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2% ================================================================
3% Chapter Ñ Lateral Ocean Physics (LDF)
4% ================================================================
5\chapter{Lateral Ocean Physics (LDF)}
6\label{LDF}
7\minitoc
8
9
10\newpage
11$\ $\newline    % force a new ligne
12
13
14The lateral physics terms in the momentum and tracer equations have been
15described in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf} 
16and \S\ref{DYN_ldf}). In this section we further discuss each lateral physics option.
17Choosing one lateral physics scheme means for the user defining, (1) the space
18and time variations of the eddy coefficients ; (2) the direction along which the
19lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal
20surfaces); and (3) the type of operator used (harmonic, or biharmonic operators,
21and for tracers only, eddy induced advection on tracers). These three aspects
22of the lateral diffusion are set through namelist parameters and CPP keys
23(see the \textit{nam\_traldf} and \textit{nam\_dynldf} below).
24
25%-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
26\namdisplay{namtra_ldf} 
27\namdisplay{namdyn_ldf} 
28%--------------------------------------------------------------------------------------------------------------
29
30
31% ================================================================
32% Lateral Mixing Coefficients
33% ================================================================
34\section [Lateral Mixing Coefficient (\textit{ldftra}, \textit{ldfdyn})]
35        {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn}) }
36\label{LDF_coef}
37
38
39Introducing a space variation in the lateral eddy mixing coefficients changes
40the model core memory requirement, adding up to four extra three-dimensional
41arrays for the geopotential or isopycnal second order operator applied to
42momentum. Six CPP keys control the space variation of eddy coefficients:
43three for momentum and three for tracer. The three choices allow:
44a space variation in the three space directions (\key{traldf\_c3d}\key{dynldf\_c3d}),
45in the horizontal plane (\key{traldf\_c2d}\key{dynldf\_c2d}),
46or in the vertical only (\key{traldf\_c1d}\key{dynldf\_c1d}).
47The default option is a constant value over the whole ocean on both momentum and tracers.
48   
49The number of additional arrays that have to be defined and the gridpoint
50position at which they are defined depend on both the space variation chosen
51and the type of operator used. The resulting eddy viscosity and diffusivity
52coefficients can be a function of more than one variable. Changes in the
53computer code when switching from one option to another have been
54minimized by introducing the eddy coefficients as statement functions
55(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions
56are replaced by their actual meaning during the preprocessing step (CPP).
57The specification of the space variation of the coefficient is made in
58\mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files
59\textit{traldf\_cNd.h90} and \textit{dynldf\_cNd.h90}, with N=1, 2 or 3.
60The user can modify these include files as he/she wishes. The way the
61mixing coefficient are set in the reference version can be briefly described
62as follows:
63
64\subsubsection{Constant Mixing Coefficients (default option)}
65When none of the \textbf{key\_dynldf\_...} and \textbf{key\_traldf\_...} keys are
66defined, a constant value is used over the whole ocean for momentum and
67tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist
68parameters.
69
70\subsubsection{Vertically varying Mixing Coefficients (\key{traldf\_c1d} and \key{dynldf\_c1d})} 
71The 1D option is only available when using the $z$-coordinate with full step.
72Indeed in all the other types of vertical coordinate, the depth is a 3D function
73of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent
74mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation
75of the lateral mixing coefficient is introduced in which the surface value is
76\np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value,
77and the transition takes place around z=300~m with a width of 300~m
78($i.e.$ both the depth and the width of the inflection point are set to 300~m).
79This profile is hard coded in file \hf{traldf\_c1d}, but can be easily modified by users.
80
81\subsubsection{Horizontally Varying Mixing Coefficients (\key{traldf\_c2d} and \key{dynldf\_c2d})}
82By default the horizontal variation of the eddy coefficient depends on the local mesh
83size and the type of operator used:
84\begin{equation} \label{Eq_title}
85  A_l = \left\{     
86   \begin{aligned}
87         & \frac{\max(e_1,e_2)}{e_{max}} A_o^l           & \text{for laplacian operator } \\
88         & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l          & \text{for bilaplacian operator } 
89   \end{aligned}    \right.
90\end{equation}
91where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked
92ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer)
93namelist parameter. This variation is intended to reflect the lesser need for subgrid
94scale eddy mixing where the grid size is smaller in the domain. It was introduced in
95the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}.
96Note that such a grid scale dependance of mixing coefficients significantly increase
97the range of stability of model configurations presenting large changes in grid pacing
98such as global ocean models. Indeed, in such a case, a constant mixing coefficient
99can lead to a blow up of the model due to large coefficient compare to the smallest
100grid size (see \S\ref{STP_forward_imp}), especially when using a bilaplacian operator.
101
102Other formulations can be introduced by the user for a given configuration.
103For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian
104viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ 
105north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s
106at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification
107can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}.
108Similar modified horizontal variations can be found with the Antarctic or Arctic
109sub-domain options of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} 
110defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}).
111
112\subsubsection{Space Varying Mixing Coefficients (\key{traldf\_c3d} and \key{dynldf\_c3d})}
113
114The 3D space variation of the mixing coefficient is simply the combination of the
1151D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with
116a grid size dependence of the magnitude of the coefficient.
117
118\subsubsection{Space and Time Varying Mixing Coefficients}
119
120There is no default specification of space and time varying mixing coefficient.
121The only case available is specific to the ORCA2 and ORCA05 global ocean
122configurations (\key{orca\_r2} or \key{orca\_r05}). It provides only a tracer
123mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and
124eddy induced velocity (ORCA05) that depends on the local growth rate of
125baroclinic instability. This specification is actually used when an ORCA key
126and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
127
128$\ $\newline    % force a new ligne
129
130A space variation in the eddy coefficient appeals several remarks:
131
132(1) the momentum diffusion operator acting along model level surfaces is
133written in terms of curl and divergent components of the horizontal current
134(see \S\ref{PE_ldf}). Although the eddy coefficient can be set to different values
135in these two terms, this option is not available.
136
137(2) with an horizontally varying viscosity, the quadratic integral constraints
138on enstrophy and on the square of the horizontal divergence for operators
139acting along model-surfaces are no longer satisfied
140(Appendix~\ref{Apdx_dynldf_properties}).
141
142(3) for isopycnal diffusion on momentum or tracers, an additional purely
143horizontal background diffusion with uniform coefficient can be added by
144setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal
145eddy viscosity or diffusivity coefficient (namelist parameters whose default
146values are $0$). However, the technique used to compute the isopycnal
147slopes is intended to get rid of such a background diffusion, since it introduces
148spurious diapycnal diffusion (see {\S\ref{LDF_slp}).
149
150(4) when an eddy induced advection term is used (\key{traldf\_eiv}), $A^{eiv}$,
151the eddy induced coefficient has to be defined. Its space variations are controlled
152by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$ 
153\textbf{key\_traldf\_cNd}).
154
155(5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value.
156
157(6) it is possible to use both the laplacian and biharmonic operators concurrently.
158
159(7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap} =
160\np{ln\_dynldf\_bilap} = false). This is recommended when using the UBS advection
161scheme on momentum (\np{ln\_dynadv\_ubs} = true, see \ref{DYN_adv_ubs})
162and can be useful for testing purposes.
163
164% ================================================================
165% Direction of lateral Mixing
166% ================================================================
167\section  [Direction of Lateral Mixing (\textit{ldfslp})]
168      {Direction of Lateral Mixing (\mdl{ldfslp})}
169\label{LDF_slp}
170
171%%%
172\gmcomment{  we should emphasize here that the implementation is a rather old one.
173Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. }
174
175A direction for lateral mixing has to be defined when the desired operator does
176not act along the model levels. This occurs when $(a)$ horizontal mixing is
177required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor})
178in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required
179whatever the vertical coordinate is. This direction of mixing is defined by its
180slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the
181quantity to be diffused. For a tracer, this leads to the following four slopes :
182$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while
183for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for
184$u$ and  $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.
185
186%gm% add here afigure of the slope in i-direction
187
188\subsection{slopes for tracer geopotential mixing in the $s$-coordinate}
189
190In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and
191$r_2$ are the slopes between the geopotential and computational surfaces.
192Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso} 
193when the diffusive fluxes in the three directions are set to zero and $T$ is
194assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the
195depth of a $T$-point.
196%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
197
198\begin{equation} \label{Eq_ldfslp_geo}
199\begin{aligned}
200 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
201           \;\delta_{i+1/2}[z_t]
202      &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t]
203\\
204 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 
205           \;\delta_{j+1/2} [z_t]
206      &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t]
207\\
208 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2}
209      &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]
210 \\
211 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2}
212      &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
213 \\
214\end{aligned}
215\end{equation}
216
217%gm%  caution I'm not sure the simplification was a good idea!
218
219These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True,
220and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True.
221
222\subsection{Slopes for tracer iso-neutral mixing}\label{LDF_slp_iso}
223In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral
224and computational surfaces. Their formulation does not depend on the vertical
225coordinate used. Their discrete formulation is found using the fact that the
226diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density)
227vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the
228diffusive fluxes in the three directions to zero leads to the following definition for
229the neutral slopes:
230
231\begin{equation} \label{Eq_ldfslp_iso}
232\begin{split}
233 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
234                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}
235\\
236 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
237                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}
238\\
239 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; 
240         \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
241             {\delta_{k+1/2}[\rho]}
242\\
243 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; 
244         \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
245             {\delta_{k+1/2}[\rho]}
246\\
247\end{split}
248\end{equation}
249
250%gm% rewrite this as the explanation is not very clear !!!
251%In practice, \eqref{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 \eqref{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.
252
253%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).
254
255%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.
256
257As the mixing is performed along neutral surfaces, the gradient of $\rho$ in
258\eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which,
259in decibars, is approximated by the depth in meters in the model). Therefore
260\eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is
261needed depending on the vertical coordinate used:
262
263\begin{description}
264
265\item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities
266appearing in the $i$ and $j$ derivatives  are taken at the same depth, thus
267the $in situ$ density can be used. This is not the case for the vertical
268derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ 
269is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following
270\citet{McDougall1987} (see \S\ref{TRA_bn2}).
271
272\item[$z$-coordinate with partial step : ] this case is identical to the full step
273case except that at partial step level, the \emph{horizontal} density gradient
274is evaluated as described in \S\ref{TRA_zpshde}.
275
276\item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,
277there is no specific treatment for iso-neutral mixing in the $s$-coordinate.
278In other words, iso-neutral mixing will only be accurately represented with a
279linear equation of state (\np{nn\_eos}=1 or 2). In the case of a "true" equation
280of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} 
281will include a pressure dependent part, leading to the wrong evaluation of
282the neutral slopes.
283
284%gm%
285Note: The solution for $s$-coordinate passes trough the use of different
286(and better) expression for the constraint on iso-neutral fluxes. Following
287\citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral
288diffusive flux of locally referenced potential density, we stay in the $T$-$S$ 
289plane and consider the balance between the neutral direction diffusive fluxes
290of potential temperature and salinity:
291\begin{equation}
292\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
293\end{equation}
294%gm{  where vector F is ....}
295
296This constraint leads to the following definition for the slopes:
297
298\begin{equation} \label{Eq_ldfslp_iso2}
299\begin{split}
300 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
301      {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
302      {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
303       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }
304\\
305 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
306      {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
307      {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
308       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }
309\\
310 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
311      {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
312       -\beta_\;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
313      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
314\\
315 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
316      {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
317       -\beta_\;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
318      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
319\\
320\end{split}
321\end{equation}
322where $\alpha$ and $\beta$, the thermal expansion and saline contraction
323coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three
324velocity points. In order to save computation time, they should be approximated
325by the mean of their values at $T$-points (for example in the case of $\alpha$
326$\alpha_u=\overline{\alpha_T}^{i+1/2}$$\alpha_v=\overline{\alpha_T}^{j+1/2}$ 
327and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).
328
329Note that such a formulation could be also used in the $z$-coordinate and
330$z$-coordinate with partial steps cases.
331
332\end{description}
333
334This implementation is a rather old one. It is similar to the one proposed
335by Cox [1987], except for the background horizontal diffusion. Indeed,
336the Cox implementation of isopycnal diffusion in GFDL-type models requires
337a minimum background horizontal diffusion for numerical stability reasons.
338To overcome this problem, several techniques have been proposed in which
339the numerical schemes of the ocean model are modified \citep{Weaver_Eby_JPO97,
340Griffies_al_JPO98}. Here, another strategy has been chosen \citep{Lazar_PhD97}:
341a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents
342the development of grid point noise generated by the iso-neutral diffusion
343operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an iso-neutral diffusion scheme
344without additional background horizontal mixing. This technique can be viewed
345as a diffusion operator that acts along large-scale (2~$\Delta$x)
346\gmcomment{2deltax doesnt seem very large scale} 
347iso-neutral surfaces. The diapycnal diffusion required for numerical stability is
348thus minimized and its net effect on the flow is quite small when compared to
349the effect of an horizontal background mixing.
350
351Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
352contrary to the \citet{Griffies_al_JPO98} operator which has that property.
353
354%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
355\begin{figure}[!ht]      \begin{center}
356\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_LDF_ZDF1.pdf}
357\caption {    \label{Fig_LDF_ZDF1}
358averaging procedure for isopycnal slope computation.}
359\end{center}    \end{figure}
360%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
361
362%There are three additional questions about the slope calculation.
363%First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.
364%Second, numerical stability issues also require a bound on slopes.
365%Third, the question of boundary condition specified on slopes...
366
367%from griffies: chapter 13.1....
368
369
370
371% In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04},
372% the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly
373% to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the
374% surface motivates this flattening of isopycnals near the surface).
375
376For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also
377be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear
378fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter
379decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the
380surface motivates this flattening of isopycnals near the surface).
381
382%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
383\begin{figure}[!ht]     \begin{center}
384\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_eiv_slp.pdf}
385\caption {     \label{Fig_eiv_slp}
386Vertical profile of the slope used for lateral mixing in the mixed layer :
387\textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
388which has to be adjusted at the surface boundary (i.e. it must tend to zero at the
389surface since there is no mixing across the air-sea interface: wall boundary
390condition). Nevertheless, the profile between the surface zero value and the interior
391iso-neutral one is unknown, and especially the value at the base of the mixed layer ;
392\textit{(b)} profile of slope using a linear tapering of the slope near the surface and
393imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in
394\NEMO: a linear decrease of the slope from zero at the surface to its ocean interior
395value computed just below the mixed layer. Note the huge change in the slope at the
396base of the mixed layer between  \textit{(b)}  and \textit{(c)}.}
397\end{center}   \end{figure}
398%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
399
400\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs  tapering the coefficient.}
401
402\subsection{slopes for momentum iso-neutral mixing}
403
404The iso-neutral diffusion operator on momentum is the same as the one used on
405tracers but applied to each component of the velocity separately (see
406\eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the
407surface along which the diffusion operator acts and the surface of computation
408($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the
409$u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component.
410They are computed from the slopes used for tracer diffusion, $i.e.$ 
411\eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :
412
413\begin{equation} \label{Eq_ldfslp_dyn}
414\begin{aligned}
415&r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
416&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&   r_{2t}\ &= \overline{r_{2v}}^{\,j} \\
417&r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
418&r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
419\end{aligned}
420\end{equation}
421
422The major issue remaining is in the specification of the boundary conditions.
423The same boundary conditions are chosen as those used for lateral
424diffusion along model level surfaces, i.e. using the shear computed along
425the model levels and with no additional friction at the ocean bottom (see
426{\S\ref{LBC_coast}).
427
428
429% ================================================================
430% Eddy Induced Mixing
431% ================================================================
432\section  [Eddy Induced Velocity (\textit{traadv\_eiv}, \textit{ldfeiv})]
433      {Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})}
434\label{LDF_eiv}
435
436When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
437an eddy induced tracer advection term is added, the formulation of which
438depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral
439mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$ 
440\eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo} 
441+ \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
442\begin{equation} \label{Eq_ldfeiv}
443\begin{split}
444 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
445v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
446w^* & = \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\} \\
447\end{split}
448\end{equation}
449where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set
450through \np{rn\_aeiv}, a \textit{nam\_traldf} namelist parameter.
451The three components of the eddy induced velocity are computed and add
452to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a
453separate computation of the advective trends associated with the eiv velocity,
454since it allows us to take advantage of all the advection schemes offered for
455the tracers (see \S\ref{TRA_adv}) and not just the $2^{nd}$ order advection
456scheme as in previous releases of OPA \citep{Madec1998}. This is particularly
457useful for passive tracers where \emph{positivity} of the advection scheme is
458of paramount importance.
459
460At the surface, lateral and bottom boundaries, the eddy induced velocity,
461and thus the advective eddy fluxes of heat and salt, are set to zero.
462
463
464
465
466
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