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1
2% ================================================================
3% Chapter � 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{\ngn{nam\_traldf}} and \textit{\ngn{nam\_dynldf}} below). Note
24that this chapter describes the default implementation of iso-neutral
25tracer mixing, and Griffies's implementation, which is used if
26\np{traldf\_grif}=true, is described in Appdx\ref{sec:triad}
27
28%-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
29\namdisplay{namtra_ldf} 
30\namdisplay{namdyn_ldf} 
31%--------------------------------------------------------------------------------------------------------------
32
33
34% ================================================================
35% Lateral Mixing Coefficients
36% ================================================================
37\section [Lateral Mixing Coefficient (\textit{ldftra}, \textit{ldfdyn})]
38        {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn}) }
39\label{LDF_coef}
40
41
42Introducing a space variation in the lateral eddy mixing coefficients changes
43the model core memory requirement, adding up to four extra three-dimensional
44arrays for the geopotential or isopycnal second order operator applied to
45momentum. Six CPP keys control the space variation of eddy coefficients:
46three for momentum and three for tracer. The three choices allow:
47a space variation in the three space directions (\key{traldf\_c3d}\key{dynldf\_c3d}),
48in the horizontal plane (\key{traldf\_c2d}\key{dynldf\_c2d}),
49or in the vertical only (\key{traldf\_c1d}\key{dynldf\_c1d}).
50The default option is a constant value over the whole ocean on both momentum and tracers.
51   
52The number of additional arrays that have to be defined and the gridpoint
53position at which they are defined depend on both the space variation chosen
54and the type of operator used. The resulting eddy viscosity and diffusivity
55coefficients can be a function of more than one variable. Changes in the
56computer code when switching from one option to another have been
57minimized by introducing the eddy coefficients as statement functions
58(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions
59are replaced by their actual meaning during the preprocessing step (CPP).
60The specification of the space variation of the coefficient is made in
61\mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files
62\textit{traldf\_cNd.h90} and \textit{dynldf\_cNd.h90}, with N=1, 2 or 3.
63The user can modify these include files as he/she wishes. The way the
64mixing coefficient are set in the reference version can be briefly described
65as follows:
66
67\subsubsection{Constant Mixing Coefficients (default option)}
68When none of the \textbf{key\_dynldf\_...} and \textbf{key\_traldf\_...} keys are
69defined, a constant value is used over the whole ocean for momentum and
70tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist
71parameters.
72
73\subsubsection{Vertically varying Mixing Coefficients (\key{traldf\_c1d} and \key{dynldf\_c1d})} 
74The 1D option is only available when using the $z$-coordinate with full step.
75Indeed in all the other types of vertical coordinate, the depth is a 3D function
76of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent
77mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation
78of the lateral mixing coefficient is introduced in which the surface value is
79\np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value,
80and the transition takes place around z=300~m with a width of 300~m
81($i.e.$ both the depth and the width of the inflection point are set to 300~m).
82This profile is hard coded in file \hf{traldf\_c1d}, but can be easily modified by users.
83
84\subsubsection{Horizontally Varying Mixing Coefficients (\key{traldf\_c2d} and \key{dynldf\_c2d})}
85By default the horizontal variation of the eddy coefficient depends on the local mesh
86size and the type of operator used:
87\begin{equation} \label{Eq_title}
88  A_l = \left\{     
89   \begin{aligned}
90         & \frac{\max(e_1,e_2)}{e_{max}} A_o^l           & \text{for laplacian operator } \\
91         & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l          & \text{for bilaplacian operator } 
92   \end{aligned}    \right.
93\end{equation}
94where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked
95ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer)
96namelist parameter. This variation is intended to reflect the lesser need for subgrid
97scale eddy mixing where the grid size is smaller in the domain. It was introduced in
98the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}.
99Note that such a grid scale dependance of mixing coefficients significantly increase
100the range of stability of model configurations presenting large changes in grid pacing
101such as global ocean models. Indeed, in such a case, a constant mixing coefficient
102can lead to a blow up of the model due to large coefficient compare to the smallest
103grid size (see \S\ref{STP_forward_imp}), especially when using a bilaplacian operator.
104
105Other formulations can be introduced by the user for a given configuration.
106For example, in the ORCA2 global ocean model (see Configurations), the laplacian
107viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ 
108north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s
109at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification
110can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}.
111Similar modified horizontal variations can be found with the Antarctic or Arctic
112sub-domain options of ORCA2 and ORCA05 (see \&namcfg namelist).
113
114\subsubsection{Space Varying Mixing Coefficients (\key{traldf\_c3d} and \key{dynldf\_c3d})}
115
116The 3D space variation of the mixing coefficient is simply the combination of the
1171D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with
118a grid size dependence of the magnitude of the coefficient.
119
120\subsubsection{Space and Time Varying Mixing Coefficients}
121
122There is no default specification of space and time varying mixing coefficient.
123The only case available is specific to the ORCA2 and ORCA05 global ocean
124configurations. It provides only a tracer
125mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and
126eddy induced velocity (ORCA05) that depends on the local growth rate of
127baroclinic instability. This specification is actually used when an ORCA key
128and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
129
130$\ $\newline    % force a new ligne
131
132The following points are relevant when the eddy coefficient varies spatially:
133
134(1) the momentum diffusion operator acting along model level surfaces is
135written in terms of curl and divergent components of the horizontal current
136(see \S\ref{PE_ldf}). Although the eddy coefficient could be set to different values
137in these two terms, this option is not currently available.
138
139(2) with an horizontally varying viscosity, the quadratic integral constraints
140on enstrophy and on the square of the horizontal divergence for operators
141acting along model-surfaces are no longer satisfied
142(Appendix~\ref{Apdx_dynldf_properties}).
143
144(3) for isopycnal diffusion on momentum or tracers, an additional purely
145horizontal background diffusion with uniform coefficient can be added by
146setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal
147eddy viscosity or diffusivity coefficient (namelist parameters whose default
148values are $0$). However, the technique used to compute the isopycnal
149slopes is intended to get rid of such a background diffusion, since it introduces
150spurious diapycnal diffusion (see {\S\ref{LDF_slp}).
151
152(4) when an eddy induced advection term is used (\key{traldf\_eiv}), $A^{eiv}$,
153the eddy induced coefficient has to be defined. Its space variations are controlled
154by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$ 
155\textbf{key\_traldf\_cNd}).
156
157(5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value.
158
159(6) it is possible to use both the laplacian and biharmonic operators concurrently.
160
161(7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap} =
162\np{ln\_dynldf\_bilap} = false). This is recommended when using the UBS advection
163scheme on momentum (\np{ln\_dynadv\_ubs} = true, see \ref{DYN_adv_ubs})
164and can be useful for testing purposes.
165
166% ================================================================
167% Direction of lateral Mixing
168% ================================================================
169\section  [Direction of Lateral Mixing (\textit{ldfslp})]
170      {Direction of Lateral Mixing (\mdl{ldfslp})}
171\label{LDF_slp}
172
173%%%
174\gmcomment{  we should emphasize here that the implementation is a rather old one.
175Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. }
176
177A direction for lateral mixing has to be defined when the desired operator does
178not act along the model levels. This occurs when $(a)$ horizontal mixing is
179required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor})
180in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required
181whatever the vertical coordinate is. This direction of mixing is defined by its
182slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the
183quantity to be diffused. For a tracer, this leads to the following four slopes :
184$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while
185for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for
186$u$ and  $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.
187
188%gm% add here afigure of the slope in i-direction
189
190\subsection{slopes for tracer geopotential mixing in the $s$-coordinate}
191
192In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and
193$r_2$ are the slopes between the geopotential and computational surfaces.
194Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso} 
195when the diffusive fluxes in the three directions are set to zero and $T$ is
196assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the
197depth of a $T$-point.
198%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
199
200\begin{equation} \label{Eq_ldfslp_geo}
201\begin{aligned}
202 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
203           \;\delta_{i+1/2}[z_t]
204      &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t]
205\\
206 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 
207           \;\delta_{j+1/2} [z_t]
208      &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t]
209\\
210 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2}
211      &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]
212 \\
213 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2}
214      &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
215 \\
216\end{aligned}
217\end{equation}
218
219%gm%  caution I'm not sure the simplification was a good idea!
220
221These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True,
222and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True.
223
224\subsection{Slopes for tracer iso-neutral mixing}\label{LDF_slp_iso}
225In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral
226and computational surfaces. Their formulation does not depend on the vertical
227coordinate used. Their discrete formulation is found using the fact that the
228diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density)
229vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the
230diffusive fluxes in the three directions to zero leads to the following definition for
231the neutral slopes:
232
233\begin{equation} \label{Eq_ldfslp_iso}
234\begin{split}
235 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
236                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}
237\\
238 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
239                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}
240\\
241 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; 
242         \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
243             {\delta_{k+1/2}[\rho]}
244\\
245 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; 
246         \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
247             {\delta_{k+1/2}[\rho]}
248\\
249\end{split}
250\end{equation}
251
252%gm% rewrite this as the explanation is not very clear !!!
253%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.
254
255%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).
256
257%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.
258
259As the mixing is performed along neutral surfaces, the gradient of $\rho$ in
260\eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which,
261in decibars, is approximated by the depth in meters in the model). Therefore
262\eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is
263needed depending on the vertical coordinate used:
264
265\begin{description}
266
267\item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities
268appearing in the $i$ and $j$ derivatives  are taken at the same depth, thus
269the $in situ$ density can be used. This is not the case for the vertical
270derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ 
271is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following
272\citet{McDougall1987} (see \S\ref{TRA_bn2}).
273
274\item[$z$-coordinate with partial step : ] this case is identical to the full step
275case except that at partial step level, the \emph{horizontal} density gradient
276is evaluated as described in \S\ref{TRA_zpshde}.
277
278\item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,
279iso-neutral mixing is only employed for $s$-coordinates if the
280Griffies scheme is used (\np{traldf\_grif}=true; see Appdx \ref{sec:triad}).
281In other words, iso-neutral mixing will only be accurately represented with a
282linear equation of state (\np{nn\_eos}=1 or 2). In the case of a "true" equation
283of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} 
284will include a pressure dependent part, leading to the wrong evaluation of
285the neutral slopes.
286
287%gm%
288Note: The solution for $s$-coordinate passes trough the use of different
289(and better) expression for the constraint on iso-neutral fluxes. Following
290\citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral
291diffusive flux of locally referenced potential density, we stay in the $T$-$S$ 
292plane and consider the balance between the neutral direction diffusive fluxes
293of potential temperature and salinity:
294\begin{equation}
295\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
296\end{equation}
297%gm{  where vector F is ....}
298
299This constraint leads to the following definition for the slopes:
300
301\begin{equation} \label{Eq_ldfslp_iso2}
302\begin{split}
303 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
304      {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
305      {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
306       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }
307\\
308 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
309      {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
310      {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
311       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }
312\\
313 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
314      {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
315       -\beta_\;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
316      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
317\\
318 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
319      {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
320       -\beta_\;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
321      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
322\\
323\end{split}
324\end{equation}
325where $\alpha$ and $\beta$, the thermal expansion and saline contraction
326coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three
327velocity points. In order to save computation time, they should be approximated
328by the mean of their values at $T$-points (for example in the case of $\alpha$
329$\alpha_u=\overline{\alpha_T}^{i+1/2}$$\alpha_v=\overline{\alpha_T}^{j+1/2}$ 
330and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).
331
332Note that such a formulation could be also used in the $z$-coordinate and
333$z$-coordinate with partial steps cases.
334
335\end{description}
336
337This implementation is a rather old one. It is similar to the one
338proposed by Cox [1987], except for the background horizontal
339diffusion. Indeed, the Cox implementation of isopycnal diffusion in
340GFDL-type models requires a minimum background horizontal diffusion
341for numerical stability reasons.  To overcome this problem, several
342techniques have been proposed in which the numerical schemes of the
343ocean model are modified \citep{Weaver_Eby_JPO97,
344  Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if
345\np{traldf\_grif\_iso} is set true; see Appdx \ref{sec:triad}. Here,
346another strategy is presented \citep{Lazar_PhD97}: a local
347filtering of the iso-neutral slopes (made on 9 grid-points) prevents
348the development of grid point noise generated by the iso-neutral
349diffusion operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an
350iso-neutral diffusion scheme without additional background horizontal
351mixing. This technique can be viewed as a diffusion operator that acts
352along large-scale (2~$\Delta$x) \gmcomment{2deltax doesnt seem very
353  large scale} iso-neutral surfaces. The diapycnal diffusion required
354for numerical stability is thus minimized and its net effect on the
355flow is quite small when compared to the effect of an horizontal
356background mixing.
357
358Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
359contrary to the \citet{Griffies_al_JPO98} operator which has that property.
360
361%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
362\begin{figure}[!ht]      \begin{center}
363\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_LDF_ZDF1.pdf}
364\caption {    \label{Fig_LDF_ZDF1}
365averaging procedure for isopycnal slope computation.}
366\end{center}    \end{figure}
367%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
368
369%There are three additional questions about the slope calculation.
370%First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.
371%Second, numerical stability issues also require a bound on slopes.
372%Third, the question of boundary condition specified on slopes...
373
374%from griffies: chapter 13.1....
375
376
377
378% In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04},
379% the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly
380% to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the
381% surface motivates this flattening of isopycnals near the surface).
382
383For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also
384be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear
385fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter
386decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the
387surface motivates this flattening of isopycnals near the surface).
388
389%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
390\begin{figure}[!ht]     \begin{center}
391\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_eiv_slp.pdf}
392\caption {     \label{Fig_eiv_slp}
393Vertical profile of the slope used for lateral mixing in the mixed layer :
394\textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
395which has to be adjusted at the surface boundary (i.e. it must tend to zero at the
396surface since there is no mixing across the air-sea interface: wall boundary
397condition). Nevertheless, the profile between the surface zero value and the interior
398iso-neutral one is unknown, and especially the value at the base of the mixed layer ;
399\textit{(b)} profile of slope using a linear tapering of the slope near the surface and
400imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in
401\NEMO: a linear decrease of the slope from zero at the surface to its ocean interior
402value computed just below the mixed layer. Note the huge change in the slope at the
403base of the mixed layer between  \textit{(b)}  and \textit{(c)}.}
404\end{center}   \end{figure}
405%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
406
407\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs  tapering the coefficient.}
408
409\subsection{slopes for momentum iso-neutral mixing}
410
411The iso-neutral diffusion operator on momentum is the same as the one used on
412tracers but applied to each component of the velocity separately (see
413\eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the
414surface along which the diffusion operator acts and the surface of computation
415($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the
416$u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component.
417They are computed from the slopes used for tracer diffusion, $i.e.$ 
418\eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :
419
420\begin{equation} \label{Eq_ldfslp_dyn}
421\begin{aligned}
422&r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
423&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&   r_{2t}\ &= \overline{r_{2v}}^{\,j} \\
424&r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
425&r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
426\end{aligned}
427\end{equation}
428
429The major issue remaining is in the specification of the boundary conditions.
430The same boundary conditions are chosen as those used for lateral
431diffusion along model level surfaces, i.e. using the shear computed along
432the model levels and with no additional friction at the ocean bottom (see
433{\S\ref{LBC_coast}).
434
435
436% ================================================================
437% Eddy Induced Mixing
438% ================================================================
439\section  [Eddy Induced Velocity (\textit{traadv\_eiv}, \textit{ldfeiv})]
440      {Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})}
441\label{LDF_eiv}
442
443When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
444an eddy induced tracer advection term is added, the formulation of which
445depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral
446mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$ 
447\eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo} 
448+ \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
449\begin{equation} \label{Eq_ldfeiv}
450\begin{split}
451 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
452v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
453w^* & = \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\} \\
454\end{split}
455\end{equation}
456where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set
457through \np{rn\_aeiv}, a \textit{nam\_traldf} namelist parameter.
458The three components of the eddy induced velocity are computed and add
459to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a
460separate computation of the advective trends associated with the eiv velocity,
461since it allows us to take advantage of all the advection schemes offered for
462the tracers (see \S\ref{TRA_adv}) and not just the $2^{nd}$ order advection
463scheme as in previous releases of OPA \citep{Madec1998}. This is particularly
464useful for passive tracers where \emph{positivity} of the advection scheme is
465of paramount importance.
466
467At the surface, lateral and bottom boundaries, the eddy induced velocity,
468and thus the advective eddy fluxes of heat and salt, are set to zero.
469
470
471
472
473
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