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1\documentclass[NEMO_book]{subfiles}
2\begin{document}
3
4% ================================================================
5% Chapter ———  Lateral Ocean Physics (LDF)
6% ================================================================
7\chapter{Lateral Ocean Physics (LDF)}
8\label{LDF}
9\minitoc
10
11
12\newpage
13$\$\newline    % force a new ligne
14
15
16The lateral physics terms in the momentum and tracer equations have been
17described in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf}
18and \S\ref{DYN_ldf}). In this section we further discuss each lateral physics option.
19Choosing one lateral physics scheme means for the user defining,
20(1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term) ;
21(2) the direction along which the lateral diffusive fluxes are evaluated (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 \textit{\ngn{nam\_traldf}} and \textit{\ngn{nam\_dynldf}} below).
25Note that this chapter describes the standard implementation of iso-neutral
26tracer mixing, and Griffies's implementation, which is used if
28
29%-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
30\namdisplay{namtra_ldf}
31\namdisplay{namdyn_ldf}
32%--------------------------------------------------------------------------------------------------------------
33
34
35% ================================================================
36% Direction of lateral Mixing
37% ================================================================
38\section  [Direction of Lateral Mixing (\textit{ldfslp})]
39      {Direction of Lateral Mixing (\protect\mdl{ldfslp})}
40\label{LDF_slp}
41
42%%%
43\gmcomment{  we should emphasize here that the implementation is a rather old one.
44Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. }
45
46A direction for lateral mixing has to be defined when the desired operator does
47not act along the model levels. This occurs when $(a)$ horizontal mixing is
48required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor})
49in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required
50whatever the vertical coordinate is. This direction of mixing is defined by its
51slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the
52quantity to be diffused. For a tracer, this leads to the following four slopes :
53$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while
54for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for
55$u$ and  $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.
56
57%gm% add here afigure of the slope in i-direction
58
59\subsection{slopes for tracer geopotential mixing in the $s$-coordinate}
60
61In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and
62$r_2$ are the slopes between the geopotential and computational surfaces.
63Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso}
64when the diffusive fluxes in the three directions are set to zero and $T$ is
65assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the
66depth of a $T$-point.
67%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
68
69\begin{equation} \label{Eq_ldfslp_geo}
70\begin{aligned}
71 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
72           \;\delta_{i+1/2}[z_t]
73      &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] \ \ \
74\\
75 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)}
76           \;\delta_{j+1/2} [z_t]
77      &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] \ \ \
78\\
79 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2}
80      &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]
81 \\
82 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2}
83      &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
84 \\
85\end{aligned}
86\end{equation}
87
88%gm%  caution I'm not sure the simplification was a good idea!
89
90These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True,
91and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True.
92
93\subsection{Slopes for tracer iso-neutral mixing}\label{LDF_slp_iso}
94In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral
95and computational surfaces. Their formulation does not depend on the vertical
96coordinate used. Their discrete formulation is found using the fact that the
97diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density)
98vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the
99diffusive fluxes in the three directions to zero leads to the following definition for
100the neutral slopes:
101
102\begin{equation} \label{Eq_ldfslp_iso}
103\begin{split}
104 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
105                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}
106\\
107 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
108                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}
109\\
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\\
114 r_{2w} &= \frac{e_{3w}}{e_{2w}}\;
115         \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
116             {\delta_{k+1/2}[\rho]}
117\\
118\end{split}
119\end{equation}
120
121%gm% rewrite this as the explanation is not very clear !!!
122%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.
123
124%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).
125
126%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.
127
128As the mixing is performed along neutral surfaces, the gradient of $\rho$ in
129\eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which,
130in decibars, is approximated by the depth in meters in the model). Therefore
131\eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is
132needed depending on the vertical coordinate used:
133
134\begin{description}
135
136\item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities
137appearing in the $i$ and $j$ derivatives  are taken at the same depth, thus
138the $in situ$ density can be used. This is not the case for the vertical
139derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$
140is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following
141\citet{McDougall1987} (see \S\ref{TRA_bn2}).
142
143\item[$z$-coordinate with partial step : ] this case is identical to the full step
144case except that at partial step level, the \emph{horizontal} density gradient
145is evaluated as described in \S\ref{TRA_zpshde}.
146
147\item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,
148iso-neutral mixing is only employed for $s$-coordinates if the
149Griffies scheme is used (\np{traldf\_grif}=true; see Appdx \ref{sec:triad}).
150In other words, iso-neutral mixing will only be accurately represented with a
151linear equation of state (\np{nn\_eos}=1 or 2). In the case of a "true" equation
152of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso}
153will include a pressure dependent part, leading to the wrong evaluation of
154the neutral slopes.
155
156%gm%
157Note: The solution for $s$-coordinate passes trough the use of different
158(and better) expression for the constraint on iso-neutral fluxes. Following
159\citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral
160diffusive flux of locally referenced potential density, we stay in the $T$-$S$
161plane and consider the balance between the neutral direction diffusive fluxes
162of potential temperature and salinity:
163\begin{equation}
164\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
165\end{equation}
166%gm{  where vector F is ....}
167
168This constraint leads to the following definition for the slopes:
169
170\begin{equation} \label{Eq_ldfslp_iso2}
171\begin{split}
172 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
173      {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
174      {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
175       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }
176\\
177 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
178      {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
179      {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
180       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }
181\\
182 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
183      {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
184       -\beta_\;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
185      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
186\\
187 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
188      {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
189       -\beta_\;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
190      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
191\\
192\end{split}
193\end{equation}
194where $\alpha$ and $\beta$, the thermal expansion and saline contraction
195coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three
196velocity points. In order to save computation time, they should be approximated
197by the mean of their values at $T$-points (for example in the case of $\alpha$
198$\alpha_u=\overline{\alpha_T}^{i+1/2}$$\alpha_v=\overline{\alpha_T}^{j+1/2}$
199and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).
200
201Note that such a formulation could be also used in the $z$-coordinate and
202$z$-coordinate with partial steps cases.
203
204\end{description}
205
206This implementation is a rather old one. It is similar to the one
207proposed by Cox [1987], except for the background horizontal
208diffusion. Indeed, the Cox implementation of isopycnal diffusion in
209GFDL-type models requires a minimum background horizontal diffusion
210for numerical stability reasons.  To overcome this problem, several
211techniques have been proposed in which the numerical schemes of the
212ocean model are modified \citep{Weaver_Eby_JPO97,
213  Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if
214\np{traldf\_grif\_iso} is set true; see Appdx \ref{sec:triad}. Here,
215another strategy is presented \citep{Lazar_PhD97}: a local
216filtering of the iso-neutral slopes (made on 9 grid-points) prevents
217the development of grid point noise generated by the iso-neutral
218diffusion operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an
219iso-neutral diffusion scheme without additional background horizontal
220mixing. This technique can be viewed as a diffusion operator that acts
221along large-scale (2~$\Delta$x) \gmcomment{2deltax doesnt seem very
222  large scale} iso-neutral surfaces. The diapycnal diffusion required
223for numerical stability is thus minimized and its net effect on the
224flow is quite small when compared to the effect of an horizontal
225background mixing.
226
227Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
228contrary to the \citet{Griffies_al_JPO98} operator which has that property.
229
230%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
231\begin{figure}[!ht]      \begin{center}
232\includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1}
233\caption {    \protect\label{Fig_LDF_ZDF1}
234averaging procedure for isopycnal slope computation.}
235\end{center}    \end{figure}
236%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
237
239%First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.
240%Second, numerical stability issues also require a bound on slopes.
241%Third, the question of boundary condition specified on slopes...
242
243%from griffies: chapter 13.1....
244
245
246
247% In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04},
248% the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly
249% to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the
250% surface motivates this flattening of isopycnals near the surface).
251
252For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also
253be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear
254fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter
255decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the
256surface motivates this flattening of isopycnals near the surface).
257
258%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
259\begin{figure}[!ht]     \begin{center}
260\includegraphics[width=0.70\textwidth]{Fig_eiv_slp}
261\caption {     \protect\label{Fig_eiv_slp}
262Vertical profile of the slope used for lateral mixing in the mixed layer :
263\textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
264which has to be adjusted at the surface boundary (i.e. it must tend to zero at the
265surface since there is no mixing across the air-sea interface: wall boundary
266condition). Nevertheless, the profile between the surface zero value and the interior
267iso-neutral one is unknown, and especially the value at the base of the mixed layer ;
268\textit{(b)} profile of slope using a linear tapering of the slope near the surface and
269imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in
270\NEMO: a linear decrease of the slope from zero at the surface to its ocean interior
271value computed just below the mixed layer. Note the huge change in the slope at the
272base of the mixed layer between  \textit{(b)}  and \textit{(c)}.}
273\end{center}   \end{figure}
274%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
275
276\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs  tapering the coefficient.}
277
278\subsection{slopes for momentum iso-neutral mixing}
279
280The iso-neutral diffusion operator on momentum is the same as the one used on
281tracers but applied to each component of the velocity separately (see
282\eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the
283surface along which the diffusion operator acts and the surface of computation
284($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the
285$u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component.
286They are computed from the slopes used for tracer diffusion, $i.e.$
287\eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :
288
289\begin{equation} \label{Eq_ldfslp_dyn}
290\begin{aligned}
291&r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
292&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&   r_{2t}\ &= \overline{r_{2v}}^{\,j} \\
293&r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
294&r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
295\end{aligned}
296\end{equation}
297
298The major issue remaining is in the specification of the boundary conditions.
299The same boundary conditions are chosen as those used for lateral
300diffusion along model level surfaces, i.e. using the shear computed along
301the model levels and with no additional friction at the ocean bottom (see
302\S\ref{LBC_coast}).
303
304
305% ================================================================
306% Lateral Mixing Operator
307% ================================================================
308\section [Lateral Mixing Operators (\textit{ldftra}, \textit{ldfdyn})]
309        {Lateral Mixing Operators (\protect\mdl{traldf}, \protect\mdl{traldf}) }
310\label{LDF_op}
311
312
313
314% ================================================================
315% Lateral Mixing Coefficients
316% ================================================================
317\section [Lateral Mixing Coefficient (\textit{ldftra}, \textit{ldfdyn})]
318        {Lateral Mixing Coefficient (\protect\mdl{ldftra}, \protect\mdl{ldfdyn}) }
319\label{LDF_coef}
320
321Introducing a space variation in the lateral eddy mixing coefficients changes
322the model core memory requirement, adding up to four extra three-dimensional
323arrays for the geopotential or isopycnal second order operator applied to
324momentum. Six CPP keys control the space variation of eddy coefficients:
325three for momentum and three for tracer. The three choices allow:
326a space variation in the three space directions (\key{traldf\_c3d}\key{dynldf\_c3d}),
327in the horizontal plane (\key{traldf\_c2d}\key{dynldf\_c2d}),
328or in the vertical only (\key{traldf\_c1d}\key{dynldf\_c1d}).
329The default option is a constant value over the whole ocean on both momentum and tracers.
330
331The number of additional arrays that have to be defined and the gridpoint
332position at which they are defined depend on both the space variation chosen
333and the type of operator used. The resulting eddy viscosity and diffusivity
334coefficients can be a function of more than one variable. Changes in the
335computer code when switching from one option to another have been
336minimized by introducing the eddy coefficients as statement functions
337(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions
338are replaced by their actual meaning during the preprocessing step (CPP).
339The specification of the space variation of the coefficient is made in
340\mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files
341\textit{traldf\_cNd.h90} and \textit{dynldf\_cNd.h90}, with N=1, 2 or 3.
342The user can modify these include files as he/she wishes. The way the
343mixing coefficient are set in the reference version can be briefly described
344as follows:
345
346\subsubsection{Constant Mixing Coefficients (default option)}
347When none of the \textbf{key\_dynldf\_...} and \textbf{key\_traldf\_...} keys are
348defined, a constant value is used over the whole ocean for momentum and
349tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist
350parameters.
351
352\subsubsection{Vertically varying Mixing Coefficients (\protect\key{traldf\_c1d} and \key{dynldf\_c1d})}
353The 1D option is only available when using the $z$-coordinate with full step.
354Indeed in all the other types of vertical coordinate, the depth is a 3D function
355of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent
356mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation
357of the lateral mixing coefficient is introduced in which the surface value is
358\np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value,
359and the transition takes place around z=300~m with a width of 300~m
360($i.e.$ both the depth and the width of the inflection point are set to 300~m).
361This profile is hard coded in file \hf{traldf\_c1d}, but can be easily modified by users.
362
363\subsubsection{Horizontally Varying Mixing Coefficients (\protect\key{traldf\_c2d} and \protect\key{dynldf\_c2d})}
364By default the horizontal variation of the eddy coefficient depends on the local mesh
365size and the type of operator used:
366\begin{equation} \label{Eq_title}
367  A_l = \left\{
368   \begin{aligned}
369         & \frac{\max(e_1,e_2)}{e_{max}} A_o^l           & \text{for laplacian operator } \\
370         & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l          & \text{for bilaplacian operator }
371   \end{aligned}    \right.
372\end{equation}
373where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked
374ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer)
375namelist parameter. This variation is intended to reflect the lesser need for subgrid
376scale eddy mixing where the grid size is smaller in the domain. It was introduced in
377the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}.
378Note that such a grid scale dependance of mixing coefficients significantly increase
379the range of stability of model configurations presenting large changes in grid pacing
380such as global ocean models. Indeed, in such a case, a constant mixing coefficient
381can lead to a blow up of the model due to large coefficient compare to the smallest
382grid size (see \S\ref{STP_forward_imp}), especially when using a bilaplacian operator.
383
384Other formulations can be introduced by the user for a given configuration.
385For example, in the ORCA2 global ocean model (see Configurations), the laplacian
386viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$
387north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s
389can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}.
390Similar modified horizontal variations can be found with the Antarctic or Arctic
391sub-domain options of ORCA2 and ORCA05 (see \&namcfg namelist).
392
393\subsubsection{Space Varying Mixing Coefficients (\protect\key{traldf\_c3d} and \key{dynldf\_c3d})}
394
395The 3D space variation of the mixing coefficient is simply the combination of the
3961D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with
397a grid size dependence of the magnitude of the coefficient.
398
399\subsubsection{Space and Time Varying Mixing Coefficients}
400
401There is no default specification of space and time varying mixing coefficient.
402The only case available is specific to the ORCA2 and ORCA05 global ocean
403configurations. It provides only a tracer
404mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and
405eddy induced velocity (ORCA05) that depends on the local growth rate of
406baroclinic instability. This specification is actually used when an ORCA key
407and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
408
409$\$\newline    % force a new ligne
410
411The following points are relevant when the eddy coefficient varies spatially:
412
413(1) the momentum diffusion operator acting along model level surfaces is
414written in terms of curl and divergent components of the horizontal current
415(see \S\ref{PE_ldf}). Although the eddy coefficient could be set to different values
416in these two terms, this option is not currently available.
417
418(2) with an horizontally varying viscosity, the quadratic integral constraints
419on enstrophy and on the square of the horizontal divergence for operators
420acting along model-surfaces are no longer satisfied
421(Appendix~\ref{Apdx_dynldf_properties}).
422
423(3) for isopycnal diffusion on momentum or tracers, an additional purely
424horizontal background diffusion with uniform coefficient can be added by
425setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal
426eddy viscosity or diffusivity coefficient (namelist parameters whose default
427values are $0$). However, the technique used to compute the isopycnal
428slopes is intended to get rid of such a background diffusion, since it introduces
429spurious diapycnal diffusion (see \S\ref{LDF_slp}).
430
431(4) when an eddy induced advection term is used (\key{traldf\_eiv}), $A^{eiv}$,
432the eddy induced coefficient has to be defined. Its space variations are controlled
433by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$
434\textbf{key\_traldf\_cNd}).
435
436(5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value.
437
438(6) it is possible to use both the laplacian and biharmonic operators concurrently.
439
440(7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap} =
441\np{ln\_dynldf\_bilap} = false). This is recommended when using the UBS advection
443and can be useful for testing purposes.
444
445% ================================================================
446% Eddy Induced Mixing
447% ================================================================
448\section  [Eddy Induced Velocity (\textit{traadv\_eiv}, \textit{ldfeiv})]
449      {Eddy Induced Velocity (\protect\mdl{traadv\_eiv}, \protect\mdl{ldfeiv})}
450\label{LDF_eiv}
451
452%%gm  from Triad appendix  : to be incorporated....
453\gmcomment{
454Values of iso-neutral diffusivity and GM coefficient are set as
455described in \S\ref{LDF_coef}. If none of the keys \key{traldf\_cNd},
456N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and
457GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and
458\np{rn\_aeiv\_0}. If 2D-varying coefficients are set with
459\key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal
460scale factor according to \eqref{Eq_title} \footnote{Except in global ORCA
461  $0.5^{\circ}$ runs with \key{traldf\_eiv}, where
462  $A_l$ is set like $A_e$ but with a minimum vale of
463  $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$}. In idealised setups with
464\key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv}
465is set in the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is
466instead set from the Held-Larichev parameterisation\footnote{In this
467  case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further
468  reduced by a factor $|f/f_{20}|$, where $f_{20}$ is the value of $f$
469  at $20^{\circ}$~N} (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored
470unless it is zero.
471}
472
473When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
474an eddy induced tracer advection term is added, the formulation of which
475depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral
476mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$
477\eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo}
478+ \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
479\begin{equation} \label{Eq_ldfeiv}
480\begin{split}
481 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
482v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
483w^* & = \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\} \\
484\end{split}
485\end{equation}
486where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set
487through \np{rn\_aeiv}, a \textit{nam\_traldf} namelist parameter.
488The three components of the eddy induced velocity are computed and add
489to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a
490separate computation of the advective trends associated with the eiv velocity,
491since it allows us to take advantage of all the advection schemes offered for
492the tracers (see \S\ref{TRA_adv}) and not just the $2^{nd}$ order advection
493scheme as in previous releases of OPA \citep{Madec1998}. This is particularly
494useful for passive tracers where \emph{positivity} of the advection scheme is
495of paramount importance.
496
497At the surface, lateral and bottom boundaries, the eddy induced velocity,
498and thus the advective eddy fluxes of heat and salt, are set to zero.
499
500
501
502
503\end{document}
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