Changeset 11831 for NEMO/branches/2019/dev_r11085_ASINTER-05_Brodeau_Advanced_Bulk/doc/latex/NEMO/subfiles/chap_LDF.tex
- Timestamp:
- 2019-10-29T18:14:49+01:00 (4 years ago)
- Location:
- NEMO/branches/2019/dev_r11085_ASINTER-05_Brodeau_Advanced_Bulk/doc
- Files:
-
- 5 edited
Legend:
- Unmodified
- Added
- Removed
-
NEMO/branches/2019/dev_r11085_ASINTER-05_Brodeau_Advanced_Bulk/doc
- Property svn:ignore deleted
-
Property
svn:externals
set to
^/utils/badges badges
^/utils/logos logos
-
NEMO/branches/2019/dev_r11085_ASINTER-05_Brodeau_Advanced_Bulk/doc/latex
- Property svn:ignore deleted
-
NEMO/branches/2019/dev_r11085_ASINTER-05_Brodeau_Advanced_Bulk/doc/latex/NEMO
- Property svn:ignore deleted
-
Property
svn:externals
set to
^/utils/figures/NEMO figures
-
NEMO/branches/2019/dev_r11085_ASINTER-05_Brodeau_Advanced_Bulk/doc/latex/NEMO/subfiles
- Property svn:ignore
-
old new 1 *.aux 2 *.bbl 3 *.blg 4 *.dvi 5 *.fdb* 6 *.fls 7 *.idx 1 *.ind 8 2 *.ilg 9 *.ind10 *.log11 *.maf12 *.mtc*13 *.out14 *.pdf15 *.toc16 _minted-*
-
- Property svn:ignore
-
NEMO/branches/2019/dev_r11085_ASINTER-05_Brodeau_Advanced_Bulk/doc/latex/NEMO/subfiles/chap_LDF.tex
r10442 r11831 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 11 \minitoc 12 13 \newpage 14 15 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:PE_zdf} and 8 \thispagestyle{plain} 9 10 \chaptertoc 11 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 \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-------------------------------------------- 30 31 \nlst{namtra_ldf} 32 33 \nlst{namdyn_ldf} 34 %-------------------------------------------------------------------------------------------------------------- 35 36 37 % ================================================================ 38 % Direction of lateral Mixing 39 % ================================================================ 40 \section{Direction of lateral mixing (\protect\mdl{ldfslp})} 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} 43 \label{sec:LDF_op} 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}. 45 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}, 52 see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes. 53 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 58 Laplacian and Bilaplacian operators for the same variable. 59 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. 64 We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. 65 66 %% ================================================================================================= 67 \section[Direction of lateral mixing (\textit{ldfslp.F90})]{Direction of lateral mixing (\protect\mdl{ldfslp})} 41 68 \label{sec:LDF_slp} 42 69 43 %%% 44 \gmcomment{ 70 \cmtgm{ 45 71 we should emphasize here that the implementation is a rather old one. 46 Better work can be achieved by using \citet{ Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme.72 Better work can be achieved by using \citet{griffies.gnanadesikan.ea_JPO98, griffies_bk04} iso-neutral scheme. 47 73 } 48 74 49 75 A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. 50 76 This occurs when $(a)$ horizontal mixing is required on tracer or momentum 51 (\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, 52 78 and $(b)$ isoneutral mixing is required whatever the vertical coordinate is. 53 79 This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of 54 80 the cell of the quantity to be diffused. 55 81 For a tracer, this leads to the following four slopes: 56 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq: tra_ldf_iso}),82 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), 57 83 while for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and 58 $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 59 60 %gm% add here afigure of the slope in i-direction 61 84 $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 85 86 \cmtgm{Add here afigure of the slope in i-direction} 87 88 %% ================================================================================================= 62 89 \subsection{Slopes for tracer geopotential mixing in the $s$-coordinate} 63 90 64 In $s$-coordinates, geopotential mixing (\ie horizontal mixing) $r_1$ and $r_2$ are the slopes between91 In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between 65 92 the geopotential and computational surfaces. 66 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 67 94 the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform, 68 \ie a linear function of $z_T$, the depth of a $T$-point. 69 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 70 71 \begin{equation} 72 \label{eq:ldfslp_geo} 95 \ie\ a linear function of $z_T$, the depth of a $T$-point. 96 \cmtgm{Steven : My version is obviously wrong since 97 I'm left with an arbitrary constant which is the local vertical temperature gradient} 98 99 \begin{equation} 100 \label{eq:LDF_slp_geo} 73 101 \begin{aligned} 74 102 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} … … 85 113 \end{equation} 86 114 87 %gm% caution I'm not sure the simplification was a good idea! 88 89 These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}\forcode{ = .true.}rue, 90 and either \np{ln\_traldf\_hor}\forcode{ = .true.} or \np{ln\_dynldf\_hor}\forcode{ = .true.}. 91 115 \cmtgm{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[=.true.]{ln_sco}{ln\_sco}, 118 and either \np[=.true.]{ln_traldf_hor}{ln\_traldf\_hor} or \np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}. 119 120 %% ================================================================================================= 92 121 \subsection{Slopes for tracer iso-neutral mixing} 93 122 \label{subsec:LDF_slp_iso} … … 96 125 Their formulation does not depend on the vertical coordinate used. 97 126 Their discrete formulation is found using the fact that the diffusive fluxes of 98 locally referenced potential density (\ie $in situ$ density) vanish.99 So, substituting $T$ by $\rho$ in \autoref{eq: tra_ldf_iso} and setting the diffusive fluxes in127 locally referenced potential density (\ie\ $in situ$ density) vanish. 128 So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in 100 129 the three directions to zero leads to the following definition for the neutral slopes: 101 130 102 131 \begin{equation} 103 \label{eq: ldfslp_iso}132 \label{eq:LDF_slp_iso} 104 133 \begin{split} 105 134 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} … … 116 145 \end{equation} 117 146 118 %gm% rewrite this as the explanation is not very clear !!! 119 %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.120 121 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{ McDougall1987}, 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).122 123 %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.124 125 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq: ldfslp_iso} has to147 \cmtgm{rewrite this as the explanation is not very clear !!!} 148 %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. 149 150 %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). 151 152 %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. 153 154 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to 126 155 be evaluated at the same local pressure 127 156 (which, in decibars, is approximated by the depth in meters in the model). 128 Therefore \autoref{eq: ldfslp_iso} cannot be used as such,157 Therefore \autoref{eq:LDF_slp_iso} cannot be used as such, 129 158 but further transformation is needed depending on the vertical coordinate used: 130 159 131 160 \begin{description} 132 133 \item[$z$-coordinate with full step: ] 134 in \autoref{eq:ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, 161 \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, 135 162 thus the $in situ$ density can be used. 136 163 This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, 137 where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{McDougall1987} 138 (see \autoref{subsec:TRA_bn2}). 139 140 \item[$z$-coordinate with partial step: ] 141 this case is identical to the full step case except that at partial step level, 164 where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87} 165 (see \autoref{subsec:TRA_bn2}). 166 \item [$z$-coordinate with partial step:] this case is identical to the full step case except that at partial step level, 142 167 the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}. 143 144 \item[$s$- or hybrid $s$-$z$- coordinate: ] 145 in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 146 the Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.}; 147 see Appdx \autoref{apdx:triad}). 168 \item [$s$- or hybrid $s$-$z$- coordinate:] in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 169 the Griffies scheme is used (\np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}; 170 see \autoref{apdx:TRIADS}). 148 171 In other words, iso-neutral mixing will only be accurately represented with a linear equation of state 149 (\np {nn\_eos}\forcode{ = 1..2}).150 In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq: ldfslp_iso}172 (\np[=.true.]{ln_seos}{ln\_seos}). 173 In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso} 151 174 will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. 152 175 153 %gm%154 176 Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for 155 177 the constraint on iso-neutral fluxes. 156 Following \citet{ Griffies_Bk04}, instead of specifying directly that there is a zero neutral diffusive flux of178 Following \citet{griffies_bk04}, instead of specifying directly that there is a zero neutral diffusive flux of 157 179 locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between 158 180 the neutral direction diffusive fluxes of potential temperature and salinity: … … 160 182 \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) 161 183 \] 162 % gm{where vector F is ....}184 \cmtgm{where vector F is ....} 163 185 164 186 This constraint leads to the following definition for the slopes: 165 187 166 188 \[ 167 % \label{eq: ldfslp_iso2}189 % \label{eq:LDF_slp_iso2} 168 190 \begin{split} 169 191 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac … … 193 215 194 216 Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases. 195 196 217 \end{description} 197 218 198 219 This implementation is a rather old one. 199 It is similar to the one proposed by Cox [1987], except for the background horizontal diffusion.200 Indeed, the Coximplementation of isopycnal diffusion in GFDL-type models requires220 It is similar to the one proposed by \citet{cox_OM87}, except for the background horizontal diffusion. 221 Indeed, the \citet{cox_OM87} implementation of isopycnal diffusion in GFDL-type models requires 201 222 a minimum background horizontal diffusion for numerical stability reasons. 202 223 To overcome this problem, several techniques have been proposed in which the numerical schemes of 203 the ocean model are modified \citep{ Weaver_Eby_JPO97, Griffies_al_JPO98}.204 Griffies's scheme is now available in \NEMO if \np{traldf\_grif\_iso} is set true; see Appdx \autoref{apdx:triad}.205 Here, another strategy is presented \citep{ Lazar_PhD97}:224 the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. 225 Griffies's scheme is now available in \NEMO\ if \np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}; see \autoref{apdx:TRIADS}. 226 Here, another strategy is presented \citep{lazar_phd97}: 206 227 a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of 207 228 grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}). 208 229 This allows an iso-neutral diffusion scheme without additional background horizontal mixing. 209 230 This technique can be viewed as a diffusion operator that acts along large-scale 210 (2~$\Delta$x) \ gmcomment{2deltax doesnt seem very large scale} iso-neutral surfaces.231 (2~$\Delta$x) \cmtgm{2deltax doesnt seem very large scale} iso-neutral surfaces. 211 232 The diapycnal diffusion required for numerical stability is thus minimized and its net effect on the flow is quite small when compared to the effect of an horizontal background mixing. 212 233 213 234 Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, 214 contrary to the \citet{Griffies_al_JPO98} operator which has that property. 215 216 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 235 contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property. 236 217 237 \begin{figure}[!ht] 218 \begin{center} 219 \includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1} 220 \caption { 221 \protect\label{fig:LDF_ZDF1} 222 averaging procedure for isopycnal slope computation. 223 } 224 \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} 225 242 \end{figure} 226 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 227 228 %There are three additional questions about the slope calculation. 229 %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. 230 %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. 231 247 %Third, the question of boundary condition specified on slopes... 232 248 233 249 %from griffies: chapter 13.1.... 234 250 235 236 237 % In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04}, 238 % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly 239 % 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 240 254 % surface motivates this flattening of isopycnals near the surface). 241 255 242 For numerical stability reasons \citep{ Cox1987, Griffies_Bk04}, the slopes must also be bounded by243 $1/100$everywhere.256 For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by 257 the namelist scalar \np{rn_slpmax}{rn\_slpmax} (usually $1/100$) everywhere. 244 258 This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to 245 259 $1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean 246 260 (the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface). 247 248 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 261 \colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.} 262 249 263 \begin{figure}[!ht] 250 \begin{center} 251 \includegraphics[width=0.70\textwidth]{Fig_eiv_slp} 252 \caption{ 253 \protect\label{fig:eiv_slp} 254 Vertical profile of the slope used for lateral mixing in the mixed layer: 255 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 256 which has to be adjusted at the surface boundary 257 \ie it must tend to zero at the surface since there is no mixing across the air-sea interface: 258 wall boundary condition). 259 Nevertheless, the profile between the surface zero value and the interior iso-neutral one is unknown, 260 and especially the value at the base of the mixed layer; 261 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and 262 imposing a maximum slope of 1/100; 263 \textit{(c)} profile of slope actually used in \NEMO: a linear decrease of the slope from 264 zero at the surface to its ocean interior value computed just below the mixed layer. 265 Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}. 266 } 267 \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} 268 283 \end{figure} 269 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>270 284 271 285 \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.} 272 286 287 %% ================================================================================================= 273 288 \subsection{Slopes for momentum iso-neutral mixing} 274 289 275 290 The iso-neutral diffusion operator on momentum is the same as the one used on tracers but 276 291 applied to each component of the velocity separately 277 (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}). 278 293 The slopes between the surface along which the diffusion operator acts and the surface of computation 279 294 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and 280 295 \textit{vw}- points for the $v$-component. 281 296 They are computed from the slopes used for tracer diffusion, 282 \ie \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso}:297 \ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}: 283 298 284 299 \[ 285 % \label{eq: ldfslp_dyn}300 % \label{eq:LDF_slp_dyn} 286 301 \begin{aligned} 287 302 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ … … 294 309 The major issue remaining is in the specification of the boundary conditions. 295 310 The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces, 296 \ie using the shear computed along the model levels and with no additional friction at the ocean bottom311 \ie\ using the shear computed along the model levels and with no additional friction at the ocean bottom 297 312 (see \autoref{sec:LBC_coast}). 298 313 299 300 % ================================================================ 301 % Lateral Mixing Operator 302 % ================================================================ 303 \section{Lateral mixing operators (\protect\mdl{traldf}, \protect\mdl{traldf}) } 304 \label{sec:LDF_op} 305 306 307 308 % ================================================================ 309 % Lateral Mixing Coefficients 310 % ================================================================ 311 \section{Lateral mixing coefficient (\protect\mdl{ldftra}, \protect\mdl{ldfdyn}) } 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})} 312 316 \label{sec:LDF_coef} 313 317 314 Introducing a space variation in the lateral eddy mixing coefficients changes the model core memory requirement, 315 adding up to four extra three-dimensional arrays for the geopotential or isopycnal second order operator applied to 316 momentum. 317 Six CPP keys control the space variation of eddy coefficients: three for momentum and three for tracer. 318 The three choices allow: 319 a space variation in the three space directions (\key{traldf\_c3d}, \key{dynldf\_c3d}), 320 in the horizontal plane (\key{traldf\_c2d}, \key{dynldf\_c2d}), 321 or in the vertical only (\key{traldf\_c1d}, \key{dynldf\_c1d}). 322 The default option is a constant value over the whole ocean on both momentum and tracers. 323 324 The number of additional arrays that have to be defined and the gridpoint position at which 325 they are defined depend on both the space variation chosen and the type of operator used. 326 The resulting eddy viscosity and diffusivity coefficients can be a function of more than one variable. 327 Changes in the computer code when switching from one option to another have been minimized by 328 introducing the eddy coefficients as statement functions 329 (include file \textit{ldftra\_substitute.h90} and \textit{ldfdyn\_substitute.h90}). 330 The functions are replaced by their actual meaning during the preprocessing step (CPP). 331 The specification of the space variation of the coefficient is made in \mdl{ldftra} and \mdl{ldfdyn}, 332 or more precisely in include files \textit{traldf\_cNd.h90} and \textit{dynldf\_cNd.h90}, with N=1, 2 or 3. 333 The user can modify these include files as he/she wishes. 334 The way the mixing coefficient are set in the reference version can be briefly described as follows: 335 336 \subsubsection{Constant mixing coefficients (default option)} 337 When none of the \key{dynldf\_...} and \key{traldf\_...} keys are defined, 338 a constant value is used over the whole ocean for momentum and tracers, 339 which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist parameters. 340 341 \subsubsection{Vertically varying mixing coefficients (\protect\key{traldf\_c1d} and \key{dynldf\_c1d})} 342 The 1D option is only available when using the $z$-coordinate with full step. 343 Indeed in all the other types of vertical coordinate, 344 the depth is a 3D function of (\textbf{i},\textbf{j},\textbf{k}) and therefore, 345 introducing depth-dependent mixing coefficients will require 3D arrays. 346 In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced in which 347 the surface value is \np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value, 348 and the transition takes place around z=300~m with a width of 300~m 349 (\ie both the depth and the width of the inflection point are set to 300~m). 350 This profile is hard coded in file \textit{traldf\_c1d.h90}, but can be easily modified by users. 351 352 \subsubsection{Horizontally varying mixing coefficients (\protect\key{traldf\_c2d} and \protect\key{dynldf\_c2d})} 353 By default the horizontal variation of the eddy coefficient depends on the local mesh size and 318 The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. 319 The way the mixing coefficients are set in the reference version can be described as follows: 320 321 %% ================================================================================================= 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, 325 the laplacian viscosity operator uses $A^l$~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and 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}). 329 330 \begin{table}[htb] 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} 342 \end{table} 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})} 346 347 If constant, mixing coefficients are set thanks to a velocity and a length scales ($U_{scl}$, $L_{scl}$) such that: 348 349 \begin{equation} 350 \label{eq:LDF_constantah} 351 A_o^l = \left\{ 352 \begin{aligned} 353 & \frac{1}{2} U_{scl} L_{scl} & \text{for laplacian operator } \\ 354 & \frac{1}{12} U_{scl} L_{scl}^3 & \text{for bilaplacian operator } 355 \end{aligned} 356 \right. 357 \end{equation} 358 359 $U_{scl}$ 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})} 363 364 In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which 365 the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value, 366 and the transition takes place around z=500~m with a width of 200~m. 367 This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. 368 369 %% ================================================================================================= 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})} 371 372 In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and 354 373 the type of operator used: 355 374 \begin{equation} 356 \label{eq: title}375 \label{eq:LDF_title} 357 376 A_l = \left\{ 358 377 \begin{aligned} 359 & \frac{ \max(e_1,e_2)}{e_{max}} A_o^l& \text{for laplacian operator } \\360 & \frac{ \max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l& \text{for bilaplacian operator }378 & \frac{1}{2} U_{scl} \max(e_1,e_2) & \text{for laplacian operator } \\ 379 & \frac{1}{12} U_{scl} \max(e_1,e_2)^{3} & \text{for bilaplacian operator } 361 380 \end{aligned} 362 381 \right. 363 382 \end{equation} 364 where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked ocean domain, 365 and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) namelist parameter. 383 where $U_{scl}$ is a user defined velocity scale (\np{rn_Ud}{rn\_Ud}, \np{rn_Uv}{rn\_Uv}). 366 384 This variation is intended to reflect the lesser need for subgrid scale eddy mixing where 367 385 the grid size is smaller in the domain. 368 It was introduced in the context of the DYNAMO modelling project \citep{ Willebrand_al_PO01}.369 Note that such a grid scale dependance of mixing coefficients significantly increase the range of stability of370 model configurations presenting large changes in grid pacing such as global ocean models.386 It was introduced in the context of the DYNAMO modelling project \citep{willebrand.barnier.ea_PO01}. 387 Note that such a grid scale dependance of mixing coefficients significantly increases the range of stability of 388 model configurations presenting large changes in grid spacing such as global ocean models. 371 389 Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to 372 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}), 373 391 especially when using a bilaplacian operator. 374 392 375 Other formulations can be introduced by the user for a given configuration. 376 For example, in the ORCA2 global ocean model (see Configurations), 377 the laplacian viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and 378 decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. 379 This modification can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. 380 Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of 381 ORCA2 and ORCA05 (see \&namcfg namelist). 382 383 \subsubsection{Space varying mixing coefficients (\protect\key{traldf\_c3d} and \key{dynldf\_c3d})} 384 385 The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases, 386 \ie a hyperbolic tangent variation with depth associated with a grid size dependence of 387 the magnitude of the coefficient. 388 389 \subsubsection{Space and time varying mixing coefficients} 390 391 There is no default specification of space and time varying mixing coefficient. 392 The only case available is specific to the ORCA2 and ORCA05 global ocean configurations. 393 It provides only a tracer mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and 394 eddy induced velocity (ORCA05) that depends on the local growth rate of baroclinic instability. 395 This specification is actually used when an ORCA key and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined. 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})} 397 398 The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above, 399 \ie\ a hyperbolic tangent variation with depth associated with a grid size dependence of 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})} 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 $12$): 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 ?} 406 407 \begin{equation} 408 \label{eq:LDF_flowah} 409 A_l = \left\{ 410 \begin{aligned} 411 & \frac{1}{12} \lvert U \rvert e & \text{for laplacian operator } \\ 412 & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } 413 \end{aligned} 414 \right. 415 \end{equation} 416 417 %% ================================================================================================= 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 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.: 422 423 \begin{equation} 424 \label{eq:LDF_smag1} 425 \begin{split} 426 T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2 } \\ 427 L_{smag} & = \frac{1}{\pi}\frac{e_1 e_2}{e_1 + e_2} 428 \end{split} 429 \end{equation} 430 431 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} 435 A_{smag} = \left\{ 436 \begin{aligned} 437 & C^2 T_{smag}^{-1} L_{smag}^2 & \text{for laplacian operator } \\ 438 & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4 & \text{for bilaplacian operator } 439 \end{aligned} 440 \right. 441 \end{equation} 442 443 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} 446 \begin{aligned} 447 & C_{min} \frac{1}{2} \lvert U \rvert e < A_{smag} < C_{max} \frac{e^2}{ 8\rdt} & \text{for laplacian 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 } 449 \end{aligned} 450 \end{equation} 451 452 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 %% ================================================================================================= 455 \subsection{About space and time varying mixing coefficients} 396 456 397 457 The following points are relevant when the eddy coefficient varies spatially: 398 458 399 459 (1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and 400 divergent components of the horizontal current (see \autoref{subsec: PE_ldf}).460 divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). 401 461 Although the eddy coefficient could be set to different values in these two terms, 402 this option is not currently available. 462 this option is not currently available. 403 463 404 464 (2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of 405 465 the horizontal divergence for operators acting along model-surfaces are no longer satisfied 406 (\autoref{sec:dynldf_properties}). 407 408 (3) for isopycnal diffusion on momentum or tracers, an additional purely horizontal background diffusion with 409 uniform coefficient can be added by setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, 410 a background horizontal eddy viscosity or diffusivity coefficient 411 (namelist parameters whose default values are $0$). 412 However, the technique used to compute the isopycnal slopes is intended to get rid of such a background diffusion, 413 since it introduces spurious diapycnal diffusion (see \autoref{sec:LDF_slp}). 414 415 (4) when an eddy induced advection term is used (\key{traldf\_eiv}), 416 $A^{eiv}$, the eddy induced coefficient has to be defined. 417 Its space variations are controlled by the same CPP variable as for the eddy diffusivity coefficient 418 (\ie \key{traldf\_cNd}). 419 420 (5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value. 421 422 (6) it is possible to use both the laplacian and biharmonic operators concurrently. 423 424 (7) it is possible to run without explicit lateral diffusion on momentum 425 (\np{ln\_dynldf\_lap}\forcode{ = .?.}\np{ln\_dynldf\_bilap}\forcode{ = .false.}). 426 This is recommended when using the UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.}, 427 see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes. 428 429 % ================================================================ 430 % Eddy Induced Mixing 431 % ================================================================ 432 \section{Eddy induced velocity (\protect\mdl{traadv\_eiv}, \protect\mdl{ldfeiv})} 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})} 470 433 471 \label{sec:LDF_eiv} 434 472 473 \begin{listing} 474 \nlst{namtra_eiv} 475 \caption{\forcode{&namtra_eiv}} 476 \label{lst:namtra_eiv} 477 \end{listing} 478 435 479 %%gm from Triad appendix : to be incorporated.... 436 \ gmcomment{480 \cmtgm{ 437 481 Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}. 438 482 If none of the keys \key{traldf\_cNd}, N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and 439 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}. 440 484 If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal 441 485 scale factor according to \autoref{eq:title} … … 450 494 In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$, 451 495 where $f_{20}$ is the value of $f$ at $20^{\circ}$~N 452 } (\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. 453 497 } 454 498 455 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),499 When \citet{gent.mcwilliams_JPO90} diffusion is used (\np[=.true.]{ln_ldfeiv}{ln\_ldfeiv}), 456 500 an eddy induced tracer advection term is added, 457 501 the formulation of which depends on the slopes of iso-neutral surfaces. 458 502 Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 459 \ie \autoref{eq:ldfslp_geo} is used in $z$-coordinates, 460 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates. 461 The eddy induced velocity is given by: 462 \begin{equation} 463 \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} 464 509 \begin{split} 465 510 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ … … 468 513 \end{split} 469 514 \end{equation} 470 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{rn\_aeiv}, 471 a \textit{nam\_traldf} namelist parameter. 472 The three components of the eddy induced velocity are computed and 473 add to the eulerian velocity in \mdl{traadv\_eiv}. 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. 516 The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and 517 added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed. 474 518 This has been preferred to a separate computation of the advective trends associated with the eiv velocity, 475 519 since it allows us to take advantage of all the advection schemes offered for the tracers 476 520 (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in 477 previous releases of OPA \citep{ Madec1998}.521 previous releases of OPA \citep{madec.delecluse.ea_NPM98}. 478 522 This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of 479 paramount importance. 523 paramount importance. 480 524 481 525 At the surface, lateral and bottom boundaries, the eddy induced velocity, 482 and thus the advective eddy fluxes of heat and salt, are set to zero. 483 484 \biblio 485 486 \pindex 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})} 534 \label{sec:LDF_mle} 535 536 \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. 543 544 \colorbox{yellow}{TBC} 545 546 \subinc{\input{../../global/epilogue}} 487 547 488 548 \end{document}
Note: See TracChangeset
for help on using the changeset viewer.