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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5% ================================================================
6% Chapter Lateral Ocean Physics (LDF)
7% ================================================================
8\chapter{Lateral Ocean Physics (LDF)}
9\label{chap:LDF}
10
11\chaptertoc
12
13\newpage
14
15The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_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 \nam{tra\_ldf} and \nam{dyn\_ldf} below).
25Note that this chapter describes the standard implementation of iso-neutral tracer mixing.
26Griffies's implementation, which is used if \np{ln\_traldf\_triad}\forcode{=.true.},
27is described in \autoref{apdx:TRIADS}
28
29%-----------------------------------namtra_ldf - namdyn_ldf--------------------------------------------
30
31%--------------------------------------------------------------------------------------------------------------
32
33% ================================================================
34% Lateral Mixing Operator
35% ================================================================
36\section[Lateral mixing operators]{Lateral mixing operators}
37\label{sec:LDF_op}
38We remind here the different lateral mixing operators that can be used. Further details can be found in \autoref{subsec:TRA_ldf_op} and  \autoref{sec:DYN_ldf}.
39
40\subsection[No lateral mixing (\forcode{ln_traldf_OFF} \& \forcode{ln_dynldf_OFF})]{No lateral mixing (\protect\np{ln\_traldf\_OFF} \& \protect\np{ln\_dynldf\_OFF})}
41
42It is possible to run without explicit lateral diffusion on tracers (\protect\np{ln\_traldf\_OFF}\forcode{=.true.}) and/or
43momentum (\protect\np{ln\_dynldf\_OFF}\forcode{=.true.}). The latter option is even recommended if using the
44UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{=.true.},
45see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes.
46
47\subsection[Laplacian mixing (\forcode{ln_traldf_lap} \& \forcode{ln_dynldf_lap})]{Laplacian mixing (\protect\np{ln\_traldf\_lap} \& \protect\np{ln\_dynldf\_lap})}
48Setting \protect\np{ln\_traldf\_lap}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_lap}\forcode{=.true.} enables
49a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine
50Laplacian and Bilaplacian operators for the same variable.
51
52\subsection[Bilaplacian mixing (\forcode{ln_traldf_blp} \& \forcode{ln_dynldf_blp})]{Bilaplacian mixing (\protect\np{ln\_traldf\_blp} \& \protect\np{ln\_dynldf\_blp})}
53Setting \protect\np{ln\_traldf\_blp}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_blp}\forcode{=.true.} enables
54a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice.
55We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed.
56
57% ================================================================
58% Direction of lateral Mixing
59% ================================================================
60\section[Direction of lateral mixing (\textit{ldfslp.F90})]{Direction of lateral mixing (\protect\mdl{ldfslp})}
61\label{sec:LDF_slp}
62
63%%%
64\gmcomment{
65  we should emphasize here that the implementation is a rather old one.
66  Better work can be achieved by using \citet{griffies.gnanadesikan.ea_JPO98, griffies_bk04} iso-neutral scheme.
67}
68
69A direction for lateral mixing has to be defined when the desired operator does not act along the model levels.
70This occurs when $(a)$ horizontal mixing is required on tracer or momentum
71(\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates,
72and $(b)$ isoneutral mixing is required whatever the vertical coordinate is.
73This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of
74the cell of the quantity to be diffused.
75For a tracer, this leads to the following four slopes:
76$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}),
77while for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and
78$r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.
79
80%gm% add here afigure of the slope in i-direction
81
82\subsection{Slopes for tracer geopotential mixing in the $s$-coordinate}
83
84In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between
85the geopotential and computational surfaces.
86Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when
87the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform,
88\ie\ a linear function of $z_T$, the depth of a $T$-point.
89%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
90
91\begin{equation}
92  \label{eq:LDF_slp_geo}
93  \begin{aligned}
94    r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
95    \;\delta_{i+1/2}[z_t]
96    &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] \ \ \ \\
97    r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)}
98    \;\delta_{j+1/2} [z_t]
99    &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] \ \ \ \\
100    r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2}
101    &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}\\
102    r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2}
103    &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
104  \end{aligned}
105\end{equation}
106
107%gm%  caution I'm not sure the simplification was a good idea!
108
109These slopes are computed once in \rou{ldf\_slp\_init} when \np{ln\_sco}\forcode{=.true.},
110and either \np{ln\_traldf\_hor}\forcode{=.true.} or \np{ln\_dynldf\_hor}\forcode{=.true.}.
111
112\subsection{Slopes for tracer iso-neutral mixing}
113\label{subsec:LDF_slp_iso}
114
115In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces.
116Their formulation does not depend on the vertical coordinate used.
117Their discrete formulation is found using the fact that the diffusive fluxes of
118locally referenced potential density (\ie\ $in situ$ density) vanish.
119So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in
120the three directions to zero leads to the following definition for the neutral slopes:
121
122\begin{equation}
123  \label{eq:LDF_slp_iso}
124  \begin{split}
125    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
126    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} \\
127    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
128    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} \\
129    r_{1w} &= \frac{e_{3w}}{e_{1w}}\;
130    \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
131    {\delta_{k+1/2}[\rho]} \\
132    r_{2w} &= \frac{e_{3w}}{e_{2w}}\;
133    \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
134    {\delta_{k+1/2}[\rho]}
135  \end{split}
136\end{equation}
137
138%gm% rewrite this as the explanation is not very clear !!!
139%In practice, \autoref{eq:LDF_slp_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:LDF_slp_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.
140
141%By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq:LDF_slp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters).
142
143%In the $z$-coordinate, the derivative of the  \autoref{eq:LDF_slp_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.
144
145As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to
146be evaluated at the same local pressure
147(which, in decibars, is approximated by the depth in meters in the model).
148Therefore \autoref{eq:LDF_slp_iso} cannot be used as such,
149but further transformation is needed depending on the vertical coordinate used:
150
151\begin{description}
152
153\item[$z$-coordinate with full step: ]
154  in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth,
155  thus the $in situ$ density can be used.
156  This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$,
157  where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87}
158  (see \autoref{subsec:TRA_bn2}).
159
160\item[$z$-coordinate with partial step: ]
161  this case is identical to the full step case except that at partial step level,
162  the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}.
163
164\item[$s$- or hybrid $s$-$z$- coordinate: ]
165  in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if
166  the Griffies scheme is used (\np{ln\_traldf\_triad}\forcode{=.true.};
167  see \autoref{apdx:TRIADS}).
168  In other words, iso-neutral mixing will only be accurately represented with a linear equation of state
169  (\np{ln\_seos}\forcode{=.true.}).
170  In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso}
171  will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes.
172
173%gm%
174  Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for
175  the constraint on iso-neutral fluxes.
176  Following \citet{griffies_bk04}, instead of specifying directly that there is a zero neutral diffusive flux of
177  locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between
178  the neutral direction diffusive fluxes of potential temperature and salinity:
179  \[
180    \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
181  \]
182  % gm{  where vector F is ....}
183
184This constraint leads to the following definition for the slopes:
185
186\[
187  % \label{eq:LDF_slp_iso2}
188  \begin{split}
189    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
190    {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
191    {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
192      -\beta_u  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } \\
193    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
194    {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
195    {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
196      -\beta_v  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }    \\
197    r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
198    {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
199      -\beta_w  \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
200    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\
201    r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
202    {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
203      -\beta_w  \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
204    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\
205  \end{split}
206\]
207where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in
208\autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points.
209In order to save computation time, they should be approximated by the mean of their values at $T$-points
210(for example in the case of $\alpha$:
211$\alpha_u=\overline{\alpha_T}^{i+1/2}$$\alpha_v=\overline{\alpha_T}^{j+1/2}$ and
212$\alpha_w=\overline{\alpha_T}^{k+1/2}$).
213
214Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases.
215
216\end{description}
217
218This implementation is a rather old one.
219It is similar to the one proposed by \citet{cox_OM87}, except for the background horizontal diffusion.
220Indeed, the \citet{cox_OM87} implementation of isopycnal diffusion in GFDL-type models requires
221a minimum background horizontal diffusion for numerical stability reasons.
222To overcome this problem, several techniques have been proposed in which the numerical schemes of
223the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}.
224Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:TRIADS}.
225Here, another strategy is presented \citep{lazar_phd97}:
226a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of
227grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}).
228This allows an iso-neutral diffusion scheme without additional background horizontal mixing.
229This technique can be viewed as a diffusion operator that acts along large-scale
230(2~$\Delta$x) \gmcomment{2deltax doesnt seem very large scale} iso-neutral surfaces.
231The 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.
232
233Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
234contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property.
235
236%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
237\begin{figure}[!ht]
238  \centering
239  \includegraphics[width=0.66\textwidth]{Fig_LDF_ZDF1}
240  \caption{Averaging procedure for isopycnal slope computation}
241  \label{fig:LDF_ZDF1}
242\end{figure}
243%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
244
245%There are three additional questions about the slope calculation.
246%First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.
247%Second, numerical stability issues also require a bound on slopes.
248%Third, the question of boundary condition specified on slopes...
249
250%from griffies: chapter 13.1....
251
252
253
254% In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04},
255% the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly
256% to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the
257% surface motivates this flattening of isopycnals near the surface).
258
259For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by
260the namelist scalar \np{rn\_slpmax} (usually $1/100$) everywhere.
261This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to
262$1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean
263(the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface).
264\colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.}
265
266%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
267\begin{figure}[!ht]
268  \centering
269  \includegraphics[width=0.66\textwidth]{Fig_eiv_slp}
270  \caption[Vertical profile of the slope used for lateral mixing in the mixed layer]{
271    Vertical profile of the slope used for lateral mixing in the mixed layer:
272    \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
273    which has to be adjusted at the surface boundary
274    \ie\ it must tend to zero at the surface since there is no mixing across the air-sea interface:
275    wall boundary condition).
276    Nevertheless,
277    the profile between the surface zero value and the interior iso-neutral one is unknown,
278    and especially the value at the base of the mixed layer;
279    \textit{(b)} profile of slope using a linear tapering of the slope near the surface and
280    imposing a maximum slope of 1/100;
281    \textit{(c)} profile of slope actually used in \NEMO:
282    a linear decrease of the slope from zero at the surface to
283    its ocean interior value computed just below the mixed layer.
284    Note the huge change in the slope at the base of the mixed layer between
285    \textit{(b)} and \textit{(c)}.}
286  \label{fig:LDF_eiv_slp}
287\end{figure}
288%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
289
290\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.}
291
292\subsection{Slopes for momentum iso-neutral mixing}
293
294The iso-neutral diffusion operator on momentum is the same as the one used on tracers but
295applied to each component of the velocity separately
296(see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}).
297The slopes between the surface along which the diffusion operator acts and the surface of computation
298($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and
299\textit{vw}- points for the $v$-component.
300They are computed from the slopes used for tracer diffusion,
301\ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}:
302
303\[
304  % \label{eq:LDF_slp_dyn}
305  \begin{aligned}
306    &r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
307    &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&  r_{2t}\ &= \overline{r_{2v}}^{\,j} \\
308    &r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
309    &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
310  \end{aligned}
311\]
312
313The major issue remaining is in the specification of the boundary conditions.
314The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces,
315\ie\ using the shear computed along the model levels and with no additional friction at the ocean bottom
316(see \autoref{sec:LBC_coast}).
317
318
319% ================================================================
320% Lateral Mixing Coefficients
321% ================================================================
322\section[Lateral mixing coefficient (\forcode{nn_aht_ijk_t} \& \forcode{nn_ahm_ijk_t})]{Lateral mixing coefficient (\protect\np{nn\_aht\_ijk\_t} \& \protect\np{nn\_ahm\_ijk\_t})}
323\label{sec:LDF_coef}
324
325The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}.
326The way the mixing coefficients are set in the reference version can be described as follows:
327
328\subsection[Mixing coefficients read from file (\forcode{=-20, -30})]{ Mixing coefficients read from file (\protect\np{nn\_aht\_ijk\_t}\forcode{=-20, -30} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=-20, -30})}
329
330Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model,
331the laplacian viscosity operator uses $A^l$~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and
332decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}.
333Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05.
334The provided fields can either be 2d (\np{nn\_aht\_ijk\_t}\forcode{=-20}, \np{nn\_ahm\_ijk\_t}\forcode{=-20}) or 3d (\np{nn\_aht\_ijk\_t}\forcode{=-30}\np{nn\_ahm\_ijk\_t}\forcode{=-30}). They must be given at U, V points for tracers and T, F points for momentum (see \autoref{tab:LDF_files}).
335
336%-------------------------------------------------TABLE---------------------------------------------------
337\begin{table}[htb]
338  \centering
339  \begin{tabular}{|l|l|l|l|}
340    \hline
341    Namelist parameter                       & Input filename                               & dimensions & variable names                      \\  \hline
342    \np{nn\_ahm\_ijk\_t}\forcode{=-20}     & \forcode{eddy_viscosity_2D.nc }            &     $(i,j)$         & \forcode{ahmt_2d, ahmf_2d}  \\  \hline
343    \np{nn\_aht\_ijk\_t}\forcode{=-20}           & \forcode{eddy_diffusivity_2D.nc }           &     $(i,j)$           & \forcode{ahtu_2d, ahtv_2d}    \\   \hline
344    \np{nn\_ahm\_ijk\_t}\forcode{=-30}        & \forcode{eddy_viscosity_3D.nc }            &     $(i,j,k)$          & \forcode{ahmt_3d, ahmf_3d}  \\  \hline
345    \np{nn\_aht\_ijk\_t}\forcode{=-30}     & \forcode{eddy_diffusivity_3D.nc }           &     $(i,j,k)$         & \forcode{ahtu_3d, ahtv_3d}    \\   \hline
346  \end{tabular}
347  \caption{Description of expected input files if mixing coefficients are read from NetCDF files}
348  \label{tab:LDF_files}
349\end{table}
350%--------------------------------------------------------------------------------------------------------------
351
352\subsection[Constant mixing coefficients (\forcode{=0})]{ Constant mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=0} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=0})}
353
354If constant, mixing coefficients are set thanks to a velocity and a length scales ($U_{scl}$, $L_{scl}$) such that:
355
356\begin{equation}
357  \label{eq:LDF_constantah}
358  A_o^l = \left\{
359    \begin{aligned}
360      & \frac{1}{2} U_{scl} L_{scl}            & \text{for laplacian operator } \\
361      & \frac{1}{12} U_{scl} L_{scl}^3                    & \text{for bilaplacian operator }
362    \end{aligned}
363  \right.
364\end{equation}
365
366 $U_{scl}$ and $L_{scl}$ are given by the namelist parameters \np{rn\_Ud}, \np{rn\_Uv}, \np{rn\_Ld} and \np{rn\_Lv}.
367
368\subsection[Vertically varying mixing coefficients (\forcode{=10})]{Vertically varying mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=10} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=10})}
369
370In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which
371the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value,
372and the transition takes place around z=500~m with a width of 200~m.
373This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users.
374
375\subsection[Mesh size dependent mixing coefficients (\forcode{=20})]{Mesh size dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=20} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=20})}
376
377In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and
378the type of operator used:
379\begin{equation}
380  \label{eq:LDF_title}
381  A_l = \left\{
382    \begin{aligned}
383      & \frac{1}{2} U_{scl}  \max(e_1,e_2)         & \text{for laplacian operator } \\
384      & \frac{1}{12} U_{scl}  \max(e_1,e_2)^{3}             & \text{for bilaplacian operator }
385    \end{aligned}
386  \right.
387\end{equation}
388where $U_{scl}$ is a user defined velocity scale (\np{rn\_Ud}, \np{rn\_Uv}).
389This variation is intended to reflect the lesser need for subgrid scale eddy mixing where
390the grid size is smaller in the domain.
391It was introduced in the context of the DYNAMO modelling project \citep{willebrand.barnier.ea_PO01}.
392Note that such a grid scale dependance of mixing coefficients significantly increases the range of stability of
393model configurations presenting large changes in grid spacing such as global ocean models.
394Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to
395large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}),
396especially when using a bilaplacian operator.
397
398\colorbox{yellow}{CASE \np{nn\_aht\_ijk\_t} = 21 to be added}
399
400\subsection[Mesh size and depth dependent mixing coefficients (\forcode{=30})]{Mesh size and depth dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=30} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=30})}
401
402The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above,
403\ie\ a hyperbolic tangent variation with depth associated with a grid size dependence of
404the magnitude of the coefficient.
405
406\subsection[Velocity dependent mixing coefficients (\forcode{=31})]{Flow dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=31} \& \protect\np{nn\_ahm\_ijk\_t}\forcode{=31})}
407In that case, the eddy coefficient is proportional to the local velocity magnitude so that the Reynolds number $Re =  \lvert U \rvert  e / A_l$ is constant (and here hardcoded to $12$):
408\colorbox{yellow}{JC comment: The Reynolds is effectively set to 12 in the code for both operators but shouldn't it be 2 for Laplacian ?}
409
410\begin{equation}
411  \label{eq:LDF_flowah}
412  A_l = \left\{
413    \begin{aligned}
414      & \frac{1}{12} \lvert U \rvert e          & \text{for laplacian operator } \\
415      & \frac{1}{12} \lvert U \rvert e^3             & \text{for bilaplacian operator }
416    \end{aligned}
417  \right.
418\end{equation}
419
420\subsection[Deformation rate dependent viscosities (\forcode{nn_ahm_ijk_t=32})]{Deformation rate dependent viscosities (\protect\np{nn\_ahm\_ijk\_t}\forcode{=32})}
421
422This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a
423characteristic time scale $T_{smag}$ the deformation rate and for the lengthscale $L_{smag}$ the maximum wavenumber possible on the horizontal grid, e.g.:
424
425\begin{equation}
426  \label{eq:LDF_smag1}
427  \begin{split}
428    T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^} \\
429    L_{smag} & = \frac{1}{\pi}\frac{e_1 e_2}{e_1 + e_2}
430  \end{split}
431\end{equation}
432
433Introducing a user defined constant $C$ (given in the namelist as \np{rn\_csmc}), one can deduce the mixing coefficients as follows:
434
435\begin{equation}
436  \label{eq:LDF_smag2}
437  A_{smag} = \left\{
438    \begin{aligned}
439      & C^2  T_{smag}^{-1}  L_{smag}^2       & \text{for laplacian operator } \\
440      & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4        & \text{for bilaplacian operator }
441    \end{aligned}
442  \right.
443\end{equation}
444
445For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that:
446\begin{equation}
447  \label{eq:LDF_smag3}
448    \begin{aligned}
449      & C_{min} \frac{1}{2}   \lvert U \rvert  e    < A_{smag} < C_{max} \frac{e^2}{   8\rdt}                 & \text{for laplacian operator } \\
450      & C_{min} \frac{1}{12} \lvert U \rvert  e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt}                 & \text{for bilaplacian operator }
451    \end{aligned}
452\end{equation}
453
454where $C_{min}$ and $C_{max}$ are adimensional namelist parameters given by \np{rn\_minfac} and \np{rn\_maxfac} respectively.
455
456\subsection{About space and time varying mixing coefficients}
457
458The following points are relevant when the eddy coefficient varies spatially:
459
460(1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and
461divergent components of the horizontal current (see \autoref{subsec:MB_ldf}).
462Although the eddy coefficient could be set to different values in these two terms,
463this option is not currently available.
464
465(2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of
466the horizontal divergence for operators acting along model-surfaces are no longer satisfied
467(\autoref{sec:INVARIANTS_dynldf_properties}).
468
469% ================================================================
470% Eddy Induced Mixing
471% ================================================================
472\section[Eddy induced velocity (\forcode{ln_ldfeiv})]{Eddy induced velocity (\protect\np{ln\_ldfeiv})}
473
474\label{sec:LDF_eiv}
475
476%--------------------------------------------namtra_eiv---------------------------------------------------
477
478\begin{listing}
479  \nlst{namtra_eiv}
480  \caption{\forcode{&namtra_eiv}}
481  \label{lst:namtra_eiv}
482\end{listing}
483
484%--------------------------------------------------------------------------------------------------------------
485
486
487%%gm  from Triad appendix  : to be incorporated....
488\gmcomment{
489  Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}.
490  If none of the keys \key{traldf\_cNd}, N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and
491  GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and \np{rn\_aeiv\_0}.
492  If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal
493  scale factor according to \autoref{eq:title}
494  \footnote{
495    Except in global ORCA  $0.5^{\circ}$ runs with \key{traldf\_eiv},
496    where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$
497  }.
498  In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in
499  the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from
500  the Held-Larichev parameterisation
501  \footnote{
502    In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$,
503    where $f_{20}$ is the value of $f$ at $20^{\circ}$~N
504  } (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored unless it is zero.
505}
506
507When  \citet{gent.mcwilliams_JPO90} diffusion is used (\np{ln\_ldfeiv}\forcode{=.true.}),
508an eddy induced tracer advection term is added,
509the formulation of which depends on the slopes of iso-neutral surfaces.
510Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces,
511\ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates,
512and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates.
513
514If isopycnal mixing is used in the standard way, \ie\ \np{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by:
515\begin{equation}
516  \label{eq:LDF_eiv}
517  \begin{split}
518    u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
519    v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
520    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\} \\
521  \end{split}
522\end{equation}
523where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn\_aei\_ijk\_t} \nam{tra\_eiv} namelist parameter.
524The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and
525added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed.
526This has been preferred to a separate computation of the advective trends associated with the eiv velocity,
527since it allows us to take advantage of all the advection schemes offered for the tracers
528(see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in
529previous releases of OPA \citep{madec.delecluse.ea_NPM98}.
530This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of
531paramount importance.
532
533At the surface, lateral and bottom boundaries, the eddy induced velocity,
534and thus the advective eddy fluxes of heat and salt, are set to zero.
535The value of the eddy induced mixing coefficient and its space variation is controlled in a similar way as for lateral mixing coefficient described in the preceding subsection (\np{nn\_aei\_ijk\_t}, \np{rn\_Ue}, \np{rn\_Le} namelist parameters).
536\colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added}
537
538In case of setting \np{ln\_traldf\_triad}\forcode{ = .true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:TRIADS}.
539
540% ================================================================
541% Mixed layer eddies
542% ================================================================
543\section[Mixed layer eddies (\forcode{ln_mle})]{Mixed layer eddies (\protect\np{ln\_mle})}
544\label{sec:LDF_mle}
545
546%--------------------------------------------namtra_eiv---------------------------------------------------
547
548\begin{listing}
549  \nlst{namtra_mle}
550  \caption{\forcode{&namtra_mle}}
551  \label{lst:namtra_mle}
552\end{listing}
553
554%--------------------------------------------------------------------------------------------------------------
555
556If  \np{ln\_mle}\forcode{=.true.} in \nam{tra\_mle} namelist, a parameterization of the mixing due to unresolved mixed layer instabilities is activated (\citet{foxkemper.ferrari_JPO08}). Additional transport is computed in \rou{ldf\_mle\_trp} and added to the eulerian transport in \rou{tra\_adv} as done for eddy induced advection.
557
558\colorbox{yellow}{TBC}
559
560\biblio
561
562\pindex
563
564\end{document}
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