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1
2% ================================================================
3% Chapter Ñ Lateral Ocean Physics (LDF)
4% ================================================================
5\chapter{Lateral Ocean Physics (LDF)}
6\label{LDF}
7\minitoc
8
9$\ $\newline    % force a new ligne
10
11The lateral physics terms in the momentum and tracer equations have been
12described in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf} 
13and \S\ref{DYN_ldf}). In this section we further discuss each lateral physics option.
14Choosing one lateral physics scheme means for the user defining, (1) the space
15and time variations of the eddy coefficients ; (2) the direction along which the
16lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal
17surfaces); and (3) the type of operator used (harmonic, or biharmonic operators,
18and for tracers only, eddy induced advection on tracers). These three aspects
19of the lateral diffusion are set through namelist parameters and CPP keys
20(see the nam\_traldf and nam\_dynldf below).
21
22%-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
23\namdisplay{nam_traldf} 
24\namdisplay{nam_dynldf} 
25%--------------------------------------------------------------------------------------------------------------
26
27
28% ================================================================
29% Lateral Mixing Coefficients
30% ================================================================
31\section {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn)} }
32\label{LDF_coef}
33
34
35Introducing a space variation in the lateral eddy mixing coefficients changes
36the model core memory requirement, adding up to four extra three-dimensional
37arrays for the geopotential or isopycnal second order operator applied to
38momentum. Six CPP keys control the space variation of eddy coefficients:
39three for momentum and three for tracer. The three choices allow:
40a space variation in the three space directions, in the horizontal plane,
41or in the vertical only. The default option is a constant value over the whole
42ocean on both momentum and tracers.
43
44The number of additional arrays that have to be defined and the gridpoint
45position at which they are defined depend on both the space variation chosen
46and the type of operator used. The resulting eddy viscosity and diffusivity
47coefficients can be a function of more than one variable. Changes in the
48computer code when switching from one option to another have been
49minimized by introducing the eddy coefficients as statement functions
50(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions
51are replaced by their actual meaning during the preprocessing step (CPP).
52The specification of the space variation of the coefficient is made in
53\mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files
54\textit{ldftra\_cNd.h90} and \textit{ldfdyn\_cNd.h90}, with N=1, 2 or 3.
55The user can modify these include files as he/she wishes. The way the
56mixing coefficient are set in the reference version can be briefly described
57as follows:
58
59\subsubsection{Constant Mixing Coefficients (default option)}
60When none of the \textbf{key\_ldfdyn\_...} and \textbf{key\_ldftra\_...} keys are
61defined, a constant value is used over the whole ocean for momentum and
62tracers, which is specified through the \np{ahm0} and \np{aht0} namelist
63parameters.
64
65\subsubsection{Vertically varying Mixing Coefficients (\key{ldftra\_c1d} and \key{ldfdyn\_c1d})} 
66The 1D option is only available when using the $z$-coordinate with full step.
67Indeed in all the other types of vertical coordinate, the depth is a 3D function
68of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent
69mixing coefficients will require 3D arrays, $i.e.$ \key{ldftra\_c3d} and \key{ldftra\_c3d}
70In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced
71in which the surface value is \np{aht0} (\np{ahm0}), the bottom value is 1/4 of
72the surface value, and the transition takes place around z=300~m with a width
73of 300~m ($i.e.$ both the depth and the width of the inflection point are set to 300~m).
74This profile is hard coded in file \hf{ldftra\_c1d}, but can be easily modified by users.
75
76\subsubsection{Horizontally Varying Mixing Coefficients (\key{ldftra\_c2d} and \key{ldfdyn\_c2d})}
77By default the horizontal variation of the eddy coefficient depends on the local mesh
78size and the type of operator used:
79\begin{equation} \label{Eq_title}
80  A_l = \left\{     
81   \begin{aligned}
82         & \frac{\max(e_1,e_2)}{e_{max}} A_o^l           & \text{for laplacian operator } \\
83         & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^& \text{for bilaplacian operator } 
84   \end{aligned}    \right.
85\quad \text{comments}
86\end{equation}
87where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked
88ocean domain, and $A_o^l$ is the \np{ahm0} (momentum) or \np{aht0} (tracer)
89namelist parameter. This variation is intended to reflect the lesser need for subgrid
90scale eddy mixing where the grid size is smaller in the domain. It was introduced in
91the context of the DYNAMO modelling project \citep{Willebrand2001}.
92%%%
93\gmcomment { not only that! stability reasons: with non uniform grid size, it is common
94to face a blow up of the model due to to large diffusive coefficient compare to the smallest
95grid size... this is especially true for bilaplacian (to be added in the text!)  }
96
97Other formulations can be introduced by the user for a given configuration.
98For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian
99viscosity operator uses \np{ahm0}~=~$4.10^4 m^2/s$ poleward of 20$^{\circ}$ 
100north and south and decreases linearly to \np{aht0}~=~$2.10^3 m^2/s$ 
101at the equator \citep{Madec1996, Delecluse_Madec_Bk00}. This modification
102can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}.
103Similar modified horizontal variations can be found with the Antarctic or Arctic
104sub-domain options of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} 
105defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}).
106
107\subsubsection{Space Varying Mixing Coefficients (\key{ldftra\_c3d} and \key{ldfdyn\_c3d})}
108
109The 3D space variation of the mixing coefficient is simply the combination of the
1101D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with
111a grid size dependence of the magnitude of the coefficient.
112
113\subsubsection{Space and Time Varying Mixing Coefficients}
114
115There is no default specification of space and time varying mixing coefficient.
116The only case available is specific to the ORCA2 and ORCA05 global ocean
117configurations (\key{orca\_r2} or \key{orca\_r05}). It provides only a tracer
118mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and
119eddy induced velocity (ORCA05) that depends on the local growth rate of
120baroclinic instability. This specification is actually used when an ORCA key
121and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
122
123A space variation in the eddy coefficient appeals several remarks:
124
125(1) the momentum diffusion operator acting along model level surfaces is
126written in terms of curl and divergent components of the horizontal current
127(see \S\ref{PE_ldf}). Although the eddy coefficient can be set to different values
128in these two terms, this option is not available.
129
130(2) with an horizontally varying viscosity, the quadratic integral constraints
131on enstrophy and on the square of the horizontal divergence for operators
132acting along model-surfaces are no longer satisfied
133(Appendix~\ref{Apdx_dynldf_properties}).
134
135(3) for isopycnal diffusion on momentum or tracers, an additional purely
136horizontal background diffusion with uniform coefficient can be added by
137setting a non zero value of \np{ahmb0} or \np{ahtb0}, a background horizontal
138eddy viscosity or diffusivity coefficient (namelist parameters whose default
139values are $0$). However, the technique used to compute the isopycnal
140slopes is intended to get rid of such a background diffusion, since it introduces
141spurious diapycnal diffusion (see {\S\ref{LDF_slp}).
142
143(4) when an eddy induced advection term is used (\key{trahdf\_eiv}), $A^{eiv}$,
144the eddy induced coefficient has to be defined. Its space variations are controlled
145by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$ 
146\textbf{key\_traldf\_cNd}).
147
148(5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value.
149
150
151% ================================================================
152% Direction of lateral Mixing
153% ================================================================
154\section  [Direction of Lateral Mixing (\textit{ldfslp})]
155      {Direction of Lateral Mixing (\mdl{ldfslp})}
156\label{LDF_slp}
157
158%%%
159\gmcomment{  we should emphasize here that the implementation is a rather old one.
160Better work can be achieved by using \citet{Griffies1998, Griffies2004} iso-neutral scheme. }
161
162A direction for lateral mixing has to be defined when the desired operator does
163not act along the model levels. This occurs when $(a)$ horizontal mixing is
164required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor})
165in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required
166whatever the vertical coordinate is. This direction of mixing is defined by its
167slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the
168quantity to be diffused. For a tracer, this leads to the following four slopes :
169$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while
170for momentum the slopes are  $r_{1T}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for
171$u$ and  $r_{1f}$, $r_{1vw}$, $r_{2T}$, $r_{2vw}$ for $v$.
172
173%gm% add here afigure of the slope in i-direction
174
175\subsection{slopes for tracer geopotential mixing in the $s$-coordinate}
176
177In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and
178$r_2$ are the slopes between the geopotential and computational surfaces.
179Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso} 
180when the diffusive fluxes in the three directions are set to zero and $T$ is
181assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the
182depth of a $T$-point.
183%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
184
185\begin{equation} \label{Eq_ldfslp_geo}
186\begin{aligned}
187 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
188           \;\delta_{i+1/2}[z_T]
189      &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_T]
190\\
191 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 
192           \;\delta_{j+1/2} [z_T]
193      &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_T]
194\\
195 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_T]}}^{\,i,\,k+1/2}
196      &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]
197 \\
198 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_T]}}^{\,j,\,k+1/2}
199      &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
200 \\
201\end{aligned}
202\end{equation}
203
204%gm%  caution I'm not sure the simplification was a good idea!
205
206These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True,
207and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True.
208
209\subsection{slopes for tracer iso-neutral mixing}
210In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral
211and computational surfaces. Their formulation does not depend on the vertical
212coordinate used. Their discrete formulation is found using the fact that the
213diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density)
214vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the
215diffusive fluxes in the three directions to zero leads to the following definition for
216the neutral slopes:
217
218\begin{equation} \label{Eq_ldfslp_iso}
219\begin{split}
220 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
221                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}
222\\
223 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
224                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}
225\\
226 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; 
227         \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
228             {\delta_{k+1/2}[\rho]}
229\\
230 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; 
231         \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
232             {\delta_{k+1/2}[\rho]}
233\\
234\end{split}
235\end{equation}
236
237%gm% rewrite this as the explanation is not very clear !!!
238%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.
239
240%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).
241
242%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.
243
244As the mixing is performed along neutral surfaces, the gradient of $\rho$ in
245\eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which,
246in decibars, is approximated by the depth in meters in the model). Therefore
247\eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is
248needed depending on the vertical coordinate used:
249
250\begin{description}
251
252\item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities
253appearing in the $i$ and $j$ derivatives  are taken at the same depth, thus
254the $in situ$ density can be used. This is not the case for the vertical
255derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ 
256is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following
257\citet{McDougall1987} (see \S\ref{TRA_bn2}).
258
259\item[$z$-coordinate with partial step : ] this case is identical to the full step
260case except that at partial step level, the \emph{horizontal} density gradient
261is evaluated as described in \S\ref{TRA_zpshde}.
262
263\item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,
264there is no specific treatment for iso-neutral mixing in the $s$-coordinate.
265In other words, iso-neutral mixing will only be accurately represented with a
266linear equation of state (\np{neos}=1 or 2). In the case of a "true" equation
267of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} 
268will include a pressure dependent part, leading to the wrong evaluation of
269the neutral slopes.
270
271%gm%
272Note: The solution for $s$-coordinate passes trough the use of different
273(and better) expression for the constraint on iso-neutral fluxes. Following
274\citet{Griffies2004}, instead of specifying directly that there is a zero neutral
275diffusive flux of locally referenced potential density, we stay in the $T$-$S$ 
276plane and consider the balance between the neutral direction diffusive fluxes
277of potential temperature and salinity:
278\begin{equation}
279\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
280\end{equation}
281%gm{  where vector F is ....}
282
283This constraint leads to the following definition for the slopes:
284
285\begin{equation} \label{Eq_ldfslp_iso2}
286\begin{split}
287 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
288      {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
289      {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
290       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }
291\\
292 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
293      {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
294      {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
295       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }
296\\
297 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
298      {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
299       -\beta_\;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
300      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
301\\
302 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
303      {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
304       -\beta_\;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
305      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
306\\
307\end{split}
308\end{equation}
309where $\alpha$ and $\beta$, the thermal expansion and saline contraction
310coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three
311velocity points. In order to save computation time, they should be approximated
312by the mean of their values at $T$-points (for example in the case of $\alpha$
313$\alpha_u=\overline{\alpha_T}^{i+1/2}$$\alpha_v=\overline{\alpha_T}^{j+1/2}$ 
314and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).
315
316Note that such a formulation could be also used in the $z$-coordinate and
317$z$-coordinate with partial steps cases.
318
319\end{description}
320
321This implementation is a rather old one. It is similar to the one proposed
322by Cox [1987], except for the background horizontal diffusion. Indeed,
323the Cox implementation of isopycnal diffusion in GFDL-type models requires
324a minimum background horizontal diffusion for numerical stability reasons.
325To overcome this problem, several techniques have been proposed in which
326the numerical schemes of the ocean model are modified \citep{Weaver1997,
327Griffies1998}. Here, another strategy has been chosen \citep{Lazar1997}:
328a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents
329the development of grid point noise generated by the iso-neutral diffusion
330operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an iso-neutral diffusion scheme
331without additional background horizontal mixing. This technique can be viewed
332as a diffusion operator that acts along large-scale (2~$\Delta$x)
333\gmcomment{2deltax doesnt seem very large scale} 
334iso-neutral surfaces. The diapycnal diffusion required for numerical stability is
335thus minimized and its net effect on the flow is quite small when compared to
336the effect of an horizontal background mixing.
337
338Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
339contrary to the \citet{Griffies1998} operator which has that property.
340
341%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
342\begin{figure}[!ht] \label{Fig_LDF_ZDF1}  \begin{center}
343\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_LDF_ZDF1.pdf}
344\caption {averaging procedure for isopycnal slope computation.}
345\end{center}   \end{figure}
346%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
347
348%There are three additional questions about the slope calculation.
349%First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.
350%Second, numerical stability issues also require a bound on slopes.
351%Third, the question of boundary condition specified on slopes...
352
353%from griffies: chapter 13.1....
354
355
356
357In addition and also for numerical stability reasons \citep{Cox1987, Griffies2004},
358the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly
359to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the
360surface motivates this flattening of isopycnals near the surface).
361
362For numerical stability reasons \citep{Cox1987, Griffies2004}, the slopes must also
363be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear
364fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter
365decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the
366surface motivates this flattening of isopycnals near the surface).
367
368%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
369\begin{figure}[!ht] \label{Fig_eiv_slp}  \begin{center}
370\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_eiv_slp.pdf}
371\caption {Vertical profile of the slope used for lateral mixing in the mixed layer :
372\textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
373which has to be adjusted at the surface boundary (i.e. it must tend to zero at the
374surface since there is no mixing across the air-sea interface: wall boundary
375condition). Nevertheless, the profile between the surface zero value and the interior
376iso-neutral one is unknown, and especially the value at the base of the mixed layer ;
377\textit{(b)} profile of slope using a linear tapering of the slope near the surface and
378imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in
379\NEMO: a linear decrease of the slope from zero at the surface to its ocean interior
380value computed just below the mixed layer. Note the huge change in the slope at the
381base of the mixed layer between  \textit{(b)}  and \textit{(c)}.}
382\end{center}   \end{figure}
383%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
384
385\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs  tapering the coefficient.}
386
387\subsection{slopes for momentum iso-neutral mixing}
388
389The iso-neutral diffusion operator on momentum is the same as the one used on
390tracers but applied to each component of the velocity separately (see
391\eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the
392surface along which the diffusion operator acts and the surface of computation
393($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the
394$u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component.
395They are computed from the slopes used for tracer diffusion, $i.e.$ 
396\eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :
397
398\begin{equation} \label{Eq_ldfslp_dyn}
399\begin{aligned}
400&r_{1T}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
401&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&   r_{2T}\ &= \overline{r_{2v}}^{\,j} \\
402&r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
403&r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
404\end{aligned}
405\end{equation}
406
407The major issue remaining is in the specification of the boundary conditions.
408The same boundary conditions are chosen as those used for lateral
409diffusion along model level surfaces, i.e. using the shear computed along
410the model levels and with no additional friction at the ocean bottom (see
411{\S\ref{LBC_coast}).
412
413
414% ================================================================
415% Eddy Induced Mixing
416% ================================================================
417\section  [Eddy Induced Velocity (\textit{traadv\_eiv}, \textit{ldfeiv})]
418      {Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})}
419\label{LDF_eiv}
420
421When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
422an eddy induced tracer advection term is added, the formulation of which
423depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral
424mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$ 
425\eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo} 
426+ \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
427\begin{equation} \label{Eq_ldfeiv}
428\begin{split}
429 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
430v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
431w^* & = \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\} \\
432\end{split}
433\end{equation}
434where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set
435through \np{aeiv}, a \textit{nam\_traldf} namelist parameter.
436The three components of the eddy induced velocity are computed and add
437to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a
438separate computation of the advective trends associated with the eiv velocity,
439since it allows us to take advantage of all the advection schemes offered for
440the tracers (see \S\ref{TRA_adv}) and not just the $2^{nd}$ order advection
441scheme as in previous releases of OPA \citep{Madec1998}. This is particularly
442useful for passive tracers where \emph{positivity} of the advection scheme is
443of paramount importance.
444
445At the surface, lateral and bottom boundaries, the eddy induced velocity,
446and thus the advective eddy fluxes of heat and salt, are set to zero.
447
448
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450
451
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