Changeset 11598 for NEMO/trunk/doc/latex/NEMO/subfiles/apdx_triads.tex
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- 2019-09-25T22:00:42+02:00 (5 years ago)
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NEMO/trunk/doc/latex/NEMO/subfiles/apdx_triads.tex
r11597 r11598 12 12 13 13 \begin{document} 14 14 15 \chapter{Iso-Neutral Diffusion and Eddy Advection using Triads} 15 16 \label{apdx:TRIADS} 16 17 18 \thispagestyle{plain} 19 17 20 \chaptertoc 18 21 22 \paragraph{Changes record} ~\\ 23 24 {\footnotesize 25 \begin{tabularx}{\textwidth}{l||X|X} 26 Release & Author(s) & Modifications \\ 27 \hline 28 {\em 4.0} & {\em ...} & {\em ...} \\ 29 {\em 3.6} & {\em ...} & {\em ...} \\ 30 {\em 3.4} & {\em ...} & {\em ...} \\ 31 {\em <=3.4} & {\em ...} & {\em ...} 32 \end{tabularx} 33 } 34 35 \clearpage 36 19 37 %% ================================================================================================= 20 38 \section[Choice of \forcode{namtra\_ldf} namelist parameters]{Choice of \protect\nam{tra_ldf}{tra\_ldf} namelist parameters} 21 22 39 23 40 Two scheme are available to perform the iso-neutral diffusion. … … 36 53 The options specific to the Griffies scheme include: 37 54 \begin{description} 38 \item [{\np{ln_triad_iso}{ln\_triad\_iso}}] 39 See \autoref{sec:TRIADS_taper}. 55 \item [{\np{ln_triad_iso}{ln\_triad\_iso}}] See \autoref{sec:TRIADS_taper}. 40 56 If this is set false (the default), 41 57 then `iso-neutral' mixing is accomplished within the surface mixed-layer along slopes linearly decreasing with … … 47 63 giving an almost pure horizontal diffusive tracer flux within the mixed layer. 48 64 This is similar to the tapering suggested by \citet{gerdes.koberle.ea_CD91}. See \autoref{subsec:TRIADS_Gerdes-taper} 49 \item [{\np{ln_botmix_triad}{ln\_botmix\_triad}}] 50 See \autoref{sec:TRIADS_iso_bdry}. 65 \item [{\np{ln_botmix_triad}{ln\_botmix\_triad}}] See \autoref{sec:TRIADS_iso_bdry}. 51 66 If this is set false (the default) then the lateral diffusive fluxes 52 67 associated with triads partly masked by topography are neglected. 53 68 If it is set true, however, then these lateral diffusive fluxes are applied, 54 69 giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. 55 \item [{\np{rn_sw_triad}{rn\_sw\_triad}}] 56 blah blah to be added.... 70 \item [{\np{rn_sw_triad}{rn\_sw\_triad}}] blah blah to be added.... 57 71 \end{description} 58 72 The options shared with the Standard scheme include: 59 73 \begin{description} 60 \item [{\np{ln_traldf_msc}{ln\_traldf\_msc}}] 61 \item [{\np{rn_slpmax}{rn\_slpmax}}] blah blah to be added74 \item [{\np{ln_traldf_msc}{ln\_traldf\_msc}}] blah blah to be added 75 \item [{\np{rn_slpmax}{rn\_slpmax}}] blah blah to be added 62 76 \end{description} 63 77 … … 196 210 % Instead of multiplying the mean slope calculated at the $u$-point by 197 211 % the mean vertical gradient at the $u$-point, 198 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>199 212 \begin{figure}[tb] 200 213 \centering … … 207 220 \label{fig:TRIADS_ISO_triad} 208 221 \end{figure} 209 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>210 222 They get the skew flux from the products of the vertical gradients at each $w$-point surrounding the $u$-point with 211 223 the corresponding `triad' slope calculated from the lateral density gradient across the $u$-point divided by … … 258 270 while the metrics are calculated at the $u$- and $w$-points on the arms. 259 271 260 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>261 272 \begin{figure}[tb] 262 273 \centering … … 269 280 \label{fig:TRIADS_qcells} 270 281 \end{figure} 271 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>272 282 273 283 Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:TRIADS_qcells}) with the quarter cell that is … … 549 559 where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. 550 560 The diffusion scheme satisfies the following six properties: 561 551 562 \begin{description} 552 \item [$\bullet$ horizontal diffusion] 553 The discretization of the diffusion operator recovers the traditional five-point Laplacian 563 \item [Horizontal diffusion] The discretization of the diffusion operator recovers the traditional five-point Laplacian 554 564 \autoref{eq:TRIADS_lat-normal} in the limit of flat iso-neutral direction: 555 565 \[ … … 560 570 \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0 561 571 \] 562 563 \item [$\bullet$ implicit treatment in the vertical] 564 Only tracer values associated with a single water column appear in the expression \autoref{eq:TRIADS_i33} for 572 \item [Implicit treatment in the vertical] Only tracer values associated with a single water column appear in the expression \autoref{eq:TRIADS_i33} for 565 573 the $_{33}$ fluxes, vertical fluxes driven by vertical gradients. 566 574 This is of paramount importance since it means that a time-implicit algorithm can be used to … … 576 584 \] 577 585 (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large. 578 579 \item [$\bullet$ pure iso-neutral operator] 580 The iso-neutral flux of locally referenced potential density is zero. 586 \item [Pure iso-neutral operator] The iso-neutral flux of locally referenced potential density is zero. 581 587 See \autoref{eq:TRIADS_latflux-rho} and \autoref{eq:TRIADS_vertflux-triad2}. 582 583 \item [$\bullet$ conservation of tracer] 584 The iso-neutral diffusion conserves tracer content, \ie 588 \item [Conservation of tracer] The iso-neutral diffusion conserves tracer content, \ie 585 589 \[ 586 590 % \label{eq:TRIADS_iso_property1} … … 588 592 \] 589 593 This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. 590 591 \item [$\bullet$ no increase of tracer variance] 592 The iso-neutral diffusion does not increase the tracer variance, \ie 594 \item [No increase of tracer variance] The iso-neutral diffusion does not increase the tracer variance, \ie 593 595 \[ 594 596 % \label{eq:TRIADS_iso_property2} … … 601 603 It therefore ensures that, when the diffusivity coefficient is large enough, 602 604 the field on which it is applied becomes free of grid-point noise. 603 604 \item [$\bullet$ self-adjoint operator] 605 The iso-neutral diffusion operator is self-adjoint, \ie 605 \item [Self-adjoint operator] The iso-neutral diffusion operator is self-adjoint, \ie 606 606 \begin{equation} 607 607 \label{eq:TRIADS_iso_property3} … … 655 655 (\np[=.true.]{ln_trabbl}{ln\_trabbl}, with \np[=1]{nn_bbl_ldf}{nn\_bbl\_ldf}), or for simple idealized problems. 656 656 For setups with topography without bbl mixing, \np[=.true.]{ln_botmix_triad}{ln\_botmix\_triad} may be necessary. 657 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>658 657 \begin{figure}[h] 659 658 \centering … … 679 678 \label{fig:TRIADS_bdry_triads} 680 679 \end{figure} 681 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>682 680 683 681 %% ================================================================================================= … … 808 806 \end{enumerate} 809 807 810 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>811 808 \begin{figure}[h] 812 809 \centering … … 831 828 \label{fig:TRIADS_MLB_triad} 832 829 \end{figure} 833 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>834 830 835 831 %% =================================================================================================
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