Changeset 6289 for trunk/DOC/TexFiles/Chapters/Annex_ISO.tex
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- 2016-02-05T00:47:05+01:00 (8 years ago)
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trunk/DOC/TexFiles/Chapters/Annex_ISO.tex
r4147 r6289 11 11 \namdisplay{namtra_ldf} 12 12 %--------------------------------------------------------------------------------------------------------- 13 If the namelist variable \np{ln\_traldf\_grif} is set true (and 14 \key{ldfslp} is set), \NEMO updates both active and passive tracers 15 using the Griffies triad representation of iso-neutral diffusion and 16 the eddy-induced advective skew (GM) fluxes. Otherwise (by default) the 17 filtered version of Cox's original scheme is employed 18 (\S\ref{LDF_slp}). In the present implementation of the Griffies 19 scheme, the advective skew fluxes are implemented even if 20 \key{traldf\_eiv} is not set. 13 14 Two scheme are available to perform the iso-neutral diffusion. 15 If the namelist logical \np{ln\_traldf\_triad} is set true, 16 \NEMO updates both active and passive tracers using the Griffies triad representation 17 of iso-neutral diffusion and the eddy-induced advective skew (GM) fluxes. 18 If the namelist logical \np{ln\_traldf\_iso} is set true, 19 the filtered version of Cox's original scheme (the Standard scheme) is employed (\S\ref{LDF_slp}). 20 In the present implementation of the Griffies scheme, 21 the advective skew fluxes are implemented even if \np{ln\_traldf\_eiv} is false. 21 22 22 23 Values of iso-neutral diffusivity and GM coefficient are set as 23 described in \S\ref{LDF_coef}. If none of the keys \key{traldf\_cNd}, 24 N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and 25 GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and 26 \np{rn\_aeiv\_0}. If 2D-varying coefficients are set with 27 \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal 28 scale factor according to \eqref{Eq_title} \footnote{Except in global ORCA 29 $0.5^{\circ}$ runs with \key{traldf\_eiv}, where 30 $A_l$ is set like $A_e$ but with a minimum vale of 31 $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$}. In idealised setups with 32 \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} 33 is set in the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is 34 instead set from the Held-Larichev parameterisation\footnote{In this 35 case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further 36 reduced by a factor $|f/f_{20}|$, where $f_{20}$ is the value of $f$ 37 at $20^{\circ}$~N} (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored 38 unless it is zero. 24 described in \S\ref{LDF_coef}. Note that when GM fluxes are used, 25 the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS, 26 even though the eddy advection is accomplished by means of the skew fluxes. 27 39 28 40 29 The options specific to the Griffies scheme include: 41 30 \begin{description}[font=\normalfont] 42 \item[\np{ln\_traldf\_gdia}] Default value is false. See \S\ref{sec:triad:sfdiag}. If this is set true, time-mean 43 eddy-advective (GM) velocities are output for diagnostic purposes, even 44 though the eddy advection is accomplished by means of the skew 45 fluxes. 46 \item[\np{ln\_traldf\_iso}] See \S\ref{sec:triad:taper}. If this is set false (the default), then 31 \item[\np{ln\_triad\_iso}] See \S\ref{sec:triad:taper}. If this is set false (the default), then 47 32 `iso-neutral' mixing is accomplished within the surface mixed-layer 48 33 along slopes linearly decreasing with depth from the value immediately below 49 the mixed-layer to zero (flat) at the surface (\S\ref{sec:triad:lintaper}). This is the same 50 treatment as used in the default implementation 51 \S\ref{LDF_slp_iso}; Fig.~\ref{Fig_eiv_slp}. Where 52 \np{ln\_traldf\_iso} is set true, the vertical skew flux is further 53 reduced to ensure no vertical buoyancy flux, giving an almost pure 34 the mixed-layer to zero (flat) at the surface (\S\ref{sec:triad:lintaper}). 35 This is the same treatment as used in the default implementation \S\ref{LDF_slp_iso}; Fig.~\ref{Fig_eiv_slp}. 36 Where \np{ln\_triad\_iso} is set true, the vertical skew flux is further reduced 37 to ensure no vertical buoyancy flux, giving an almost pure 54 38 horizontal diffusive tracer flux within the mixed layer. This is similar to 55 39 the tapering suggested by \citet{Gerdes1991}. See \S\ref{sec:triad:Gerdes-taper} 56 \item[\np{ln\_traldf\_botmix}] See \S\ref{sec:triad:iso_bdry}. If this 57 is set false (the default) then the lateral diffusive fluxes 58 associated with triads partly masked by topography are neglected. If 59 it is set true, however, then these lateral diffusive fluxes are 60 applied, giving smoother bottom tracer fields at the cost of 61 introducing diapycnal mixing. 40 \item[\np{ln\_botmix\_triad}] See \S\ref{sec:triad:iso_bdry}. 41 If this is set false (the default) then the lateral diffusive fluxes 42 associated with triads partly masked by topography are neglected. 43 If it is set true, however, then these lateral diffusive fluxes are applied, 44 giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. 45 \item[\np{rn\_sw\_triad}] blah blah to be added.... 46 \end{description} 47 The options shared with the Standard scheme include: 48 \begin{description}[font=\normalfont] 49 \item[\np{ln\_traldf\_msc}] blah blah to be added 50 \item[\np{rn\_slpmax}] blah blah to be added 62 51 \end{description} 63 52 \section{Triad formulation of iso-neutral diffusion} 64 53 \label{sec:triad:iso} 65 We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, but formulated within the \NEMO66 framework, using scale factors rather than grid-sizes.54 We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, 55 but formulated within the \NEMO framework, using scale factors rather than grid-sizes. 67 56 68 57 \subsection{The iso-neutral diffusion operator} … … 84 73 \mbox{with}\quad \;\;\Re = 85 74 \begin{pmatrix} 86 1&0&-r_1\mystrut \\87 0&1&-r_2\mystrut \\88 -r_1 &-r_2&r_1 ^2+r_2 ^2\mystrut75 1 & 0 & -r_1 \mystrut \\ 76 0 & 1 & -r_2 \mystrut \\ 77 -r_1 & -r_2 & r_1 ^2+r_2 ^2 \mystrut 89 78 \end{pmatrix} 90 79 \quad \text{and} \quad\grad T= 91 80 \begin{pmatrix} 92 \frac{1}{e_1} \pd[T]{i}\mystrut \\93 \frac{1}{e_2} \pd[T]{j}\mystrut \\94 \frac{1}{e_3} \pd[T]{k}\mystrut81 \frac{1}{e_1} \pd[T]{i} \mystrut \\ 82 \frac{1}{e_2} \pd[T]{j} \mystrut \\ 83 \frac{1}{e_3} \pd[T]{k} \mystrut 95 84 \end{pmatrix}. 96 85 \end{equation} … … 101 90 % {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ 102 91 % \end{array} }} \right) 103 Here \eqref{Eq_PE_iso_slopes} 92 Here \eqref{Eq_PE_iso_slopes} 104 93 \begin{align*} 105 94 r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} … … 200 189 % the mean vertical gradient at the $u$-point, 201 190 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 202 \begin{figure}[ h] \begin{center}191 \begin{figure}[tb] \begin{center} 203 192 \includegraphics[width=1.05\textwidth]{./TexFiles/Figures/Fig_GRIFF_triad_fluxes} 204 193 \caption{ \label{fig:triad:ISO_triad} … … 256 245 \ 257 246 \frac 258 {\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } 259 {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] }. 260 \end{equation} 261 In calculating the slopes of the local neutral 262 surfaces, the expansion coefficients $\alpha$ and $\beta$ are 263 evaluated at the anchor points of the triad \footnote{Note that in \eqref{eq:triad:R} we use the ratio $\alpha / \beta$ 264 instead of multiplying the temperature derivative by $\alpha$ and the 265 salinity derivative by $\beta$. This is more efficient as the ratio 266 $\alpha / \beta$ can to be evaluated directly}, while the metrics are 267 calculated at the $u$- and $w$-points on the arms. 247 { \alpha_i^k \ \delta_{i+i_p}[T^k] - \beta_i^k \ \delta_{i+i_p}[S^k] } 248 { \alpha_i^k \ \delta_{k+k_p}[T^i] - \beta_i^k \ \delta_{k+k_p}[S^i] }. 249 \end{equation} 250 In calculating the slopes of the local neutral surfaces, 251 the expansion coefficients $\alpha$ and $\beta$ are evaluated at the anchor points of the triad, 252 while the metrics are calculated at the $u$- and $w$-points on the arms. 268 253 269 254 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 270 \begin{figure}[ h] \begin{center}255 \begin{figure}[tb] \begin{center} 271 256 \includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_GRIFF_qcells} 272 257 \caption{ \label{fig:triad:qcells} … … 277 262 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 278 263 279 Each triad $\{_i^k\:_{i_p}^{k_p}\}$ is associated (Fig.~\ref{fig:triad:qcells}) with the quarter 280 cell that is the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ 281 $u$-cell and the $i,k+k_p$ $w$-cell. Expressing the slopes $s_i$ and 282 $s'_i$ in \eqref{eq:triad:i13} and \eqref{eq:triad:i31} in this notation, we have 283 e.g.\ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. Each triad slope $_i^k 284 \mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to calculate the 285 lateral flux along its $u$-arm, at $(i+i_p,k)$, and then again as an 286 $s'$ to calculate the vertical flux along its $w$-arm at 287 $(i,k+k_p)$. Each vertical area $a_i$ used to calculate the lateral 288 flux and horizontal area $a'_i$ used to calculate the vertical flux 289 can also be identified as the area across the $u$- and $w$-arms of a 290 unique triad, and we notate these areas, similarly to the triad 291 slopes, as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, 292 $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, where e.g. in \eqref{eq:triad:i13} 293 $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, and in \eqref{eq:triad:i31} 294 $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 264 Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (Fig.~\ref{fig:triad:qcells}) with the quarter 265 cell that is the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell. 266 Expressing the slopes $s_i$ and $s'_i$ in \eqref{eq:triad:i13} and \eqref{eq:triad:i31} in this notation, 267 we have $e.g.$ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. 268 Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) 269 to calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$, 270 and then again as an $s'$ to calculate the vertical flux along its $w$-arm at $(i,k+k_p)$. 271 Each vertical area $a_i$ used to calculate the lateral flux and horizontal area $a'_i$ used 272 to calculate the vertical flux can also be identified as the area across the $u$- and $w$-arms 273 of a unique triad, and we notate these areas, similarly to the triad slopes, 274 as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, 275 where $e.g.$ in \eqref{eq:triad:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, 276 and in \eqref{eq:triad:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 295 277 296 278 \subsection{The full triad fluxes} … … 667 649 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is 668 650 masked. The associated lateral fluxes (grey-black dashed line) are 669 masked if \np{ln\_botmix\_ grif}=false, but left unmasked,670 giving bottom mixing, if \np{ln\_botmix\_ grif}=true.671 672 The default option \np{ln\_botmix\_ grif}=false is suitable when the651 masked if \np{ln\_botmix\_triad}=false, but left unmasked, 652 giving bottom mixing, if \np{ln\_botmix\_triad}=true. 653 654 The default option \np{ln\_botmix\_triad}=false is suitable when the 673 655 bbl mixing option is enabled (\key{trabbl}, with \np{nn\_bbl\_ldf}=1), 674 656 or for simple idealized problems. For setups with topography without 675 bbl mixing, \np{ln\_botmix\_ grif}=true may be necessary.657 bbl mixing, \np{ln\_botmix\_triad}=true may be necessary. 676 658 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 677 659 \begin{figure}[h] \begin{center} … … 690 672 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point 691 673 is masked. The associated lateral fluxes (grey-black dashed 692 line) are masked if \np{botmix\_ grif}=.false., but left693 unmasked, giving bottom mixing, if \np{botmix\_ grif}=.true.}674 line) are masked if \np{botmix\_triad}=.false., but left 675 unmasked, giving bottom mixing, if \np{botmix\_triad}=.true.} 694 676 \end{center} \end{figure} 695 677 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 931 913 it to the Eulerian velocity prior to computing the tracer 932 914 advection. This is implemented if \key{traldf\_eiv} is set in the 933 default implementation, where \np{ln\_traldf\_ grif} is set915 default implementation, where \np{ln\_traldf\_triad} is set 934 916 false. This allows us to take advantage of all the advection schemes 935 917 offered for the tracers (see \S\ref{TRA_adv}) and not just a $2^{nd}$ … … 938 920 paramount importance. 939 921 940 However, when \np{ln\_traldf\_ grif} is set true, \NEMO instead922 However, when \np{ln\_traldf\_triad} is set true, \NEMO instead 941 923 implements eddy induced advection according to the so-called skew form 942 924 \citep{Griffies_JPO98}. It is based on a transformation of the advective fluxes … … 1137 1119 and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the 1138 1120 $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ 1139 $u$-point is masked. The namelist parameter \np{ln\_botmix\_ grif} has1121 $u$-point is masked. The namelist parameter \np{ln\_botmix\_triad} has 1140 1122 no effect on the eddy-induced skew-fluxes. 1141 1123
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