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r994 r999 610 610 % ================================================================ 611 611 \section{Conservation Properties on Lateral Momentum Physics} 612 \label{Apdx_ C.3}612 \label{Apdx_dynldf_properties} 613 613 614 614 
trunk/DOC/TexFiles/Chapters/Chap_LDF.tex
r998 r999 9 9 $\ $\newline % force a new ligne 10 10 11 The lateral physics on momentum and tracer equations have been given in 12 \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf} and \S\ref{DYN_ldf}). 13 In this section we further discuss the choices that underlie each lateral physics option. 14 Choosing one lateral physics means for the user defining, (1) the space and time 15 variations of the eddy coefficients ; (2) the direction along which the lateral diffusive 16 fluxes are evaluated (model level, geopotential or isopycnal surfaces); and (3) the 17 type of operator used (harmonic, or biharmonic operators, and for tracers only, eddy 18 induced advection on tracers). These three aspects of the lateral diffusion are set 19 through namelist parameters and CPP keys (see the nam\_traldf and nam\_dynldf 20 below). 11 The lateral physics terms in the momentum and tracer equations have been 12 described in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf} 13 and \S\ref{DYN_ldf}). In this section we further discuss each lateral physics option. 14 Choosing one lateral physics scheme means for the user defining, (1) the space 15 and time variations of the eddy coefficients ; (2) the direction along which the 16 lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal 17 surfaces); and (3) the type of operator used (harmonic, or biharmonic operators, 18 and for tracers only, eddy induced advection on tracers). These three aspects 19 of the lateral diffusion are set through namelist parameters and CPP keys 20 (see the nam\_traldf and nam\_dynldf below). 21 21 22 %nam_traldf  nam_dynldf 22 23 \namdisplay{nam_traldf} … … 28 29 % Lateral Mixing Coefficients 29 30 % ================================================================ 30 \section {Lateral Mixing Coefficient (\textbf{key\_ldftra\_c.d}  \textbf{key\_ldfdyn\_c.d)} }31 \section {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn)} } 31 32 \label{LDF_coef} 32 33 33 34 34 35 Introducing a space variation in the lateral eddy mixing coefficients changes 35 the model core memory requirement, adding up to four threedimensional36 arrays for geopotential or isopycnal second order operator applied to37 momentum. Six cpp keys control the space variation of eddy38 coefficients: three for momentum and three for tracer. They39 a llow to specify a space variation in the three space directions, in the40 horizontal plane, or in the vertical only. The default option is a constant41 value over the whole ocean onmomentum and tracers.36 the model core memory requirement, adding up to four extra threedimensional 37 arrays for the geopotential or isopycnal second order operator applied to 38 momentum. Six CPP keys control the space variation of eddy coefficients: 39 three for momentum and three for tracer. The three choices allow: 40 a space variation in the three space directions, in the horizontal plane, 41 or in the vertical only. The default option is a constant value over the whole 42 ocean on both momentum and tracers. 42 43 43 44 The number of additional arrays that have to be defined and the gridpoint 44 45 position at which they are defined depend on both the space variation chosen 45 and the type of operator used. The resulting eddy viscosity and 46 diffusivity coefficients can be either single or multiple valued functions. 47 Changes in the computer code when switching from one option to another have 48 been minimized by introducing the eddy coefficients as statement function 49 (include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions are replaced by their actual meaning during the preprocessing step (cpp capability). The specification of the space variation of the coefficient is settled in \mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files \textit{ldftra\_cNd.h90} and \textit{ldfdyn\_cNd.h90}, with N=1, 2 or 3. The user can change these include files following his desiderata. The way the mixing coefficient are set in the reference version can be briefly described as follows: 46 and the type of operator used. The resulting eddy viscosity and diffusivity 47 coefficients can be a function of more than one variable. Changes in the 48 computer code when switching from one option to another have been 49 minimized by introducing the eddy coefficients as statement functions 50 (include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions 51 are replaced by their actual meaning during the preprocessing step (CPP). 52 The specification of the space variation of the coefficient is made in 53 \mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files 54 \textit{ldftra\_cNd.h90} and \textit{ldfdyn\_cNd.h90}, with N=1, 2 or 3. 55 The user can modify these include files as he/she wishes. The way the 56 mixing coefficient are set in the reference version can be briefly described 57 as follows: 50 58 51 59 \subsubsection{Constant Mixing Coefficients (default option)} 52 When none of the \textbf{key\_ldfdyn\_...} and \textbf{key\_ldftra\_...} keys are defined, a constant value over the whole ocean on momentum and tracers that is specified through 53 \np{ahm0} and \np{aht0} namelist parameters. 60 When none of the \textbf{key\_ldfdyn\_...} and \textbf{key\_ldftra\_...} keys are 61 defined, a constant value is used over the whole ocean for momentum and 62 tracers, which is specified through the \np{ahm0} and \np{aht0} namelist 63 parameters. 54 64 55 65 \subsubsection{Vertically varying Mixing Coefficients (\key{ldftra\_c1d} and \key{ldfdyn\_c1d})} 56 The 1D option is only available in $z$coordinate with full step. Indeed in all the other type of vertical coordinate, the depth is a 3D function of (\textbf{i},\textbf{j},\textbf{j}) and therefore, introducing depthdependant mixing coefficients will requires 3D arrays, $i.e.$ \key{ldftra\_c3d} and \key{ldftra\_c3d}. In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced in which the surface value is \np{aht0} (\np{ahm0}), the bottom value is 1/4 of the surface value, and the transition is round z=300~m with a width of 300~m ($i.e.$ both the depth and the width of the inflection point are set to 300~m). This profile is hard coded in \hf{ldftra\_c1d} file, but can be easily modified by users. 66 The 1D option is only available when using the $z$coordinate with full step. 67 Indeed in all the other types of vertical coordinate, the depth is a 3D function 68 of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depthdependent 69 mixing coefficients will require 3D arrays, $i.e.$ \key{ldftra\_c3d} and \key{ldftra\_c3d}. 70 In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced 71 in which the surface value is \np{aht0} (\np{ahm0}), the bottom value is 1/4 of 72 the surface value, and the transition takes place around z=300~m with a width 73 of 300~m ($i.e.$ both the depth and the width of the inflection point are set to 300~m). 74 This profile is hard coded in file \hf{ldftra\_c1d}, but can be easily modified by users. 57 75 58 76 \subsubsection{Horizontally Varying Mixing Coefficients (\key{ldftra\_c2d} and \key{ldfdyn\_c2d})} 59 60 By default the horizontal variation of the eddy coefficient depend on the local mesh size and the type of operator used: 77 By default the horizontal variation of the eddy coefficient depends on the local mesh size and the type of operator used: 61 78 \begin{equation} \label{Eq_title} 62 79 A_l = \left\{ … … 67 84 \quad \text{comments} 68 85 \end{equation} 69 where $e_{max}$ is the max of $e_1$ and $e_2$ taken over the whole masked ocean domain, and $A_o^l$ is \np{ahm0} (momentum) or \np{aht0} (tracer) namelist parameters. This variation is intended to reflect the lesser need for subgrid scale eddy mixing where the grid size is smaller in the domain. It was introduced in the context of the DYNAMO modelling project \citep{Willebrand2001}.86 where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked ocean domain, and $A_o^l$ is the \np{ahm0} (momentum) or \np{aht0} (tracer) namelist parameter. This variation is intended to reflect the lesser need for subgrid scale eddy mixing where the grid size is smaller in the domain. It was introduced in the context of the DYNAMO modelling project \citep{Willebrand2001}. 70 87 %%% 71 88 \gmcomment { not only that! stability reasons: with non uniform grid size, it is common to face a blow up of the model due to to large diffusive coefficient compare to the smallest grid size... this is especially true for bilaplacian (to be added in the text!) } 72 89 73 Other formulations can be introduced by the user for a given configuration. For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian viscous operator uses \np{ahm0}~=~$4.10^4 m^2.s^{1}$ poleward of 20$^{\circ}$ north and south and decreases to \np{aht0}~=~$2.10^3 m^2.s^{1}$ at the equator \citep{Madec1996, Delecluse_Madec_Bk00}. This specification can be found in \rou{ldf\_dyn\_c2d\_orca} routine defined in \mdl{ldfdyn\_c2d}. Similar specific horizontal variation can be found for Antarctic or Arctic subdomain of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}). 90 Other formulations can be introduced by the user for a given configuration. 91 For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian 92 viscosity operator uses \np{ahm0}~=~$4.10^4 m^2/s$ poleward of 20$^{\circ}$ 93 north and south and decreases linearly to \np{aht0}~=~$2.10^3 m^2/s$ 94 at the equator \citep{Madec1996, Delecluse_Madec_Bk00}. This modification 95 can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. 96 Similar modified horizontal variations can be found with the Antarctic or Arctic 97 subdomain options of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} 98 defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}). 74 99 75 100 \subsubsection{Space Varying Mixing Coefficients (\key{ldftra\_c3d} and \key{ldfdyn\_c3d})} 76 101 77 The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with a grid size dependence of the magnitude of the coefficient. 78 79 \subsubsection{Space and time Varying Mixing Coefficients} 80 81 There is no default specification of space and time varying mixing coefficient. The only case available is specific to ORCA2 and ORCA05 global ocean configurations (\key{orca\_r2} or \key{orca\_r05}). It provides only a tracer mixing coefficient for eddy induced velocity (ORCA2) or both isoneutral and eddy induced velocity (ORCA05) that depends on the local growth rate of baroclinic instability. This specification is actually used when a ORCA key plus \key{traldf\_eiv} plus \key{traldf\_c2d} are defined. 102 The 3D space variation of the mixing coefficient is simply the combination of the 103 1D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with 104 a grid size dependence of the magnitude of the coefficient. 105 106 \subsubsection{Space and Time Varying Mixing Coefficients} 107 108 There is no default specification of space and time varying mixing coefficient. 109 The only case available is specific to the ORCA2 and ORCA05 global ocean 110 configurations (\key{orca\_r2} or \key{orca\_r05}). It provides only a tracer 111 mixing coefficient for eddy induced velocity (ORCA2) or both isoneutral and 112 eddy induced velocity (ORCA05) that depends on the local growth rate of 113 baroclinic instability. This specification is actually used when an ORCA key 114 and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined. 82 115 83 116 A space variation in the eddy coefficient appeals several remarks: 84 117 85 (1) the momentum diffusi veoperator acting along model level surfaces is118 (1) the momentum diffusion operator acting along model level surfaces is 86 119 written in terms of curl and divergent components of the horizontal current 87 120 (see \S\ref{PE_ldf}). Although the eddy coefficient can be set to different values 88 121 in these two terms, this option is not available. 89 122 90 (2) with a horizontalvarying viscosity, the quadratic integral constraints123 (2) with an horizontally varying viscosity, the quadratic integral constraints 91 124 on enstrophy and on the square of the horizontal divergence for operators 92 acting along modelsurfaces are no more satisfied (\colorbox{yellow}{Appendix C}). 125 acting along modelsurfaces are no longer satisfied 126 (Appendix~\ref{Apdx_dynldf_properties}). 93 127 94 128 (3) for isopycnal diffusion on momentum or tracers, an additional purely 95 129 horizontal background diffusion with uniform coefficient can be added by 96 setting a non zero value of \np{ahmb0} or \np{ahtb0}, a background horizontal eddy 97 viscosity or diffusivity coefficient (\textbf{namelist parameters} which default value are $0$). Nevertheless, the technique used to compute the isopycnal slopes allows to get rid of such a background diffusion which introduces spurious diapycnal diffusion (see {\S\ref{LDF_slp}). 98 99 (4) when an eddy induced advection is used (\key{trahdf\_eiv}), $A^{eiv}$ , the eddy induced coefficient has to be defined. Its space variations are controlled by the same CPP variable as for the eddy diffusivity coefficient (i.e. \textbf{key\_traldf\_cNd}). 100 101 (5) the eddy coefficient associated to a biharmonic operator must be set to a \emph{negative} value. 130 setting a non zero value of \np{ahmb0} or \np{ahtb0}, a background horizontal 131 eddy viscosity or diffusivity coefficient (namelist parameters whose default 132 values are $0$). However, the technique used to compute the isopycnal 133 slopes is intended to get rid of such a background diffusion, since it introduces 134 spurious diapycnal diffusion (see {\S\ref{LDF_slp}). 135 136 (4) when an eddy induced advection term is used (\key{trahdf\_eiv}), $A^{eiv}$, 137 the eddy induced coefficient has to be defined. Its space variations are controlled 138 by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$ 139 \textbf{key\_traldf\_cNd}). 140 141 (5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value. 102 142 103 143 … … 112 152 \gmcomment{ we should emphasize here that the implementation is a rather old one. Better work can be achieved by using \citet{Griffies1998, Griffies2004} isoneutral scheme. } 113 153 114 115 A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. This occurs when $(a)$ horizontal mixing is required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$ or mixed $s$$z$coordinate, and $(b)$ isoneutral mixing is required whatever the vertical coordinate is. This direction of mixing is defined by its slopes in the \textbf{i} and \textbf{j}directions at the face of the cell of the quantity to be diffused. For tracer, this leads to the following four slopes : $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while for momentum the slopes are $r_{1T}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and $r_{1f}$, $r_{1vw}$, $r_{2T}$, $r_{2vw}$ for $v$. 154 A direction for lateral mixing has to be defined when the desired operator does 155 not act along the model levels. This occurs when $(a)$ horizontal mixing is 156 required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) 157 in $s$ or mixed $s$$z$ coordinates, and $(b)$ isoneutral mixing is required 158 whatever the vertical coordinate is. This direction of mixing is defined by its 159 slopes in the \textbf{i} and \textbf{j}directions at the face of the cell of the 160 quantity to be diffused. For a tracer, this leads to the following four slopes : 161 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while 162 for momentum the slopes are $r_{1T}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for 163 $u$ and $r_{1f}$, $r_{1vw}$, $r_{2T}$, $r_{2vw}$ for $v$. 164 116 165 %gm% add here afigure of the slope in idirection 117 166 118 \subsection{slopes for tracer geopotential mixing in $s$coordinate} 119 120 In $s$coordinates, geopotential mixing ($i.e.$ horizontal one) $r_1$ and $r_2$ are the slopes between the geopotential and computational surfaces. Their discrete formulation is found by locally 121 vanishing the diffusive fluxes when $T$ is horizontally uniform, i.e. by replacing in \eqref{Eq_tra_ldf_iso} $T$ by $z_T$, the depth of $T$point, and setting to zero the diffusive fluxes in the three directions. This leads to the following expression for the slopes: 167 \subsection{slopes for tracer geopotential mixing in the $s$coordinate} 168 169 In $s$coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and 170 $r_2$ are the slopes between the geopotential and computational surfaces. 171 Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso} 172 when the diffusive fluxes in the three directions are set to zero and $T$ is 173 assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the 174 depth of a $T$point. 175 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 122 176 123 177 \begin{equation} \label{Eq_ldfslp_geo} … … 141 195 142 196 %gm% caution I'm not sure the simplification was a good idea! 143 These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=T and \np{ln\_traldf\_hor}=T or \np{ln\_dynldf\_hor}=T. 197 198 These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True, 199 and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True. 144 200 145 201 \subsection{slopes for tracer isoneutral mixing} 146 In isoneutral mixing $r_1$ and $r_2$ are the slopes between the isoneutral and computational 147 surfaces. Their formulation does not depend on the vertical coordinate used. Their discrete formulation is found using the fact that the diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density) vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting to zero diffusive fluxes in the three directions leads to the following definition for the neutral slopes: 202 In isoneutral mixing $r_1$ and $r_2$ are the slopes between the isoneutral 203 and computational surfaces. Their formulation does not depend on the vertical 204 coordinate used. Their discrete formulation is found using the fact that the 205 diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density) 206 vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the 207 diffusive fluxes in the three directions to zero leads to the following definition for 208 the neutral slopes: 148 209 149 210 \begin{equation} \label{Eq_ldfslp_iso} … … 171 232 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{McDougall1987}, therefore in \eqref{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). 172 233 173 %In $z$coordinate, the derivative of the \eqref{Eq_ldfslp_iso} numerator is evaluated at a same depth ($T$level which is also $u$ and $v$levels), so the $in situ$ density can be used for its evaluation. 174 175 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \eqref{Eq_ldfslp_iso} have to be evaluated at the same local pressure (which, in decibars, is approximated by the depth in meters in the model). Therefore \eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is needed depending on the vertical coordinate used: 234 %In the $z$coordinate, the derivative of the \eqref{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. 235 236 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in 237 \eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which, 238 in decibars, is approximated by the depth in meters in the model). Therefore 239 \eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is 240 needed depending on the vertical coordinate used: 176 241 177 242 \begin{description} 178 243 179 \item[$z$coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, thus the $in situ$ density can be used. it is not the case for the vertical derivatives. $\delta_{k+1/2}[\rho]$ is replaced by $\rho N^2/g$, where $N^2$ is the local BruntVais\"{a}l\"{a} frequency evaluated following \citet{McDougall1987} (see \S\ref{TRA_bn2}). 180 181 \item[$z$coordinate with partial step : ] the technique is identical to the full step case except that at partial step level, the \emph{horizontal} density gradient is evaluated as described in \S\ref{TRA_zpshde}. 182 183 \item[$s$ or hybrid $s$$z$ coordinate : ] in the current release of \NEMO, there is no specific treatment for isoneutral mixing in $s$coordinate. In other word, isoneutral mixing will only be accurately represented with a linear equation of state (\np{neos}=1 or 2). In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} will include a pressure dependent part, leading to a wrong evaluation of the neutral slopes. 244 \item[$z$coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities 245 appearing in the $i$ and $j$ derivatives are taken at the same depth, thus 246 the $in situ$ density can be used. This is not the case for the vertical 247 derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $\rho N^2/g$, where $N^2$ 248 is the local BruntVais\"{a}l\"{a} frequency evaluated following 249 \citet{McDougall1987} (see \S\ref{TRA_bn2}). 250 251 \item[$z$coordinate with partial step : ] this case is identical to the full step 252 case except that at partial step level, the \emph{horizontal} density gradient 253 is evaluated as described in \S\ref{TRA_zpshde}. 254 255 \item[$s$ or hybrid $s$$z$ coordinate : ] in the current release of \NEMO, 256 there is no specific treatment for isoneutral mixing in the $s$coordinate. 257 In other words, isoneutral mixing will only be accurately represented with a 258 linear equation of state (\np{neos}=1 or 2). In the case of a "true" equation 259 of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} 260 will include a pressure dependent part, leading to the wrong evaluation of 261 the neutral slopes. 184 262 185 263 %gm% … … 188 266 \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) 189 267 \end{equation} 268 %gm{ where vector F is ....} 190 269 191 270 This constraint leads to the following definition for the slopes: … … 215 294 \end{split} 216 295 \end{equation} 217 where $\alpha$ and $\beta$, the thermal expansion and saline contracion coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three velocity point. Inorder to save computation time, they should be approximated by the mean of their values at $T$points (for example in the case of $\alpha$: $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$ and $\alpha_w=\overline{\alpha_T}^{k+1/2}$). 218 219 Note that such a formulation could be also used in $z$ and $zps$ cases. 220 296 where $\alpha$ and $\beta$, the thermal expansion and saline contraction 297 coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three 298 velocity points. In order to save computation time, they should be approximated 299 by the mean of their values at $T$points (for example in the case of $\alpha$: 300 $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$ 301 and $\alpha_w=\overline{\alpha_T}^{k+1/2}$). 302 303 Note that such a formulation could be also used in the $z$coordinate and 304 $z$coordinate with partial steps cases. 221 305 222 306 \end{description} 223 307 224 This implementation is a rather old one. It is similar to the one proposed by Cox [1987], except for 225 the background horizontal diffusion. Indeed, the Cox implementation of isopycnal diffusion in GFDLtype models requires a minimum background horizontal diffusion for numerical stability reasons. To overcome this problem, several techniques have been proposed in which the numerical 226 schemes of the OGCM are modified \citep{Weaver1997, Griffies1998}. 227 Here, another strategy has been chosen \citep{Lazar1997}: a local filtering of the isoneutral slopes (made on 9 gridpoints) prevents the development of grid point noise generated by the isoneutral diffusive operator (Fig.~\ref{Fig_LDF_ZDF1}). 228 This allows an isoneutral diffusion scheme without additional background horizontal mixing. This technique can be viewed as a diffusive operator that acts along largescale (2~$\Delta$x) isoneutral surfaces. 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 a horizontal background mixing. 308 This implementation is a rather old one. It is similar to the one proposed 309 by Cox [1987], except for the background horizontal diffusion. Indeed, 310 the Cox implementation of isopycnal diffusion in GFDLtype models requires 311 a minimum background horizontal diffusion for numerical stability reasons. 312 To overcome this problem, several techniques have been proposed in which 313 the numerical schemes of the ocean model are modified \citep{Weaver1997, 314 Griffies1998}. Here, another strategy has been chosen \citep{Lazar1997}: 315 a local filtering of the isoneutral slopes (made on 9 gridpoints) prevents 316 the development of grid point noise generated by the isoneutral diffusion 317 operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an isoneutral diffusion scheme 318 without additional background horizontal mixing. This technique can be viewed 319 as a diffusion operator that acts along largescale (2~$\Delta$x) 320 \gmcomment{2deltax doesnt seem very large scale} 321 isoneutral surfaces. The diapycnal diffusion required for numerical stability is 322 thus minimized and its net effect on the flow is quite small when compared to 323 the effect of an horizontal background mixing. 229 324 230 325 Nevertheless, this isoneutral operator does not ensure that variance cannot increase, contrary to the \citet{Griffies1998} operator which has that property. … … 237 332 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 238 333 239 %There is three additional questions about the slope calculation. First the expression of the rotation tensor used have been obtain assuming the "small slope" approximation, so a bound has to be specified on slopes. Second, numerical stability issues also require a bound on slopes. Third, the question of boundary condition spefified on slopes...334 %There are three additional questions about the slope calculation. First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. Second, numerical stability issues also require a bound on slopes. Third, the question of boundary condition specified on slopes... 240 335 241 336 %from griffies: chapter 13.1.... … … 247 342 flattening of isopycnals near the surface). 248 343 344 For numerical stability reasons \citep{Cox1987, Griffies2004}, the slopes must also 345 be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear 346 fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter 347 decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the 348 surface motivates this flattening of isopycnals near the surface). 349 249 350 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 250 351 \begin{figure}[!ht] \label{Fig_eiv_slp} \begin{center} 251 352 \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_eiv_slp.pdf} 252 \caption {Vertical profile of the slope used for lateral mixing in the mixed layer : \textit{(a)} in the real ocean the slope is the isoneutral slope in the ocean interior and their have to adjust to the surface boundary (i.e. tend to zero at the surface as there is no mixing across the airsea interface: wall boundary condition). Nevertheless, the profile between surface zero value and interior isoneutral one is unknown, and especially the value at the based of the mixed layer ; \textit{(b)} profile of slope using a linear tapering of the slope near the surface and imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actuelly used in \NEMO: linear decrease of the slope from zero at the surface to its ocean interior value computed just below the mixed layer. Note the huge change in the slope at the based of the mixed layer between \textit{(b)} and \textit{(c)}. 253 .} 353 \caption {Vertical profile of the slope used for lateral mixing in the mixed layer : 354 \textit{(a)} in the real ocean the slope is the isoneutral slope in the ocean interior, 355 which has to be adjusted at the surface boundary (i.e. it must tend to zero at the 356 surface since there is no mixing across the airsea interface: wall boundary 357 condition). Nevertheless, the profile between the surface zero value and the interior 358 isoneutral one is unknown, and especially the value at the base of the mixed layer ; 359 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and 360 imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in 361 \NEMO: a linear decrease of the slope from zero at the surface to its ocean interior 362 value computed just below the mixed layer. Note the huge change in the slope at the 363 base of the mixed layer between \textit{(b)} and \textit{(c)}.} 254 364 \end{center} \end{figure} 255 365 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 259 369 \subsection{slopes for momentum isoneutral mixing} 260 370 261 The diffusive isoneutral operator on momentum is the same as the on used on tracer but applied to each component of the velocity (see \eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the surface along which the diffusive operator acts and the surface of computation ($z$ or $s$surfaces) are defined at $T$, $f$, and \textit{uw}points for the $u$component, and $fT$, \textit{vw}points for the $v$component. They are computed as follows from the slopes used for tracer diffusion, i.e. \eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} : 371 The isoneutral diffusion operator on momentum is the same as the one used on 372 tracers but applied to each component of the velocity separately (see 373 \eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the 374 surface along which the diffusion operator acts and the surface of computation 375 ($z$ or $s$surfaces) are defined at $T$, $f$, and \textit{uw} points for the 376 $u$component, and $T$, $f$ and \textit{vw} points for the $v$component. 377 They are computed from the slopes used for tracer diffusion, $i.e.$ 378 \eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} : 262 379 263 380 \begin{equation} \label{Eq_ldfslp_dyn} … … 270 387 \end{equation} 271 388 272 The major issue remain s in the specification of the boundary conditions. The273 choice made consists in keeping the same boundary conditions asfor lateral389 The major issue remaining is in the specification of the boundary conditions. 390 The same boundary conditions are chosen as those used for lateral 274 391 diffusion along model level surfaces, i.e. using the shear computed along 275 392 the model levels and with no additional friction at the ocean bottom (see … … 284 401 \label{LDF_eiv} 285 402 286 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), an eddy induced tracer advection term is added, the formulation of which depends on the slopes of isoneutral surfaces. Contrary to isoneutral mixing, the slopes use here are referenced to the geopotential surfaces, i.e. \eqref{Eq_ldfslp_geo} is used in $z$coordinates, and the sum \eqref{Eq_ldfslp_geo} + \eqref{Eq_ldfslp_iso} in $s$coordinates. The eddy induced velocity is given by: 403 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), 404 an eddy induced tracer advection term is added, the formulation of which 405 depends on the slopes of isoneutral surfaces. Contrary to the case of isoneutral 406 mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$ \eqref{Eq_ldfslp_geo} is used in $z$coordinates, and the sum \eqref{Eq_ldfslp_geo} 407 + \eqref{Eq_ldfslp_iso} in $s$coordinates. The eddy induced velocity is given by: 287 408 \begin{equation} \label{Eq_ldfeiv} 288 409 \begin{split} … … 292 413 \end{split} 293 414 \end{equation} 294 where $A^{eiv}$ is the eddy induced velocity coefficient set through \np{aeiv}, a \textit{nam\_traldf} namelist parameter. 295 The three components of the eddy induced velocity are computed and add to the eulerian velocity in the mdl{traadv\_eiv}. This has been preferred to a separate computation of the advective trends associated to the eiv velocity as it allows to take advantage of all the advection schemes offered for the tracers (see \S\ref{TRA_adv}) and not only the $2^{nd}$ order advection scheme as in previous release of OPA \citep{Madec1998}. This is particularly useful for passive tracers where \emph{positivity}of the advection scheme is of paramount importance. 296 297 At surface, lateral and bottom boundaries, the eddy induced velocity and thus the 298 advective eddy fluxes of heat and salt are set to zero. 299 300 301 302 303 415 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set 416 through \np{aeiv}, a \textit{nam\_traldf} namelist parameter. 417 The three components of the eddy induced velocity are computed and add 418 to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a 419 separate computation of the advective trends associated with the eiv velocity, 420 since it allows us to take advantage of all the advection schemes offered for 421 the tracers (see \S\ref{TRA_adv}) and not just the $2^{nd}$ order advection 422 scheme as in previous releases of OPA \citep{Madec1998}. This is particularly 423 useful for passive tracers where \emph{positivity} of the advection scheme is 424 of paramount importance. 425 426 At the surface, lateral and bottom boundaries, the eddy induced velocity, 427 and thus the advective eddy fluxes of heat and salt, are set to zero. 428 429 430 431 432
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