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5\chapter{Lateral Ocean Physics (LDF)}
12\paragraph{Changes record} ~\\
15  \begin{tabularx}{\textwidth}{l||X|X}
16    Release & Author(s) & Modifications \\
17    \hline
18    {\em   4.0} & {\em ...} & {\em ...} \\
19    {\em   3.6} & {\em ...} & {\em ...} \\
20    {\em   3.4} & {\em ...} & {\em ...} \\
21    {\em <=3.4} & {\em ...} & {\em ...}
22  \end{tabularx}
27The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and
28their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}).
29In this section we further discuss each lateral physics option.
30Choosing one lateral physics scheme means for the user defining,
31(1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term);
32(2) the direction along which the lateral diffusive fluxes are evaluated
33(model level, geopotential or isopycnal surfaces); and
34(3) the space and time variations of the eddy coefficients.
35These three aspects of the lateral diffusion are set through namelist parameters
36(see the \nam{tra_ldf}{tra\_ldf} and \nam{dyn_ldf}{dyn\_ldf} below).
37Note that this chapter describes the standard implementation of iso-neutral tracer mixing.
38Griffies's implementation, which is used if \np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad},
39is described in \autoref{apdx:TRIADS}
41%% =================================================================================================
42\section[Lateral mixing operators]{Lateral mixing operators}
44We 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}.
46%% =================================================================================================
47\subsection[No lateral mixing (\forcode{ln_traldf_OFF} \& \forcode{ln_dynldf_OFF})]{No lateral mixing (\protect\np{ln_traldf_OFF}{ln\_traldf\_OFF} \& \protect\np{ln_dynldf_OFF}{ln\_dynldf\_OFF})}
49It is possible to run without explicit lateral diffusion on tracers (\protect\np[=.true.]{ln_traldf_OFF}{ln\_traldf\_OFF}) and/or
50momentum (\protect\np[=.true.]{ln_dynldf_OFF}{ln\_dynldf\_OFF}). The latter option is even recommended if using the
51UBS advection scheme on momentum (\np[=.true.]{ln_dynadv_ubs}{ln\_dynadv\_ubs},
52see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes.
54%% =================================================================================================
55\subsection[Laplacian mixing (\forcode{ln_traldf_lap} \& \forcode{ln_dynldf_lap})]{Laplacian mixing (\protect\np{ln_traldf_lap}{ln\_traldf\_lap} \& \protect\np{ln_dynldf_lap}{ln\_dynldf\_lap})}
56Setting \protect\np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap} and/or \protect\np[=.true.]{ln_dynldf_lap}{ln\_dynldf\_lap} enables
57a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine
58Laplacian and Bilaplacian operators for the same variable.
60%% =================================================================================================
61\subsection[Bilaplacian mixing (\forcode{ln_traldf_blp} \& \forcode{ln_dynldf_blp})]{Bilaplacian mixing (\protect\np{ln_traldf_blp}{ln\_traldf\_blp} \& \protect\np{ln_dynldf_blp}{ln\_dynldf\_blp})}
62Setting \protect\np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp} and/or \protect\np[=.true.]{ln_dynldf_blp}{ln\_dynldf\_blp} enables
63a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice.
64We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed.
66%% =================================================================================================
67\section[Direction of lateral mixing (\textit{ldfslp.F90})]{Direction of lateral mixing (\protect\mdl{ldfslp})}
71  we should emphasize here that the implementation is a rather old one.
72  Better work can be achieved by using \citet{griffies.gnanadesikan.ea_JPO98, griffies_bk04} iso-neutral scheme.
75A direction for lateral mixing has to be defined when the desired operator does not act along the model levels.
76This occurs when $(a)$ horizontal mixing is required on tracer or momentum
77(\np{ln_traldf_hor}{ln\_traldf\_hor} or \np{ln_dynldf_hor}{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates,
78and $(b)$ isoneutral mixing is required whatever the vertical coordinate is.
79This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of
80the cell of the quantity to be diffused.
81For a tracer, this leads to the following four slopes:
82$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}),
83while for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and
84$r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.
86%gm% add here afigure of the slope in i-direction
88%% =================================================================================================
89\subsection{Slopes for tracer geopotential mixing in the $s$-coordinate}
91In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between
92the geopotential and computational surfaces.
93Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when
94the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform,
95\ie\ a linear function of $z_T$, the depth of a $T$-point.
96%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
99  \label{eq:LDF_slp_geo}
100  \begin{aligned}
101    r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
102    \;\delta_{i+1/2}[z_t]
103    &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] \ \ \ \\
104    r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)}
105    \;\delta_{j+1/2} [z_t]
106    &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] \ \ \ \\
107    r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2}
108    &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}\\
109    r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2}
110    &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
111  \end{aligned}
114%gm%  caution I'm not sure the simplification was a good idea!
116These slopes are computed once in \rou{ldf\_slp\_init} when \np[=.true.]{ln_sco}{ln\_sco},
117and either \np[=.true.]{ln_traldf_hor}{ln\_traldf\_hor} or \np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}.
119%% =================================================================================================
120\subsection{Slopes for tracer iso-neutral mixing}
123In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces.
124Their formulation does not depend on the vertical coordinate used.
125Their discrete formulation is found using the fact that the diffusive fluxes of
126locally referenced potential density (\ie\ $in situ$ density) vanish.
127So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in
128the three directions to zero leads to the following definition for the neutral slopes:
131  \label{eq:LDF_slp_iso}
132  \begin{split}
133    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
134    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} \\
135    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
136    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} \\
137    r_{1w} &= \frac{e_{3w}}{e_{1w}}\;
138    \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
139    {\delta_{k+1/2}[\rho]} \\
140    r_{2w} &= \frac{e_{3w}}{e_{2w}}\;
141    \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
142    {\delta_{k+1/2}[\rho]}
143  \end{split}
146%gm% rewrite this as the explanation is not very clear !!!
147%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.
149%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).
151%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.
153As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to
154be evaluated at the same local pressure
155(which, in decibars, is approximated by the depth in meters in the model).
156Therefore \autoref{eq:LDF_slp_iso} cannot be used as such,
157but further transformation is needed depending on the vertical coordinate used:
160\item [$z$-coordinate with full step:] in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth,
161  thus the $in situ$ density can be used.
162  This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$,
163  where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87}
164  (see \autoref{subsec:TRA_bn2}).
165\item [$z$-coordinate with partial step:] this case is identical to the full step case except that at partial step level,
166  the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}.
167\item [$s$- or hybrid $s$-$z$- coordinate:] in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if
168  the Griffies scheme is used (\np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad};
169  see \autoref{apdx:TRIADS}).
170  In other words, iso-neutral mixing will only be accurately represented with a linear equation of state
171  (\np[=.true.]{ln_seos}{ln\_seos}).
172  In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso}
173  will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes.
176  Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for
177  the constraint on iso-neutral fluxes.
178  Following \citet{griffies_bk04}, instead of specifying directly that there is a zero neutral diffusive flux of
179  locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between
180  the neutral direction diffusive fluxes of potential temperature and salinity:
181  \[
182    \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
183  \]
184  % gm{  where vector F is ....}
186This constraint leads to the following definition for the slopes:
189  % \label{eq:LDF_slp_iso2}
190  \begin{split}
191    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
192    {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
193    {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
194      -\beta_u  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } \\
195    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
196    {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
197    {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
198      -\beta_v  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }    \\
199    r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
200    {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
201      -\beta_w  \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
202    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\
203    r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
204    {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
205      -\beta_w  \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
206    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\
207  \end{split}
209where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in
210\autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points.
211In order to save computation time, they should be approximated by the mean of their values at $T$-points
212(for example in the case of $\alpha$:
213$\alpha_u=\overline{\alpha_T}^{i+1/2}$$\alpha_v=\overline{\alpha_T}^{j+1/2}$ and
216Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases.
219This implementation is a rather old one.
220It is similar to the one proposed by \citet{cox_OM87}, except for the background horizontal diffusion.
221Indeed, the \citet{cox_OM87} implementation of isopycnal diffusion in GFDL-type models requires
222a minimum background horizontal diffusion for numerical stability reasons.
223To overcome this problem, several techniques have been proposed in which the numerical schemes of
224the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}.
225Griffies's scheme is now available in \NEMO\ if \np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}; see \autoref{apdx:TRIADS}.
226Here, another strategy is presented \citep{lazar_phd97}:
227a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of
228grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}).
229This allows an iso-neutral diffusion scheme without additional background horizontal mixing.
230This technique can be viewed as a diffusion operator that acts along large-scale
231(2~$\Delta$x) \gmcomment{2deltax doesnt seem very large scale} iso-neutral surfaces.
232The 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.
234Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
235contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property.
238  \centering
239  \includegraphics[width=0.66\textwidth]{LDF_ZDF1}
240  \caption{Averaging procedure for isopycnal slope computation}
241  \label{fig:LDF_ZDF1}
244%There are three additional questions about the slope calculation.
245%First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.
246%Second, numerical stability issues also require a bound on slopes.
247%Third, the question of boundary condition specified on slopes...
249%from griffies: chapter 13.1....
251% In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04},
252% the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly
253% to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the
254% surface motivates this flattening of isopycnals near the surface).
256For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by
257the namelist scalar \np{rn_slpmax}{rn\_slpmax} (usually $1/100$) everywhere.
258This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to
259$1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean
260(the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface).
261\colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.}
264  \centering
265  \includegraphics[width=0.66\textwidth]{LDF_eiv_slp}
266  \caption[Vertical profile of the slope used for lateral mixing in the mixed layer]{
267    Vertical profile of the slope used for lateral mixing in the mixed layer:
268    \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
269    which has to be adjusted at the surface boundary
270    \ie\ it must tend to zero at the surface since there is no mixing across the air-sea interface:
271    wall boundary condition).
272    Nevertheless,
273    the profile between the surface zero value and the interior iso-neutral one is unknown,
274    and especially the value at the base of the mixed layer;
275    \textit{(b)} profile of slope using a linear tapering of the slope near the surface and
276    imposing a maximum slope of 1/100;
277    \textit{(c)} profile of slope actually used in \NEMO:
278    a linear decrease of the slope from zero at the surface to
279    its ocean interior value computed just below the mixed layer.
280    Note the huge change in the slope at the base of the mixed layer between
281    \textit{(b)} and \textit{(c)}.}
282  \label{fig:LDF_eiv_slp}
285\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.}
287%% =================================================================================================
288\subsection{Slopes for momentum iso-neutral mixing}
290The iso-neutral diffusion operator on momentum is the same as the one used on tracers but
291applied to each component of the velocity separately
292(see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}).
293The slopes between the surface along which the diffusion operator acts and the surface of computation
294($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and
295\textit{vw}- points for the $v$-component.
296They are computed from the slopes used for tracer diffusion,
297\ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}:
300  % \label{eq:LDF_slp_dyn}
301  \begin{aligned}
302    &r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
303    &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&  r_{2t}\ &= \overline{r_{2v}}^{\,j} \\
304    &r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
305    &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
306  \end{aligned}
309The major issue remaining is in the specification of the boundary conditions.
310The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces,
311\ie\ using the shear computed along the model levels and with no additional friction at the ocean bottom
312(see \autoref{sec:LBC_coast}).
314%% =================================================================================================
315\section[Lateral mixing coefficient (\forcode{nn_aht_ijk_t} \& \forcode{nn_ahm_ijk_t})]{Lateral mixing coefficient (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})}
318The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}.
319The way the mixing coefficients are set in the reference version can be described as follows:
321%% =================================================================================================
322\subsection[Mixing coefficients read from file (\forcode{=-20, -30})]{ Mixing coefficients read from file (\protect\np[=-20, -30]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=-20, -30]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})}
324Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model,
325the laplacian viscosity operator uses $A^l$~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and
326decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}.
327Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05.
328The provided fields can either be 2d (\np[=-20]{nn_aht_ijk_t}{nn\_aht\_ijk\_t}, \np[=-20]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}) or 3d (\np[=-30]{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\np[=-30]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}). They must be given at U, V points for tracers and T, F points for momentum (see \autoref{tab:LDF_files}).
331  \centering
332  \begin{tabular}{|l|l|l|l|}
333    \hline
334    Namelist parameter                       & Input filename                               & dimensions & variable names                      \\  \hline
335    \np[=-20]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}     & \forcode{ }            &     $(i,j)$         & \forcode{ahmt_2d, ahmf_2d}  \\  \hline
336    \np[=-20]{nn_aht_ijk_t}{nn\_aht\_ijk\_t}           & \forcode{ }           &     $(i,j)$           & \forcode{ahtu_2d, ahtv_2d}    \\   \hline
337    \np[=-30]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}        & \forcode{ }            &     $(i,j,k)$          & \forcode{ahmt_3d, ahmf_3d}  \\  \hline
338    \np[=-30]{nn_aht_ijk_t}{nn\_aht\_ijk\_t}     & \forcode{ }           &     $(i,j,k)$         & \forcode{ahtu_3d, ahtv_3d}    \\   \hline
339  \end{tabular}
340  \caption{Description of expected input files if mixing coefficients are read from NetCDF files}
341  \label{tab:LDF_files}
344%% =================================================================================================
345\subsection[Constant mixing coefficients (\forcode{=0})]{ Constant mixing coefficients (\protect\np[=0]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=0]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})}
347If constant, mixing coefficients are set thanks to a velocity and a length scales ($U_{scl}$, $L_{scl}$) such that:
350  \label{eq:LDF_constantah}
351  A_o^l = \left\{
352    \begin{aligned}
353      & \frac{1}{2} U_{scl} L_{scl}            & \text{for laplacian operator } \\
354      & \frac{1}{12} U_{scl} L_{scl}^3                    & \text{for bilaplacian operator }
355    \end{aligned}
356  \right.
359 $U_{scl}$ and $L_{scl}$ are given by the namelist parameters \np{rn_Ud}{rn\_Ud}, \np{rn_Uv}{rn\_Uv}, \np{rn_Ld}{rn\_Ld} and \np{rn_Lv}{rn\_Lv}.
361%% =================================================================================================
362\subsection[Vertically varying mixing coefficients (\forcode{=10})]{Vertically varying mixing coefficients (\protect\np[=10]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=10]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})}
364In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which
365the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value,
366and the transition takes place around z=500~m with a width of 200~m.
367This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users.
369%% =================================================================================================
370\subsection[Mesh size dependent mixing coefficients (\forcode{=20})]{Mesh size dependent mixing coefficients (\protect\np[=20]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=20]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})}
372In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and
373the type of operator used:
375  \label{eq:LDF_title}
376  A_l = \left\{
377    \begin{aligned}
378      & \frac{1}{2} U_{scl}  \max(e_1,e_2)         & \text{for laplacian operator } \\
379      & \frac{1}{12} U_{scl}  \max(e_1,e_2)^{3}             & \text{for bilaplacian operator }
380    \end{aligned}
381  \right.
383where $U_{scl}$ is a user defined velocity scale (\np{rn_Ud}{rn\_Ud}, \np{rn_Uv}{rn\_Uv}).
384This variation is intended to reflect the lesser need for subgrid scale eddy mixing where
385the grid size is smaller in the domain.
386It was introduced in the context of the DYNAMO modelling project \citep{willebrand.barnier.ea_PO01}.
387Note that such a grid scale dependance of mixing coefficients significantly increases the range of stability of
388model configurations presenting large changes in grid spacing such as global ocean models.
389Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to
390large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}),
391especially when using a bilaplacian operator.
393\colorbox{yellow}{CASE \np{nn_aht_ijk_t}{nn\_aht\_ijk\_t} = 21 to be added}
395%% =================================================================================================
396\subsection[Mesh size and depth dependent mixing coefficients (\forcode{=30})]{Mesh size and depth dependent mixing coefficients (\protect\np[=30]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=30]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})}
398The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above,
399\ie\ a hyperbolic tangent variation with depth associated with a grid size dependence of
400the magnitude of the coefficient.
402%% =================================================================================================
403\subsection[Velocity dependent mixing coefficients (\forcode{=31})]{Flow dependent mixing coefficients (\protect\np[=31]{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np[=31]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})}
404In 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 $12$):
405\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 ?}
408  \label{eq:LDF_flowah}
409  A_l = \left\{
410    \begin{aligned}
411      & \frac{1}{12} \lvert U \rvert e          & \text{for laplacian operator } \\
412      & \frac{1}{12} \lvert U \rvert e^3             & \text{for bilaplacian operator }
413    \end{aligned}
414  \right.
417%% =================================================================================================
418\subsection[Deformation rate dependent viscosities (\forcode{nn_ahm_ijk_t=32})]{Deformation rate dependent viscosities (\protect\np[=32]{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})}
420This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a
421characteristic time scale $T_{smag}$ the deformation rate and for the lengthscale $L_{smag}$ the maximum wavenumber possible on the horizontal grid, e.g.:
424  \label{eq:LDF_smag1}
425  \begin{split}
426    T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^} \\
427    L_{smag} & = \frac{1}{\pi}\frac{e_1 e_2}{e_1 + e_2}
428  \end{split}
431Introducing a user defined constant $C$ (given in the namelist as \np{rn_csmc}{rn\_csmc}), one can deduce the mixing coefficients as follows:
434  \label{eq:LDF_smag2}
435  A_{smag} = \left\{
436    \begin{aligned}
437      & C^2  T_{smag}^{-1}  L_{smag}^2       & \text{for laplacian operator } \\
438      & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4        & \text{for bilaplacian operator }
439    \end{aligned}
440  \right.
443For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that:
445  \label{eq:LDF_smag3}
446    \begin{aligned}
447      & C_{min} \frac{1}{2}   \lvert U \rvert  e    < A_{smag} < C_{max} \frac{e^2}{   8\rdt}                 & \text{for laplacian operator } \\
448      & C_{min} \frac{1}{12} \lvert U \rvert  e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt}                 & \text{for bilaplacian operator }
449    \end{aligned}
452where $C_{min}$ and $C_{max}$ are adimensional namelist parameters given by \np{rn_minfac}{rn\_minfac} and \np{rn_maxfac}{rn\_maxfac} respectively.
454%% =================================================================================================
455\subsection{About space and time varying mixing coefficients}
457The following points are relevant when the eddy coefficient varies spatially:
459(1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and
460divergent components of the horizontal current (see \autoref{subsec:MB_ldf}).
461Although the eddy coefficient could be set to different values in these two terms,
462this option is not currently available.
464(2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of
465the horizontal divergence for operators acting along model-surfaces are no longer satisfied
468%% =================================================================================================
469\section[Eddy induced velocity (\forcode{ln_ldfeiv})]{Eddy induced velocity (\protect\np{ln_ldfeiv}{ln\_ldfeiv})}
474  \nlst{namtra_eiv}
475  \caption{\forcode{&namtra_eiv}}
476  \label{lst:namtra_eiv}
479%%gm  from Triad appendix  : to be incorporated....
481  Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}.
482  If none of the keys \key{traldf\_cNd}, N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and
483  GM diffusivity $A_e$ are directly set by \np{rn_aeih_0}{rn\_aeih\_0} and \np{rn_aeiv_0}{rn\_aeiv\_0}.
484  If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal
485  scale factor according to \autoref{eq:title}
486  \footnote{
487    Except in global ORCA  $0.5^{\circ}$ runs with \key{traldf\_eiv},
488    where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$
489  }.
490  In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in
491  the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from
492  the Held-Larichev parameterisation
493  \footnote{
494    In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$,
495    where $f_{20}$ is the value of $f$ at $20^{\circ}$~N
496  } (\mdl{ldfeiv}) and \np{rn_aeiv_0}{rn\_aeiv\_0} is ignored unless it is zero.
499When  \citet{gent.mcwilliams_JPO90} diffusion is used (\np[=.true.]{ln_ldfeiv}{ln\_ldfeiv}),
500an eddy induced tracer advection term is added,
501the formulation of which depends on the slopes of iso-neutral surfaces.
502Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces,
503\ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates,
504and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates.
506If isopycnal mixing is used in the standard way, \ie\ \np[=.false.]{ln_traldf_triad}{ln\_traldf\_triad}, the eddy induced velocity is given by:
508  \label{eq:LDF_eiv}
509  \begin{split}
510    u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
511    v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
512    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\} \\
513  \end{split}
515where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn_aei_ijk_t}{nn\_aei\_ijk\_t} \nam{tra_eiv}{tra\_eiv} namelist parameter.
516The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and
517added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed.
518This has been preferred to a separate computation of the advective trends associated with the eiv velocity,
519since it allows us to take advantage of all the advection schemes offered for the tracers
520(see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in
521previous releases of OPA \citep{madec.delecluse.ea_NPM98}.
522This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of
523paramount importance.
525At the surface, lateral and bottom boundaries, the eddy induced velocity,
526and thus the advective eddy fluxes of heat and salt, are set to zero.
527The 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}{nn\_aei\_ijk\_t}, \np{rn_Ue}{rn\_Ue}, \np{rn_Le}{rn\_Le} namelist parameters).
528\colorbox{yellow}{CASE \np{nn_aei_ijk_t}{nn\_aei\_ijk\_t} = 21 to be added}
530In case of setting \np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:TRIADS}.
532%% =================================================================================================
533\section[Mixed layer eddies (\forcode{ln_mle})]{Mixed layer eddies (\protect\np{ln_mle}{ln\_mle})}
537  \nlst{namtra_mle}
538  \caption{\forcode{&namtra_mle}}
539  \label{lst:namtra_mle}
542If  \np[=.true.]{ln_mle}{ln\_mle} in \nam{tra_mle}{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.
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