 & \frac{\max(e_1,e_2)}{e_{ref}} A_o^l & \text{for laplacian operator } \\ 402 & \frac{\max(e_1,e_2)^{3}}{e_{ref}^{3}} A_o^l & \text{for bilaplacian operator } 366 403 \end{aligned} 367 404 \right. 368 405 \end{equation} 369 where $e_{ max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked ocean domain,370 and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) namelist parameter.406 where $e_{ref}$ is a reference grid size harcoded to a $1^{\circ}$ grid size (\ie $e_{ref}\approx 111 km$), 407 and $A_o^l$ is the user defined mixing coefficient defined according to \autoref{eq:constantah}. 371 408 This variation is intended to reflect the lesser need for subgrid scale eddy mixing where 372 409 the grid size is smaller in the domain. 373 410 It was introduced in the context of the DYNAMO modelling project \citep{willebrand.barnier.ea_PO01}. 374 Note that such a grid scale dependance of mixing coefficients significantly increase the range of stability of375 model configurations presenting large changes in grid pacing such as global ocean models.411 Note that such a grid scale dependance of mixing coefficients significantly increases the range of stability of 412 model configurations presenting large changes in grid spacing such as global ocean models. 376 413 Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to 377 414 large coefficient compare to the smallest grid size (see \autoref{sec:STP_forward_imp}), 378 415 especially when using a bilaplacian operator. 379 416 380 Other formulations can be introduced by the user for a given configuration. 381 For example, in the ORCA2 global ocean model (see Configurations), 382 the laplacian viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and 383 decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}. 384 This modification can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. 385 Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of 386 ORCA2 and ORCA05 (see \&namcfg namelist). 387 388 \subsubsection[Space varying mixing coefficients (\texttt{\textbf{key\_traldf\_c3d}} and \texttt{\textbf{key\_dynldf\_c3d}})] 389 {Space varying mixing coefficients (\protect\key{traldf\_c3d} and \key{dynldf\_c3d})} 390 391 The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases, 417 \colorbox{yellow}{CASE \np{nn\_aht\_ijk\_t} = 21 to be added} 418 419 \subsection[Mesh size and depth dependent mixing coefficients (\forcode{nn_aht_ijk_t = 30}, \forcode{nn_ahm_ijk_t = 30})] 420 {Mesh size and depth dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{ = 30}, \protect\np{nn\_ahm\_ijk\_t}\forcode{ = 30})} 421 422 The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above, 392 423 \ie a hyperbolic tangent variation with depth associated with a grid size dependence of 393 424 the magnitude of the coefficient. 394 425 395 \subsubsection{Space and time varying mixing coefficients} 396 397 There is no default specification of space and time varying mixing coefficient. 398 The only case available is specific to the ORCA2 and ORCA05 global ocean configurations. 399 It provides only a tracer mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and 400 eddy induced velocity (ORCA05) that depends on the local growth rate of baroclinic instability. 401 This specification is actually used when an ORCA key and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined. 426 \subsection[Velocity dependent mixing coefficients (\forcode{nn_aht_ijk_t = 31}, \forcode{nn_ahm_ijk_t = 31})] 427 {Flow dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{ = 31}, \protect\np{nn\_ahm\_ijk\_t}\forcode{ = 31})} 428 In that case, the eddy coefficient is proportional to the local velocity magnitude so that the Reynolds number $Re = \lvert U \rvert e / A_l$ is constant (and here hardcoded to $12$): 429 \colorbox{yellow}{JC comment: The Reynolds is effectively set to 12 in the code for both operators but shouldn't it be 2 for Laplacian ?} 430 431 \begin{equation} 432 \label{eq:flowah} 433 A_l = \left\{ 434 \begin{aligned} 435 & \frac{1}{12} \lvert U \rvert e & \text{for laplacian operator } \\ 436 & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } 437 \end{aligned} 438 \right. 439 \end{equation} 440 441 \subsection[Deformation rate dependent viscosities (\forcode{nn_ahm_ijk_t = 32})] 442 {Deformation rate dependent viscosities (\protect\np{nn\_ahm\_ijk\_t}\forcode{ = 32})} 443 444 This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a 445 characteristic time scale $T_{smag}$ the deformation rate and for the lengthscale $L_{smag}$ the maximum wavenumber possible on the horizontal grid, e.g.: 446 447 \begin{equation} 448 \label{eq:smag1} 449 \begin{split} 450 T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2 } \\ 451 L_{smag} & = \frac{1}{\pi}\frac{e_1 e_2}{e_1 + e_2} 452 \end{split} 453 \end{equation} 454 455 Introducing a user defined constant $C$ (given in the namelist as \np{rn\_csmc}), one can deduce the mixing coefficients as follows: 456 457 \begin{equation} 458 \label{eq:smag2} 459 A_{smag} = \left\{ 460 \begin{aligned} 461 & C^2 T_{smag}^{-1} L_{smag}^2 & \text{for laplacian operator } \\ 462 & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4 & \text{for bilaplacian operator } 463 \end{aligned} 464 \right. 465 \end{equation} 466 467 For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:STP_forward_imp}) so that: 468 \begin{equation} 469 \label{eq:smag3} 470 \begin{aligned} 471 & C_{min} \frac{1}{2} \lvert U \rvert e < A_{smag} < C_{max} \frac{e^2}{ 8\rdt} & \text{for laplacian operator } \\ 472 & C_{min} \frac{1}{12} \lvert U \rvert e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt} & \text{for bilaplacian operator } 473 \end{aligned} 474 \end{equation} 475 476 where $C_{min}$ and $C_{max}$ are adimensional namelist parameters given by \np{rn\_minfac} and \np{rn\_maxfac} respectively. 477 478 \subsection{About space and time varying mixing coefficients} 402 479 403 480 The following points are relevant when the eddy coefficient varies spatially: … … 412 489 (\autoref{sec:dynldf_properties}). 413 490 414 (3) for isopycnal diffusion on momentum or tracers, an additional purely horizontal background diffusion with415 uniform coefficient can be added by setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0},416 a background horizontal eddy viscosity or diffusivity coefficient417 (namelist parameters whose default values are $0$).418 However, the technique used to compute the isopycnal slopes is intended to get rid of such a background diffusion,419 since it introduces spurious diapycnal diffusion (see \autoref{sec:LDF_slp}).420 421 (4) when an eddy induced advection term is used (\key{traldf\_eiv}),422 $A^{eiv}$, the eddy induced coefficient has to be defined.423 Its space variations are controlled by the same CPP variable as for the eddy diffusivity coefficient424 (\ie \key{traldf\_cNd}).425 426 (5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value.427 428 (6) it is possible to use both the laplacian and biharmonic operators concurrently.429 430 (7) it is possible to run without explicit lateral diffusion on momentum431 (\np{ln\_dynldf\_lap}\forcode{ = .?.}\np{ln\_dynldf\_bilap}\forcode{ = .false.}).432 This is recommended when using the UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.},433 see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes.434 435 491 % ================================================================ 436 492 % Eddy Induced Mixing 437 493 % ================================================================ 438 \section[Eddy induced velocity (\textit{traadv\_eiv.F90}, \textit{ldfeiv.F90})] 439 {Eddy induced velocity (\protect\mdl{traadv\_eiv}, \protect\mdl{ldfeiv})} 494 \section[Eddy induced velocity (\forcode{ln_ldfeiv = .true.})] 495 {Eddy induced velocity (\protect\np{ln\_ldfeiv}\forcode{ = .true.})} 496 440 497 \label{sec:LDF_eiv} 498 499 %--------------------------------------------namtra_eiv--------------------------------------------------- 500 501 \nlst{namtra_eiv} 502 503 %-------------------------------------------------------------------------------------------------------------- 504 441 505 442 506 %%gm from Triad appendix : to be incorporated.... … … 460 524 } 461 525 462 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),526 When \citet{gent.mcwilliams_JPO90} diffusion is used (\np{ln\_ldfeiv}\forcode{ = .true.}), 463 527 an eddy induced tracer advection term is added, 464 528 the formulation of which depends on the slopes of iso-neutral surfaces. … … 466 530 \ie \autoref{eq:ldfslp_geo} is used in $z$-coordinates, 467 531 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates. 468 The eddy induced velocity is given by: 532 533 If isopycnal mixing is used in the standard way, \ie \np{ln\_traldf\_triad}\forcode{ = .false.}, the eddy induced velocity is given by: 469 534 \begin{equation} 470 535 \label{eq:ldfeiv} … … 475 540 \end{split} 476 541 \end{equation} 477 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{rn\_aeiv}, 478 a \textit{nam\_traldf} namelist parameter. 479 The three components of the eddy induced velocity are computed and 480 add to the eulerian velocity in \mdl{traadv\_eiv}. 542 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn\_aei\_ijk\_t} \ngn{namtra\_eiv} namelist parameter. 543 The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and 544 added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed. 481 545 This has been preferred to a separate computation of the advective trends associated with the eiv velocity, 482 546 since it allows us to take advantage of all the advection schemes offered for the tracers … … 488 552 At the surface, lateral and bottom boundaries, the eddy induced velocity, 489 553 and thus the advective eddy fluxes of heat and salt, are set to zero. 554 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). 555 \colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added} 556 557 In case of setting \np{ln\_traldf\_triad}\forcode{ = .true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:triad}. 558 559 % ================================================================ 560 % Mixed layer eddies 561 % ================================================================ 562 \section[Mixed layer eddies (\forcode{ln_mle = .true.})] 563 {Mixed layer eddies (\protect\np{ln\_mle}\forcode{ = .true.})} 564 565 \label{sec:LDF_mle} 566 567 %--------------------------------------------namtra_eiv--------------------------------------------------- 568 569 \nlst{namtra_mle} 570 571 %-------------------------------------------------------------------------------------------------------------- 572 573 If \np{ln\_mle}\forcode{ = .true.} in \ngn{namtra\_mle} namelist, a parameterization of the mixing due to unresolved mixed layer instabilities is activated (\citet{foxkemper.ferrari_JPO08}). Additional transport is computed in \rou{ldf\_mle\_trp} and added to the eulerian transport in \rou{tra\_adv} as done for eddy induced advection. 574 575 \colorbox{yellow}{TBC} 490 576 491 577 \biblio
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