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branches/2015/dev_merge_2015/DOC/TexFiles/Chapters/Chap_LDF.tex
r4147 r6060 1 1 2 2 % ================================================================ 3 % Chapter �Lateral Ocean Physics (LDF)3 % Chapter ——— Lateral Ocean Physics (LDF) 4 4 % ================================================================ 5 5 \chapter{Lateral Ocean Physics (LDF)} … … 15 15 described in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf} 16 16 and \S\ref{DYN_ldf}). In this section we further discuss each lateral physics option. 17 Choosing one lateral physics scheme means for the user defining, (1) the space 18 and time variations of the eddy coefficients ; (2) the direction along which the 19 lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal 20 surfaces); and (3) the type of operator used (harmonic, or biharmonic operators, 21 and for tracers only, eddy induced advection on tracers). These three aspects 22 of the lateral diffusion are set through namelist parameters and CPP keys 23 (see the \textit{\ngn{nam\_traldf}} and \textit{\ngn{nam\_dynldf}} below). Note 24 that this chapter describes the default implementation of iso-neutral 17 Choosing one lateral physics scheme means for the user defining, 18 (1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term) ; 19 (2) the direction along which the lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal surfaces) ; and 20 (3) the space and time variations of the eddy coefficients. 21 These three aspects of the lateral diffusion are set through namelist parameters 22 (see the \textit{\ngn{nam\_traldf}} and \textit{\ngn{nam\_dynldf}} below). 23 Note that this chapter describes the standard implementation of iso-neutral 25 24 tracer mixing, and Griffies's implementation, which is used if 26 25 \np{traldf\_grif}=true, is described in Appdx\ref{sec:triad} … … 33 32 34 33 % ================================================================ 34 % Direction of lateral Mixing 35 % ================================================================ 36 \section [Direction of Lateral Mixing (\textit{ldfslp})] 37 {Direction of Lateral Mixing (\mdl{ldfslp})} 38 \label{LDF_slp} 39 40 %%% 41 \gmcomment{ we should emphasize here that the implementation is a rather old one. 42 Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. } 43 44 A direction for lateral mixing has to be defined when the desired operator does 45 not act along the model levels. This occurs when $(a)$ horizontal mixing is 46 required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) 47 in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required 48 whatever the vertical coordinate is. This direction of mixing is defined by its 49 slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the 50 quantity to be diffused. For a tracer, this leads to the following four slopes : 51 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while 52 for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for 53 $u$ and $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 54 55 %gm% add here afigure of the slope in i-direction 56 57 \subsection{slopes for tracer geopotential mixing in the $s$-coordinate} 58 59 In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and 60 $r_2$ are the slopes between the geopotential and computational surfaces. 61 Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso} 62 when the diffusive fluxes in the three directions are set to zero and $T$ is 63 assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the 64 depth of a $T$-point. 65 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 66 67 \begin{equation} \label{Eq_ldfslp_geo} 68 \begin{aligned} 69 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} 70 \;\delta_{i+1/2}[z_t] 71 &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] 72 \\ 73 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 74 \;\delta_{j+1/2} [z_t] 75 &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] 76 \\ 77 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2} 78 &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}] 79 \\ 80 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2} 81 &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] 82 \\ 83 \end{aligned} 84 \end{equation} 85 86 %gm% caution I'm not sure the simplification was a good idea! 87 88 These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True, 89 and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True. 90 91 \subsection{Slopes for tracer iso-neutral mixing}\label{LDF_slp_iso} 92 In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral 93 and computational surfaces. Their formulation does not depend on the vertical 94 coordinate used. Their discrete formulation is found using the fact that the 95 diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density) 96 vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the 97 diffusive fluxes in the three directions to zero leads to the following definition for 98 the neutral slopes: 99 100 \begin{equation} \label{Eq_ldfslp_iso} 101 \begin{split} 102 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} 103 {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} 104 \\ 105 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]} 106 {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} 107 \\ 108 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; 109 \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}} 110 {\delta_{k+1/2}[\rho]} 111 \\ 112 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; 113 \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}} 114 {\delta_{k+1/2}[\rho]} 115 \\ 116 \end{split} 117 \end{equation} 118 119 %gm% rewrite this as the explanation is not very clear !!! 120 %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. 121 122 %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). 123 124 %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. 125 126 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in 127 \eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which, 128 in decibars, is approximated by the depth in meters in the model). Therefore 129 \eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is 130 needed depending on the vertical coordinate used: 131 132 \begin{description} 133 134 \item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities 135 appearing in the $i$ and $j$ derivatives are taken at the same depth, thus 136 the $in situ$ density can be used. This is not the case for the vertical 137 derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ 138 is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following 139 \citet{McDougall1987} (see \S\ref{TRA_bn2}). 140 141 \item[$z$-coordinate with partial step : ] this case is identical to the full step 142 case except that at partial step level, the \emph{horizontal} density gradient 143 is evaluated as described in \S\ref{TRA_zpshde}. 144 145 \item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO, 146 iso-neutral mixing is only employed for $s$-coordinates if the 147 Griffies scheme is used (\np{traldf\_grif}=true; see Appdx \ref{sec:triad}). 148 In other words, iso-neutral mixing will only be accurately represented with a 149 linear equation of state (\np{nn\_eos}=1 or 2). In the case of a "true" equation 150 of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} 151 will include a pressure dependent part, leading to the wrong evaluation of 152 the neutral slopes. 153 154 %gm% 155 Note: The solution for $s$-coordinate passes trough the use of different 156 (and better) expression for the constraint on iso-neutral fluxes. Following 157 \citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral 158 diffusive flux of locally referenced potential density, we stay in the $T$-$S$ 159 plane and consider the balance between the neutral direction diffusive fluxes 160 of potential temperature and salinity: 161 \begin{equation} 162 \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) 163 \end{equation} 164 %gm{ where vector F is ....} 165 166 This constraint leads to the following definition for the slopes: 167 168 \begin{equation} \label{Eq_ldfslp_iso2} 169 \begin{split} 170 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac 171 {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]} 172 {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k} 173 -\beta_u \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } 174 \\ 175 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac 176 {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]} 177 {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k} 178 -\beta_v \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} } 179 \\ 180 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac 181 {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2} 182 -\beta_w \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} } 183 {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} 184 \\ 185 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac 186 {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2} 187 -\beta_w \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} } 188 {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} 189 \\ 190 \end{split} 191 \end{equation} 192 where $\alpha$ and $\beta$, the thermal expansion and saline contraction 193 coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three 194 velocity points. In order to save computation time, they should be approximated 195 by the mean of their values at $T$-points (for example in the case of $\alpha$: 196 $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$ 197 and $\alpha_w=\overline{\alpha_T}^{k+1/2}$). 198 199 Note that such a formulation could be also used in the $z$-coordinate and 200 $z$-coordinate with partial steps cases. 201 202 \end{description} 203 204 This implementation is a rather old one. It is similar to the one 205 proposed by Cox [1987], except for the background horizontal 206 diffusion. Indeed, the Cox implementation of isopycnal diffusion in 207 GFDL-type models requires a minimum background horizontal diffusion 208 for numerical stability reasons. To overcome this problem, several 209 techniques have been proposed in which the numerical schemes of the 210 ocean model are modified \citep{Weaver_Eby_JPO97, 211 Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if 212 \np{traldf\_grif\_iso} is set true; see Appdx \ref{sec:triad}. Here, 213 another strategy is presented \citep{Lazar_PhD97}: a local 214 filtering of the iso-neutral slopes (made on 9 grid-points) prevents 215 the development of grid point noise generated by the iso-neutral 216 diffusion operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an 217 iso-neutral diffusion scheme without additional background horizontal 218 mixing. This technique can be viewed as a diffusion operator that acts 219 along large-scale (2~$\Delta$x) \gmcomment{2deltax doesnt seem very 220 large scale} iso-neutral surfaces. The diapycnal diffusion required 221 for numerical stability is thus minimized and its net effect on the 222 flow is quite small when compared to the effect of an horizontal 223 background mixing. 224 225 Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, 226 contrary to the \citet{Griffies_al_JPO98} operator which has that property. 227 228 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 229 \begin{figure}[!ht] \begin{center} 230 \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_LDF_ZDF1.pdf} 231 \caption { \label{Fig_LDF_ZDF1} 232 averaging procedure for isopycnal slope computation.} 233 \end{center} \end{figure} 234 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 235 236 %There are three additional questions about the slope calculation. 237 %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. 238 %Second, numerical stability issues also require a bound on slopes. 239 %Third, the question of boundary condition specified on slopes... 240 241 %from griffies: chapter 13.1.... 242 243 244 245 % In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04}, 246 % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly 247 % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the 248 % surface motivates this flattening of isopycnals near the surface). 249 250 For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also 251 be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear 252 fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter 253 decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the 254 surface motivates this flattening of isopycnals near the surface). 255 256 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 257 \begin{figure}[!ht] \begin{center} 258 \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_eiv_slp.pdf} 259 \caption { \label{Fig_eiv_slp} 260 Vertical profile of the slope used for lateral mixing in the mixed layer : 261 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 262 which has to be adjusted at the surface boundary (i.e. it must tend to zero at the 263 surface since there is no mixing across the air-sea interface: wall boundary 264 condition). Nevertheless, the profile between the surface zero value and the interior 265 iso-neutral one is unknown, and especially the value at the base of the mixed layer ; 266 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and 267 imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in 268 \NEMO: a linear decrease of the slope from zero at the surface to its ocean interior 269 value computed just below the mixed layer. Note the huge change in the slope at the 270 base of the mixed layer between \textit{(b)} and \textit{(c)}.} 271 \end{center} \end{figure} 272 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 273 274 \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.} 275 276 \subsection{slopes for momentum iso-neutral mixing} 277 278 The iso-neutral diffusion operator on momentum is the same as the one used on 279 tracers but applied to each component of the velocity separately (see 280 \eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the 281 surface along which the diffusion operator acts and the surface of computation 282 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the 283 $u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component. 284 They are computed from the slopes used for tracer diffusion, $i.e.$ 285 \eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} : 286 287 \begin{equation} \label{Eq_ldfslp_dyn} 288 \begin{aligned} 289 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ 290 &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2t}\ &= \overline{r_{2v}}^{\,j} \\ 291 &r_{1uw} = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ & r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\ 292 &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&& r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\ 293 \end{aligned} 294 \end{equation} 295 296 The major issue remaining is in the specification of the boundary conditions. 297 The same boundary conditions are chosen as those used for lateral 298 diffusion along model level surfaces, i.e. using the shear computed along 299 the model levels and with no additional friction at the ocean bottom (see 300 {\S\ref{LBC_coast}). 301 302 303 % ================================================================ 304 % Lateral Mixing Operator 305 % ================================================================ 306 \section [Lateral Mixing Operators (\textit{ldftra}, \textit{ldfdyn})] 307 {Lateral Mixing Operators (\mdl{traldf}, \mdl{traldf}) } 308 \label{LDF_op} 309 310 311 312 % ================================================================ 35 313 % Lateral Mixing Coefficients 36 314 % ================================================================ … … 38 316 {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn}) } 39 317 \label{LDF_coef} 40 41 318 42 319 Introducing a space variation in the lateral eddy mixing coefficients changes … … 165 442 166 443 % ================================================================ 167 % Direction of lateral Mixing168 % ================================================================169 \section [Direction of Lateral Mixing (\textit{ldfslp})]170 {Direction of Lateral Mixing (\mdl{ldfslp})}171 \label{LDF_slp}172 173 %%%174 \gmcomment{ we should emphasize here that the implementation is a rather old one.175 Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. }176 177 A direction for lateral mixing has to be defined when the desired operator does178 not act along the model levels. This occurs when $(a)$ horizontal mixing is179 required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor})180 in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required181 whatever the vertical coordinate is. This direction of mixing is defined by its182 slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the183 quantity to be diffused. For a tracer, this leads to the following four slopes :184 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while185 for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for186 $u$ and $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.187 188 %gm% add here afigure of the slope in i-direction189 190 \subsection{slopes for tracer geopotential mixing in the $s$-coordinate}191 192 In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and193 $r_2$ are the slopes between the geopotential and computational surfaces.194 Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso}195 when the diffusive fluxes in the three directions are set to zero and $T$ is196 assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the197 depth of a $T$-point.198 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}199 200 \begin{equation} \label{Eq_ldfslp_geo}201 \begin{aligned}202 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}203 \;\delta_{i+1/2}[z_t]204 &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t]205 \\206 r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)}207 \;\delta_{j+1/2} [z_t]208 &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t]209 \\210 r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2}211 &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]212 \\213 r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2}214 &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]215 \\216 \end{aligned}217 \end{equation}218 219 %gm% caution I'm not sure the simplification was a good idea!220 221 These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True,222 and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True.223 224 \subsection{Slopes for tracer iso-neutral mixing}\label{LDF_slp_iso}225 In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral226 and computational surfaces. Their formulation does not depend on the vertical227 coordinate used. Their discrete formulation is found using the fact that the228 diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density)229 vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the230 diffusive fluxes in the three directions to zero leads to the following definition for231 the neutral slopes:232 233 \begin{equation} \label{Eq_ldfslp_iso}234 \begin{split}235 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}236 {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}237 \\238 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}239 {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}240 \\241 r_{1w} &= \frac{e_{3w}}{e_{1w}}\;242 \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}243 {\delta_{k+1/2}[\rho]}244 \\245 r_{2w} &= \frac{e_{3w}}{e_{2w}}\;246 \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}247 {\delta_{k+1/2}[\rho]}248 \\249 \end{split}250 \end{equation}251 252 %gm% rewrite this as the explanation is not very clear !!!253 %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.254 255 %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).256 257 %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.258 259 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in260 \eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which,261 in decibars, is approximated by the depth in meters in the model). Therefore262 \eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is263 needed depending on the vertical coordinate used:264 265 \begin{description}266 267 \item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities268 appearing in the $i$ and $j$ derivatives are taken at the same depth, thus269 the $in situ$ density can be used. This is not the case for the vertical270 derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$271 is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following272 \citet{McDougall1987} (see \S\ref{TRA_bn2}).273 274 \item[$z$-coordinate with partial step : ] this case is identical to the full step275 case except that at partial step level, the \emph{horizontal} density gradient276 is evaluated as described in \S\ref{TRA_zpshde}.277 278 \item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,279 iso-neutral mixing is only employed for $s$-coordinates if the280 Griffies scheme is used (\np{traldf\_grif}=true; see Appdx \ref{sec:triad}).281 In other words, iso-neutral mixing will only be accurately represented with a282 linear equation of state (\np{nn\_eos}=1 or 2). In the case of a "true" equation283 of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso}284 will include a pressure dependent part, leading to the wrong evaluation of285 the neutral slopes.286 287 %gm%288 Note: The solution for $s$-coordinate passes trough the use of different289 (and better) expression for the constraint on iso-neutral fluxes. Following290 \citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral291 diffusive flux of locally referenced potential density, we stay in the $T$-$S$292 plane and consider the balance between the neutral direction diffusive fluxes293 of potential temperature and salinity:294 \begin{equation}295 \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)296 \end{equation}297 %gm{ where vector F is ....}298 299 This constraint leads to the following definition for the slopes:300 301 \begin{equation} \label{Eq_ldfslp_iso2}302 \begin{split}303 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac304 {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}305 {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}306 -\beta_u \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }307 \\308 r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac309 {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}310 {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}311 -\beta_v \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }312 \\313 r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac314 {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}315 -\beta_w \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }316 {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}317 \\318 r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac319 {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}320 -\beta_w \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }321 {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}322 \\323 \end{split}324 \end{equation}325 where $\alpha$ and $\beta$, the thermal expansion and saline contraction326 coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three327 velocity points. In order to save computation time, they should be approximated328 by the mean of their values at $T$-points (for example in the case of $\alpha$:329 $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$330 and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).331 332 Note that such a formulation could be also used in the $z$-coordinate and333 $z$-coordinate with partial steps cases.334 335 \end{description}336 337 This implementation is a rather old one. It is similar to the one338 proposed by Cox [1987], except for the background horizontal339 diffusion. Indeed, the Cox implementation of isopycnal diffusion in340 GFDL-type models requires a minimum background horizontal diffusion341 for numerical stability reasons. To overcome this problem, several342 techniques have been proposed in which the numerical schemes of the343 ocean model are modified \citep{Weaver_Eby_JPO97,344 Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if345 \np{traldf\_grif\_iso} is set true; see Appdx \ref{sec:triad}. Here,346 another strategy is presented \citep{Lazar_PhD97}: a local347 filtering of the iso-neutral slopes (made on 9 grid-points) prevents348 the development of grid point noise generated by the iso-neutral349 diffusion operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an350 iso-neutral diffusion scheme without additional background horizontal351 mixing. This technique can be viewed as a diffusion operator that acts352 along large-scale (2~$\Delta$x) \gmcomment{2deltax doesnt seem very353 large scale} iso-neutral surfaces. The diapycnal diffusion required354 for numerical stability is thus minimized and its net effect on the355 flow is quite small when compared to the effect of an horizontal356 background mixing.357 358 Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,359 contrary to the \citet{Griffies_al_JPO98} operator which has that property.360 361 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>362 \begin{figure}[!ht] \begin{center}363 \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_LDF_ZDF1.pdf}364 \caption { \label{Fig_LDF_ZDF1}365 averaging procedure for isopycnal slope computation.}366 \end{center} \end{figure}367 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>368 369 %There are three additional questions about the slope calculation.370 %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.371 %Second, numerical stability issues also require a bound on slopes.372 %Third, the question of boundary condition specified on slopes...373 374 %from griffies: chapter 13.1....375 376 377 378 % In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04},379 % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly380 % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the381 % surface motivates this flattening of isopycnals near the surface).382 383 For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also384 be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear385 fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter386 decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the387 surface motivates this flattening of isopycnals near the surface).388 389 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>390 \begin{figure}[!ht] \begin{center}391 \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_eiv_slp.pdf}392 \caption { \label{Fig_eiv_slp}393 Vertical profile of the slope used for lateral mixing in the mixed layer :394 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,395 which has to be adjusted at the surface boundary (i.e. it must tend to zero at the396 surface since there is no mixing across the air-sea interface: wall boundary397 condition). Nevertheless, the profile between the surface zero value and the interior398 iso-neutral one is unknown, and especially the value at the base of the mixed layer ;399 \textit{(b)} profile of slope using a linear tapering of the slope near the surface and400 imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in401 \NEMO: a linear decrease of the slope from zero at the surface to its ocean interior402 value computed just below the mixed layer. Note the huge change in the slope at the403 base of the mixed layer between \textit{(b)} and \textit{(c)}.}404 \end{center} \end{figure}405 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>406 407 \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.}408 409 \subsection{slopes for momentum iso-neutral mixing}410 411 The iso-neutral diffusion operator on momentum is the same as the one used on412 tracers but applied to each component of the velocity separately (see413 \eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the414 surface along which the diffusion operator acts and the surface of computation415 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the416 $u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component.417 They are computed from the slopes used for tracer diffusion, $i.e.$418 \eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :419 420 \begin{equation} \label{Eq_ldfslp_dyn}421 \begin{aligned}422 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\423 &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2t}\ &= \overline{r_{2v}}^{\,j} \\424 &r_{1uw} = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ & r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\425 &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&& r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\426 \end{aligned}427 \end{equation}428 429 The major issue remaining is in the specification of the boundary conditions.430 The same boundary conditions are chosen as those used for lateral431 diffusion along model level surfaces, i.e. using the shear computed along432 the model levels and with no additional friction at the ocean bottom (see433 {\S\ref{LBC_coast}).434 435 436 % ================================================================437 444 % Eddy Induced Mixing 438 445 % ================================================================
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