Changeset 11598 for NEMO/trunk/doc/latex/NEMO/subfiles/apdx_algos.tex

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2019-09-25T22:00:42+02:00 (16 months ago)
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Add template of versioning record at the beginning of chapters

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 r11597 \begin{document} \chapter{Note on some algorithms} \label{apdx:ALGOS} \thispagestyle{plain} \chaptertoc \paragraph{Changes record} ~\\ {\footnotesize \begin{tabularx}{\textwidth}{l||X|X} Release & Author(s) & Modifications \\ \hline {\em   4.0} & {\em ...} & {\em ...} \\ {\em   3.6} & {\em ...} & {\em ...} \\ {\em   3.4} & {\em ...} & {\em ...} \\ {\em <=3.4} & {\em ...} & {\em ...} \end{tabularx} } \clearpage This appendix some on going consideration on algorithms used or planned to be used in \NEMO. This expression of the iso-neutral diffusion has been chosen in order to satisfy the following six properties: \begin{description} \item [$\bullet$ horizontal diffusion] The discretization of the diffusion operator recovers the traditional five-point Laplacian in the limit of flat iso-neutral direction: \item [Horizontal diffusion] The discretization of the diffusion operator recovers the traditional five-point Laplacian in the limit of flat iso-neutral direction: $% \label{eq:ALGOS_Gf_property1a} { _i^k \mathbb{R}_{i_p}^{k_p} }=0$ \item [$\bullet$ implicit treatment in the vertical] In the diagonal term associated with the vertical divergence of the iso-neutral fluxes \item [Implicit treatment in the vertical] In the diagonal term associated with the vertical divergence of the iso-neutral fluxes \ie\ the term associated with a second order vertical derivative) appears only tracer values associated with a single water column. \] can be quite large. \item [$\bullet$ pure iso-neutral operator] The iso-neutral flux of locally referenced potential density is zero, \ie \item [Pure iso-neutral operator] The iso-neutral flux of locally referenced potential density is zero, \ie \begin{align*} % \label{eq:ALGOS_Gf_property2} This result is trivially obtained using the \autoref{eq:ALGOS_Gf_triads} applied to $T$ and $S$ and the definition of the triads' slopes \autoref{eq:ALGOS_Gf_slopes}. \item [$\bullet$ conservation of tracer] The iso-neutral diffusion term conserve the total tracer content, \ie \item [Conservation of tracer] The iso-neutral diffusion term conserve the total tracer content, \ie $% \label{eq:ALGOS_Gf_property1}$ This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. \item [$\bullet$ decrease of tracer variance] The iso-neutral diffusion term does not increase the total tracer variance, \ie \item [Decrease of tracer variance] The iso-neutral diffusion term does not increase the total tracer variance, \ie \[ % \label{eq:ALGOS_Gf_property1} It therfore ensures that, when the diffusivity coefficient is large enough, the field on which it is applied become free of grid-point noise. \item [$\bullet$ self-adjoint operator] The iso-neutral diffusion operator is self-adjoint, \ie \item [Self-adjoint operator] The iso-neutral diffusion operator is self-adjoint, \ie \[ % \label{eq:ALGOS_Gf_property1} \ie\ it does not include a diffusive component but is a "pure" advection term. $\$\newpage      %force an empty line %% ================================================================================================= \subsection{Discrete invariants of the iso-neutral diffrusion}