Changeset 11564 for NEMO/branches/2019/dev_r10973_AGRIF-01_jchanut_small_jpi_jpj/doc/latex/NEMO/subfiles/chap_LDF.tex
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NEMO/branches/2019/dev_r10973_AGRIF-01_jchanut_small_jpi_jpj/doc/latex/NEMO/subfiles/chap_LDF.tex
r10442 r11564 9 9 \label{chap:LDF} 10 10 11 \ minitoc11 \chaptertoc 12 12 13 13 \newpage 14 14 15 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq: PE_zdf} and15 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and 16 16 their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}). 17 17 In this section we further discuss each lateral physics option. … … 22 22 (3) the space and time variations of the eddy coefficients. 23 23 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} 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:TRIADS} 28 29 %-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- 30 34 31 %-------------------------------------------------------------------------------------------------------------- 35 32 33 % ================================================================ 34 % Lateral Mixing Operator 35 % ================================================================ 36 \section[Lateral mixing operators] 37 {Lateral mixing operators} 38 \label{sec:LDF_op} 39 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}. 40 41 \subsection[No lateral mixing (\forcode{ln_traldf_OFF}, \forcode{ln_dynldf_OFF})] 42 {No lateral mixing (\protect\np{ln\_traldf\_OFF}, \protect\np{ln\_dynldf\_OFF})} 43 44 It is possible to run without explicit lateral diffusion on tracers (\protect\np{ln\_traldf\_OFF}\forcode{=.true.}) and/or 45 momentum (\protect\np{ln\_dynldf\_OFF}\forcode{=.true.}). The latter option is even recommended if using the 46 UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{=.true.}, 47 see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes. 48 49 \subsection[Laplacian mixing (\forcode{ln_traldf_lap}, \forcode{ln_dynldf_lap})] 50 {Laplacian mixing (\protect\np{ln\_traldf\_lap}, \protect\np{ln\_dynldf\_lap})} 51 Setting \protect\np{ln\_traldf\_lap}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_lap}\forcode{=.true.} enables 52 a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine 53 Laplacian and Bilaplacian operators for the same variable. 54 55 \subsection[Bilaplacian mixing (\forcode{ln_traldf_blp}, \forcode{ln_dynldf_blp})] 56 {Bilaplacian mixing (\protect\np{ln\_traldf\_blp}, \protect\np{ln\_dynldf\_blp})} 57 Setting \protect\np{ln\_traldf\_blp}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_blp}\forcode{=.true.} enables 58 a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice. 59 We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. 36 60 37 61 % ================================================================ 38 62 % Direction of lateral Mixing 39 63 % ================================================================ 40 \section{Direction of lateral mixing (\protect\mdl{ldfslp})} 64 \section[Direction of lateral mixing (\textit{ldfslp.F90})] 65 {Direction of lateral mixing (\protect\mdl{ldfslp})} 41 66 \label{sec:LDF_slp} 42 67 … … 44 69 \gmcomment{ 45 70 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.71 Better work can be achieved by using \citet{griffies.gnanadesikan.ea_JPO98, griffies_bk04} iso-neutral scheme. 47 72 } 48 73 … … 54 79 the cell of the quantity to be diffused. 55 80 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}),81 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), 57 82 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 $v$. 83 $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 59 84 60 85 %gm% add here afigure of the slope in i-direction … … 62 87 \subsection{Slopes for tracer geopotential mixing in the $s$-coordinate} 63 88 64 In $s$-coordinates, geopotential mixing (\ie horizontal mixing) $r_1$ and $r_2$ are the slopes between89 In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between 65 90 the geopotential and computational surfaces. 66 Their discrete formulation is found by locally solving \autoref{eq: tra_ldf_iso} when91 Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when 67 92 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.93 \ie\ a linear function of $z_T$, the depth of a $T$-point. 69 94 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 70 95 71 96 \begin{equation} 72 \label{eq: ldfslp_geo}97 \label{eq:LDF_slp_geo} 73 98 \begin{aligned} 74 99 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} … … 85 110 \end{equation} 86 111 87 %gm% caution I'm not sure the simplification was a good idea! 88 89 These slopes are computed once in \rou{ldf slp\_init} when \np{ln\_sco}\forcode{ = .true.}rue,90 and either \np{ln\_traldf\_hor}\forcode{ = .true.} or \np{ln\_dynldf\_hor}\forcode{ = .true.}.112 %gm% caution I'm not sure the simplification was a good idea! 113 114 These slopes are computed once in \rou{ldf\_slp\_init} when \np{ln\_sco}\forcode{=.true.}, 115 and either \np{ln\_traldf\_hor}\forcode{=.true.} or \np{ln\_dynldf\_hor}\forcode{=.true.}. 91 116 92 117 \subsection{Slopes for tracer iso-neutral mixing} … … 96 121 Their formulation does not depend on the vertical coordinate used. 97 122 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 in123 locally referenced potential density (\ie\ $in situ$ density) vanish. 124 So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in 100 125 the three directions to zero leads to the following definition for the neutral slopes: 101 126 102 127 \begin{equation} 103 \label{eq: ldfslp_iso}128 \label{eq:LDF_slp_iso} 104 129 \begin{split} 105 130 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} … … 117 142 118 143 %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 to144 %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. 145 146 %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). 147 148 %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. 149 150 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to 126 151 be evaluated at the same local pressure 127 152 (which, in decibars, is approximated by the depth in meters in the model). 128 Therefore \autoref{eq: ldfslp_iso} cannot be used as such,153 Therefore \autoref{eq:LDF_slp_iso} cannot be used as such, 129 154 but further transformation is needed depending on the vertical coordinate used: 130 155 … … 132 157 133 158 \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,159 in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, 135 160 thus the $in situ$ density can be used. 136 161 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}). 162 where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87} 163 (see \autoref{subsec:TRA_bn2}). 139 164 140 165 \item[$z$-coordinate with partial step: ] … … 144 169 \item[$s$- or hybrid $s$-$z$- coordinate: ] 145 170 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}).171 the Griffies scheme is used (\np{ln\_traldf\_triad}\forcode{=.true.}; 172 see \autoref{apdx:TRIADS}). 148 173 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}174 (\np{ln\_seos}\forcode{=.true.}). 175 In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso} 151 176 will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. 152 177 153 %gm% 178 %gm% 154 179 Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for 155 180 the constraint on iso-neutral fluxes. 156 Following \citet{ Griffies_Bk04}, instead of specifying directly that there is a zero neutral diffusive flux of181 Following \citet{griffies_bk04}, instead of specifying directly that there is a zero neutral diffusive flux of 157 182 locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between 158 183 the neutral direction diffusive fluxes of potential temperature and salinity: … … 165 190 166 191 \[ 167 % \label{eq: ldfslp_iso2}192 % \label{eq:LDF_slp_iso2} 168 193 \begin{split} 169 194 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac … … 197 222 198 223 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 requires224 It is similar to the one proposed by \citet{cox_OM87}, except for the background horizontal diffusion. 225 Indeed, the \citet{cox_OM87} implementation of isopycnal diffusion in GFDL-type models requires 201 226 a minimum background horizontal diffusion for numerical stability reasons. 202 227 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}:228 the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. 229 Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:TRIADS}. 230 Here, another strategy is presented \citep{lazar_phd97}: 206 231 a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of 207 232 grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}). … … 212 237 213 238 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.239 contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property. 215 240 216 241 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 217 242 \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} 243 \centering 244 \includegraphics[width=\textwidth]{Fig_LDF_ZDF1} 245 \caption{Averaging procedure for isopycnal slope computation} 246 \label{fig:LDF_ZDF1} 225 247 \end{figure} 226 248 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 227 249 228 %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. 250 %There are three additional questions about the slope calculation. 251 %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. 252 %Second, numerical stability issues also require a bound on slopes. 231 253 %Third, the question of boundary condition specified on slopes... 232 254 … … 235 257 236 258 237 % 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 259 % In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04}, 260 % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly 261 % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the 240 262 % surface motivates this flattening of isopycnals near the surface). 241 263 242 For numerical stability reasons \citep{ Cox1987, Griffies_Bk04}, the slopes must also be bounded by243 $1/100$everywhere.264 For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by 265 the namelist scalar \np{rn\_slpmax} (usually $1/100$) everywhere. 244 266 This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to 245 267 $1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean 246 268 (the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface). 269 \colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.} 247 270 248 271 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 249 272 \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} 273 \centering 274 \includegraphics[width=\textwidth]{Fig_eiv_slp} 275 \caption[Vertical profile of the slope used for lateral mixing in the mixed layer]{ 276 Vertical profile of the slope used for lateral mixing in the mixed layer: 277 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 278 which has to be adjusted at the surface boundary 279 \ie\ it must tend to zero at the surface since there is no mixing across the air-sea interface: 280 wall boundary condition). 281 Nevertheless, 282 the profile between the surface zero value and the interior iso-neutral one is unknown, 283 and especially the value at the base of the mixed layer; 284 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and 285 imposing a maximum slope of 1/100; 286 \textit{(c)} profile of slope actually used in \NEMO: 287 a linear decrease of the slope from zero at the surface to 288 its ocean interior value computed just below the mixed layer. 289 Note the huge change in the slope at the base of the mixed layer between 290 \textit{(b)} and \textit{(c)}.} 291 \label{fig:LDF_eiv_slp} 268 292 \end{figure} 269 293 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 275 299 The iso-neutral diffusion operator on momentum is the same as the one used on tracers but 276 300 applied to each component of the velocity separately 277 (see \autoref{eq: dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}).301 (see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). 278 302 The slopes between the surface along which the diffusion operator acts and the surface of computation 279 303 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and 280 304 \textit{vw}- points for the $v$-component. 281 305 They are computed from the slopes used for tracer diffusion, 282 \ie \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso}:306 \ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}: 283 307 284 308 \[ 285 % \label{eq: ldfslp_dyn}309 % \label{eq:LDF_slp_dyn} 286 310 \begin{aligned} 287 311 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ … … 294 318 The major issue remaining is in the specification of the boundary conditions. 295 319 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 bottom320 \ie\ using the shear computed along the model levels and with no additional friction at the ocean bottom 297 321 (see \autoref{sec:LBC_coast}). 298 322 299 323 300 324 % ================================================================ 301 % Lateral Mixing Operator302 % ================================================================303 \section{Lateral mixing operators (\protect\mdl{traldf}, \protect\mdl{traldf}) }304 \label{sec:LDF_op}305 306 307 308 % ================================================================309 325 % Lateral Mixing Coefficients 310 326 % ================================================================ 311 \section{Lateral mixing coefficient (\protect\mdl{ldftra}, \protect\mdl{ldfdyn}) } 327 \section[Lateral mixing coefficient (\forcode{nn_aht_ijk_t}, \forcode{nn_ahm_ijk_t})] 328 {Lateral mixing coefficient (\protect\np{nn\_aht\_ijk\_t}, \protect\np{nn\_ahm\_ijk\_t})} 312 329 \label{sec:LDF_coef} 313 330 314 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 331 The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. 332 The way the mixing coefficients are set in the reference version can be described as follows: 333 334 \subsection[Mixing coefficients read from file (\forcode{nn_aht_ijk_t=-20, -30}, \forcode{nn_ahm_ijk_t=-20,-30})] 335 { Mixing coefficients read from file (\protect\np{nn\_aht\_ijk\_t}\forcode{=-20, -30}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=-20, -30})} 336 337 Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model, 338 the laplacian viscosity operator uses $A^l$~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and 339 decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}. 340 Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05. 341 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}). 342 343 %-------------------------------------------------TABLE--------------------------------------------------- 344 \begin{table}[htb] 345 \centering 346 \begin{tabular}{|l|l|l|l|} 347 \hline 348 Namelist parameter & Input filename & dimensions & variable names \\ \hline 349 \np{nn\_ahm\_ijk\_t}\forcode{=-20} & \forcode{eddy_viscosity_2D.nc } & $(i,j)$ & \forcode{ahmt_2d, ahmf_2d} \\ \hline 350 \np{nn\_aht\_ijk\_t}\forcode{=-20} & \forcode{eddy_diffusivity_2D.nc } & $(i,j)$ & \forcode{ahtu_2d, ahtv_2d} \\ \hline 351 \np{nn\_ahm\_ijk\_t}\forcode{=-30} & \forcode{eddy_viscosity_3D.nc } & $(i,j,k)$ & \forcode{ahmt_3d, ahmf_3d} \\ \hline 352 \np{nn\_aht\_ijk\_t}\forcode{=-30} & \forcode{eddy_diffusivity_3D.nc } & $(i,j,k)$ & \forcode{ahtu_3d, ahtv_3d} \\ \hline 353 \end{tabular} 354 \caption{Description of expected input files if mixing coefficients are read from NetCDF files} 355 \label{tab:LDF_files} 356 \end{table} 357 %-------------------------------------------------------------------------------------------------------------- 358 359 \subsection[Constant mixing coefficients (\forcode{nn_aht_ijk_t=0}, \forcode{nn_ahm_ijk_t=0})] 360 { Constant mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=0}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=0})} 361 362 If constant, mixing coefficients are set thanks to a velocity and a length scales ($U_{scl}$, $L_{scl}$) such that: 363 364 \begin{equation} 365 \label{eq:LDF_constantah} 366 A_o^l = \left\{ 367 \begin{aligned} 368 & \frac{1}{2} U_{scl} L_{scl} & \text{for laplacian operator } \\ 369 & \frac{1}{12} U_{scl} L_{scl}^3 & \text{for bilaplacian operator } 370 \end{aligned} 371 \right. 372 \end{equation} 373 374 $U_{scl}$ and $L_{scl}$ are given by the namelist parameters \np{rn\_Ud}, \np{rn\_Uv}, \np{rn\_Ld} and \np{rn\_Lv}. 375 376 \subsection[Vertically varying mixing coefficients (\forcode{nn_aht_ijk_t=10}, \forcode{nn_ahm_ijk_t=10})] 377 {Vertically varying mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=10}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=10})} 378 379 In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which 380 the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value, 381 and the transition takes place around z=500~m with a width of 200~m. 382 This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. 383 384 \subsection[Mesh size dependent mixing coefficients (\forcode{nn_aht_ijk_t=20}, \forcode{nn_ahm_ijk_t=20})] 385 {Mesh size dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=20}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=20})} 386 387 In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and 354 388 the type of operator used: 355 389 \begin{equation} 356 \label{eq: title}390 \label{eq:LDF_title} 357 391 A_l = \left\{ 358 392 \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 }393 & \frac{1}{2} U_{scl} \max(e_1,e_2) & \text{for laplacian operator } \\ 394 & \frac{1}{12} U_{scl} \max(e_1,e_2)^{3} & \text{for bilaplacian operator } 361 395 \end{aligned} 362 396 \right. 363 397 \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. 398 where $U_{scl}$ is a user defined velocity scale (\np{rn\_Ud}, \np{rn\_Uv}). 366 399 This variation is intended to reflect the lesser need for subgrid scale eddy mixing where 367 400 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.401 It was introduced in the context of the DYNAMO modelling project \citep{willebrand.barnier.ea_PO01}. 402 Note that such a grid scale dependance of mixing coefficients significantly increases the range of stability of 403 model configurations presenting large changes in grid spacing such as global ocean models. 371 404 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}),405 large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}), 373 406 especially when using a bilaplacian operator. 374 407 375 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. 408 \colorbox{yellow}{CASE \np{nn\_aht\_ijk\_t} = 21 to be added} 409 410 \subsection[Mesh size and depth dependent mixing coefficients (\forcode{nn_aht_ijk_t=30}, \forcode{nn_ahm_ijk_t=30})] 411 {Mesh size and depth dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=30}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=30})} 412 413 The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above, 414 \ie\ a hyperbolic tangent variation with depth associated with a grid size dependence of 415 the magnitude of the coefficient. 416 417 \subsection[Velocity dependent mixing coefficients (\forcode{nn_aht_ijk_t=31}, \forcode{nn_ahm_ijk_t=31})] 418 {Flow dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=31}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=31})} 419 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 $12$): 420 \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 ?} 421 422 \begin{equation} 423 \label{eq:LDF_flowah} 424 A_l = \left\{ 425 \begin{aligned} 426 & \frac{1}{12} \lvert U \rvert e & \text{for laplacian operator } \\ 427 & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } 428 \end{aligned} 429 \right. 430 \end{equation} 431 432 \subsection[Deformation rate dependent viscosities (\forcode{nn_ahm_ijk_t=32})] 433 {Deformation rate dependent viscosities (\protect\np{nn\_ahm\_ijk\_t}\forcode{=32})} 434 435 This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a 436 characteristic time scale $T_{smag}$ the deformation rate and for the lengthscale $L_{smag}$ the maximum wavenumber possible on the horizontal grid, e.g.: 437 438 \begin{equation} 439 \label{eq:LDF_smag1} 440 \begin{split} 441 T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2 } \\ 442 L_{smag} & = \frac{1}{\pi}\frac{e_1 e_2}{e_1 + e_2} 443 \end{split} 444 \end{equation} 445 446 Introducing a user defined constant $C$ (given in the namelist as \np{rn\_csmc}), one can deduce the mixing coefficients as follows: 447 448 \begin{equation} 449 \label{eq:LDF_smag2} 450 A_{smag} = \left\{ 451 \begin{aligned} 452 & C^2 T_{smag}^{-1} L_{smag}^2 & \text{for laplacian operator } \\ 453 & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4 & \text{for bilaplacian operator } 454 \end{aligned} 455 \right. 456 \end{equation} 457 458 For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that: 459 \begin{equation} 460 \label{eq:LDF_smag3} 461 \begin{aligned} 462 & C_{min} \frac{1}{2} \lvert U \rvert e < A_{smag} < C_{max} \frac{e^2}{ 8\rdt} & \text{for laplacian operator } \\ 463 & C_{min} \frac{1}{12} \lvert U \rvert e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt} & \text{for bilaplacian operator } 464 \end{aligned} 465 \end{equation} 466 467 where $C_{min}$ and $C_{max}$ are adimensional namelist parameters given by \np{rn\_minfac} and \np{rn\_maxfac} respectively. 468 469 \subsection{About space and time varying mixing coefficients} 396 470 397 471 The following points are relevant when the eddy coefficient varies spatially: 398 472 399 473 (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}).474 divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). 401 475 Although the eddy coefficient could be set to different values in these two terms, 402 this option is not currently available. 476 this option is not currently available. 403 477 404 478 (2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of 405 479 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. 480 (\autoref{sec:INVARIANTS_dynldf_properties}). 428 481 429 482 % ================================================================ 430 483 % Eddy Induced Mixing 431 484 % ================================================================ 432 \section{Eddy induced velocity (\protect\mdl{traadv\_eiv}, \protect\mdl{ldfeiv})} 485 \section[Eddy induced velocity (\forcode{ln_ldfeiv=.true.})] 486 {Eddy induced velocity (\protect\np{ln\_ldfeiv}\forcode{=.true.})} 487 433 488 \label{sec:LDF_eiv} 489 490 %--------------------------------------------namtra_eiv--------------------------------------------------- 491 492 \begin{listing} 493 \nlst{namtra_eiv} 494 \caption{\texttt{namtra\_eiv}} 495 \label{lst:namtra_eiv} 496 \end{listing} 497 498 %-------------------------------------------------------------------------------------------------------------- 499 434 500 435 501 %%gm from Triad appendix : to be incorporated.... … … 453 519 } 454 520 455 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),521 When \citet{gent.mcwilliams_JPO90} diffusion is used (\np{ln\_ldfeiv}\forcode{=.true.}), 456 522 an eddy induced tracer advection term is added, 457 523 the formulation of which depends on the slopes of iso-neutral surfaces. 458 524 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} 525 \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates, 526 and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates. 527 528 If isopycnal mixing is used in the standard way, \ie\ \np{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by: 529 \begin{equation} 530 \label{eq:LDF_eiv} 464 531 \begin{split} 465 532 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ … … 468 535 \end{split} 469 536 \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}. 537 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn\_aei\_ijk\_t} \nam{tra\_eiv} namelist parameter. 538 The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and 539 added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed. 474 540 This has been preferred to a separate computation of the advective trends associated with the eiv velocity, 475 541 since it allows us to take advantage of all the advection schemes offered for the tracers 476 542 (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in 477 previous releases of OPA \citep{ Madec1998}.543 previous releases of OPA \citep{madec.delecluse.ea_NPM98}. 478 544 This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of 479 paramount importance. 545 paramount importance. 480 546 481 547 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. 548 and thus the advective eddy fluxes of heat and salt, are set to zero. 549 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). 550 \colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added} 551 552 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:TRIADS}. 553 554 % ================================================================ 555 % Mixed layer eddies 556 % ================================================================ 557 \section[Mixed layer eddies (\forcode{ln_mle=.true.})] 558 {Mixed layer eddies (\protect\np{ln\_mle}\forcode{=.true.})} 559 560 \label{sec:LDF_mle} 561 562 %--------------------------------------------namtra_eiv--------------------------------------------------- 563 564 \begin{listing} 565 \nlst{namtra_mle} 566 \caption{\texttt{namtra\_mle}} 567 \label{lst:namtra_mle} 568 \end{listing} 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. 573 574 \colorbox{yellow}{TBC} 483 575 484 576 \biblio
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