Changeset 9407 for branches/2017/dev_merge_2017/DOC/tex_sub/chap_LDF.tex
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branches/2017/dev_merge_2017/DOC/tex_sub/chap_LDF.tex
r9394 r9407 6 6 % ================================================================ 7 7 \chapter{Lateral Ocean Physics (LDF)} 8 \label{ LDF}8 \label{chap:LDF} 9 9 \minitoc 10 10 … … 15 15 16 16 The lateral physics terms in the momentum and tracer equations have been 17 described in \ S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf}18 and \ S\ref{DYN_ldf}). In this section we further discuss each lateral physics option.17 described in \autoref{eq:PE_zdf} and their discrete formulation in \autoref{sec:TRA_ldf} 18 and \autoref{sec:DYN_ldf}). In this section we further discuss each lateral physics option. 19 19 Choosing one lateral physics scheme means for the user defining, 20 20 (1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term) ; … … 25 25 Note that this chapter describes the standard implementation of iso-neutral 26 26 tracer mixing, and Griffies's implementation, which is used if 27 \np{traldf\_grif}\forcode{ = .true.}, is described in Appdx\ ref{sec:triad}27 \np{traldf\_grif}\forcode{ = .true.}, is described in Appdx\autoref{apdx:triad} 28 28 29 29 %-----------------------------------nam_traldf - nam_dynldf-------------------------------------------- … … 37 37 % ================================================================ 38 38 \section{Direction of lateral mixing (\protect\mdl{ldfslp})} 39 \label{ LDF_slp}39 \label{sec:LDF_slp} 40 40 41 41 %%% … … 50 50 slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the 51 51 quantity to be diffused. For a tracer, this leads to the following four slopes : 52 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \ eqref{Eq_tra_ldf_iso}), while52 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:tra_ldf_iso}), while 53 53 for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for 54 54 $u$ and $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. … … 60 60 In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and 61 61 $r_2$ are the slopes between the geopotential and computational surfaces. 62 Their discrete formulation is found by locally solving \ eqref{Eq_tra_ldf_iso}62 Their discrete formulation is found by locally solving \autoref{eq:tra_ldf_iso} 63 63 when the diffusive fluxes in the three directions are set to zero and $T$ is 64 64 assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the … … 66 66 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 67 67 68 \begin{equation} \label{ Eq_ldfslp_geo}68 \begin{equation} \label{eq:ldfslp_geo} 69 69 \begin{aligned} 70 70 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} … … 91 91 92 92 \subsection{Slopes for tracer iso-neutral mixing} 93 \label{ LDF_slp_iso}93 \label{subsec:LDF_slp_iso} 94 94 In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral 95 95 and computational surfaces. Their formulation does not depend on the vertical 96 96 coordinate used. Their discrete formulation is found using the fact that the 97 97 diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density) 98 vanish. So, substituting $T$ by $\rho$ in \ eqref{Eq_tra_ldf_iso} and setting the98 vanish. So, substituting $T$ by $\rho$ in \autoref{eq:tra_ldf_iso} and setting the 99 99 diffusive fluxes in the three directions to zero leads to the following definition for 100 100 the neutral slopes: 101 101 102 \begin{equation} \label{ Eq_ldfslp_iso}102 \begin{equation} \label{eq:ldfslp_iso} 103 103 \begin{split} 104 104 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} … … 120 120 121 121 %gm% rewrite this as the explanation is not very clear !!! 122 %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.123 124 %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).125 126 %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.122 %In practice, \autoref{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 \autoref{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. 123 124 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{McDougall1987}, therefore in \autoref{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). 125 126 %In the $z$-coordinate, the derivative of the \autoref{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. 127 127 128 128 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in 129 \ eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which,129 \autoref{eq:ldfslp_iso} has to be evaluated at the same local pressure (which, 130 130 in decibars, is approximated by the depth in meters in the model). Therefore 131 \ eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is131 \autoref{eq:ldfslp_iso} cannot be used as such, but further transformation is 132 132 needed depending on the vertical coordinate used: 133 133 134 134 \begin{description} 135 135 136 \item[$z$-coordinate with full step : ] in \ eqref{Eq_ldfslp_iso} the densities136 \item[$z$-coordinate with full step : ] in \autoref{eq:ldfslp_iso} the densities 137 137 appearing in the $i$ and $j$ derivatives are taken at the same depth, thus 138 138 the $in situ$ density can be used. This is not the case for the vertical 139 139 derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ 140 140 is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following 141 \citet{McDougall1987} (see \ S\ref{TRA_bn2}).141 \citet{McDougall1987} (see \autoref{subsec:TRA_bn2}). 142 142 143 143 \item[$z$-coordinate with partial step : ] this case is identical to the full step 144 144 case except that at partial step level, the \emph{horizontal} density gradient 145 is evaluated as described in \ S\ref{TRA_zpshde}.145 is evaluated as described in \autoref{sec:TRA_zpshde}. 146 146 147 147 \item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO, 148 148 iso-neutral mixing is only employed for $s$-coordinates if the 149 Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.}; see Appdx \ ref{sec:triad}).149 Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.}; see Appdx \autoref{apdx:triad}). 150 150 In other words, iso-neutral mixing will only be accurately represented with a 151 151 linear equation of state (\np{nn\_eos}\forcode{ = 1..2}). In the case of a "true" equation 152 of state, the evaluation of $i$ and $j$ derivatives in \ eqref{Eq_ldfslp_iso}152 of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso} 153 153 will include a pressure dependent part, leading to the wrong evaluation of 154 154 the neutral slopes. … … 168 168 This constraint leads to the following definition for the slopes: 169 169 170 \begin{equation} \label{ Eq_ldfslp_iso2}170 \begin{equation} \label{eq:ldfslp_iso2} 171 171 \begin{split} 172 172 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac … … 193 193 \end{equation} 194 194 where $\alpha$ and $\beta$, the thermal expansion and saline contraction 195 coefficients introduced in \ S\ref{TRA_bn2}, have to be evaluated at the three195 coefficients introduced in \autoref{subsec:TRA_bn2}, have to be evaluated at the three 196 196 velocity points. In order to save computation time, they should be approximated 197 197 by the mean of their values at $T$-points (for example in the case of $\alpha$: … … 212 212 ocean model are modified \citep{Weaver_Eby_JPO97, 213 213 Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if 214 \np{traldf\_grif\_iso} is set true; see Appdx \ ref{sec:triad}. Here,214 \np{traldf\_grif\_iso} is set true; see Appdx \autoref{apdx:triad}. Here, 215 215 another strategy is presented \citep{Lazar_PhD97}: a local 216 216 filtering of the iso-neutral slopes (made on 9 grid-points) prevents 217 217 the development of grid point noise generated by the iso-neutral 218 diffusion operator ( Fig.~\ref{Fig_LDF_ZDF1}). This allows an218 diffusion operator (\autoref{fig:LDF_ZDF1}). This allows an 219 219 iso-neutral diffusion scheme without additional background horizontal 220 220 mixing. This technique can be viewed as a diffusion operator that acts … … 231 231 \begin{figure}[!ht] \begin{center} 232 232 \includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1} 233 \caption { \protect\label{ Fig_LDF_ZDF1}233 \caption { \protect\label{fig:LDF_ZDF1} 234 234 averaging procedure for isopycnal slope computation.} 235 235 \end{center} \end{figure} … … 259 259 \begin{figure}[!ht] \begin{center} 260 260 \includegraphics[width=0.70\textwidth]{Fig_eiv_slp} 261 \caption { \protect\label{ Fig_eiv_slp}261 \caption { \protect\label{fig:eiv_slp} 262 262 Vertical profile of the slope used for lateral mixing in the mixed layer : 263 263 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, … … 280 280 The iso-neutral diffusion operator on momentum is the same as the one used on 281 281 tracers but applied to each component of the velocity separately (see 282 \ eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the282 \autoref{eq:dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). The slopes between the 283 283 surface along which the diffusion operator acts and the surface of computation 284 284 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the 285 285 $u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component. 286 286 They are computed from the slopes used for tracer diffusion, $i.e.$ 287 \ eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :288 289 \begin{equation} \label{ Eq_ldfslp_dyn}287 \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso} : 288 289 \begin{equation} \label{eq:ldfslp_dyn} 290 290 \begin{aligned} 291 291 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ … … 300 300 diffusion along model level surfaces, i.e. using the shear computed along 301 301 the model levels and with no additional friction at the ocean bottom (see 302 \ S\ref{LBC_coast}).302 \autoref{sec:LBC_coast}). 303 303 304 304 … … 307 307 % ================================================================ 308 308 \section{Lateral mixing operators (\protect\mdl{traldf}, \protect\mdl{traldf}) } 309 \label{ LDF_op}309 \label{sec:LDF_op} 310 310 311 311 … … 315 315 % ================================================================ 316 316 \section{Lateral mixing coefficient (\protect\mdl{ldftra}, \protect\mdl{ldfdyn}) } 317 \label{ LDF_coef}317 \label{sec:LDF_coef} 318 318 319 319 Introducing a space variation in the lateral eddy mixing coefficients changes … … 362 362 By default the horizontal variation of the eddy coefficient depends on the local mesh 363 363 size and the type of operator used: 364 \begin{equation} \label{ Eq_title}364 \begin{equation} \label{eq:title} 365 365 A_l = \left\{ 366 366 \begin{aligned} … … 378 378 such as global ocean models. Indeed, in such a case, a constant mixing coefficient 379 379 can lead to a blow up of the model due to large coefficient compare to the smallest 380 grid size (see \ S\ref{STP_forward_imp}), especially when using a bilaplacian operator.380 grid size (see \autoref{sec:STP_forward_imp}), especially when using a bilaplacian operator. 381 381 382 382 Other formulations can be introduced by the user for a given configuration. … … 411 411 (1) the momentum diffusion operator acting along model level surfaces is 412 412 written in terms of curl and divergent components of the horizontal current 413 (see \ S\ref{PE_ldf}). Although the eddy coefficient could be set to different values413 (see \autoref{subsec:PE_ldf}). Although the eddy coefficient could be set to different values 414 414 in these two terms, this option is not currently available. 415 415 … … 417 417 on enstrophy and on the square of the horizontal divergence for operators 418 418 acting along model-surfaces are no longer satisfied 419 ( Appendix~\ref{Apdx_dynldf_properties}).419 (\autoref{sec:dynldf_properties}). 420 420 421 421 (3) for isopycnal diffusion on momentum or tracers, an additional purely … … 425 425 values are $0$). However, the technique used to compute the isopycnal 426 426 slopes is intended to get rid of such a background diffusion, since it introduces 427 spurious diapycnal diffusion (see \ S\ref{LDF_slp}).427 spurious diapycnal diffusion (see \autoref{sec:LDF_slp}). 428 428 429 429 (4) when an eddy induced advection term is used (\key{traldf\_eiv}), $A^{eiv}$, … … 438 438 (7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap}\forcode{ = 439 439 }\np{ln\_dynldf\_bilap}\forcode{ = .false.}). This is recommended when using the UBS advection 440 scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.}, see \ ref{DYN_adv_ubs})440 scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.}, see \autoref{subsec:DYN_adv_ubs}) 441 441 and can be useful for testing purposes. 442 442 … … 445 445 % ================================================================ 446 446 \section{Eddy induced velocity (\protect\mdl{traadv\_eiv}, \protect\mdl{ldfeiv})} 447 \label{ LDF_eiv}447 \label{sec:LDF_eiv} 448 448 449 449 %%gm from Triad appendix : to be incorporated.... 450 450 \gmcomment{ 451 451 Values of iso-neutral diffusivity and GM coefficient are set as 452 described in \ S\ref{LDF_coef}. If none of the keys \key{traldf\_cNd},452 described in \autoref{sec:LDF_coef}. If none of the keys \key{traldf\_cNd}, 453 453 N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and 454 454 GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and 455 455 \np{rn\_aeiv\_0}. If 2D-varying coefficients are set with 456 456 \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal 457 scale factor according to \ eqref{Eq_title} \footnote{Except in global ORCA457 scale factor according to \autoref{eq:title} \footnote{Except in global ORCA 458 458 $0.5^{\circ}$ runs with \key{traldf\_eiv}, where 459 459 $A_l$ is set like $A_e$ but with a minimum vale of … … 472 472 depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral 473 473 mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$ 474 \ eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo}475 + \ eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:476 \begin{equation} \label{ Eq_ldfeiv}474 \autoref{eq:ldfslp_geo} is used in $z$-coordinates, and the sum \autoref{eq:ldfslp_geo} 475 + \autoref{eq:ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by: 476 \begin{equation} \label{eq:ldfeiv} 477 477 \begin{split} 478 478 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ … … 487 487 separate computation of the advective trends associated with the eiv velocity, 488 488 since it allows us to take advantage of all the advection schemes offered for 489 the tracers (see \ S\ref{TRA_adv}) and not just the $2^{nd}$ order advection489 the tracers (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection 490 490 scheme as in previous releases of OPA \citep{Madec1998}. This is particularly 491 491 useful for passive tracers where \emph{positivity} of the advection scheme is
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