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r11435 r11692 3 3 \begin{document} 4 4 5 % ================================================================6 % Chapter Lateral Ocean Physics (LDF)7 % ================================================================8 5 \chapter{Lateral Ocean Physics (LDF)} 9 6 \label{chap:LDF} 10 7 8 \thispagestyle{plain} 9 11 10 \chaptertoc 12 11 13 \newpage 14 15 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:PE_zdf} and 12 \paragraph{Changes record} ~\\ 13 14 {\footnotesize 15 \begin{tabularx}{\textwidth}{l||X|X} 16 Release & Author(s) & Modifications \\ 17 \hline 18 {\em 4.0} & {\em ...} & {\em ...} \\ 19 {\em 3.6} & {\em ...} & {\em ...} \\ 20 {\em 3.4} & {\em ...} & {\em ...} \\ 21 {\em <=3.4} & {\em ...} & {\em ...} 22 \end{tabularx} 23 } 24 25 \clearpage 26 27 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and 16 28 their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}). 17 29 In this section we further discuss each lateral physics option. … … 22 34 (3) the space and time variations of the eddy coefficients. 23 35 These three aspects of the lateral diffusion are set through namelist parameters 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. 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 31 \nlst{namtra_ldf} 32 33 \nlst{namdyn_ldf} 34 %-------------------------------------------------------------------------------------------------------------- 35 36 % ================================================================ 37 % Lateral Mixing Operator 38 % ================================================================ 39 \section[Lateral mixing operators] 40 {Lateral mixing operators} 36 (see the \nam{tra_ldf}{tra\_ldf} and \nam{dyn_ldf}{dyn\_ldf} below). 37 Note that this chapter describes the standard implementation of iso-neutral tracer mixing. 38 Griffies's implementation, which is used if \np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}, 39 is described in \autoref{apdx:TRIADS} 40 41 %% ================================================================================================= 42 \section[Lateral mixing operators]{Lateral mixing operators} 41 43 \label{sec:LDF_op} 42 44 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 45 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/or48 momentum (\protect\np {ln\_dynldf\_OFF}\forcode{ = .true.}). The latter option is even recommended if using the49 UBS advection scheme on momentum (\np {ln\_dynadv\_ubs}\forcode{ = .true.},46 %% ================================================================================================= 47 \subsection[No lateral mixing (\forcode{ln_traldf_OFF} \& \forcode{ln_dynldf_OFF})]{No lateral mixing (\protect\np{ln_traldf_OFF}{ln\_traldf\_OFF} \& \protect\np{ln_dynldf_OFF}{ln\_dynldf\_OFF})} 48 49 It is possible to run without explicit lateral diffusion on tracers (\protect\np[=.true.]{ln_traldf_OFF}{ln\_traldf\_OFF}) and/or 50 momentum (\protect\np[=.true.]{ln_dynldf_OFF}{ln\_dynldf\_OFF}). The latter option is even recommended if using the 51 UBS advection scheme on momentum (\np[=.true.]{ln_dynadv_ubs}{ln\_dynadv\_ubs}, 50 52 see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes. 51 53 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.} enables55 a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine 54 %% ================================================================================================= 55 \subsection[Laplacian mixing (\forcode{ln_traldf_lap} \& \forcode{ln_dynldf_lap})]{Laplacian mixing (\protect\np{ln_traldf_lap}{ln\_traldf\_lap} \& \protect\np{ln_dynldf_lap}{ln\_dynldf\_lap})} 56 Setting \protect\np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap} and/or \protect\np[=.true.]{ln_dynldf_lap}{ln\_dynldf\_lap} enables 57 a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine 56 58 Laplacian and Bilaplacian operators for the same variable. 57 59 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.} enables61 a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice. 60 %% ================================================================================================= 61 \subsection[Bilaplacian mixing (\forcode{ln_traldf_blp} \& \forcode{ln_dynldf_blp})]{Bilaplacian mixing (\protect\np{ln_traldf_blp}{ln\_traldf\_blp} \& \protect\np{ln_dynldf_blp}{ln\_dynldf\_blp})} 62 Setting \protect\np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp} and/or \protect\np[=.true.]{ln_dynldf_blp}{ln\_dynldf\_blp} enables 63 a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice. 62 64 We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. 63 65 64 % ================================================================ 65 % Direction of lateral Mixing 66 % ================================================================ 67 \section[Direction of lateral mixing (\textit{ldfslp.F90})] 68 {Direction of lateral mixing (\protect\mdl{ldfslp})} 66 %% ================================================================================================= 67 \section[Direction of lateral mixing (\textit{ldfslp.F90})]{Direction of lateral mixing (\protect\mdl{ldfslp})} 69 68 \label{sec:LDF_slp} 70 69 71 %%%72 70 \gmcomment{ 73 71 we should emphasize here that the implementation is a rather old one. … … 77 75 A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. 78 76 This occurs when $(a)$ horizontal mixing is required on tracer or momentum 79 (\np{ln \_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates,77 (\np{ln_traldf_hor}{ln\_traldf\_hor} or \np{ln_dynldf_hor}{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates, 80 78 and $(b)$ isoneutral mixing is required whatever the vertical coordinate is. 81 79 This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of 82 80 the cell of the quantity to be diffused. 83 81 For a tracer, this leads to the following four slopes: 84 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq: tra_ldf_iso}),82 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), 85 83 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 $v$. 84 $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 87 85 88 86 %gm% add here afigure of the slope in i-direction 89 87 88 %% ================================================================================================= 90 89 \subsection{Slopes for tracer geopotential mixing in the $s$-coordinate} 91 90 92 91 In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between 93 92 the geopotential and computational surfaces. 94 Their discrete formulation is found by locally solving \autoref{eq: tra_ldf_iso} when93 Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when 95 94 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. 95 \ie\ a linear function of $z_T$, the depth of a $T$-point. 97 96 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 98 97 99 98 \begin{equation} 100 \label{eq: ldfslp_geo}99 \label{eq:LDF_slp_geo} 101 100 \begin{aligned} 102 101 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} … … 113 112 \end{equation} 114 113 115 %gm% caution I'm not sure the simplification was a good idea! 116 117 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.}. 119 114 %gm% caution I'm not sure the simplification was a good idea! 115 116 These slopes are computed once in \rou{ldf\_slp\_init} when \np[=.true.]{ln_sco}{ln\_sco}, 117 and either \np[=.true.]{ln_traldf_hor}{ln\_traldf\_hor} or \np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}. 118 119 %% ================================================================================================= 120 120 \subsection{Slopes for tracer iso-neutral mixing} 121 121 \label{subsec:LDF_slp_iso} … … 125 125 Their discrete formulation is found using the fact that the diffusive fluxes of 126 126 locally referenced potential density (\ie\ $in situ$ density) vanish. 127 So, substituting $T$ by $\rho$ in \autoref{eq: tra_ldf_iso} and setting the diffusive fluxes in127 So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in 128 128 the three directions to zero leads to the following definition for the neutral slopes: 129 129 130 130 \begin{equation} 131 \label{eq: ldfslp_iso}131 \label{eq:LDF_slp_iso} 132 132 \begin{split} 133 133 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} … … 145 145 146 146 %gm% rewrite this as the explanation is not very clear !!! 147 %In practice, \autoref{eq: ldfslp_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:ldfslp_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.148 149 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq: ldfslp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters).150 151 %In the $z$-coordinate, the derivative of the \autoref{eq: ldfslp_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.152 153 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq: ldfslp_iso} has to147 %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. 148 149 %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). 150 151 %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. 152 153 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to 154 154 be evaluated at the same local pressure 155 155 (which, in decibars, is approximated by the depth in meters in the model). 156 Therefore \autoref{eq: ldfslp_iso} cannot be used as such,156 Therefore \autoref{eq:LDF_slp_iso} cannot be used as such, 157 157 but further transformation is needed depending on the vertical coordinate used: 158 158 159 159 \begin{description} 160 161 \item[$z$-coordinate with full step: ] 162 in \autoref{eq:ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, 160 \item [$z$-coordinate with full step:] in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, 163 161 thus the $in situ$ density can be used. 164 162 This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, 165 163 where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87} 166 (see \autoref{subsec:TRA_bn2}). 167 168 \item[$z$-coordinate with partial step: ] 169 this case is identical to the full step case except that at partial step level, 164 (see \autoref{subsec:TRA_bn2}). 165 \item [$z$-coordinate with partial step:] this case is identical to the full step case except that at partial step level, 170 166 the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}. 171 172 \item[$s$- or hybrid $s$-$z$- coordinate: ] 173 in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 174 the Griffies scheme is used (\np{ln\_traldf\_triad}\forcode{ = .true.}; 175 see \autoref{apdx:triad}). 167 \item [$s$- or hybrid $s$-$z$- coordinate:] in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 168 the Griffies scheme is used (\np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}; 169 see \autoref{apdx:TRIADS}). 176 170 In other words, iso-neutral mixing will only be accurately represented with a linear equation of state 177 (\np {ln\_seos}\forcode{ = .true.}).178 In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq: ldfslp_iso}171 (\np[=.true.]{ln_seos}{ln\_seos}). 172 In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso} 179 173 will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. 180 174 181 %gm% 175 %gm% 182 176 Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for 183 177 the constraint on iso-neutral fluxes. … … 193 187 194 188 \[ 195 % \label{eq: ldfslp_iso2}189 % \label{eq:LDF_slp_iso2} 196 190 \begin{split} 197 191 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac … … 221 215 222 216 Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases. 223 224 217 \end{description} 225 218 … … 230 223 To overcome this problem, several techniques have been proposed in which the numerical schemes of 231 224 the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. 232 Griffies's scheme is now available in \NEMO\ if \np {ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:triad}.225 Griffies's scheme is now available in \NEMO\ if \np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}; see \autoref{apdx:TRIADS}. 233 226 Here, another strategy is presented \citep{lazar_phd97}: 234 227 a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of … … 240 233 241 234 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. 243 244 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 235 contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property. 236 245 237 \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} 238 \centering 239 \includegraphics[width=0.66\textwidth]{LDF_ZDF1} 240 \caption{Averaging procedure for isopycnal slope computation} 241 \label{fig:LDF_ZDF1} 253 242 \end{figure} 254 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 255 256 %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. 243 244 %There are three additional questions about the slope calculation. 245 %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. 246 %Second, numerical stability issues also require a bound on slopes. 259 247 %Third, the question of boundary condition specified on slopes... 260 248 261 249 %from griffies: chapter 13.1.... 262 250 263 264 265 % 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 251 % In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04}, 252 % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly 253 % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the 268 254 % surface motivates this flattening of isopycnals near the surface). 269 255 270 256 For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by 271 the namelist scalar \np{rn \_slpmax} (usually $1/100$) everywhere.257 the namelist scalar \np{rn_slpmax}{rn\_slpmax} (usually $1/100$) everywhere. 272 258 This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to 273 259 $1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean … … 275 261 \colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.} 276 262 277 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>278 263 \begin{figure}[!ht] 279 \begin{center} 280 \includegraphics[width=\textwidth]{Fig_eiv_slp} 281 \caption{ 282 \protect\label{fig: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} 264 \centering 265 \includegraphics[width=0.66\textwidth]{LDF_eiv_slp} 266 \caption[Vertical profile of the slope used for lateral mixing in the mixed layer]{ 267 Vertical profile of the slope used for lateral mixing in the mixed layer: 268 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 269 which has to be adjusted at the surface boundary 270 \ie\ it must tend to zero at the surface since there is no mixing across the air-sea interface: 271 wall boundary condition). 272 Nevertheless, 273 the profile between the surface zero value and the interior iso-neutral one is unknown, 274 and especially the value at the base of the mixed layer; 275 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and 276 imposing a maximum slope of 1/100; 277 \textit{(c)} profile of slope actually used in \NEMO: 278 a linear decrease of the slope from zero at the surface to 279 its ocean interior value computed just below the mixed layer. 280 Note the huge change in the slope at the base of the mixed layer between 281 \textit{(b)} and \textit{(c)}.} 282 \label{fig:LDF_eiv_slp} 297 283 \end{figure} 298 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>299 284 300 285 \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.} 301 286 287 %% ================================================================================================= 302 288 \subsection{Slopes for momentum iso-neutral mixing} 303 289 304 290 The iso-neutral diffusion operator on momentum is the same as the one used on tracers but 305 291 applied to each component of the velocity separately 306 (see \autoref{eq: dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}).292 (see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). 307 293 The slopes between the surface along which the diffusion operator acts and the surface of computation 308 294 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and 309 295 \textit{vw}- points for the $v$-component. 310 296 They are computed from the slopes used for tracer diffusion, 311 \ie\ \autoref{eq: ldfslp_geo} and \autoref{eq:ldfslp_iso}:297 \ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}: 312 298 313 299 \[ 314 % \label{eq: ldfslp_dyn}300 % \label{eq:LDF_slp_dyn} 315 301 \begin{aligned} 316 302 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ … … 326 312 (see \autoref{sec:LBC_coast}). 327 313 328 329 % ================================================================ 330 % Lateral Mixing Coefficients 331 % ================================================================ 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})} 314 %% ================================================================================================= 315 \section[Lateral mixing coefficient (\forcode{nn_aht_ijk_t} \& \forcode{nn_ahm_ijk_t})]{Lateral mixing coefficient (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} 334 316 \label{sec:LDF_coef} 335 317 336 The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. 318 The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. 337 319 The way the mixing coefficients are set in the reference version can be described as follows: 338 320 339 \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, 321 %% ================================================================================================= 322 \subsection[Mixing coefficients read from file (\forcode{=-20, -30})]{ Mixing coefficients read from file (\protect\np[=-20, -30]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=-20, -30]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} 323 324 Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model, 343 325 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--------------------------------------------------- 326 decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}. 327 Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05. 328 The provided fields can either be 2d (\np[=-20]{nn_aht_ijk_t}{nn\_aht\_ijk\_t}, \np[=-20]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}) or 3d (\np[=-30]{nn_aht_ijk_t}{nn\_aht\_ijk\_t}, \np[=-30]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}). They must be given at U, V points for tracers and T, F points for momentum (see \autoref{tab:LDF_files}). 329 349 330 \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} 331 \centering 332 \begin{tabular}{|l|l|l|l|} 333 \hline 334 Namelist parameter & Input filename & dimensions & variable names \\ \hline 335 \np[=-20]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t} & \forcode{eddy_viscosity_2D.nc } & $(i,j)$ & \forcode{ahmt_2d, ahmf_2d} \\ \hline 336 \np[=-20]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} & \forcode{eddy_diffusivity_2D.nc } & $(i,j)$ & \forcode{ahtu_2d, ahtv_2d} \\ \hline 337 \np[=-30]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t} & \forcode{eddy_viscosity_3D.nc } & $(i,j,k)$ & \forcode{ahmt_3d, ahmf_3d} \\ \hline 338 \np[=-30]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} & \forcode{eddy_diffusivity_3D.nc } & $(i,j,k)$ & \forcode{ahtu_3d, ahtv_3d} \\ \hline 339 \end{tabular} 340 \caption{Description of expected input files if mixing coefficients are read from NetCDF files} 341 \label{tab:LDF_files} 364 342 \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})} 343 344 %% ================================================================================================= 345 \subsection[Constant mixing coefficients (\forcode{=0})]{ Constant mixing coefficients (\protect\np[=0]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=0]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} 369 346 370 347 If constant, mixing coefficients are set thanks to a velocity and a length scales ($U_{scl}$, $L_{scl}$) such that: 371 348 372 349 \begin{equation} 373 \label{eq: constantah}350 \label{eq:LDF_constantah} 374 351 A_o^l = \left\{ 375 352 \begin{aligned} … … 380 357 \end{equation} 381 358 382 $U_{scl}$ 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})}359 $U_{scl}$ and $L_{scl}$ are given by the namelist parameters \np{rn_Ud}{rn\_Ud}, \np{rn_Uv}{rn\_Uv}, \np{rn_Ld}{rn\_Ld} and \np{rn_Lv}{rn\_Lv}. 360 361 %% ================================================================================================= 362 \subsection[Vertically varying mixing coefficients (\forcode{=10})]{Vertically varying mixing coefficients (\protect\np[=10]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=10]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} 386 363 387 364 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,365 the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value, 389 366 and the transition takes place around z=500~m with a width of 200~m. 390 367 This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. 391 368 392 \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})}369 %% ================================================================================================= 370 \subsection[Mesh size dependent mixing coefficients (\forcode{=20})]{Mesh size dependent mixing coefficients (\protect\np[=20]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=20]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} 394 371 395 372 In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and 396 373 the type of operator used: 397 374 \begin{equation} 398 \label{eq: title}375 \label{eq:LDF_title} 399 376 A_l = \left\{ 400 377 \begin{aligned} … … 404 381 \right. 405 382 \end{equation} 406 where $U_{scl}$ is a user defined velocity scale (\np{rn \_Ud}, \np{rn\_Uv}).383 where $U_{scl}$ is a user defined velocity scale (\np{rn_Ud}{rn\_Ud}, \np{rn_Uv}{rn\_Uv}). 407 384 This variation is intended to reflect the lesser need for subgrid scale eddy mixing where 408 385 the grid size is smaller in the domain. … … 411 388 model configurations presenting large changes in grid spacing such as global ocean models. 412 389 Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to 413 large coefficient compare to the smallest grid size (see \autoref{sec: STP_forward_imp}),390 large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}), 414 391 especially when using a bilaplacian operator. 415 392 416 \colorbox{yellow}{CASE \np{nn \_aht\_ijk\_t} = 21 to be added}417 418 \subsection[Mesh size and depth dependent mixing coefficients (\forcode{nn_aht_ijk_t = 30}, \forcode{nn_ahm_ijk_t = 30})] 419 {Mesh size and depth dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{ = 30}, \protect\np{nn\_ahm\_ijk\_t}\forcode{ = 30})}393 \colorbox{yellow}{CASE \np{nn_aht_ijk_t}{nn\_aht\_ijk\_t} = 21 to be added} 394 395 %% ================================================================================================= 396 \subsection[Mesh size and depth dependent mixing coefficients (\forcode{=30})]{Mesh size and depth dependent mixing coefficients (\protect\np[=30]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=30]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} 420 397 421 398 The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above, 422 399 \ie\ a hyperbolic tangent variation with depth associated with a grid size dependence of 423 the magnitude of the coefficient. 424 425 \subsection[Velocity dependent mixing coefficients (\forcode{nn_aht_ijk_t = 31}, \forcode{nn_ahm_ijk_t = 31})] 426 {Flow dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{ = 31}, \protect\np{nn\_ahm\_ijk\_t}\forcode{ = 31})}400 the magnitude of the coefficient. 401 402 %% ================================================================================================= 403 \subsection[Velocity dependent mixing coefficients (\forcode{=31})]{Flow dependent mixing coefficients (\protect\np[=31]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=31]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} 427 404 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 $12$): 428 405 \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 ?} 429 406 430 407 \begin{equation} 431 \label{eq: flowah}408 \label{eq:LDF_flowah} 432 409 A_l = \left\{ 433 410 \begin{aligned} 434 411 & \frac{1}{12} \lvert U \rvert e & \text{for laplacian operator } \\ 435 & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } 412 & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } 436 413 \end{aligned} 437 414 \right. 438 415 \end{equation} 439 416 440 \subsection[Deformation rate dependent viscosities (\forcode{nn_ahm_ijk_t = 32})] 441 {Deformation rate dependent viscosities (\protect\np{nn\_ahm\_ijk\_t}\forcode{ = 32})}442 443 This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a 417 %% ================================================================================================= 418 \subsection[Deformation rate dependent viscosities (\forcode{nn_ahm_ijk_t=32})]{Deformation rate dependent viscosities (\protect\np[=32]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} 419 420 This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a 444 421 characteristic time scale $T_{smag}$ the deformation rate and for the lengthscale $L_{smag}$ the maximum wavenumber possible on the horizontal grid, e.g.: 445 422 446 423 \begin{equation} 447 \label{eq: smag1}424 \label{eq:LDF_smag1} 448 425 \begin{split} 449 426 T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2 } \\ … … 452 429 \end{equation} 453 430 454 Introducing a user defined constant $C$ (given in the namelist as \np{rn \_csmc}), one can deduce the mixing coefficients as follows:455 456 \begin{equation} 457 \label{eq: smag2}431 Introducing a user defined constant $C$ (given in the namelist as \np{rn_csmc}{rn\_csmc}), one can deduce the mixing coefficients as follows: 432 433 \begin{equation} 434 \label{eq:LDF_smag2} 458 435 A_{smag} = \left\{ 459 436 \begin{aligned} 460 437 & 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 } 438 & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4 & \text{for bilaplacian operator } 462 439 \end{aligned} 463 440 \right. 464 441 \end{equation} 465 442 466 For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec: STP_forward_imp}) so that:467 \begin{equation} 468 \label{eq: smag3}443 For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that: 444 \begin{equation} 445 \label{eq:LDF_smag3} 469 446 \begin{aligned} 470 447 & 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 } 448 & C_{min} \frac{1}{12} \lvert U \rvert e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt} & \text{for bilaplacian operator } 472 449 \end{aligned} 473 450 \end{equation} 474 451 475 where $C_{min}$ and $C_{max}$ are adimensional namelist parameters given by \np{rn\_minfac} and \np{rn\_maxfac} respectively. 476 452 where $C_{min}$ and $C_{max}$ are adimensional namelist parameters given by \np{rn_minfac}{rn\_minfac} and \np{rn_maxfac}{rn\_maxfac} respectively. 453 454 %% ================================================================================================= 477 455 \subsection{About space and time varying mixing coefficients} 478 456 … … 480 458 481 459 (1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and 482 divergent components of the horizontal current (see \autoref{subsec: PE_ldf}).460 divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). 483 461 Although the eddy coefficient could be set to different values in these two terms, 484 this option is not currently available. 462 this option is not currently available. 485 463 486 464 (2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of 487 465 the horizontal divergence for operators acting along model-surfaces are no longer satisfied 488 (\autoref{sec:dynldf_properties}). 489 490 % ================================================================ 491 % Eddy Induced Mixing 492 % ================================================================ 493 \section[Eddy induced velocity (\forcode{ln_ldfeiv = .true.})] 494 {Eddy induced velocity (\protect\np{ln\_ldfeiv}\forcode{ = .true.})} 466 (\autoref{sec:INVARIANTS_dynldf_properties}). 467 468 %% ================================================================================================= 469 \section[Eddy induced velocity (\forcode{ln_ldfeiv})]{Eddy induced velocity (\protect\np{ln_ldfeiv}{ln\_ldfeiv})} 495 470 496 471 \label{sec:LDF_eiv} 497 472 498 %--------------------------------------------namtra_eiv--------------------------------------------------- 499 500 \nlst{namtra_eiv} 501 502 %-------------------------------------------------------------------------------------------------------------- 503 473 \begin{listing} 474 \nlst{namtra_eiv} 475 \caption{\forcode{&namtra_eiv}} 476 \label{lst:namtra_eiv} 477 \end{listing} 504 478 505 479 %%gm from Triad appendix : to be incorporated.... … … 507 481 Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}. 508 482 If none of the keys \key{traldf\_cNd}, N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and 509 GM diffusivity $A_e$ are directly set by \np{rn \_aeih\_0} and \np{rn\_aeiv\_0}.483 GM diffusivity $A_e$ are directly set by \np{rn_aeih_0}{rn\_aeih\_0} and \np{rn_aeiv_0}{rn\_aeiv\_0}. 510 484 If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal 511 485 scale factor according to \autoref{eq:title} … … 520 494 In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$, 521 495 where $f_{20}$ is the value of $f$ at $20^{\circ}$~N 522 } (\mdl{ldfeiv}) and \np{rn \_aeiv\_0} is ignored unless it is zero.496 } (\mdl{ldfeiv}) and \np{rn_aeiv_0}{rn\_aeiv\_0} is ignored unless it is zero. 523 497 } 524 498 525 When \citet{gent.mcwilliams_JPO90} diffusion is used (\np {ln\_ldfeiv}\forcode{ = .true.}),499 When \citet{gent.mcwilliams_JPO90} diffusion is used (\np[=.true.]{ln_ldfeiv}{ln\_ldfeiv}), 526 500 an eddy induced tracer advection term is added, 527 501 the formulation of which depends on the slopes of iso-neutral surfaces. 528 502 Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 529 \ie\ \autoref{eq: ldfslp_geo} is used in $z$-coordinates,530 and the sum \autoref{eq: ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates.531 532 If isopycnal mixing is used in the standard way, \ie\ \np {ln\_traldf\_triad}\forcode{ = .false.}, the eddy induced velocity is given by:533 \begin{equation} 534 \label{eq: ldfeiv}503 \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates, 504 and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates. 505 506 If isopycnal mixing is used in the standard way, \ie\ \np[=.false.]{ln_traldf_triad}{ln\_traldf\_triad}, the eddy induced velocity is given by: 507 \begin{equation} 508 \label{eq:LDF_eiv} 535 509 \begin{split} 536 510 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ … … 539 513 \end{split} 540 514 \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.515 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn_aei_ijk_t}{nn\_aei\_ijk\_t} \nam{tra_eiv}{tra\_eiv} namelist parameter. 542 516 The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and 543 517 added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed. … … 547 521 previous releases of OPA \citep{madec.delecluse.ea_NPM98}. 548 522 This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of 549 paramount importance. 523 paramount importance. 550 524 551 525 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). 554 \colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added} 555 556 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}. 557 558 % ================================================================ 559 % Mixed layer eddies 560 % ================================================================ 561 \section[Mixed layer eddies (\forcode{ln_mle = .true.})] 562 {Mixed layer eddies (\protect\np{ln\_mle}\forcode{ = .true.})} 563 526 and thus the advective eddy fluxes of heat and salt, are set to zero. 527 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}{nn\_aei\_ijk\_t}, \np{rn_Ue}{rn\_Ue}, \np{rn_Le}{rn\_Le} namelist parameters). 528 \colorbox{yellow}{CASE \np{nn_aei_ijk_t}{nn\_aei\_ijk\_t} = 21 to be added} 529 530 In case of setting \np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:TRIADS}. 531 532 %% ================================================================================================= 533 \section[Mixed layer eddies (\forcode{ln_mle})]{Mixed layer eddies (\protect\np{ln_mle}{ln\_mle})} 564 534 \label{sec:LDF_mle} 565 535 566 %--------------------------------------------namtra_eiv--------------------------------------------------- 567 568 \nlst{namtra_mle}569 570 %-------------------------------------------------------------------------------------------------------------- 571 572 If \np {ln\_mle}\forcode{ = .true.} in \nam{tra\_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.536 \begin{listing} 537 \nlst{namtra_mle} 538 \caption{\forcode{&namtra_mle}} 539 \label{lst:namtra_mle} 540 \end{listing} 541 542 If \np[=.true.]{ln_mle}{ln\_mle} in \nam{tra_mle}{tra\_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. 573 543 574 544 \colorbox{yellow}{TBC} 575 545 576 \biblio 577 578 \pindex 546 \onlyinsubfile{\input{../../global/epilogue}} 579 547 580 548 \end{document}
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