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5% ================================================================
6% Chapter Lateral Ocean Physics (LDF)
7% ================================================================
8\chapter{Lateral Ocean Physics (LDF)}
15The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:PE_zdf} and
16their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}).
17In this section we further discuss each lateral physics option.
18Choosing one lateral physics scheme means for the user defining,
19(1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term);
20(2) the direction along which the lateral diffusive fluxes are evaluated
21(model level, geopotential or isopycnal surfaces); and
22(3) the space and time variations of the eddy coefficients.
23These three aspects of the lateral diffusion are set through namelist parameters
24(see the \ngn{nam\_traldf} and \ngn{nam\_dynldf} below).
25Note that this chapter describes the standard implementation of iso-neutral tracer mixing,
26and Griffies's implementation, which is used if \np{traldf\_grif}\forcode{ = .true.},
27is described in Appdx\autoref{apdx:triad}
29%-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
37% ================================================================
38% Direction of lateral Mixing
39% ================================================================
40\section[Direction of lateral mixing (\textit{ldfslp.F90})]
41{Direction of lateral mixing (\protect\mdl{ldfslp})}
46  we should emphasize here that the implementation is a rather old one.
47  Better work can be achieved by using \citet{griffies.gnanadesikan.ea_JPO98, griffies_bk04} iso-neutral scheme.
50A direction for lateral mixing has to be defined when the desired operator does not act along the model levels.
51This occurs when $(a)$ horizontal mixing is required on tracer or momentum
52(\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates,
53and $(b)$ isoneutral mixing is required whatever the vertical coordinate is.
54This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of
55the cell of the quantity to be diffused.
56For a tracer, this leads to the following four slopes:
57$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:tra_ldf_iso}),
58while for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and
59$r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.
61%gm% add here afigure of the slope in i-direction
63\subsection{Slopes for tracer geopotential mixing in the $s$-coordinate}
65In $s$-coordinates, geopotential mixing (\ie horizontal mixing) $r_1$ and $r_2$ are the slopes between
66the geopotential and computational surfaces.
67Their discrete formulation is found by locally solving \autoref{eq:tra_ldf_iso} when
68the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform,
69\ie a linear function of $z_T$, the depth of a $T$-point.
70%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
73  \label{eq:ldfslp_geo}
74  \begin{aligned}
75    r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
76    \;\delta_{i+1/2}[z_t]
77    &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] \ \ \ \\
78    r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)}
79    \;\delta_{j+1/2} [z_t]
80    &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] \ \ \ \\
81    r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2}
82    &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}\\
83    r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2}
84    &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
85  \end{aligned}
88%gm%  caution I'm not sure the simplification was a good idea!
90These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}\forcode{ = .true.}rue,
91and either \np{ln\_traldf\_hor}\forcode{ = .true.} or \np{ln\_dynldf\_hor}\forcode{ = .true.}.
93\subsection{Slopes for tracer iso-neutral mixing}
96In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces.
97Their formulation does not depend on the vertical coordinate used.
98Their discrete formulation is found using the fact that the diffusive fluxes of
99locally referenced potential density (\ie $in situ$ density) vanish.
100So, substituting $T$ by $\rho$ in \autoref{eq:tra_ldf_iso} and setting the diffusive fluxes in
101the three directions to zero leads to the following definition for the neutral slopes:
104  \label{eq:ldfslp_iso}
105  \begin{split}
106    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
107    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} \\
108    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
109    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} \\
110    r_{1w} &= \frac{e_{3w}}{e_{1w}}\;
111    \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
112    {\delta_{k+1/2}[\rho]} \\
113    r_{2w} &= \frac{e_{3w}}{e_{2w}}\;
114    \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
115    {\delta_{k+1/2}[\rho]}
116  \end{split}
119%gm% rewrite this as the explanation is not very clear !!!
120%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.
122%By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, 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).
124%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.
126As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:ldfslp_iso} has to
127be evaluated at the same local pressure
128(which, in decibars, is approximated by the depth in meters in the model).
129Therefore \autoref{eq:ldfslp_iso} cannot be used as such,
130but further transformation is needed depending on the vertical coordinate used:
134\item[$z$-coordinate with full step: ]
135  in \autoref{eq:ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth,
136  thus the $in situ$ density can be used.
137  This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$,
138  where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87}
139  (see \autoref{subsec:TRA_bn2}).
141\item[$z$-coordinate with partial step: ]
142  this case is identical to the full step case except that at partial step level,
143  the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}.
145\item[$s$- or hybrid $s$-$z$- coordinate: ]
146  in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if
147  the Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.};
148  see Appdx \autoref{apdx:triad}).
149  In other words, iso-neutral mixing will only be accurately represented with a linear equation of state
150  (\np{nn\_eos}\forcode{ = 1..2}).
151  In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso}
152  will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes.
155  Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for
156  the constraint on iso-neutral fluxes.
157  Following \citet{griffies_bk04}, instead of specifying directly that there is a zero neutral diffusive flux of
158  locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between
159  the neutral direction diffusive fluxes of potential temperature and salinity:
160  \[
161    \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
162  \]
163  % gm{  where vector F is ....}
165This constraint leads to the following definition for the slopes:
168  % \label{eq:ldfslp_iso2}
169  \begin{split}
170    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
171    {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
172    {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
173      -\beta_u  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } \\
174    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
175    {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
176    {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
177      -\beta_v  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }    \\
178    r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
179    {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
180      -\beta_w  \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
181    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\
182    r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
183    {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
184      -\beta_w  \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
185    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\
186  \end{split}
188where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in
189\autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points.
190In order to save computation time, they should be approximated by the mean of their values at $T$-points
191(for example in the case of $\alpha$:
192$\alpha_u=\overline{\alpha_T}^{i+1/2}$$\alpha_v=\overline{\alpha_T}^{j+1/2}$ and
195Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases.
199This implementation is a rather old one.
200It is similar to the one proposed by Cox [1987], except for the background horizontal diffusion.
201Indeed, the Cox implementation of isopycnal diffusion in GFDL-type models requires
202a minimum background horizontal diffusion for numerical stability reasons.
203To overcome this problem, several techniques have been proposed in which the numerical schemes of
204the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}.
205Griffies's scheme is now available in \NEMO if \np{traldf\_grif\_iso} is set true; see Appdx \autoref{apdx:triad}.
206Here, another strategy is presented \citep{lazar_phd97}:
207a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of
208grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}).
209This allows an iso-neutral diffusion scheme without additional background horizontal mixing.
210This technique can be viewed as a diffusion operator that acts along large-scale
211(2~$\Delta$x) \gmcomment{2deltax doesnt seem very large scale} iso-neutral surfaces.
212The diapycnal diffusion required for numerical stability is thus minimized and its net effect on the flow is quite small when compared to the effect of an horizontal background mixing.
214Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
215contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property.
219  \begin{center}
220    \includegraphics[width=\textwidth]{Fig_LDF_ZDF1}
221    \caption {
222      \protect\label{fig:LDF_ZDF1}
223      averaging procedure for isopycnal slope computation.
224    }
225  \end{center}
229%There are three additional questions about the slope calculation.
230%First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.
231%Second, numerical stability issues also require a bound on slopes.
232%Third, the question of boundary condition specified on slopes...
234%from griffies: chapter 13.1....
238% In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04},
239% the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly
240% to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the
241% surface motivates this flattening of isopycnals near the surface).
243For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by
244$1/100$ everywhere.
245This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to
246$1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean
247(the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface).
251  \begin{center}
252    \includegraphics[width=\textwidth]{Fig_eiv_slp}
253    \caption{
254      \protect\label{fig:eiv_slp}
255      Vertical profile of the slope used for lateral mixing in the mixed layer:
256      \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
257      which has to be adjusted at the surface boundary
258      \ie it must tend to zero at the surface since there is no mixing across the air-sea interface:
259      wall boundary condition).
260      Nevertheless, the profile between the surface zero value and the interior iso-neutral one is unknown,
261      and especially the value at the base of the mixed layer;
262      \textit{(b)} profile of slope using a linear tapering of the slope near the surface and
263      imposing a maximum slope of 1/100;
264      \textit{(c)} profile of slope actually used in \NEMO: a linear decrease of the slope from
265      zero at the surface to its ocean interior value computed just below the mixed layer.
266      Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}.
267    }
268  \end{center}
272\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.}
274\subsection{Slopes for momentum iso-neutral mixing}
276The iso-neutral diffusion operator on momentum is the same as the one used on tracers but
277applied to each component of the velocity separately
278(see \autoref{eq:dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}).
279The slopes between the surface along which the diffusion operator acts and the surface of computation
280($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and
281\textit{vw}- points for the $v$-component.
282They are computed from the slopes used for tracer diffusion,
283\ie \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso}:
286  % \label{eq:ldfslp_dyn}
287  \begin{aligned}
288    &r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
289    &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&  r_{2t}\ &= \overline{r_{2v}}^{\,j} \\
290    &r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
291    &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
292  \end{aligned}
295The major issue remaining is in the specification of the boundary conditions.
296The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces,
297\ie using the shear computed along the model levels and with no additional friction at the ocean bottom
298(see \autoref{sec:LBC_coast}).
301% ================================================================
302% Lateral Mixing Operator
303% ================================================================
304\section[Lateral mixing operators (\textit{traldf.F90})]
305{Lateral mixing operators (\protect\mdl{traldf}, \protect\mdl{traldf})}
310% ================================================================
311% Lateral Mixing Coefficients
312% ================================================================
313\section[Lateral mixing coefficient (\textit{ldftra.F90}, \textit{ldfdyn.F90})]
314{Lateral mixing coefficient (\protect\mdl{ldftra}, \protect\mdl{ldfdyn})}
317Introducing a space variation in the lateral eddy mixing coefficients changes the model core memory requirement,
318adding up to four extra three-dimensional arrays for the geopotential or isopycnal second order operator applied to
320Six CPP keys control the space variation of eddy coefficients: three for momentum and three for tracer.
321The three choices allow:
322a space variation in the three space directions (\key{traldf\_c3d}\key{dynldf\_c3d}),
323in the horizontal plane (\key{traldf\_c2d}\key{dynldf\_c2d}),
324or in the vertical only (\key{traldf\_c1d}\key{dynldf\_c1d}).
325The default option is a constant value over the whole ocean on both momentum and tracers.
327The number of additional arrays that have to be defined and the gridpoint position at which
328they are defined depend on both the space variation chosen and the type of operator used.
329The resulting eddy viscosity and diffusivity coefficients can be a function of more than one variable.
330Changes in the computer code when switching from one option to another have been minimized by
331introducing the eddy coefficients as statement functions
332(include file \textit{ldftra\_substitute.h90} and \textit{ldfdyn\_substitute.h90}).
333The functions are replaced by their actual meaning during the preprocessing step (CPP).
334The specification of the space variation of the coefficient is made in \mdl{ldftra} and \mdl{ldfdyn},
335or more precisely in include files \textit{traldf\_cNd.h90} and \textit{dynldf\_cNd.h90}, with N=1, 2 or 3.
336The user can modify these include files as he/she wishes.
337The way the mixing coefficient are set in the reference version can be briefly described as follows:
339\subsubsection{Constant mixing coefficients (default option)}
340When none of the \key{dynldf\_...} and \key{traldf\_...} keys are defined,
341a constant value is used over the whole ocean for momentum and tracers,
342which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist parameters.
344\subsubsection[Vertically varying mixing coefficients (\texttt{\textbf{key\_traldf\_c1d}} and \texttt{\textbf{key\_dynldf\_c1d}})]
345{Vertically varying mixing coefficients (\protect\key{traldf\_c1d} and \key{dynldf\_c1d})}
346The 1D option is only available when using the $z$-coordinate with full step.
347Indeed in all the other types of vertical coordinate,
348the depth is a 3D function of (\textbf{i},\textbf{j},\textbf{k}) and therefore,
349introducing depth-dependent mixing coefficients will require 3D arrays.
350In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced in which
351the surface value is \np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value,
352and the transition takes place around z=300~m with a width of 300~m
353(\ie both the depth and the width of the inflection point are set to 300~m).
354This profile is hard coded in file \textit{traldf\_c1d.h90}, but can be easily modified by users.
356\subsubsection[Horizontally varying mixing coefficients (\texttt{\textbf{key\_traldf\_c2d}} and \texttt{\textbf{key\_dynldf\_c2d}})]
357{Horizontally varying mixing coefficients (\protect\key{traldf\_c2d} and \protect\key{dynldf\_c2d})}
358By default the horizontal variation of the eddy coefficient depends on the local mesh size and
359the type of operator used:
361  \label{eq:title}
362  A_l = \left\{
363    \begin{aligned}
364      & \frac{\max(e_1,e_2)}{e_{max}} A_o^l           & \text{for laplacian operator } \\
365      & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l          & \text{for bilaplacian operator }
366    \end{aligned}
367  \right.
369where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked ocean domain,
370and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) namelist parameter.
371This variation is intended to reflect the lesser need for subgrid scale eddy mixing where
372the grid size is smaller in the domain.
373It was introduced in the context of the DYNAMO modelling project \citep{willebrand.barnier.ea_PO01}.
374Note that such a grid scale dependance of mixing coefficients significantly increase the range of stability of
375model configurations presenting large changes in grid pacing such as global ocean models.
376Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to
377large coefficient compare to the smallest grid size (see \autoref{sec:STP_forward_imp}),
378especially when using a bilaplacian operator.
380Other formulations can be introduced by the user for a given configuration.
381For example, in the ORCA2 global ocean model (see Configurations),
382the laplacian viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and
383decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}.
384This modification can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}.
385Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of
386ORCA2 and ORCA05 (see \&namcfg namelist).
388\subsubsection[Space varying mixing coefficients (\texttt{\textbf{key\_traldf\_c3d}} and \texttt{\textbf{key\_dynldf\_c3d}})]
389{Space varying mixing coefficients (\protect\key{traldf\_c3d} and \key{dynldf\_c3d})}
391The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases,
392\ie a hyperbolic tangent variation with depth associated with a grid size dependence of
393the magnitude of the coefficient.
395\subsubsection{Space and time varying mixing coefficients}
397There is no default specification of space and time varying mixing coefficient.
398The only case available is specific to the ORCA2 and ORCA05 global ocean configurations.
399It provides only a tracer mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and
400eddy induced velocity (ORCA05) that depends on the local growth rate of baroclinic instability.
401This specification is actually used when an ORCA key and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
403The following points are relevant when the eddy coefficient varies spatially:
405(1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and
406divergent components of the horizontal current (see \autoref{subsec:PE_ldf}).
407Although the eddy coefficient could be set to different values in these two terms,
408this option is not currently available.
410(2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of
411the horizontal divergence for operators acting along model-surfaces are no longer satisfied
414(3) for isopycnal diffusion on momentum or tracers, an additional purely horizontal background diffusion with
415uniform coefficient can be added by setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0},
416a background horizontal eddy viscosity or diffusivity coefficient
417(namelist parameters whose default values are $0$).
418However, the technique used to compute the isopycnal slopes is intended to get rid of such a background diffusion,
419since it introduces spurious diapycnal diffusion (see \autoref{sec:LDF_slp}).
421(4) when an eddy induced advection term is used (\key{traldf\_eiv}),
422$A^{eiv}$, the eddy induced coefficient has to be defined.
423Its space variations are controlled by the same CPP variable as for the eddy diffusivity coefficient
424(\ie \key{traldf\_cNd}).
426(5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value.
428(6) it is possible to use both the laplacian and biharmonic operators concurrently.
430(7) it is possible to run without explicit lateral diffusion on momentum
431(\np{ln\_dynldf\_lap}\forcode{ = .?.}\np{ln\_dynldf\_bilap}\forcode{ = .false.}).
432This is recommended when using the UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.},
433see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes.
435% ================================================================
436% Eddy Induced Mixing
437% ================================================================
438\section[Eddy induced velocity (\textit{traadv\_eiv.F90}, \textit{ldfeiv.F90})]
439{Eddy induced velocity (\protect\mdl{traadv\_eiv}, \protect\mdl{ldfeiv})}
442%%gm  from Triad appendix  : to be incorporated....
444  Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}.
445  If none of the keys \key{traldf\_cNd}, N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and
446  GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and \np{rn\_aeiv\_0}.
447  If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal
448  scale factor according to \autoref{eq:title}
449  \footnote{
450    Except in global ORCA  $0.5^{\circ}$ runs with \key{traldf\_eiv},
451    where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$
452  }.
453  In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in
454  the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from
455  the Held-Larichev parameterisation
456  \footnote{
457    In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$,
458    where $f_{20}$ is the value of $f$ at $20^{\circ}$~N
459  } (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored unless it is zero.
462When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
463an eddy induced tracer advection term is added,
464the formulation of which depends on the slopes of iso-neutral surfaces.
465Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces,
466\ie \autoref{eq:ldfslp_geo} is used in $z$-coordinates,
467and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates.
468The eddy induced velocity is given by:
470  \label{eq:ldfeiv}
471  \begin{split}
472    u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
473    v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
474    w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + \delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\
475  \end{split}
477where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{rn\_aeiv},
478a \textit{nam\_traldf} namelist parameter.
479The three components of the eddy induced velocity are computed and
480add to the eulerian velocity in \mdl{traadv\_eiv}.
481This has been preferred to a separate computation of the advective trends associated with the eiv velocity,
482since it allows us to take advantage of all the advection schemes offered for the tracers
483(see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in
484previous releases of OPA \citep{madec.delecluse.ea_NPM98}.
485This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of
486paramount importance.
488At the surface, lateral and bottom boundaries, the eddy induced velocity,
489and thus the advective eddy fluxes of heat and salt, are set to zero.
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