Changeset 999


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Timestamp:
2008-05-28T15:12:25+02:00 (13 years ago)
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gm
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trunk - DOC - LDF update from steven

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trunk/DOC/TexFiles/Chapters
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  • trunk/DOC/TexFiles/Chapters/Annex_C.tex

    r994 r999  
    610610% ================================================================ 
    611611\section{Conservation Properties on Lateral Momentum Physics} 
    612 \label{Apdx_C.3} 
     612\label{Apdx_dynldf_properties} 
    613613 
    614614 
  • trunk/DOC/TexFiles/Chapters/Chap_LDF.tex

    r998 r999  
    99$\ $\newline    % force a new ligne 
    1010 
    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). 
     11The lateral physics terms in the momentum and tracer equations have been  
     12described in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf}  
     13and \S\ref{DYN_ldf}). In this section we further discuss each lateral physics option.  
     14Choosing one lateral physics scheme means for the user defining, (1) the space  
     15and time variations of the eddy coefficients ; (2) the direction along which the  
     16lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal  
     17surfaces); and (3) the type of operator used (harmonic, or biharmonic operators,  
     18and for tracers only, eddy induced advection on tracers). These three aspects  
     19of the lateral diffusion are set through namelist parameters and CPP keys  
     20(see the nam\_traldf and nam\_dynldf below). 
     21 
    2122%-----------------------------------nam_traldf - nam_dynldf-------------------------------------------- 
    2223\namdisplay{nam_traldf}  
     
    2829% Lateral Mixing Coefficients 
    2930% ================================================================ 
    30 \section{Lateral Mixing Coefficient (\textbf{key\_ldftra\_c.d} - \textbf{key\_ldfdyn\_c.d)} } 
     31\section {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn)} } 
    3132\label{LDF_coef} 
    3233 
    3334 
    3435Introducing a space variation in the lateral eddy mixing coefficients changes  
    35 the model core memory requirement, adding up to four three-dimensional  
    36 arrays for geopotential or isopycnal second order operator applied to  
    37 momentum. Six cpp keys control the space variation of eddy  
    38 coefficients: three for momentum and three for tracer. They  
    39 allow to specify a space variation in the three space directions, in the  
    40 horizontal plane, or in the vertical only. The default option is a constant  
    41 value over the whole ocean on momentum and tracers.  
     36the model core memory requirement, adding up to four extra three-dimensional  
     37arrays for the geopotential or isopycnal second order operator applied to  
     38momentum. Six CPP keys control the space variation of eddy coefficients:  
     39three for momentum and three for tracer. The three choices allow:  
     40a space variation in the three space directions, in the horizontal plane,  
     41or in the vertical only. The default option is a constant value over the whole  
     42ocean on both momentum and tracers.  
    4243 
    4344The number of additional arrays that have to be defined and the gridpoint  
    4445position 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: 
     46and the type of operator used. The resulting eddy viscosity and diffusivity  
     47coefficients can be a function of more than one variable. Changes in the  
     48computer code when switching from one option to another have been  
     49minimized by introducing the eddy coefficients as statement functions 
     50(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions  
     51are replaced by their actual meaning during the preprocessing step (CPP).  
     52The 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.  
     55The user can modify these include files as he/she wishes. The way the  
     56mixing coefficient are set in the reference version can be briefly described  
     57as follows: 
    5058 
    5159\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. 
     60When none of the \textbf{key\_ldfdyn\_...} and \textbf{key\_ldftra\_...} keys are  
     61defined, a constant value is used over the whole ocean for momentum and  
     62tracers, which is specified through the \np{ahm0} and \np{aht0} namelist  
     63parameters. 
    5464 
    5565\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 depth-dependant 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. 
     66The 1D option is only available when using the $z$-coordinate with full step.  
     67Indeed in all the other types of vertical coordinate, the depth is a 3D function  
     68of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent  
     69mixing coefficients will require 3D arrays, $i.e.$ \key{ldftra\_c3d} and \key{ldftra\_c3d}.   
     70In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced  
     71in which the surface value is \np{aht0} (\np{ahm0}), the bottom value is 1/4 of  
     72the surface value, and the transition takes place around z=300~m with a width  
     73of 300~m ($i.e.$ both the depth and the width of the inflection point are set to 300~m).  
     74This profile is hard coded in file \hf{ldftra\_c1d}, but can be easily modified by users. 
    5775 
    5876\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: 
     77By default the horizontal variation of the eddy coefficient depends on the local mesh size and the type of operator used: 
    6178\begin{equation} \label{Eq_title} 
    6279  A_l = \left\{      
     
    6784\quad \text{comments} 
    6885\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}.  
     86where $e_{max}$ 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}.  
    7087%%% 
    7188\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!)  } 
    7289 
    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 sub-domain of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}). 
     90Other formulations can be introduced by the user for a given configuration.  
     91For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian  
     92viscosity operator uses \np{ahm0}~=~$4.10^4 m^2/s$ poleward of 20$^{\circ}$  
     93north and south and decreases linearly to \np{aht0}~=~$2.10^3 m^2/s$  
     94at the equator \citep{Madec1996, Delecluse_Madec_Bk00}. This modification  
     95can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}.  
     96Similar modified horizontal variations can be found with the Antarctic or Arctic  
     97sub-domain options of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic}  
     98defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}). 
    7499 
    75100\subsubsection{Space Varying Mixing Coefficients (\key{ldftra\_c3d} and \key{ldfdyn\_c3d})} 
    76101 
    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 iso-neutral 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. 
     102The 3D space variation of the mixing coefficient is simply the combination of the  
     1031D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with  
     104a grid size dependence of the magnitude of the coefficient.  
     105 
     106\subsubsection{Space and Time Varying Mixing Coefficients} 
     107 
     108There is no default specification of space and time varying mixing coefficient.  
     109The only case available is specific to the ORCA2 and ORCA05 global ocean  
     110configurations (\key{orca\_r2} or \key{orca\_r05}). It provides only a tracer  
     111mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and  
     112eddy induced velocity (ORCA05) that depends on the local growth rate of  
     113baroclinic instability. This specification is actually used when an ORCA key  
     114and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined. 
    82115 
    83116A space variation in the eddy coefficient appeals several remarks: 
    84117 
    85 (1) the momentum diffusive operator acting along model level surfaces is  
     118(1) the momentum diffusion operator acting along model level surfaces is  
    86119written in terms of curl and divergent components of the horizontal current  
    87120(see \S\ref{PE_ldf}). Although the eddy coefficient can be set to different values  
    88121in these two terms, this option is not available.  
    89122 
    90 (2) with a horizontal varying viscosity, the quadratic integral constraints  
     123(2) with an horizontally varying viscosity, the quadratic integral constraints  
    91124on enstrophy and on the square of the horizontal divergence for operators  
    92 acting along model-surfaces are no more satisfied (\colorbox{yellow}{Appendix C}). 
     125acting along model-surfaces are no longer satisfied  
     126(Appendix~\ref{Apdx_dynldf_properties}). 
    93127 
    94128(3) for isopycnal diffusion on momentum or tracers, an additional purely  
    95129horizontal 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. 
     130setting a non zero value of \np{ahmb0} or \np{ahtb0}, a background horizontal  
     131eddy viscosity or diffusivity coefficient (namelist parameters whose default  
     132values are $0$). However, the technique used to compute the isopycnal  
     133slopes is intended to get rid of such a background diffusion, since it introduces  
     134spurious diapycnal diffusion (see {\S\ref{LDF_slp}). 
     135 
     136(4) when an eddy induced advection term is used (\key{trahdf\_eiv}), $A^{eiv}$,  
     137the eddy induced coefficient has to be defined. Its space variations are controlled  
     138by 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. 
    102142 
    103143 
     
    112152\gmcomment{  we should emphasize here that the implementation is a rather old one. Better work can be achieved by using \citet{Griffies1998, Griffies2004} iso-neutral scheme. } 
    113153 
    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 $v$.  
     154A direction for lateral mixing has to be defined when the desired operator does  
     155not act along the model levels. This occurs when $(a)$ horizontal mixing is  
     156required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor})  
     157in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required  
     158whatever the vertical coordinate is. This direction of mixing is defined by its  
     159slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the  
     160quantity 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  
     162for 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 
    116165%gm% add here afigure of the slope in i-direction 
    117166 
    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 
     169In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and  
     170$r_2$ are the slopes between the geopotential and computational surfaces.  
     171Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso}  
     172when the diffusive fluxes in the three directions are set to zero and $T$ is  
     173assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the  
     174depth 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} 
    122176 
    123177\begin{equation} \label{Eq_ldfslp_geo} 
     
    141195 
    142196%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 
     198These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True,  
     199and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True.  
    144200 
    145201\subsection{slopes for tracer iso-neutral mixing} 
    146 In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral 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: 
     202In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral  
     203and computational surfaces. Their formulation does not depend on the vertical  
     204coordinate used. Their discrete formulation is found using the fact that the  
     205diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density)  
     206vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the  
     207diffusive fluxes in the three directions to zero leads to the following definition for  
     208the neutral slopes: 
    148209 
    149210\begin{equation} \label{Eq_ldfslp_iso} 
     
    171232%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). 
    172233 
    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 
     236As 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,  
     238in 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  
     240needed depending on the vertical coordinate used: 
    176241 
    177242\begin{description} 
    178243 
    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 Brunt-Vais\"{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 iso-neutral mixing in $s$-coordinate. In other word, iso-neutral 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  
     245appearing in the $i$ and $j$ derivatives  are taken at the same depth, thus  
     246the $in situ$ density can be used. This is not the case for the vertical  
     247derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$  
     248is the local Brunt-Vais\"{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  
     252case except that at partial step level, the \emph{horizontal} density gradient  
     253is evaluated as described in \S\ref{TRA_zpshde}. 
     254 
     255\item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,  
     256there is no specific treatment for iso-neutral mixing in the $s$-coordinate.  
     257In other words, iso-neutral mixing will only be accurately represented with a  
     258linear equation of state (\np{neos}=1 or 2). In the case of a "true" equation  
     259of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso}  
     260will include a pressure dependent part, leading to the wrong evaluation of  
     261the neutral slopes. 
    184262 
    185263%gm%  
     
    188266\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) 
    189267\end{equation} 
     268%gm{  where vector F is ....} 
    190269 
    191270This constraint leads to the following definition for the slopes: 
     
    215294\end{split} 
    216295\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  
     296where $\alpha$ and $\beta$, the thermal expansion and saline contraction  
     297coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three  
     298velocity points. In order to save computation time, they should be approximated  
     299by the mean of their values at $T$-points (for example in the case of $\alpha$:   
     300$\alpha_u=\overline{\alpha_T}^{i+1/2}$,  $\alpha_v=\overline{\alpha_T}^{j+1/2}$  
     301and $\alpha_w=\overline{\alpha_T}^{k+1/2}$). 
     302 
     303Note that such a formulation could be also used in the $z$-coordinate and  
     304$z$-coordinate with partial steps cases. 
    221305 
    222306\end{description} 
    223307 
    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 GFDL-type 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 iso-neutral slopes (made on 9 grid-points) prevents the development of grid point noise generated by the iso-neutral diffusive operator (Fig.~\ref{Fig_LDF_ZDF1}).  
    228 This allows an iso-neutral diffusion scheme without additional background horizontal mixing. This technique can be viewed as a diffusive operator that acts along large-scale (2~$\Delta$x) iso-neutral 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.  
     308This implementation is a rather old one. It is similar to the one proposed  
     309by Cox [1987], except for the background horizontal diffusion. Indeed,  
     310the Cox implementation of isopycnal diffusion in GFDL-type models requires  
     311a minimum background horizontal diffusion for numerical stability reasons.  
     312To overcome this problem, several techniques have been proposed in which  
     313the numerical schemes of the ocean model are modified \citep{Weaver1997,  
     314Griffies1998}. Here, another strategy has been chosen \citep{Lazar1997}:  
     315a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents  
     316the development of grid point noise generated by the iso-neutral diffusion  
     317operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an iso-neutral diffusion scheme  
     318without additional background horizontal mixing. This technique can be viewed  
     319as a diffusion operator that acts along large-scale (2~$\Delta$x)  
     320\gmcomment{2deltax doesnt seem very large scale}  
     321iso-neutral surfaces. The diapycnal diffusion required for numerical stability is  
     322thus minimized and its net effect on the flow is quite small when compared to  
     323the effect of an horizontal background mixing.  
    229324 
    230325Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, contrary to the \citet{Griffies1998} operator which has that property.  
     
    237332%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    238333 
    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... 
    240335 
    241336%from griffies: chapter 13.1.... 
     
    247342flattening of isopycnals near the surface). 
    248343 
     344For numerical stability reasons \citep{Cox1987, Griffies2004}, the slopes must also  
     345be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear  
     346fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter  
     347decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the  
     348surface motivates this flattening of isopycnals near the surface). 
     349 
    249350%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    250351\begin{figure}[!ht] \label{Fig_eiv_slp}  \begin{center} 
    251352\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 iso-neutral 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 air-sea interface: wall boundary condition). Nevertheless, the profile between surface zero value and interior iso-neutral 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 iso-neutral slope in the ocean interior,  
     355which has to be adjusted at the surface boundary (i.e. it must tend to zero at the  
     356surface since there is no mixing across the air-sea interface: wall boundary  
     357condition). Nevertheless, the profile between the surface zero value and the interior  
     358iso-neutral 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  
     360imposing 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  
     362value computed just below the mixed layer. Note the huge change in the slope at the  
     363base of the mixed layer between  \textit{(b)}  and \textit{(c)}.} 
    254364\end{center}   \end{figure} 
    255365%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    259369\subsection{slopes for momentum iso-neutral mixing} 
    260370 
    261 The diffusive iso-neutral 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 $f-T$-, \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} : 
     371The iso-neutral diffusion operator on momentum is the same as the one used on  
     372tracers 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  
     374surface 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.  
     377They are computed from the slopes used for tracer diffusion, $i.e.$  
     378\eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} : 
    262379 
    263380\begin{equation} \label{Eq_ldfslp_dyn} 
     
    270387\end{equation} 
    271388 
    272 The major issue remains in the specification of the boundary conditions. The  
    273 choice made consists in keeping the same boundary conditions as for lateral  
     389The major issue remaining is in the specification of the boundary conditions.  
     390The same boundary conditions are chosen as those used for lateral  
    274391diffusion along model level surfaces, i.e. using the shear computed along  
    275392the model levels and with no additional friction at the ocean bottom (see  
     
    284401\label{LDF_eiv} 
    285402 
    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 iso-neutral surfaces. Contrary to iso-neutral 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: 
     403When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),  
     404an eddy induced tracer advection term is added, the formulation of which  
     405depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral  
     406mixing, 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:  
    287408\begin{equation} \label{Eq_ldfeiv} 
    288409\begin{split} 
     
    292413\end{split} 
    293414\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  
     415where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set  
     416through \np{aeiv}, a \textit{nam\_traldf} namelist parameter.  
     417The three components of the eddy induced velocity are computed and add  
     418to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a  
     419separate computation of the advective trends associated with the eiv velocity,  
     420since it allows us to take advantage of all the advection schemes offered for  
     421the tracers (see \S\ref{TRA_adv}) and not just the $2^{nd}$ order advection  
     422scheme as in previous releases of OPA \citep{Madec1998}. This is particularly  
     423useful for passive tracers where \emph{positivity} of the advection scheme is  
     424of paramount importance.  
     425 
     426At the surface, lateral and bottom boundaries, the eddy induced velocity,  
     427and thus the advective eddy fluxes of heat and salt, are set to zero.  
     428 
     429 
     430 
     431 
     432 
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