Changeset 9393 for branches/2017/dev_merge_2017/DOC/tex_sub/annex_iso.tex
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branches/2017/dev_merge_2017/DOC/tex_sub/annex_iso.tex
r9392 r9393 4 4 % Iso-neutral diffusion : 5 5 % ================================================================ 6 \chapter[Iso-neutral diffusion and eddy advection using 7 triads]{Iso-neutral diffusion and eddy advection using triads} 6 \chapter{Iso-neutral diffusion and eddy advection using triads} 8 7 \label{sec:triad} 9 8 \minitoc … … 15 14 16 15 Two scheme are available to perform the iso-neutral diffusion. 17 If the namelist logical \np{ln _traldf_triad} is set true,16 If the namelist logical \np{ln\_traldf\_triad} is set true, 18 17 \NEMO updates both active and passive tracers using the Griffies triad representation 19 18 of iso-neutral diffusion and the eddy-induced advective skew (GM) fluxes. 20 If the namelist logical \np{ln _traldf_iso} is set true,19 If the namelist logical \np{ln\_traldf\_iso} is set true, 21 20 the filtered version of Cox's original scheme (the Standard scheme) is employed (\S\ref{LDF_slp}). 22 21 In the present implementation of the Griffies scheme, 23 the advective skew fluxes are implemented even if \np{ln _traldf_eiv} is false.22 the advective skew fluxes are implemented even if \np{ln\_traldf\_eiv} is false. 24 23 25 24 Values of iso-neutral diffusivity and GM coefficient are set as … … 31 30 The options specific to the Griffies scheme include: 32 31 \begin{description}[font=\normalfont] 33 \item[\np{ln _triad_iso}] See \S\ref{sec:triad:taper}. If this is set false (the default), then32 \item[\np{ln\_triad\_iso}] See \S\ref{sec:triad:taper}. If this is set false (the default), then 34 33 `iso-neutral' mixing is accomplished within the surface mixed-layer 35 34 along slopes linearly decreasing with depth from the value immediately below 36 35 the mixed-layer to zero (flat) at the surface (\S\ref{sec:triad:lintaper}). 37 36 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 reduced37 Where \np{ln\_triad\_iso} is set true, the vertical skew flux is further reduced 39 38 to ensure no vertical buoyancy flux, giving an almost pure 40 39 horizontal diffusive tracer flux within the mixed layer. This is similar to 41 40 the tapering suggested by \citet{Gerdes1991}. See \S\ref{sec:triad:Gerdes-taper} 42 \item[\np{ln _botmix_triad}] See \S\ref{sec:triad:iso_bdry}.41 \item[\np{ln\_botmix\_triad}] See \S\ref{sec:triad:iso_bdry}. 43 42 If this is set false (the default) then the lateral diffusive fluxes 44 43 associated with triads partly masked by topography are neglected. 45 44 If it is set true, however, then these lateral diffusive fluxes are applied, 46 45 giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. 47 \item[\np{rn _sw_triad}] blah blah to be added....46 \item[\np{rn\_sw\_triad}] blah blah to be added.... 48 47 \end{description} 49 48 The options shared with the Standard scheme include: 50 49 \begin{description}[font=\normalfont] 51 \item[\np{ln _traldf_msc}] blah blah to be added52 \item[\np{rn _slpmax}] blah blah to be added50 \item[\np{ln\_traldf\_msc}] blah blah to be added 51 \item[\np{rn\_slpmax}] blah blah to be added 53 52 \end{description} 53 54 54 \section{Triad formulation of iso-neutral diffusion} 55 55 \label{sec:triad:iso} … … 57 57 but formulated within the \NEMO framework, using scale factors rather than grid-sizes. 58 58 59 \subsection{ The iso-neutral diffusion operator}59 \subsection{Iso-neutral diffusion operator} 60 60 The iso-neutral second order tracer diffusive operator for small 61 61 angles between iso-neutral surfaces and geopotentials is given by … … 147 147 $w$-points but involves horizontal gradients defined at $u$-points. 148 148 149 \subsection{ The standard discretization}149 \subsection{Standard discretization} 150 150 The straightforward approach to discretize the lateral skew flux 151 151 \eqref{eq:triad:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 … … 185 185 ($i.e.$ they enter the computation of density), but it does not work 186 186 for a passive tracer. 187 187 188 \subsection{Expression of the skew-flux in terms of triad slopes} 188 189 \citep{Griffies_al_JPO98} introduce a different discretization of the … … 278 279 and in \eqref{eq:triad:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 279 280 280 \subsection{ The full triad fluxes}281 \subsection{Full triad fluxes} 281 282 A key property of iso-neutral diffusion is that it should not affect 282 283 the (locally referenced) density. In particular there should be no … … 368 369 \end{pmatrix}. 369 370 \end{flalign} 371 370 372 \subsection{Ensuring the scheme does not increase tracer variance} 371 373 \label{sec:triad:variance} … … 471 473 \right) 472 474 \] 475 473 476 \subsection{Triad volumes in Griffes's scheme and in \NEMO} 474 477 To complete the discretization we now need only specify the triad … … 633 636 RHS of \eqref{eq:triad:iso_property3}. 634 637 \end{description} 638 635 639 \subsection{Treatment of the triads at the boundaries}\label{sec:triad:iso_bdry} 636 640 The triad slope can only be defined where both the grid boxes centred at … … 651 655 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is 652 656 masked. The associated lateral fluxes (grey-black dashed line) are 653 masked if \ forcode{ln_botmix_triad= .false.}, but left unmasked,654 giving bottom mixing, if \ forcode{ln_botmix_triad= .true.}.655 656 The default option \ forcode{ln_botmix_triad= .false.} is suitable when the657 bbl mixing option is enabled (\key{trabbl}, with \ forcode{nn_bbl_ldf= 1}),657 masked if \np{ln\_botmix\_triad}\forcode{ = .false.}, but left unmasked, 658 giving bottom mixing, if \np{ln\_botmix\_triad}\forcode{ = .true.}. 659 660 The default option \np{ln\_botmix\_triad}\forcode{ = .false.} is suitable when the 661 bbl mixing option is enabled (\key{trabbl}, with \np{nn\_bbl\_ldf}\forcode{ = 1}), 658 662 or for simple idealized problems. For setups with topography without 659 bbl mixing, \ forcode{ln_botmix_triad= .true.} may be necessary.663 bbl mixing, \np{ln\_botmix\_triad}\forcode{ = .true.} may be necessary. 660 664 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 661 665 \begin{figure}[h] \begin{center} … … 674 678 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point 675 679 is masked. The associated lateral fluxes (grey-black dashed 676 line) are masked if \ protect\np{botmix_triad}=.false., but left677 unmasked, giving bottom mixing, if \ protect\np{botmix_triad}=.true.}680 line) are masked if \np{botmix\_triad}\forcode{ = .false.}, but left 681 unmasked, giving bottom mixing, if \np{botmix\_triad}\forcode{ = .true.}} 678 682 \end{center} \end{figure} 679 683 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 684 680 685 \subsection{ Limiting of the slopes within the interior}\label{sec:triad:limit} 681 686 As discussed in \S\ref{LDF_slp_iso}, iso-neutral slopes relative to … … 703 708 iso-neutral density flux that drives dianeutral mixing. In particular this iso-neutral density flux 704 709 is always downwards, and so acts to reduce gravitational potential energy. 710 705 711 \subsection{Tapering within the surface mixed layer}\label{sec:triad:taper} 706 707 712 Additional tapering of the iso-neutral fluxes is necessary within the 708 713 surface mixed layer. When the Griffies triads are used, we offer two 709 714 options for this. 715 710 716 \subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:triad:lintaper} 711 717 This is the option activated by the default choice 712 \ forcode{ln_triad_iso= .false.}. Slopes $\tilde{r}_i$ relative to718 \np{ln\_triad\_iso}\forcode{ = .false.}. Slopes $\tilde{r}_i$ relative to 713 719 geopotentials are tapered linearly from their value immediately below the mixed layer to zero at the 714 720 surface, as described in option (c) of Fig.~\ref{Fig_eiv_slp}, to values … … 839 845 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 840 846 841 \subsubsection{Additional truncation of skew iso-neutral flux 842 components} 847 \subsubsection{Additional truncation of skew iso-neutral flux components} 843 848 \label{sec:triad:Gerdes-taper} 844 The alternative option is activated by setting \np{ln _triad_iso} =849 The alternative option is activated by setting \np{ln\_triad\_iso} = 845 850 true. This retains the same tapered slope $\rML$ described above for the 846 851 calculation of the $_{33}$ term of the iso-neutral diffusion tensor (the … … 884 889 \section{Eddy induced advection formulated as a skew flux}\label{sec:triad:skew-flux} 885 890 886 \subsection{ The continuous skew flux formulation}\label{sec:triad:continuous-skew-flux}891 \subsection{Continuous skew flux formulation}\label{sec:triad:continuous-skew-flux} 887 892 888 893 When Gent and McWilliams's [1990] diffusion is used, … … 917 922 it to the Eulerian velocity prior to computing the tracer 918 923 advection. This is implemented if \key{traldf\_eiv} is set in the 919 default implementation, where \np{ln _traldf_triad} is set924 default implementation, where \np{ln\_traldf\_triad} is set 920 925 false. This allows us to take advantage of all the advection schemes 921 926 offered for the tracers (see \S\ref{TRA_adv}) and not just a $2^{nd}$ … … 924 929 paramount importance. 925 930 926 However, when \np{ln _traldf_triad} is set true, \NEMO instead931 However, when \np{ln\_traldf\_triad} is set true, \NEMO instead 927 932 implements eddy induced advection according to the so-called skew form 928 933 \citep{Griffies_JPO98}. It is based on a transformation of the advective fluxes … … 989 994 preserves the tracer variance. 990 995 991 \subsection{ The discrete skew flux formulation}996 \subsection{Discrete skew flux formulation} 992 997 The skew fluxes in (\ref{eq:triad:eiv_skew_physical}, \ref{eq:triad:eiv_skew_ijk}), like the off-diagonal terms 993 998 (\ref{eq:triad:i13c}, \ref{eq:triad:i31c}) of the small angle diffusion tensor, are best … … 1030 1035 operator as it uses the same definition for the slopes. It also 1031 1036 ensures the following two key properties. 1037 1032 1038 \subsubsection{No change in tracer variance} 1033 1039 The discretization conserves tracer variance, $i.e.$ it does not … … 1123 1129 and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the 1124 1130 $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ 1125 $u$-point is masked. The namelist parameter \np{ln _botmix_triad} has1131 $u$-point is masked. The namelist parameter \np{ln\_botmix\_triad} has 1126 1132 no effect on the eddy-induced skew-fluxes. 1127 1133 1128 \subsection{ 1134 \subsection{Limiting of the slopes within the interior}\label{sec:triad:limitskew} 1129 1135 Presently, the iso-neutral slopes $\tilde{r}_i$ relative 1130 1136 to geopotentials are limited to be less than $1/100$, exactly as in … … 1138 1144 option (c) of Fig.~\ref{Fig_eiv_slp}. This linear tapering for the 1139 1145 slopes used to calculate the eddy-induced fluxes is 1140 unaffected by the value of \np{ln _triad_iso}.1146 unaffected by the value of \np{ln\_triad\_iso}. 1141 1147 1142 1148 The justification for this linear slope tapering is that, for $A_e$ … … 1153 1159 1154 1160 \subsection{Streamfunction diagnostics}\label{sec:triad:sfdiag} 1155 Where the namelist parameter \ forcode{ln_traldf_gdia= .true.}, diagnosed1161 Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.}, diagnosed 1156 1162 mean eddy-induced velocities are output. Each time step, 1157 1163 streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at
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