Changeset 11336


Ignore:
Timestamp:
2019-07-24T12:30:54+02:00 (9 months ago)
Author:
jchanut
Message:

Update LDF chapter, #2216

Location:
NEMO/trunk/doc/latex/NEMO
Files:
2 edited

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  • NEMO/trunk/doc/latex/NEMO/main/bibliography.bib

    r11335 r11336  
    928928} 
    929929 
     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 
    930944@article{         galperin.kantha.ea_JAS88, 
    931945  title         = "A quasi-equilibrium turbulent energy model for geophysical 
     
    26392653} 
    26402654 
     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 
    26412669@article{         song.haidvogel_JCP94, 
    26422670  title         = "A semi-implicit ocean circulation model using a 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex

    r11179 r11336  
    2323These three aspects of the lateral diffusion are set through namelist parameters 
    2424(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-------------------------------------------- 
     25Note that this chapter describes the standard implementation of iso-neutral tracer mixing.  
     26Griffies's implementation, which is used if \np{ln\_traldf\_triad}\forcode{ = .true.}, 
     27is described in \autoref{apdx:triad} 
     28 
     29%-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- 
    3030 
    3131\nlst{namtra_ldf}  
     
    3434%-------------------------------------------------------------------------------------------------------------- 
    3535 
     36% ================================================================ 
     37% Lateral Mixing Operator 
     38% ================================================================ 
     39\section[Lateral mixing operators] 
     40{Lateral mixing operators} 
     41\label{sec:LDF_op} 
     42We 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 
     47It is possible to run without explicit lateral diffusion on tracers (\protect\np{ln\_traldf\_OFF}\forcode{ = .true.}) and/or  
     48momentum (\protect\np{ln\_dynldf\_OFF}\forcode{ = .true.}). The latter option is even recommended if using the  
     49UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.}, 
     50see \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})} 
     54Setting \protect\np{ln\_traldf\_lap}\forcode{ = .true.} and/or \protect\np{ln\_dynldf\_lap}\forcode{ = .true.} enables  
     55a second order diffusion on tracers and momentum respectively. Note that in \NEMO 4, one can not combine  
     56Laplacian 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})} 
     60Setting \protect\np{ln\_traldf\_blp}\forcode{ = .true.} and/or \protect\np{ln\_dynldf\_blp}\forcode{ = .true.} enables  
     61a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice.  
     62We stress again that from \NEMO 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. 
    3663 
    3764% ================================================================ 
     
    88115%gm%  caution I'm not sure the simplification was a good idea!  
    89116 
    90 These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}\forcode{ = .true.}rue, 
     117These slopes are computed once in \rou{ldf\_slp\_init} when \np{ln\_sco}\forcode{ = .true.}, 
    91118and either \np{ln\_traldf\_hor}\forcode{ = .true.} or \np{ln\_dynldf\_hor}\forcode{ = .true.}.  
    92119 
     
    145172\item[$s$- or hybrid $s$-$z$- coordinate: ] 
    146173  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}). 
    149176  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.}). 
    151178  In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso} 
    152179  will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. 
     
    198225 
    199226This 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 Cox implementation of isopycnal diffusion in GFDL-type models requires 
     227It is similar to the one proposed by \citet{cox_OM87}, except for the background horizontal diffusion. 
     228Indeed, the \citet{cox_OM87} implementation of isopycnal diffusion in GFDL-type models requires 
    202229a minimum background horizontal diffusion for numerical stability reasons. 
    203230To overcome this problem, several techniques have been proposed in which the numerical schemes of 
    204231the 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}. 
     232Griffies's scheme is now available in \NEMO if \np{ln\_traldf\_triad}=\forcode{= .true.}; see \autoref{apdx:triad}. 
    206233Here, another strategy is presented \citep{lazar_phd97}: 
    207234a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of 
     
    242269 
    243270For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by 
    244 $1/100$ everywhere. 
     271the namelist scalar \np{rn\_slpmax} (usually $1/100$) everywhere. 
    245272This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to 
    246273$1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean 
    247274(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.} 
    248276 
    249277%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    298326(see \autoref{sec:LBC_coast}). 
    299327 
    300  
    301 % ================================================================ 
    302 % Lateral Mixing Operator 
    303 % ================================================================ 
    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  
    309328    
    310329% ================================================================ 
    311330% Lateral Mixing Coefficients 
    312331% ================================================================ 
    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})} 
    315334\label{sec:LDF_coef} 
    316335 
    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 
     336The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}.  
     337The 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 
     342Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model,  
     343the laplacian viscosity operator uses $A^l$~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and 
     344decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}.  
     345Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05.  
     346The 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 
     370If 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 
     387In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which 
     388the surface value is given by \autoref{eq:constantah}, the bottom value is 1/4 of the surface value, 
     389and the transition takes place around z=500~m with a width of 200~m. 
     390This 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 
     395In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and 
    359396the type of operator used: 
    360397\begin{equation} 
     
    362399  A_l = \left\{ 
    363400    \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 } 
    366403    \end{aligned} 
    367404  \right. 
    368405\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. 
     406where $e_{ref}$ is a reference grid size harcoded to a $1^{\circ}$ grid size (\ie $e_{ref}\approx 111 km$), 
     407and $A_o^l$ is the user defined mixing coefficient defined according to  \autoref{eq:constantah}. 
    371408This variation is intended to reflect the lesser need for subgrid scale eddy mixing where 
    372409the grid size is smaller in the domain. 
    373410It 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 of 
    375 model configurations presenting large changes in grid pacing such as global ocean models. 
     411Note that such a grid scale dependance of mixing coefficients significantly increases the range of stability of 
     412model configurations presenting large changes in grid spacing such as global ocean models. 
    376413Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to 
    377414large coefficient compare to the smallest grid size (see \autoref{sec:STP_forward_imp}), 
    378415especially when using a bilaplacian operator. 
    379416 
    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 
     422The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above, 
    392423\ie a hyperbolic tangent variation with depth associated with a grid size dependence of 
    393424the magnitude of the coefficient.  
    394425 
    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})} 
     428In 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 
     444This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a  
     445characteristic 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 
     455Introducing 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 
     467For 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 
     476where $C_{min}$ 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} 
    402479 
    403480The following points are relevant when the eddy coefficient varies spatially: 
     
    412489(\autoref{sec:dynldf_properties}). 
    413490 
    414 (3) for isopycnal diffusion on momentum or tracers, an additional purely horizontal background diffusion with 
    415 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 coefficient 
    417 (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 coefficient 
    424 (\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 momentum 
    431 (\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  
    435491% ================================================================ 
    436492% Eddy Induced Mixing 
    437493% ================================================================ 
    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 
    440497\label{sec:LDF_eiv} 
     498 
     499%--------------------------------------------namtra_eiv--------------------------------------------------- 
     500 
     501\nlst{namtra_eiv} 
     502 
     503%-------------------------------------------------------------------------------------------------------------- 
     504 
    441505 
    442506%%gm  from Triad appendix  : to be incorporated.... 
     
    460524} 
    461525 
    462 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), 
     526When  \citet{gent.mcwilliams_JPO90} diffusion is used (\np{ln\_ldfeiv}\forcode{ = .true.}), 
    463527an eddy induced tracer advection term is added, 
    464528the formulation of which depends on the slopes of iso-neutral surfaces. 
     
    466530\ie \autoref{eq:ldfslp_geo} is used in $z$-coordinates, 
    467531and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates. 
    468 The eddy induced velocity is given by:  
     532 
     533If isopycnal mixing is used in the standard way, \ie \np{ln\_traldf\_triad}\forcode{ = .false.}, the eddy induced velocity is given by:  
    469534\begin{equation} 
    470535  \label{eq:ldfeiv} 
     
    475540  \end{split} 
    476541\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}. 
     542where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn\_aei\_ijk\_t} \ngn{namtra\_eiv} namelist parameter.  
     543The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and 
     544added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed. 
    481545This has been preferred to a separate computation of the advective trends associated with the eiv velocity, 
    482546since it allows us to take advantage of all the advection schemes offered for the tracers 
     
    488552At the surface, lateral and bottom boundaries, the eddy induced velocity, 
    489553and thus the advective eddy fluxes of heat and salt, are set to zero.  
     554The 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 
     557In 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 
     573If  \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} 
    490576 
    491577\biblio 
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