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r10146 r10354 14 14 15 15 16 The lateral physics terms in the momentum and tracer equations have been 17 described in \autoref{eq:PE_zdf} and their discrete formulation in \autoref{sec:TRA_ldf} 18 and \autoref{sec:DYN_ldf}). In this section we further discuss each lateral physics option. 19 Choosing one lateral physics scheme means for the user defining, 20 (1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term) ; 21 (2) the direction along which the lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal surfaces) ; and 22 (3) the space and time variations of the eddy coefficients. 23 These three aspects of the lateral diffusion are set through namelist parameters 24 (see the \textit{\ngn{nam\_traldf}} and \textit{\ngn{nam\_dynldf}} below). 25 Note that this chapter describes the standard implementation of iso-neutral 26 tracer mixing, and Griffies's implementation, which is used if 27 \np{traldf\_grif}\forcode{ = .true.}, is described in Appdx\autoref{apdx:triad} 16 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:PE_zdf} and 17 their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}). 18 In this section we further discuss each lateral physics option. 19 Choosing one lateral physics scheme means for the user defining, 20 (1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term); 21 (2) the direction along which the lateral diffusive fluxes are evaluated 22 (model level, geopotential or isopycnal surfaces); and 23 (3) the space and time variations of the eddy coefficients. 24 These three aspects of the lateral diffusion are set through namelist parameters 25 (see the \textit{\ngn{nam\_traldf}} and \textit{\ngn{nam\_dynldf}} below). 26 Note that this chapter describes the standard implementation of iso-neutral tracer mixing, 27 and Griffies's implementation, which is used if \np{traldf\_grif}\forcode{ = .true.}, 28 is described in Appdx\autoref{apdx:triad} 28 29 29 30 %-----------------------------------nam_traldf - nam_dynldf-------------------------------------------- … … 45 46 Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. } 46 47 47 A direction for lateral mixing has to be defined when the desired operator does 48 not act along the model levels. This occurs when $(a)$ horizontal mixing is 49 required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) 50 in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required 51 whatever the vertical coordinate is. This direction of mixing is defined by its 52 slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the 53 quantity to be diffused. For a tracer, this leads to the following four slopes : 54 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:tra_ldf_iso}), while55 for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for 56 $ u$ and $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.48 A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. 49 This occurs when $(a)$ horizontal mixing is required on tracer or momentum 50 (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates, 51 and $(b)$ isoneutral mixing is required whatever the vertical coordinate is. 52 This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of 53 the cell of the quantity to be diffused. 54 For a tracer, this leads to the following four slopes: 55 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:tra_ldf_iso}), 56 while for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and 57 $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 57 58 58 59 %gm% add here afigure of the slope in i-direction … … 60 61 \subsection{Slopes for tracer geopotential mixing in the $s$-coordinate} 61 62 62 In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and 63 $r_2$ are the slopes between the geopotential and computational surfaces. 64 Their discrete formulation is found by locally solving \autoref{eq:tra_ldf_iso} 65 when the diffusive fluxes in the three directions are set to zero and $T$ is 66 assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the 67 depth of a $T$-point. 63 In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and $r_2$ are the slopes between 64 the geopotential and computational surfaces. 65 Their discrete formulation is found by locally solving \autoref{eq:tra_ldf_iso} when 66 the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform, 67 $i.e.$ a linear function of $z_T$, the depth of a $T$-point. 68 68 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 69 69 … … 89 89 %gm% caution I'm not sure the simplification was a good idea! 90 90 91 These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}\forcode{ = .true.}rue, 92 and either \np{ln\_traldf\_hor}\forcode{ = .true.} rue or \np{ln\_dynldf\_hor}\forcode{ = .true.}rue.91 These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}\forcode{ = .true.}rue, 92 and either \np{ln\_traldf\_hor}\forcode{ = .true.} or \np{ln\_dynldf\_hor}\forcode{ = .true.}. 93 93 94 94 \subsection{Slopes for tracer iso-neutral mixing} 95 95 \label{subsec:LDF_slp_iso} 96 In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral 97 and computational surfaces. Their formulation does not depend on the vertical 98 coordinate used. Their discrete formulation is found using the fact that the 99 diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density) 100 vanish. So, substituting $T$ by $\rho$ in \autoref{eq:tra_ldf_iso} and setting the 101 diffusive fluxes in the three directions to zero leads to the following definition for 102 the neutral slopes: 96 In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces. 97 Their formulation does not depend on the vertical coordinate used. 98 Their discrete formulation is found using the fact that the diffusive fluxes of 99 locally referenced potential density ($i.e.$ $in situ$ density) vanish. 100 So, substituting $T$ by $\rho$ in \autoref{eq:tra_ldf_iso} and setting the diffusive fluxes in 101 the three directions to zero leads to the following definition for the neutral slopes: 103 102 104 103 \begin{equation} \label{eq:ldfslp_iso} … … 128 127 %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. 129 128 130 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in 131 \autoref{eq:ldfslp_iso} has to be evaluated at the same local pressure (which, 132 in decibars, is approximated by the depth in meters in the model). Therefore 133 \autoref{eq:ldfslp_iso} cannot be used as such, but further transformation is 134 needed depending on the vertical coordinate used:129 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:ldfslp_iso} has to 130 be evaluated at the same local pressure 131 (which, in decibars, is approximated by the depth in meters in the model). 132 Therefore \autoref{eq:ldfslp_iso} cannot be used as such, 133 but further transformation is needed depending on the vertical coordinate used: 135 134 136 135 \begin{description} 137 136 138 \item[$z$-coordinate with full step : ] in \autoref{eq:ldfslp_iso} the densities139 appearing in the $i$ and $j$ derivatives are taken at the same depth, thus 140 the $in situ$ density can be used. This is not the case for the vertical 141 derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ 142 is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following 143 \citet{McDougall1987}(see \autoref{subsec:TRA_bn2}).144 145 \item[$z$-coordinate with partial step : ] this case is identical to the full step146 case except that at partial step level, the \emph{horizontal} density gradient 147 is evaluated as described in \autoref{sec:TRA_zpshde}.148 149 \item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,150 iso-neutral mixing is only employed for $s$-coordinates if the 151 Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.}; see Appdx \autoref{apdx:triad}). 152 In other words, iso-neutral mixing will only be accurately represented with a 153 linear equation of state (\np{nn\_eos}\forcode{ = 1..2}). In the case of a "true" equation 154 of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso} 155 will include a pressure dependent part, leading to the wrong evaluation of 156 the neutral slopes.137 \item[$z$-coordinate with full step: ] 138 in \autoref{eq:ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, 139 thus the $in situ$ density can be used. 140 This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, 141 where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{McDougall1987} 142 (see \autoref{subsec:TRA_bn2}). 143 144 \item[$z$-coordinate with partial step: ] 145 this case is identical to the full step case except that at partial step level, 146 the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}. 147 148 \item[$s$- or hybrid $s$-$z$- coordinate: ] 149 in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 150 the Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.}; 151 see Appdx \autoref{apdx:triad}). 152 In other words, iso-neutral mixing will only be accurately represented with a linear equation of state 153 (\np{nn\_eos}\forcode{ = 1..2}). 154 In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso} 155 will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. 157 156 158 157 %gm% 159 Note: The solution for $s$-coordinate passes trough the use of different 160 (and better) expression for the constraint on iso-neutral fluxes. Following 161 \citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral 162 diffusive flux of locally referenced potential density, we stay in the $T$-$S$ 163 plane and consider the balance between the neutral direction diffusive fluxes 164 of potential temperature and salinity: 158 Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for 159 the constraint on iso-neutral fluxes. 160 Following \citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral diffusive flux of 161 locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between 162 the neutral direction diffusive fluxes of potential temperature and salinity: 165 163 \begin{equation} 166 164 \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) … … 194 192 \end{split} 195 193 \end{equation} 196 where $\alpha$ and $\beta$, the thermal expansion and saline contraction 197 coefficients introduced in \autoref{subsec:TRA_bn2}, have to be evaluated at the three 198 velocity points. In order to save computation time, they should be approximated 199 by the mean of their values at $T$-points (for example in the case of $\alpha$: 200 $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$ 201 and $\alpha_w=\overline{\alpha_T}^{k+1/2}$). 202 203 Note that such a formulation could be also used in the $z$-coordinate and 204 $z$-coordinate with partial steps cases. 194 where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in 195 \autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points. 196 In order to save computation time, they should be approximated by the mean of their values at $T$-points 197 (for example in the case of $\alpha$: 198 $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$ and 199 $\alpha_w=\overline{\alpha_T}^{k+1/2}$). 200 201 Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases. 205 202 206 203 \end{description} 207 204 208 This implementation is a rather old one. It is similar to the one 209 proposed by Cox [1987], except for the background horizontal 210 diffusion. Indeed, the Cox implementation of isopycnal diffusion in 211 GFDL-type models requires a minimum background horizontal diffusion 212 for numerical stability reasons. To overcome this problem, several 213 techniques have been proposed in which the numerical schemes of the 214 ocean model are modified \citep{Weaver_Eby_JPO97, 215 Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if 216 \np{traldf\_grif\_iso} is set true; see Appdx \autoref{apdx:triad}. Here, 217 another strategy is presented \citep{Lazar_PhD97}: a local 218 filtering of the iso-neutral slopes (made on 9 grid-points) prevents 219 the development of grid point noise generated by the iso-neutral 220 diffusion operator (\autoref{fig:LDF_ZDF1}). This allows an 221 iso-neutral diffusion scheme without additional background horizontal 222 mixing. This technique can be viewed as a diffusion operator that acts 223 along large-scale (2~$\Delta$x) \gmcomment{2deltax doesnt seem very 224 large scale} iso-neutral surfaces. The diapycnal diffusion required 225 for numerical stability is thus minimized and its net effect on the 226 flow is quite small when compared to the effect of an horizontal 227 background mixing. 228 229 Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, 205 This implementation is a rather old one. 206 It is similar to the one proposed by Cox [1987], except for the background horizontal diffusion. 207 Indeed, the Cox implementation of isopycnal diffusion in GFDL-type models requires 208 a minimum background horizontal diffusion for numerical stability reasons. 209 To overcome this problem, several techniques have been proposed in which the numerical schemes of 210 the ocean model are modified \citep{Weaver_Eby_JPO97, Griffies_al_JPO98}. 211 Griffies's scheme is now available in \NEMO if \np{traldf\_grif\_iso} is set true; see Appdx \autoref{apdx:triad}. 212 Here, another strategy is presented \citep{Lazar_PhD97}: 213 a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of 214 grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}). 215 This allows an iso-neutral diffusion scheme without additional background horizontal mixing. 216 This technique can be viewed as a diffusion operator that acts along large-scale 217 (2~$\Delta$x) \gmcomment{2deltax doesnt seem very large scale} iso-neutral surfaces. 218 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. 219 220 Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, 230 221 contrary to the \citet{Griffies_al_JPO98} operator which has that property. 231 222 … … 234 225 \includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1} 235 226 \caption { \protect\label{fig:LDF_ZDF1} 236 averaging procedure for isopycnal slope computation.}227 averaging procedure for isopycnal slope computation.} 237 228 \end{center} \end{figure} 238 229 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 252 243 % surface motivates this flattening of isopycnals near the surface). 253 244 254 For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also 255 be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear 256 fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter 257 decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the 258 surface motivates this flattening of isopycnals near the surface).245 For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also be bounded by 246 $1/100$ everywhere. 247 This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to 248 $1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean 249 (the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface). 259 250 260 251 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 261 \begin{figure}[!ht] \begin{center} 262 \includegraphics[width=0.70\textwidth]{Fig_eiv_slp} 263 \caption { \protect\label{fig:eiv_slp} 264 Vertical profile of the slope used for lateral mixing in the mixed layer : 265 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 266 which has to be adjusted at the surface boundary (i.e. it must tend to zero at the 267 surface since there is no mixing across the air-sea interface: wall boundary 268 condition). Nevertheless, the profile between the surface zero value and the interior 269 iso-neutral one is unknown, and especially the value at the base of the mixed layer ; 270 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and 271 imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in 272 \NEMO: a linear decrease of the slope from zero at the surface to its ocean interior 273 value computed just below the mixed layer. Note the huge change in the slope at the 274 base of the mixed layer between \textit{(b)} and \textit{(c)}.} 275 \end{center} \end{figure} 252 \begin{figure}[!ht] 253 \begin{center} 254 \includegraphics[width=0.70\textwidth]{Fig_eiv_slp} 255 \caption { \protect\label{fig:eiv_slp} 256 Vertical profile of the slope used for lateral mixing in the mixed layer: 257 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 258 which has to be adjusted at the surface boundary 259 (i.e. it must tend to zero at the surface since there is no mixing across the air-sea interface: 260 wall boundary condition). 261 Nevertheless, the profile between the surface zero value and the interior iso-neutral one is unknown, 262 and especially the value at the base of the mixed layer; 263 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and 264 imposing a maximum slope of 1/100; 265 \textit{(c)} profile of slope actually used in \NEMO: a linear decrease of the slope from 266 zero at the surface to its ocean interior value computed just below the mixed layer. 267 Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}.} 268 \end{center} 269 \end{figure} 276 270 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 277 271 278 \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs 272 \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.} 279 273 280 274 \subsection{Slopes for momentum iso-neutral mixing} 281 275 282 The iso-neutral diffusion operator on momentum is the same as the one used on 283 tracers but applied to each component of the velocity separately (see 284 \autoref{eq:dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). The slopes between the 285 surface along which the diffusion operator acts and the surface of computation 286 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the 287 $u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component. 288 They are computed from the slopes used for tracer diffusion, $i.e.$289 \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso} :276 The iso-neutral diffusion operator on momentum is the same as the one used on tracers but 277 applied to each component of the velocity separately 278 (see \autoref{eq:dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). 279 The slopes between the surface along which the diffusion operator acts and the surface of computation 280 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and 281 \textit{vw}- points for the $v$-component. 282 They are computed from the slopes used for tracer diffusion, 283 $i.e.$ \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso} : 290 284 291 285 \begin{equation} \label{eq:ldfslp_dyn} … … 298 292 \end{equation} 299 293 300 The major issue remaining is in the specification of the boundary conditions. 301 The same boundary conditions are chosen as those used for lateral 302 diffusion along model level surfaces, i.e. using the shear computed along 303 the model levels and with no additional friction at the ocean bottom (see 304 \autoref{sec:LBC_coast}). 294 The major issue remaining is in the specification of the boundary conditions. 295 The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces, 296 $i.e.$ using the shear computed along the model levels and with no additional friction at the ocean bottom 297 (see \autoref{sec:LBC_coast}). 305 298 306 299 … … 319 312 \label{sec:LDF_coef} 320 313 321 Introducing a space variation in the lateral eddy mixing coefficients changes 322 the model core memory requirement, adding up to four extra three-dimensional323 arrays for the geopotential or isopycnal second order operator applied to 324 momentum. Six CPP keys control the space variation of eddy coefficients: 325 three for momentum and three for tracer. The three choices allow: 326 a space variation in the three space directions (\key{traldf\_c3d}, \key{dynldf\_c3d}), 327 in the horizontal plane (\key{traldf\_c2d}, \key{dynldf\_c2d}), 328 or in the vertical only (\key{traldf\_c1d}, \key{dynldf\_c1d}). 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}). 329 322 The default option is a constant value over the whole ocean on both momentum and tracers. 330 323 331 The number of additional arrays that have to be defined and the gridpoint 332 position at which they are defined depend on both the space variation chosen 333 and the type of operator used. The resulting eddy viscosity and diffusivity 334 coefficients can be a function of more than one variable. Changes in the 335 computer code when switching from one option to another have been 336 minimized by introducing the eddy coefficients as statement functions 337 (include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions 338 are replaced by their actual meaning during the preprocessing step (CPP). 339 The specification of the space variation of the coefficient is made in 340 \mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files 341 \hf{traldf\_cNd} and \hf{dynldf\_cNd}, with N=1, 2 or 3. 342 The user can modify these include files as he/she wishes. The way the 343 mixing coefficient are set in the reference version can be briefly described 344 as follows: 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 \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). 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 \hf{traldf\_cNd} and \hf{dynldf\_cNd}, 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: 345 335 346 336 \subsubsection{Constant mixing coefficients (default option)} 347 When none of the \key{dynldf\_...} and \key{traldf\_...} keys are 348 defined, a constant value is used over the whole ocean for momentum and 349 tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist 350 parameters. 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. 351 340 352 341 \subsubsection{Vertically varying mixing coefficients (\protect\key{traldf\_c1d} and \key{dynldf\_c1d})} 353 The 1D option is only available when using the $z$-coordinate with full step. 354 Indeed in all the other types of vertical coordinate, the depth is a 3D function355 of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent 356 mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation 357 of the lateral mixing coefficient is introduced in which the surface value is 358 \np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value, 359 and the transition takes place around z=300~m with a width of 300~m 360 ($i.e.$ both the depth and the width of the inflection point are set to 300~m). 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 ($i.e.$ both the depth and the width of the inflection point are set to 300~m). 361 350 This profile is hard coded in file \hf{traldf\_c1d}, but can be easily modified by users. 362 351 363 352 \subsubsection{Horizontally varying mixing coefficients (\protect\key{traldf\_c2d} and \protect\key{dynldf\_c2d})} 364 By default the horizontal variation of the eddy coefficient depends on the local mesh 365 size andthe type of operator used:353 By default the horizontal variation of the eddy coefficient depends on the local mesh size and 354 the type of operator used: 366 355 \begin{equation} \label{eq:title} 367 356 A_l = \left\{ … … 371 360 \end{aligned} \right. 372 361 \end{equation} 373 where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked 374 ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) 375 namelist parameter. This variation is intended to reflect the lesser need for subgrid 376 scale eddy mixing where the grid size is smaller in the domain. It was introduced in 377 the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}. 378 Note that such a grid scale dependance of mixing coefficients significantly increase 379 the range of stability of model configurations presenting large changes in grid pacing 380 such as global ocean models. Indeed, in such a case, a constant mixing coefficient 381 can lead to a blow up of the model due to large coefficient compare to the smallest 382 grid size (see \autoref{sec:STP_forward_imp}), especially when using a bilaplacian operator. 383 384 Other formulations can be introduced by the user for a given configuration. 385 For example, in the ORCA2 global ocean model (see Configurations), the laplacian 386 viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ 387 north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s 388 at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification 389 can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. 390 Similar modified horizontal variations can be found with the Antarctic or Arctic 391 sub-domain options of ORCA2 and ORCA05 (see \&namcfg namelist). 362 where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked ocean domain, 363 and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) namelist parameter. 364 This variation is intended to reflect the lesser need for subgrid scale eddy mixing where 365 the grid size is smaller in the domain. 366 It was introduced in the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}. 367 Note that such a grid scale dependance of mixing coefficients significantly increase the range of stability of 368 model configurations presenting large changes in grid pacing such as global ocean models. 369 Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to 370 large coefficient compare to the smallest grid size (see \autoref{sec:STP_forward_imp}), 371 especially when using a bilaplacian operator. 372 373 Other formulations can be introduced by the user for a given configuration. 374 For example, in the ORCA2 global ocean model (see Configurations), 375 the laplacian viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and 376 decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. 377 This modification can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. 378 Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of 379 ORCA2 and ORCA05 (see \&namcfg namelist). 392 380 393 381 \subsubsection{Space varying mixing coefficients (\protect\key{traldf\_c3d} and \key{dynldf\_c3d})} 394 382 395 The 3D space variation of the mixing coefficient is simply the combination of the 396 1D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with 397 a grid size dependence ofthe magnitude of the coefficient.383 The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases, 384 $i.e.$ a hyperbolic tangent variation with depth associated with a grid size dependence of 385 the magnitude of the coefficient. 398 386 399 387 \subsubsection{Space and time varying mixing coefficients} 400 388 401 389 There is no default specification of space and time varying mixing coefficient. 402 The only case available is specific to the ORCA2 and ORCA05 global ocean 403 configurations. It provides only a tracer 404 mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and 405 eddy induced velocity (ORCA05) that depends on the local growth rate of 406 baroclinic instability. This specification is actually used when an ORCA key 407 and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined. 390 The only case available is specific to the ORCA2 and ORCA05 global ocean configurations. 391 It provides only a tracer mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and 392 eddy induced velocity (ORCA05) that depends on the local growth rate of baroclinic instability. 393 This specification is actually used when an ORCA key and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined. 408 394 409 395 $\ $\newline % force a new ligne … … 411 397 The following points are relevant when the eddy coefficient varies spatially: 412 398 413 (1) the momentum diffusion operator acting along model level surfaces is 414 written in terms of curl and divergent components of the horizontal current 415 (see \autoref{subsec:PE_ldf}). Although the eddy coefficient could be set to different values 416 in these two terms, this option is not currently available. 417 418 (2) with an horizontally varying viscosity, the quadratic integral constraints 419 on enstrophy and on the square of the horizontal divergence for operators 420 acting along model-surfaces are no longer satisfied 399 (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}). 401 Although the eddy coefficient could be set to different values in these two terms, 402 this option is not currently available. 403 404 (2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of 405 the horizontal divergence for operators acting along model-surfaces are no longer satisfied 421 406 (\autoref{sec:dynldf_properties}). 422 407 423 (3) for isopycnal diffusion on momentum or tracers, an additional purely 424 horizontal background diffusion with uniform coefficient can be added by 425 setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal 426 eddy viscosity or diffusivity coefficient (namelist parameters whose default 427 values are $0$). However, the technique used to compute the isopycnal 428 slopes is intended to get rid of such a background diffusion, since it introduces 429 spurious diapycnal diffusion (see \autoref{sec:LDF_slp}). 430 431 (4) when an eddy induced advection term is used (\key{traldf\_eiv}), $A^{eiv}$, 432 the eddy induced coefficient has to be defined. Its space variations are controlled 433 by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$ 434 \key{traldf\_cNd}). 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 ($i.e.$ \key{traldf\_cNd}). 435 419 436 420 (5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value. … … 438 422 (6) it is possible to use both the laplacian and biharmonic operators concurrently. 439 423 440 (7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap}\forcode{ =441 }\np{ln\_dynldf\_bilap}\forcode{ = .false.}). This is recommended when using the UBS advection 442 scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.}, see \autoref{subsec:DYN_adv_ubs}) 443 and can be useful for testing purposes.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. 444 428 445 429 % ================================================================ … … 451 435 %%gm from Triad appendix : to be incorporated.... 452 436 \gmcomment{ 453 Values of iso-neutral diffusivity and GM coefficient are set as 454 described in \autoref{sec:LDF_coef}. If none of the keys \key{traldf\_cNd}, 455 N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and 456 GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and 457 \np{rn\_aeiv\_0}. If 2D-varying coefficients are set with 458 \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal 459 scale factor according to \autoref{eq:title} \footnote{Except in global ORCA 460 $0.5^{\circ}$ runs with \key{traldf\_eiv}, where 461 $A_l$ is set like $A_e$ but with a minimum vale of 462 $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$}. In idealised setups with 463 \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} 464 is set in the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is 465 instead set from the Held-Larichev parameterisation\footnote{In this 466 case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further 467 reduced by a factor $|f/f_{20}|$, where $f_{20}$ is the value of $f$ 468 at $20^{\circ}$~N} (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored 469 unless it is zero. 437 Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}. 438 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}. 440 If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal 441 scale factor according to \autoref{eq:title} \footnote{ 442 Except in global ORCA $0.5^{\circ}$ runs with \key{traldf\_eiv}, 443 where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$ 444 }. 445 In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in 446 the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from 447 the Held-Larichev parameterisation \footnote{ 448 In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$, 449 where $f_{20}$ is the value of $f$ at $20^{\circ}$~N 450 } (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored unless it is zero. 470 451 } 471 452 472 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), 473 an eddy induced tracer advection term is added, the formulation of which 474 depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral 475 mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$ 476 \autoref{eq:ldfslp_geo} is used in $z$-coordinates, and the sum \autoref{eq:ldfslp_geo} 477 + \autoref{eq:ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by: 453 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), 454 an eddy induced tracer advection term is added, 455 the formulation of which depends on the slopes of iso-neutral surfaces. 456 Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 457 $i.e.$ \autoref{eq:ldfslp_geo} is used in $z$-coordinates, 458 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates. 459 The eddy induced velocity is given by: 478 460 \begin{equation} \label{eq:ldfeiv} 479 461 \begin{split} … … 483 465 \end{split} 484 466 \end{equation} 485 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set 486 through \np{rn\_aeiv}, a \textit{nam\_traldf} namelist parameter. 487 The three components of the eddy induced velocity are computed and add488 to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a 489 separate computation of the advective trends associated with the eiv velocity, 490 since it allows us to take advantage of all the advection schemes offered for 491 the tracers (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection 492 scheme as in previous releases of OPA \citep{Madec1998}. This is particularly 493 useful for passive tracers where \emph{positivity} of the advection scheme is 494 ofparamount importance.495 496 At the surface, lateral and bottom boundaries, the eddy induced velocity, 467 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{rn\_aeiv}, 468 a \textit{nam\_traldf} namelist parameter. 469 The three components of the eddy induced velocity are computed and 470 add to the eulerian velocity in \mdl{traadv\_eiv}. 471 This has been preferred to a separate computation of the advective trends associated with the eiv velocity, 472 since it allows us to take advantage of all the advection schemes offered for the tracers 473 (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in 474 previous releases of OPA \citep{Madec1998}. 475 This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of 476 paramount importance. 477 478 At the surface, lateral and bottom boundaries, the eddy induced velocity, 497 479 and thus the advective eddy fluxes of heat and salt, are set to zero. 498 480
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