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2% ================================================================
3% Chapter Ñ Lateral Ocean Physics (LDF)
4% ================================================================
5\chapter{Lateral Ocean Physics (LDF)}
9$\ $\newline    % force a new ligne
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).
22%-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
28% ================================================================
29% Lateral Mixing Coefficients
30% ================================================================
31\section {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn)} }
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.
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:
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
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.
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}
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}.
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!)  }
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}).
107\subsubsection{Space Varying Mixing Coefficients (\key{ldftra\_c3d} and \key{ldfdyn\_c3d})}
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.
113\subsubsection{Space and Time Varying Mixing Coefficients}
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.
123A space variation in the eddy coefficient appeals several remarks:
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.
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
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}).
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 ($i.e.$ 
148(5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value.
150(6) it is possible to use both the laplacian and biharmonic operators concurrently.
152(7) for testing purposes it is possible to run without lateral diffusion on momentum.
155% ================================================================
156% Direction of lateral Mixing
157% ================================================================
158\section  [Direction of Lateral Mixing (\textit{ldfslp})]
159      {Direction of Lateral Mixing (\mdl{ldfslp})}
163\gmcomment{  we should emphasize here that the implementation is a rather old one.
164Better work can be achieved by using \citet{Griffies1998, Griffies2004} iso-neutral scheme. }
166A direction for lateral mixing has to be defined when the desired operator does
167not act along the model levels. This occurs when $(a)$ horizontal mixing is
168required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor})
169in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required
170whatever the vertical coordinate is. This direction of mixing is defined by its
171slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the
172quantity to be diffused. For a tracer, this leads to the following four slopes :
173$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while
174for momentum the slopes are  $r_{1T}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for
175$u$ and  $r_{1f}$, $r_{1vw}$, $r_{2T}$, $r_{2vw}$ for $v$.
177%gm% add here afigure of the slope in i-direction
179\subsection{slopes for tracer geopotential mixing in the $s$-coordinate}
181In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and
182$r_2$ are the slopes between the geopotential and computational surfaces.
183Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso} 
184when the diffusive fluxes in the three directions are set to zero and $T$ is
185assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the
186depth of a $T$-point.
187%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
189\begin{equation} \label{Eq_ldfslp_geo}
191 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
192           \;\delta_{i+1/2}[z_T]
193      &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_T]
195 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 
196           \;\delta_{j+1/2} [z_T]
197      &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_T]
199 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_T]}}^{\,i,\,k+1/2}
200      &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]
201 \\
202 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_T]}}^{\,j,\,k+1/2}
203      &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
204 \\
208%gm%  caution I'm not sure the simplification was a good idea!
210These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True,
211and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True.
213\subsection{slopes for tracer iso-neutral mixing}
214In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral
215and computational surfaces. Their formulation does not depend on the vertical
216coordinate used. Their discrete formulation is found using the fact that the
217diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density)
218vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the
219diffusive fluxes in the three directions to zero leads to the following definition for
220the neutral slopes:
222\begin{equation} \label{Eq_ldfslp_iso}
224 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
225                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}
227 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
228                        {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}
230 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; 
231         \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
232             {\delta_{k+1/2}[\rho]}
234 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; 
235         \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
236             {\delta_{k+1/2}[\rho]}
241%gm% rewrite this as the explanation is not very clear !!!
242%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.
244%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).
246%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.
248As the mixing is performed along neutral surfaces, the gradient of $\rho$ in
249\eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which,
250in decibars, is approximated by the depth in meters in the model). Therefore
251\eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is
252needed depending on the vertical coordinate used:
256\item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities
257appearing in the $i$ and $j$ derivatives  are taken at the same depth, thus
258the $in situ$ density can be used. This is not the case for the vertical
259derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ 
260is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following
261\citet{McDougall1987} (see \S\ref{TRA_bn2}).
263\item[$z$-coordinate with partial step : ] this case is identical to the full step
264case except that at partial step level, the \emph{horizontal} density gradient
265is evaluated as described in \S\ref{TRA_zpshde}.
267\item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,
268there is no specific treatment for iso-neutral mixing in the $s$-coordinate.
269In other words, iso-neutral mixing will only be accurately represented with a
270linear equation of state (\np{neos}=1 or 2). In the case of a "true" equation
271of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} 
272will include a pressure dependent part, leading to the wrong evaluation of
273the neutral slopes.
276Note: The solution for $s$-coordinate passes trough the use of different
277(and better) expression for the constraint on iso-neutral fluxes. Following
278\citet{Griffies2004}, instead of specifying directly that there is a zero neutral
279diffusive flux of locally referenced potential density, we stay in the $T$-$S$ 
280plane and consider the balance between the neutral direction diffusive fluxes
281of potential temperature and salinity:
283\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
285%gm{  where vector F is ....}
287This constraint leads to the following definition for the slopes:
289\begin{equation} \label{Eq_ldfslp_iso2}
291 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
292      {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
293      {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
294       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }
296 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
297      {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
298      {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
299       -\beta_\;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }
301 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
302      {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
303       -\beta_\;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
304      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
306 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
307      {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
308       -\beta_\;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
309      {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
313where $\alpha$ and $\beta$, the thermal expansion and saline contraction
314coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three
315velocity points. In order to save computation time, they should be approximated
316by the mean of their values at $T$-points (for example in the case of $\alpha$
318and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).
320Note that such a formulation could be also used in the $z$-coordinate and
321$z$-coordinate with partial steps cases.
325This implementation is a rather old one. It is similar to the one proposed
326by Cox [1987], except for the background horizontal diffusion. Indeed,
327the Cox implementation of isopycnal diffusion in GFDL-type models requires
328a minimum background horizontal diffusion for numerical stability reasons.
329To overcome this problem, several techniques have been proposed in which
330the numerical schemes of the ocean model are modified \citep{Weaver1997,
331Griffies1998}. Here, another strategy has been chosen \citep{Lazar1997}:
332a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents
333the development of grid point noise generated by the iso-neutral diffusion
334operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an iso-neutral diffusion scheme
335without additional background horizontal mixing. This technique can be viewed
336as a diffusion operator that acts along large-scale (2~$\Delta$x)
337\gmcomment{2deltax doesnt seem very large scale} 
338iso-neutral surfaces. The diapycnal diffusion required for numerical stability is
339thus minimized and its net effect on the flow is quite small when compared to
340the effect of an horizontal background mixing.
342Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
343contrary to the \citet{Griffies1998} operator which has that property.
346\begin{figure}[!ht] \label{Fig_LDF_ZDF1}  \begin{center}
348\caption {averaging procedure for isopycnal slope computation.}
349\end{center}   \end{figure}
352%There are three additional questions about the slope calculation.
353%First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.
354%Second, numerical stability issues also require a bound on slopes.
355%Third, the question of boundary condition specified on slopes...
357%from griffies: chapter 13.1....
361In addition and also for numerical stability reasons \citep{Cox1987, Griffies2004},
362the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly
363to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the
364surface motivates this flattening of isopycnals near the surface).
366For numerical stability reasons \citep{Cox1987, Griffies2004}, the slopes must also
367be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear
368fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter
369decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the
370surface motivates this flattening of isopycnals near the surface).
373\begin{figure}[!ht] \label{Fig_eiv_slp}  \begin{center}
375\caption {Vertical profile of the slope used for lateral mixing in the mixed layer :
376\textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
377which has to be adjusted at the surface boundary (i.e. it must tend to zero at the
378surface since there is no mixing across the air-sea interface: wall boundary
379condition). Nevertheless, the profile between the surface zero value and the interior
380iso-neutral one is unknown, and especially the value at the base of the mixed layer ;
381\textit{(b)} profile of slope using a linear tapering of the slope near the surface and
382imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in
383\NEMO: a linear decrease of the slope from zero at the surface to its ocean interior
384value computed just below the mixed layer. Note the huge change in the slope at the
385base of the mixed layer between  \textit{(b)}  and \textit{(c)}.}
386\end{center}   \end{figure}
389\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs  tapering the coefficient.}
391\subsection{slopes for momentum iso-neutral mixing}
393The iso-neutral diffusion operator on momentum is the same as the one used on
394tracers but applied to each component of the velocity separately (see
395\eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the
396surface along which the diffusion operator acts and the surface of computation
397($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the
398$u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component.
399They are computed from the slopes used for tracer diffusion, $i.e.$ 
400\eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :
402\begin{equation} \label{Eq_ldfslp_dyn}
404&r_{1T}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
405&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&   r_{2T}\ &= \overline{r_{2v}}^{\,j} \\
406&r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
407&r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
411The major issue remaining is in the specification of the boundary conditions.
412The same boundary conditions are chosen as those used for lateral
413diffusion along model level surfaces, i.e. using the shear computed along
414the model levels and with no additional friction at the ocean bottom (see
418% ================================================================
419% Eddy Induced Mixing
420% ================================================================
421\section  [Eddy Induced Velocity (\textit{traadv\_eiv}, \textit{ldfeiv})]
422      {Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})}
425When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
426an eddy induced tracer advection term is added, the formulation of which
427depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral
428mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$ 
429\eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo} 
430+ \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
431\begin{equation} \label{Eq_ldfeiv}
433 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
434v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
435w^* & = \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\} \\
438where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set
439through \np{aeiv}, a \textit{nam\_traldf} namelist parameter.
440The three components of the eddy induced velocity are computed and add
441to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a
442separate computation of the advective trends associated with the eiv velocity,
443since it allows us to take advantage of all the advection schemes offered for
444the tracers (see \S\ref{TRA_adv}) and not just the $2^{nd}$ order advection
445scheme as in previous releases of OPA \citep{Madec1998}. This is particularly
446useful for passive tracers where \emph{positivity} of the advection scheme is
447of paramount importance.
449At the surface, lateral and bottom boundaries, the eddy induced velocity,
450and thus the advective eddy fluxes of heat and salt, are set to zero.
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