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 | |
---|
15 | The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and |
---|
16 | their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}). |
---|
17 | In this section we further discuss each lateral physics option. |
---|
18 | Choosing 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. |
---|
23 | These three aspects of the lateral diffusion are set through namelist parameters |
---|
24 | (see the \nam{tra\_ldf} and \nam{dyn\_ldf} below). |
---|
25 | Note that this chapter describes the standard implementation of iso-neutral tracer mixing. |
---|
26 | Griffies's implementation, which is used if \np{ln\_traldf\_triad}\forcode{=.true.}, |
---|
27 | is described in \autoref{apdx:TRIADS} |
---|
28 | |
---|
29 | %-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- |
---|
30 | |
---|
31 | \nlst{namtra_ldf} |
---|
32 | |
---|
33 | \nlst{namdyn_ldf} |
---|
34 | %-------------------------------------------------------------------------------------------------------------- |
---|
35 | |
---|
36 | % ================================================================ |
---|
37 | % Lateral Mixing Operator |
---|
38 | % ================================================================ |
---|
39 | \section[Lateral mixing operators] |
---|
40 | {Lateral mixing operators} |
---|
41 | \label{sec:LDF_op} |
---|
42 | We 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}. |
---|
43 | |
---|
44 | \subsection[No lateral mixing (\forcode{ln_traldf_OFF}, \forcode{ln_dynldf_OFF})] |
---|
45 | {No lateral mixing (\protect\np{ln\_traldf\_OFF}, \protect\np{ln\_dynldf\_OFF})} |
---|
46 | |
---|
47 | It is possible to run without explicit lateral diffusion on tracers (\protect\np{ln\_traldf\_OFF}\forcode{=.true.}) and/or |
---|
48 | momentum (\protect\np{ln\_dynldf\_OFF}\forcode{=.true.}). The latter option is even recommended if using the |
---|
49 | UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{=.true.}, |
---|
50 | see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes. |
---|
51 | |
---|
52 | \subsection[Laplacian mixing (\forcode{ln_traldf_lap}, \forcode{ln_dynldf_lap})] |
---|
53 | {Laplacian mixing (\protect\np{ln\_traldf\_lap}, \protect\np{ln\_dynldf\_lap})} |
---|
54 | Setting \protect\np{ln\_traldf\_lap}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_lap}\forcode{=.true.} enables |
---|
55 | a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine |
---|
56 | Laplacian and Bilaplacian operators for the same variable. |
---|
57 | |
---|
58 | \subsection[Bilaplacian mixing (\forcode{ln_traldf_blp}, \forcode{ln_dynldf_blp})] |
---|
59 | {Bilaplacian mixing (\protect\np{ln\_traldf\_blp}, \protect\np{ln\_dynldf\_blp})} |
---|
60 | Setting \protect\np{ln\_traldf\_blp}\forcode{=.true.} and/or \protect\np{ln\_dynldf\_blp}\forcode{=.true.} enables |
---|
61 | a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice. |
---|
62 | We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. |
---|
63 | |
---|
64 | % ================================================================ |
---|
65 | % Direction of lateral Mixing |
---|
66 | % ================================================================ |
---|
67 | \section[Direction of lateral mixing (\textit{ldfslp.F90})] |
---|
68 | {Direction of lateral mixing (\protect\mdl{ldfslp})} |
---|
69 | \label{sec:LDF_slp} |
---|
70 | |
---|
71 | %%% |
---|
72 | \gmcomment{ |
---|
73 | we should emphasize here that the implementation is a rather old one. |
---|
74 | Better work can be achieved by using \citet{griffies.gnanadesikan.ea_JPO98, griffies_bk04} iso-neutral scheme. |
---|
75 | } |
---|
76 | |
---|
77 | A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. |
---|
78 | This occurs when $(a)$ horizontal mixing is required on tracer or momentum |
---|
79 | (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates, |
---|
80 | and $(b)$ isoneutral mixing is required whatever the vertical coordinate is. |
---|
81 | This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of |
---|
82 | the cell of the quantity to be diffused. |
---|
83 | For a tracer, this leads to the following four slopes: |
---|
84 | $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), |
---|
85 | while for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and |
---|
86 | $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. |
---|
87 | |
---|
88 | %gm% add here afigure of the slope in i-direction |
---|
89 | |
---|
90 | \subsection{Slopes for tracer geopotential mixing in the $s$-coordinate} |
---|
91 | |
---|
92 | In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between |
---|
93 | the geopotential and computational surfaces. |
---|
94 | Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when |
---|
95 | the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform, |
---|
96 | \ie\ a linear function of $z_T$, the depth of a $T$-point. |
---|
97 | %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} |
---|
98 | |
---|
99 | \begin{equation} |
---|
100 | \label{eq:LDF_slp_geo} |
---|
101 | \begin{aligned} |
---|
102 | r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} |
---|
103 | \;\delta_{i+1/2}[z_t] |
---|
104 | &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] \ \ \ \\ |
---|
105 | r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} |
---|
106 | \;\delta_{j+1/2} [z_t] |
---|
107 | &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] \ \ \ \\ |
---|
108 | r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2} |
---|
109 | &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}] \\ |
---|
110 | r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2} |
---|
111 | &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] |
---|
112 | \end{aligned} |
---|
113 | \end{equation} |
---|
114 | |
---|
115 | %gm% caution I'm not sure the simplification was a good idea! |
---|
116 | |
---|
117 | These slopes are computed once in \rou{ldf\_slp\_init} when \np{ln\_sco}\forcode{=.true.}, |
---|
118 | and either \np{ln\_traldf\_hor}\forcode{=.true.} or \np{ln\_dynldf\_hor}\forcode{=.true.}. |
---|
119 | |
---|
120 | \subsection{Slopes for tracer iso-neutral mixing} |
---|
121 | \label{subsec:LDF_slp_iso} |
---|
122 | |
---|
123 | In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces. |
---|
124 | Their formulation does not depend on the vertical coordinate used. |
---|
125 | Their discrete formulation is found using the fact that the diffusive fluxes of |
---|
126 | locally referenced potential density (\ie\ $in situ$ density) vanish. |
---|
127 | So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in |
---|
128 | the three directions to zero leads to the following definition for the neutral slopes: |
---|
129 | |
---|
130 | \begin{equation} |
---|
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} |
---|
144 | \end{equation} |
---|
145 | |
---|
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. |
---|
148 | |
---|
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). |
---|
150 | |
---|
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. |
---|
152 | |
---|
153 | As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to |
---|
154 | be evaluated at the same local pressure |
---|
155 | (which, in decibars, is approximated by the depth in meters in the model). |
---|
156 | Therefore \autoref{eq:LDF_slp_iso} cannot be used as such, |
---|
157 | but further transformation is needed depending on the vertical coordinate used: |
---|
158 | |
---|
159 | \begin{description} |
---|
160 | |
---|
161 | \item[$z$-coordinate with full step: ] |
---|
162 | in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, |
---|
163 | thus the $in situ$ density can be used. |
---|
164 | This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, |
---|
165 | where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87} |
---|
166 | (see \autoref{subsec:TRA_bn2}). |
---|
167 | |
---|
168 | \item[$z$-coordinate with partial step: ] |
---|
169 | this case is identical to the full step case except that at partial step level, |
---|
170 | the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}. |
---|
171 | |
---|
172 | \item[$s$- or hybrid $s$-$z$- coordinate: ] |
---|
173 | in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if |
---|
174 | the Griffies scheme is used (\np{ln\_traldf\_triad}\forcode{=.true.}; |
---|
175 | see \autoref{apdx:TRIADS}). |
---|
176 | In other words, iso-neutral mixing will only be accurately represented with a linear equation of state |
---|
177 | (\np{ln\_seos}\forcode{=.true.}). |
---|
178 | In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso} |
---|
179 | will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. |
---|
180 | |
---|
181 | %gm% |
---|
182 | Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for |
---|
183 | the constraint on iso-neutral fluxes. |
---|
184 | Following \citet{griffies_bk04}, instead of specifying directly that there is a zero neutral diffusive flux of |
---|
185 | locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between |
---|
186 | the neutral direction diffusive fluxes of potential temperature and salinity: |
---|
187 | \[ |
---|
188 | \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) |
---|
189 | \] |
---|
190 | % gm{ where vector F is ....} |
---|
191 | |
---|
192 | This constraint leads to the following definition for the slopes: |
---|
193 | |
---|
194 | \[ |
---|
195 | % \label{eq:LDF_slp_iso2} |
---|
196 | \begin{split} |
---|
197 | r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac |
---|
198 | {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]} |
---|
199 | {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k} |
---|
200 | -\beta_u \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } \\ |
---|
201 | r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac |
---|
202 | {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]} |
---|
203 | {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k} |
---|
204 | -\beta_v \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} } \\ |
---|
205 | r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac |
---|
206 | {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2} |
---|
207 | -\beta_w \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} } |
---|
208 | {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\ |
---|
209 | r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac |
---|
210 | {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2} |
---|
211 | -\beta_w \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} } |
---|
212 | {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\ |
---|
213 | \end{split} |
---|
214 | \] |
---|
215 | where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in |
---|
216 | \autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points. |
---|
217 | In order to save computation time, they should be approximated by the mean of their values at $T$-points |
---|
218 | (for example in the case of $\alpha$: |
---|
219 | $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$ and |
---|
220 | $\alpha_w=\overline{\alpha_T}^{k+1/2}$). |
---|
221 | |
---|
222 | Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases. |
---|
223 | |
---|
224 | \end{description} |
---|
225 | |
---|
226 | This implementation is a rather old one. |
---|
227 | It is similar to the one proposed by \citet{cox_OM87}, except for the background horizontal diffusion. |
---|
228 | Indeed, the \citet{cox_OM87} implementation of isopycnal diffusion in GFDL-type models requires |
---|
229 | a minimum background horizontal diffusion for numerical stability reasons. |
---|
230 | To overcome this problem, several techniques have been proposed in which the numerical schemes of |
---|
231 | the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. |
---|
232 | Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:TRIADS}. |
---|
233 | Here, another strategy is presented \citep{lazar_phd97}: |
---|
234 | a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of |
---|
235 | grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}). |
---|
236 | This allows an iso-neutral diffusion scheme without additional background horizontal mixing. |
---|
237 | This technique can be viewed as a diffusion operator that acts along large-scale |
---|
238 | (2~$\Delta$x) \gmcomment{2deltax doesnt seem very large scale} iso-neutral surfaces. |
---|
239 | The 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. |
---|
240 | |
---|
241 | Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, |
---|
242 | contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property. |
---|
243 | |
---|
244 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
245 | \begin{figure}[!ht] |
---|
246 | \begin{center} |
---|
247 | \includegraphics[width=\textwidth]{Fig_LDF_ZDF1} |
---|
248 | \caption { |
---|
249 | \protect\label{fig:LDF_ZDF1} |
---|
250 | averaging procedure for isopycnal slope computation. |
---|
251 | } |
---|
252 | \end{center} |
---|
253 | \end{figure} |
---|
254 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
255 | |
---|
256 | %There are three additional questions about the slope calculation. |
---|
257 | %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. |
---|
258 | %Second, numerical stability issues also require a bound on slopes. |
---|
259 | %Third, the question of boundary condition specified on slopes... |
---|
260 | |
---|
261 | %from griffies: chapter 13.1.... |
---|
262 | |
---|
263 | |
---|
264 | |
---|
265 | % In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04}, |
---|
266 | % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly |
---|
267 | % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the |
---|
268 | % surface motivates this flattening of isopycnals near the surface). |
---|
269 | |
---|
270 | For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by |
---|
271 | the namelist scalar \np{rn\_slpmax} (usually $1/100$) everywhere. |
---|
272 | This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to |
---|
273 | $1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean |
---|
274 | (the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface). |
---|
275 | \colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.} |
---|
276 | |
---|
277 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
278 | \begin{figure}[!ht] |
---|
279 | \begin{center} |
---|
280 | \includegraphics[width=\textwidth]{Fig_eiv_slp} |
---|
281 | \caption{ |
---|
282 | \protect\label{fig:LDF_eiv_slp} |
---|
283 | Vertical profile of the slope used for lateral mixing in the mixed layer: |
---|
284 | \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, |
---|
285 | which has to be adjusted at the surface boundary |
---|
286 | \ie\ it must tend to zero at the surface since there is no mixing across the air-sea interface: |
---|
287 | wall boundary condition). |
---|
288 | Nevertheless, the profile between the surface zero value and the interior iso-neutral one is unknown, |
---|
289 | and especially the value at the base of the mixed layer; |
---|
290 | \textit{(b)} profile of slope using a linear tapering of the slope near the surface and |
---|
291 | imposing a maximum slope of 1/100; |
---|
292 | \textit{(c)} profile of slope actually used in \NEMO: a linear decrease of the slope from |
---|
293 | zero at the surface to its ocean interior value computed just below the mixed layer. |
---|
294 | Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}. |
---|
295 | } |
---|
296 | \end{center} |
---|
297 | \end{figure} |
---|
298 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
299 | |
---|
300 | \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.} |
---|
301 | |
---|
302 | \subsection{Slopes for momentum iso-neutral mixing} |
---|
303 | |
---|
304 | The iso-neutral diffusion operator on momentum is the same as the one used on tracers but |
---|
305 | applied to each component of the velocity separately |
---|
306 | (see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). |
---|
307 | The slopes between the surface along which the diffusion operator acts and the surface of computation |
---|
308 | ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and |
---|
309 | \textit{vw}- points for the $v$-component. |
---|
310 | They are computed from the slopes used for tracer diffusion, |
---|
311 | \ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}: |
---|
312 | |
---|
313 | \[ |
---|
314 | % \label{eq:LDF_slp_dyn} |
---|
315 | \begin{aligned} |
---|
316 | &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ |
---|
317 | &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2t}\ &= \overline{r_{2v}}^{\,j} \\ |
---|
318 | &r_{1uw} = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ & r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\ |
---|
319 | &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&& r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\ |
---|
320 | \end{aligned} |
---|
321 | \] |
---|
322 | |
---|
323 | The major issue remaining is in the specification of the boundary conditions. |
---|
324 | The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces, |
---|
325 | \ie\ using the shear computed along the model levels and with no additional friction at the ocean bottom |
---|
326 | (see \autoref{sec:LBC_coast}). |
---|
327 | |
---|
328 | |
---|
329 | % ================================================================ |
---|
330 | % Lateral Mixing Coefficients |
---|
331 | % ================================================================ |
---|
332 | \section[Lateral mixing coefficient (\forcode{nn_aht_ijk_t}, \forcode{nn_ahm_ijk_t})] |
---|
333 | {Lateral mixing coefficient (\protect\np{nn\_aht\_ijk\_t}, \protect\np{nn\_ahm\_ijk\_t})} |
---|
334 | \label{sec:LDF_coef} |
---|
335 | |
---|
336 | The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. |
---|
337 | The way the mixing coefficients are set in the reference version can be described as follows: |
---|
338 | |
---|
339 | \subsection[Mixing coefficients read from file (\forcode{nn_aht_ijk_t=-20, -30}, \forcode{nn_ahm_ijk_t=-20,-30})] |
---|
340 | { Mixing coefficients read from file (\protect\np{nn\_aht\_ijk\_t}\forcode{=-20, -30}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=-20, -30})} |
---|
341 | |
---|
342 | Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model, |
---|
343 | the laplacian viscosity operator uses $A^l$~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and |
---|
344 | decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}. |
---|
345 | Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05. |
---|
346 | The 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}). |
---|
347 | |
---|
348 | %-------------------------------------------------TABLE--------------------------------------------------- |
---|
349 | \begin{table}[htb] |
---|
350 | \begin{center} |
---|
351 | \begin{tabular}{|l|l|l|l|} |
---|
352 | \hline |
---|
353 | Namelist parameter & Input filename & dimensions & variable names \\ \hline |
---|
354 | \np{nn\_ahm\_ijk\_t}\forcode{=-20} & \forcode{eddy_viscosity_2D.nc } & $(i,j)$ & \forcode{ahmt_2d, ahmf_2d} \\ \hline |
---|
355 | \np{nn\_aht\_ijk\_t}\forcode{=-20} & \forcode{eddy_diffusivity_2D.nc } & $(i,j)$ & \forcode{ahtu_2d, ahtv_2d} \\ \hline |
---|
356 | \np{nn\_ahm\_ijk\_t}\forcode{=-30} & \forcode{eddy_viscosity_3D.nc } & $(i,j,k)$ & \forcode{ahmt_3d, ahmf_3d} \\ \hline |
---|
357 | \np{nn\_aht\_ijk\_t}\forcode{=-30} & \forcode{eddy_diffusivity_3D.nc } & $(i,j,k)$ & \forcode{ahtu_3d, ahtv_3d} \\ \hline |
---|
358 | \end{tabular} |
---|
359 | \caption{ |
---|
360 | \protect\label{tab:LDF_files} |
---|
361 | Description of expected input files if mixing coefficients are read from NetCDF files. |
---|
362 | } |
---|
363 | \end{center} |
---|
364 | \end{table} |
---|
365 | %-------------------------------------------------------------------------------------------------------------- |
---|
366 | |
---|
367 | \subsection[Constant mixing coefficients (\forcode{nn_aht_ijk_t=0}, \forcode{nn_ahm_ijk_t=0})] |
---|
368 | { Constant mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=0}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=0})} |
---|
369 | |
---|
370 | If constant, mixing coefficients are set thanks to a velocity and a length scales ($U_{scl}$, $L_{scl}$) such that: |
---|
371 | |
---|
372 | \begin{equation} |
---|
373 | \label{eq:LDF_constantah} |
---|
374 | A_o^l = \left\{ |
---|
375 | \begin{aligned} |
---|
376 | & \frac{1}{2} U_{scl} L_{scl} & \text{for laplacian operator } \\ |
---|
377 | & \frac{1}{12} U_{scl} L_{scl}^3 & \text{for bilaplacian operator } |
---|
378 | \end{aligned} |
---|
379 | \right. |
---|
380 | \end{equation} |
---|
381 | |
---|
382 | $U_{scl}$ and $L_{scl}$ are given by the namelist parameters \np{rn\_Ud}, \np{rn\_Uv}, \np{rn\_Ld} and \np{rn\_Lv}. |
---|
383 | |
---|
384 | \subsection[Vertically varying mixing coefficients (\forcode{nn_aht_ijk_t=10}, \forcode{nn_ahm_ijk_t=10})] |
---|
385 | {Vertically varying mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=10}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=10})} |
---|
386 | |
---|
387 | In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which |
---|
388 | the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value, |
---|
389 | and the transition takes place around z=500~m with a width of 200~m. |
---|
390 | This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. |
---|
391 | |
---|
392 | \subsection[Mesh size dependent mixing coefficients (\forcode{nn_aht_ijk_t=20}, \forcode{nn_ahm_ijk_t=20})] |
---|
393 | {Mesh size dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=20}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=20})} |
---|
394 | |
---|
395 | In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and |
---|
396 | the type of operator used: |
---|
397 | \begin{equation} |
---|
398 | \label{eq:LDF_title} |
---|
399 | A_l = \left\{ |
---|
400 | \begin{aligned} |
---|
401 | & \frac{1}{2} U_{scl} \max(e_1,e_2) & \text{for laplacian operator } \\ |
---|
402 | & \frac{1}{12} U_{scl} \max(e_1,e_2)^{3} & \text{for bilaplacian operator } |
---|
403 | \end{aligned} |
---|
404 | \right. |
---|
405 | \end{equation} |
---|
406 | where $U_{scl}$ is a user defined velocity scale (\np{rn\_Ud}, \np{rn\_Uv}). |
---|
407 | This variation is intended to reflect the lesser need for subgrid scale eddy mixing where |
---|
408 | the grid size is smaller in the domain. |
---|
409 | It was introduced in the context of the DYNAMO modelling project \citep{willebrand.barnier.ea_PO01}. |
---|
410 | Note that such a grid scale dependance of mixing coefficients significantly increases the range of stability of |
---|
411 | model configurations presenting large changes in grid spacing such as global ocean models. |
---|
412 | Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to |
---|
413 | large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}), |
---|
414 | especially when using a bilaplacian operator. |
---|
415 | |
---|
416 | \colorbox{yellow}{CASE \np{nn\_aht\_ijk\_t} = 21 to be added} |
---|
417 | |
---|
418 | \subsection[Mesh size and depth dependent mixing coefficients (\forcode{nn_aht_ijk_t=30}, \forcode{nn_ahm_ijk_t=30})] |
---|
419 | {Mesh size and depth dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=30}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=30})} |
---|
420 | |
---|
421 | The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above, |
---|
422 | \ie\ a hyperbolic tangent variation with depth associated with a grid size dependence of |
---|
423 | the magnitude of the coefficient. |
---|
424 | |
---|
425 | \subsection[Velocity dependent mixing coefficients (\forcode{nn_aht_ijk_t=31}, \forcode{nn_ahm_ijk_t=31})] |
---|
426 | {Flow dependent mixing coefficients (\protect\np{nn\_aht\_ijk\_t}\forcode{=31}, \protect\np{nn\_ahm\_ijk\_t}\forcode{=31})} |
---|
427 | In 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 $12$): |
---|
428 | \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 ?} |
---|
429 | |
---|
430 | \begin{equation} |
---|
431 | \label{eq:LDF_flowah} |
---|
432 | A_l = \left\{ |
---|
433 | \begin{aligned} |
---|
434 | & \frac{1}{12} \lvert U \rvert e & \text{for laplacian operator } \\ |
---|
435 | & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } |
---|
436 | \end{aligned} |
---|
437 | \right. |
---|
438 | \end{equation} |
---|
439 | |
---|
440 | \subsection[Deformation rate dependent viscosities (\forcode{nn_ahm_ijk_t=32})] |
---|
441 | {Deformation rate dependent viscosities (\protect\np{nn\_ahm\_ijk\_t}\forcode{=32})} |
---|
442 | |
---|
443 | This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a |
---|
444 | characteristic time scale $T_{smag}$ the deformation rate and for the lengthscale $L_{smag}$ the maximum wavenumber possible on the horizontal grid, e.g.: |
---|
445 | |
---|
446 | \begin{equation} |
---|
447 | \label{eq:LDF_smag1} |
---|
448 | \begin{split} |
---|
449 | T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2 } \\ |
---|
450 | L_{smag} & = \frac{1}{\pi}\frac{e_1 e_2}{e_1 + e_2} |
---|
451 | \end{split} |
---|
452 | \end{equation} |
---|
453 | |
---|
454 | Introducing a user defined constant $C$ (given in the namelist as \np{rn\_csmc}), one can deduce the mixing coefficients as follows: |
---|
455 | |
---|
456 | \begin{equation} |
---|
457 | \label{eq:LDF_smag2} |
---|
458 | A_{smag} = \left\{ |
---|
459 | \begin{aligned} |
---|
460 | & C^2 T_{smag}^{-1} L_{smag}^2 & \text{for laplacian operator } \\ |
---|
461 | & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4 & \text{for bilaplacian operator } |
---|
462 | \end{aligned} |
---|
463 | \right. |
---|
464 | \end{equation} |
---|
465 | |
---|
466 | For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that: |
---|
467 | \begin{equation} |
---|
468 | \label{eq:LDF_smag3} |
---|
469 | \begin{aligned} |
---|
470 | & C_{min} \frac{1}{2} \lvert U \rvert e < A_{smag} < C_{max} \frac{e^2}{ 8\rdt} & \text{for laplacian operator } \\ |
---|
471 | & C_{min} \frac{1}{12} \lvert U \rvert e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt} & \text{for bilaplacian operator } |
---|
472 | \end{aligned} |
---|
473 | \end{equation} |
---|
474 | |
---|
475 | where $C_{min}$ and $C_{max}$ are adimensional namelist parameters given by \np{rn\_minfac} and \np{rn\_maxfac} respectively. |
---|
476 | |
---|
477 | \subsection{About space and time varying mixing coefficients} |
---|
478 | |
---|
479 | The following points are relevant when the eddy coefficient varies spatially: |
---|
480 | |
---|
481 | (1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and |
---|
482 | divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). |
---|
483 | Although the eddy coefficient could be set to different values in these two terms, |
---|
484 | this option is not currently available. |
---|
485 | |
---|
486 | (2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of |
---|
487 | the horizontal divergence for operators acting along model-surfaces are no longer satisfied |
---|
488 | (\autoref{sec:INVARIANTS_dynldf_properties}). |
---|
489 | |
---|
490 | % ================================================================ |
---|
491 | % Eddy Induced Mixing |
---|
492 | % ================================================================ |
---|
493 | \section[Eddy induced velocity (\forcode{ln_ldfeiv=.true.})] |
---|
494 | {Eddy induced velocity (\protect\np{ln\_ldfeiv}\forcode{=.true.})} |
---|
495 | |
---|
496 | \label{sec:LDF_eiv} |
---|
497 | |
---|
498 | %--------------------------------------------namtra_eiv--------------------------------------------------- |
---|
499 | |
---|
500 | \nlst{namtra_eiv} |
---|
501 | |
---|
502 | %-------------------------------------------------------------------------------------------------------------- |
---|
503 | |
---|
504 | |
---|
505 | %%gm from Triad appendix : to be incorporated.... |
---|
506 | \gmcomment{ |
---|
507 | Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}. |
---|
508 | If none of the keys \key{traldf\_cNd}, N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and |
---|
509 | GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and \np{rn\_aeiv\_0}. |
---|
510 | If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal |
---|
511 | scale factor according to \autoref{eq:title} |
---|
512 | \footnote{ |
---|
513 | Except in global ORCA $0.5^{\circ}$ runs with \key{traldf\_eiv}, |
---|
514 | where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$ |
---|
515 | }. |
---|
516 | In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in |
---|
517 | the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from |
---|
518 | the Held-Larichev parameterisation |
---|
519 | \footnote{ |
---|
520 | In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$, |
---|
521 | where $f_{20}$ is the value of $f$ at $20^{\circ}$~N |
---|
522 | } (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored unless it is zero. |
---|
523 | } |
---|
524 | |
---|
525 | When \citet{gent.mcwilliams_JPO90} diffusion is used (\np{ln\_ldfeiv}\forcode{=.true.}), |
---|
526 | an eddy induced tracer advection term is added, |
---|
527 | the formulation of which depends on the slopes of iso-neutral surfaces. |
---|
528 | Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, |
---|
529 | \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates, |
---|
530 | and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates. |
---|
531 | |
---|
532 | If isopycnal mixing is used in the standard way, \ie\ \np{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by: |
---|
533 | \begin{equation} |
---|
534 | \label{eq:LDF_eiv} |
---|
535 | \begin{split} |
---|
536 | u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ |
---|
537 | v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\ |
---|
538 | 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\} \\ |
---|
539 | \end{split} |
---|
540 | \end{equation} |
---|
541 | where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn\_aei\_ijk\_t} \nam{tra\_eiv} namelist parameter. |
---|
542 | The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and |
---|
543 | added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed. |
---|
544 | This has been preferred to a separate computation of the advective trends associated with the eiv velocity, |
---|
545 | since it allows us to take advantage of all the advection schemes offered for the tracers |
---|
546 | (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in |
---|
547 | previous releases of OPA \citep{madec.delecluse.ea_NPM98}. |
---|
548 | This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of |
---|
549 | paramount importance. |
---|
550 | |
---|
551 | At the surface, lateral and bottom boundaries, the eddy induced velocity, |
---|
552 | and thus the advective eddy fluxes of heat and salt, are set to zero. |
---|
553 | The 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). |
---|
554 | \colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added} |
---|
555 | |
---|
556 | In 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}. |
---|
557 | |
---|
558 | % ================================================================ |
---|
559 | % Mixed layer eddies |
---|
560 | % ================================================================ |
---|
561 | \section[Mixed layer eddies (\forcode{ln_mle=.true.})] |
---|
562 | {Mixed layer eddies (\protect\np{ln\_mle}\forcode{=.true.})} |
---|
563 | |
---|
564 | \label{sec:LDF_mle} |
---|
565 | |
---|
566 | %--------------------------------------------namtra_eiv--------------------------------------------------- |
---|
567 | |
---|
568 | \nlst{namtra_mle} |
---|
569 | |
---|
570 | %-------------------------------------------------------------------------------------------------------------- |
---|
571 | |
---|
572 | If \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. |
---|
573 | |
---|
574 | \colorbox{yellow}{TBC} |
---|
575 | |
---|
576 | \biblio |
---|
577 | |
---|
578 | \pindex |
---|
579 | |
---|
580 | \end{document} |
---|