Changeset 11435 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_iso.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/annex_iso.tex
r11187 r11435 18 18 \label{apdx:triad} 19 19 20 \ minitoc20 \chaptertoc 21 21 22 22 \newpage 23 23 24 \section{Choice of \protect\ngn{namtra\_ldf} namelist parameters} 24 \section[Choice of \texttt{namtra\_ldf} namelist parameters] 25 {Choice of \protect\nam{tra\_ldf} namelist parameters} 25 26 %-----------------------------------------nam_traldf------------------------------------------------------ 26 27 … … 30 31 Two scheme are available to perform the iso-neutral diffusion. 31 32 If the namelist logical \np{ln\_traldf\_triad} is set true, 32 \NEMO updates both active and passive tracers using the Griffies triad representation of iso-neutral diffusion and33 \NEMO\ updates both active and passive tracers using the Griffies triad representation of iso-neutral diffusion and 33 34 the eddy-induced advective skew (GM) fluxes. 34 35 If the namelist logical \np{ln\_traldf\_iso} is set true, … … 38 39 39 40 Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}. 40 Note that when GM fluxes are used, the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS,41 Note that when GM fluxes are used, the eddy-advective (GM) velocities are output for diagnostic purposes using XIOS, 41 42 even though the eddy advection is accomplished by means of the skew fluxes. 42 43 … … 72 73 \label{sec:iso} 73 74 74 We have implemented into \NEMO a scheme inspired by \citet{griffies.gnanadesikan.ea_JPO98},75 but formulated within the \NEMO framework, using scale factors rather than grid-sizes.75 We have implemented into \NEMO\ a scheme inspired by \citet{griffies.gnanadesikan.ea_JPO98}, 76 but formulated within the \NEMO\ framework, using scale factors rather than grid-sizes. 76 77 77 78 \subsection{Iso-neutral diffusion operator} … … 191 192 To correct this, we introduced a smoothing of the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}). 192 193 This technique works for $T$ and $S$ in so far as they are active tracers 193 (\ie they enter the computation of density), but it does not work for a passive tracer.194 (\ie\ they enter the computation of density), but it does not work for a passive tracer. 194 195 195 196 \subsection{Expression of the skew-flux in terms of triad slopes} … … 280 281 the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell. 281 282 Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:i13} and \autoref{eq:i31} in this notation, 282 we have \eg \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$.283 we have \eg\ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. 283 284 Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to 284 285 calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$, … … 288 289 and we notate these areas, similarly to the triad slopes, 289 290 as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, 290 where \eg in \autoref{eq:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$,291 where \eg\ in \autoref{eq:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, 291 292 and in \autoref{eq:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 292 293 … … 477 478 defined in terms of the distances between $T$, $u$,$f$ and $w$-points. 478 479 This is the natural discretization of \autoref{eq:cts-var}. 479 The \NEMO model, however, operates with scale factors instead of grid sizes,480 The \NEMO\ model, however, operates with scale factors instead of grid sizes, 480 481 and scale factors for the quarter cells are not defined. 481 482 Instead, therefore we simply choose … … 600 601 It is a key property for a diffusion term. 601 602 It means that it is also a dissipation term, 602 \ie it dissipates the square of the quantity on which it is applied.603 \ie\ it dissipates the square of the quantity on which it is applied. 603 604 It therefore ensures that, when the diffusivity coefficient is large enough, 604 605 the field on which it is applied becomes free of grid-point noise. … … 649 650 Similar comments apply to triads that would intersect the ocean floor (\autoref{fig:bdry_triads}b). 650 651 Note that both near bottom triad slopes \triad{i}{k}{R}{1/2}{1/2} and \triad{i+1}{k}{R}{-1/2}{1/2} are masked when 651 either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie the $i,k+1$ $u$-point is masked.652 either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked. 652 653 The associated lateral fluxes (grey-black dashed line) are masked if \np{ln\_botmix\_triad}\forcode{ = .false.}, 653 654 but left unmasked, giving bottom mixing, if \np{ln\_botmix\_triad}\forcode{ = .true.}. 654 655 655 656 The default option \np{ln\_botmix\_triad}\forcode{ = .false.} is suitable when the bbl mixing option is enabled 656 (\ key{trabbl}, with \np{nn\_bbl\_ldf}\forcode{ = 1}), or for simple idealized problems.657 (\np{ln\_trabbl}\forcode{ = .true.}, with \np{nn\_bbl\_ldf}\forcode{ = 1}), or for simple idealized problems. 657 658 For setups with topography without bbl mixing, \np{ln\_botmix\_triad}\forcode{ = .true.} may be necessary. 658 659 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 672 673 (b) Both near bottom triad slopes \triad{i}{k}{R}{1/2}{1/2} and 673 674 \triad{i+1}{k}{R}{-1/2}{1/2} are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, 674 \ie the $i,k+1$ $u$-point is masked.675 \ie\ the $i,k+1$ $u$-point is masked. 675 676 The associated lateral fluxes (grey-black dashed line) are masked if 676 \protect\np{ botmix\_triad}\forcode{ = .false.}, but left unmasked,677 giving bottom mixing, if \protect\np{ botmix\_triad}\forcode{ = .true.}677 \protect\np{ln\_botmix\_triad}\forcode{ = .false.}, but left unmasked, 678 giving bottom mixing, if \protect\np{ln\_botmix\_triad}\forcode{ = .true.} 678 679 } 679 680 \end{center} … … 687 688 iso-neutral slopes relative to geopotentials must be bounded everywhere, 688 689 both for consistency with the small-slope approximation and for numerical stability \citep{cox_OM87, griffies_bk04}. 689 The bound chosen in \NEMO is applied to each component of the slope separately and690 The bound chosen in \NEMO\ is applied to each component of the slope separately and 690 691 has a value of $1/100$ in the ocean interior. 691 692 %, ramping linearly down above 70~m depth to zero at the surface … … 765 766 where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m. 766 767 See the left side of \autoref{fig:MLB_triad}. 767 We use the $k_{10}$-gridbox instead of the surface gridbox to avoid problems \eg with thin daytime mixed-layers.768 We use the $k_{10}$-gridbox instead of the surface gridbox to avoid problems \eg\ with thin daytime mixed-layers. 768 769 Currently we use the same $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is used to 769 770 output the diagnosed mixed-layer depth $h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$, … … 781 782 % \label{eq:Rbase} 782 783 \\ 783 \intertext{with \eg the green triad}784 \intertext{with \eg\ the green triad} 784 785 {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&= 785 786 {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}. … … 828 829 ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$. 829 830 Triads with different $i_p,k_p$, denoted by different colours, 830 (\eg the green triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}831 (\eg\ the green triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.} 831 832 % } 832 833 \includegraphics[width=\textwidth]{Fig_GRIFF_MLB_triads} … … 889 890 the formulation of which depends on the slopes of iso-neutral surfaces. 890 891 Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 891 \ie \autoref{eq:ldfslp_geo} is used in $z$-coordinate,892 \ie\ \autoref{eq:ldfslp_geo} is used in $z$-coordinate, 892 893 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates. 893 894 … … 918 919 The traditional way to implement this additional advection is to add it to the Eulerian velocity prior to 919 920 computing the tracer advection. 920 This is implemented if \ key{traldf\_eiv} is set in the default implementation,921 This is implemented if \texttt{traldf\_eiv?} is set in the default implementation, 921 922 where \np{ln\_traldf\_triad} is set false. 922 923 This allows us to take advantage of all the advection schemes offered for the tracers … … 926 927 927 928 However, when \np{ln\_traldf\_triad} is set true, 928 \NEMO instead implements eddy induced advection according to the so-called skew form \citep{griffies_JPO98}.929 \NEMO\ instead implements eddy induced advection according to the so-called skew form \citep{griffies_JPO98}. 929 930 It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity. 930 931 For example in the (\textbf{i},\textbf{k}) plane, … … 1034 1035 \subsubsection{No change in tracer variance} 1035 1036 1036 The discretization conserves tracer variance, \ie it does not include a diffusive component but is a `pure' advection term.1037 The discretization conserves tracer variance, \ie\ it does not include a diffusive component but is a `pure' advection term. 1037 1038 This can be seen %either from Appendix \autoref{apdx:eiv_skew} or 1038 1039 by considering the fluxes associated with a given triad slope $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. … … 1116 1117 Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked, 1117 1118 and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when 1118 either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie the $i,k+1$ $u$-point is masked.1119 either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked. 1119 1120 The namelist parameter \np{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes. 1120 1121
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