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branches/2016/dev_INGV_UKMO_2016/DOC/TexFiles/Chapters/Annex_ISO.tex
r4147 r7351 1 \documentclass[NEMO_book]{subfiles} 2 \begin{document} 1 3 % ================================================================ 2 4 % Iso-neutral diffusion : … … 11 13 \namdisplay{namtra_ldf} 12 14 %--------------------------------------------------------------------------------------------------------- 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. 15 16 Two scheme are available to perform the iso-neutral diffusion. 17 If the namelist logical \np{ln\_traldf\_triad} is set true, 18 \NEMO updates both active and passive tracers using the Griffies triad representation 19 of iso-neutral diffusion and the eddy-induced advective skew (GM) fluxes. 20 If the namelist logical \np{ln\_traldf\_iso} is set true, 21 the filtered version of Cox's original scheme (the Standard scheme) is employed (\S\ref{LDF_slp}). 22 In the present implementation of the Griffies scheme, 23 the advective skew fluxes are implemented even if \np{ln\_traldf\_eiv} is false. 21 24 22 25 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. 26 described in \S\ref{LDF_coef}. Note that when GM fluxes are used, 27 the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS, 28 even though the eddy advection is accomplished by means of the skew fluxes. 29 39 30 40 31 The options specific to the Griffies scheme include: 41 32 \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 33 \item[\np{ln\_triad\_iso}] See \S\ref{sec:triad:taper}. If this is set false (the default), then 47 34 `iso-neutral' mixing is accomplished within the surface mixed-layer 48 35 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 36 the mixed-layer to zero (flat) at the surface (\S\ref{sec:triad:lintaper}). 37 This is the same treatment as used in the default implementation \S\ref{LDF_slp_iso}; Fig.~\ref{Fig_eiv_slp}. 38 Where \np{ln\_triad\_iso} is set true, the vertical skew flux is further reduced 39 to ensure no vertical buoyancy flux, giving an almost pure 54 40 horizontal diffusive tracer flux within the mixed layer. This is similar to 55 41 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. 42 \item[\np{ln\_botmix\_triad}] See \S\ref{sec:triad:iso_bdry}. 43 If this is set false (the default) then the lateral diffusive fluxes 44 associated with triads partly masked by topography are neglected. 45 If it is set true, however, then these lateral diffusive fluxes are applied, 46 giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. 47 \item[\np{rn\_sw\_triad}] blah blah to be added.... 48 \end{description} 49 The options shared with the Standard scheme include: 50 \begin{description}[font=\normalfont] 51 \item[\np{ln\_traldf\_msc}] blah blah to be added 52 \item[\np{rn\_slpmax}] blah blah to be added 62 53 \end{description} 63 54 \section{Triad formulation of iso-neutral diffusion} 64 55 \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.56 We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, 57 but formulated within the \NEMO framework, using scale factors rather than grid-sizes. 67 58 68 59 \subsection{The iso-neutral diffusion operator} … … 84 75 \mbox{with}\quad \;\;\Re = 85 76 \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\mystrut77 1 & 0 & -r_1 \mystrut \\ 78 0 & 1 & -r_2 \mystrut \\ 79 -r_1 & -r_2 & r_1 ^2+r_2 ^2 \mystrut 89 80 \end{pmatrix} 90 81 \quad \text{and} \quad\grad T= 91 82 \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}\mystrut83 \frac{1}{e_1} \pd[T]{i} \mystrut \\ 84 \frac{1}{e_2} \pd[T]{j} \mystrut \\ 85 \frac{1}{e_3} \pd[T]{k} \mystrut 95 86 \end{pmatrix}. 96 87 \end{equation} … … 101 92 % {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ 102 93 % \end{array} }} \right) 103 Here \eqref{Eq_PE_iso_slopes} 94 Here \eqref{Eq_PE_iso_slopes} 104 95 \begin{align*} 105 96 r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} … … 200 191 % the mean vertical gradient at the $u$-point, 201 192 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 202 \begin{figure}[ h] \begin{center}203 \includegraphics[width=1.05\textwidth]{ ./TexFiles/Figures/Fig_GRIFF_triad_fluxes}193 \begin{figure}[tb] \begin{center} 194 \includegraphics[width=1.05\textwidth]{Fig_GRIFF_triad_fluxes} 204 195 \caption{ \label{fig:triad:ISO_triad} 205 196 (a) Arrangement of triads $S_i$ and tracer gradients to … … 256 247 \ 257 248 \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. 249 { \alpha_i^k \ \delta_{i+i_p}[T^k] - \beta_i^k \ \delta_{i+i_p}[S^k] } 250 { \alpha_i^k \ \delta_{k+k_p}[T^i] - \beta_i^k \ \delta_{k+k_p}[S^i] }. 251 \end{equation} 252 In calculating the slopes of the local neutral surfaces, 253 the expansion coefficients $\alpha$ and $\beta$ are evaluated at the anchor points of the triad, 254 while the metrics are calculated at the $u$- and $w$-points on the arms. 268 255 269 256 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 270 \begin{figure}[ h] \begin{center}271 \includegraphics[width=0.80\textwidth]{ ./TexFiles/Figures/Fig_GRIFF_qcells}257 \begin{figure}[tb] \begin{center} 258 \includegraphics[width=0.80\textwidth]{Fig_GRIFF_qcells} 272 259 \caption{ \label{fig:triad:qcells} 273 260 Triad notation for quarter cells. $T$-cells are inside … … 277 264 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 278 265 279 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 $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 266 Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (Fig.~\ref{fig:triad:qcells}) with the quarter 267 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. 268 Expressing the slopes $s_i$ and $s'_i$ in \eqref{eq:triad:i13} and \eqref{eq:triad:i31} in this notation, 269 we have $e.g.$ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. 270 Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) 271 to calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$, 272 and then again as an $s'$ to calculate the vertical flux along its $w$-arm at $(i,k+k_p)$. 273 Each vertical area $a_i$ used to calculate the lateral flux and horizontal area $a'_i$ used 274 to calculate the vertical flux can also be identified as the area across the $u$- and $w$-arms 275 of a unique triad, and we notate these areas, similarly to the triad slopes, 276 as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, 277 where $e.g.$ in \eqref{eq:triad:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, 278 and in \eqref{eq:triad:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 295 279 296 280 \subsection{The full triad fluxes} … … 667 651 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is 668 652 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 the653 masked if \np{ln\_botmix\_triad}=false, but left unmasked, 654 giving bottom mixing, if \np{ln\_botmix\_triad}=true. 655 656 The default option \np{ln\_botmix\_triad}=false is suitable when the 673 657 bbl mixing option is enabled (\key{trabbl}, with \np{nn\_bbl\_ldf}=1), 674 658 or for simple idealized problems. For setups with topography without 675 bbl mixing, \np{ln\_botmix\_ grif}=true may be necessary.659 bbl mixing, \np{ln\_botmix\_triad}=true may be necessary. 676 660 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 677 661 \begin{figure}[h] \begin{center} 678 \includegraphics[width=0.60\textwidth]{ ./TexFiles/Figures/Fig_GRIFF_bdry_triads}662 \includegraphics[width=0.60\textwidth]{Fig_GRIFF_bdry_triads} 679 663 \caption{ \label{fig:triad:bdry_triads} 680 664 (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer … … 690 674 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point 691 675 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.}676 line) are masked if \np{botmix\_triad}=.false., but left 677 unmasked, giving bottom mixing, if \np{botmix\_triad}=.true.} 694 678 \end{center} \end{figure} 695 679 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 849 833 different $i_p,k_p$, denoted by different colours, (e.g. the green 850 834 triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}} 851 {\includegraphics[width=0.60\textwidth]{ ./TexFiles/Figures/Fig_GRIFF_MLB_triads}}835 {\includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads}} 852 836 \end{figure} 853 837 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 931 915 it to the Eulerian velocity prior to computing the tracer 932 916 advection. This is implemented if \key{traldf\_eiv} is set in the 933 default implementation, where \np{ln\_traldf\_ grif} is set917 default implementation, where \np{ln\_traldf\_triad} is set 934 918 false. This allows us to take advantage of all the advection schemes 935 919 offered for the tracers (see \S\ref{TRA_adv}) and not just a $2^{nd}$ … … 938 922 paramount importance. 939 923 940 However, when \np{ln\_traldf\_ grif} is set true, \NEMO instead924 However, when \np{ln\_traldf\_triad} is set true, \NEMO instead 941 925 implements eddy induced advection according to the so-called skew form 942 926 \citep{Griffies_JPO98}. It is based on a transformation of the advective fluxes … … 1137 1121 and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the 1138 1122 $i,k+1$ 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} has1123 $u$-point is masked. The namelist parameter \np{ln\_botmix\_triad} has 1140 1124 no effect on the eddy-induced skew-fluxes. 1141 1125 … … 1193 1177 \end{split} 1194 1178 \end{equation} 1179 \end{document}
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