Changeset 11512 for NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/annex_E.tex
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NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/annex_E.tex
r11263 r11512 8 8 \label{apdx:E} 9 9 10 \ minitoc10 \chaptertoc 11 11 12 12 \newpage … … 48 48 $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$. 49 49 50 This results in a dissipatively dominant (\ie hyper-diffusive) truncation error50 This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error 51 51 \citep{shchepetkin.mcwilliams_OM05}. 52 52 The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}. … … 135 135 \end{equation} 136 136 with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, 137 \ie $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$137 \ie\ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ 138 138 it comes: 139 139 \begin{equation} … … 147 147 \end{split} 148 148 \end{equation} 149 if the velocity is uniform (\ie $|u|=cst$) then the diffusive flux is149 if the velocity is uniform (\ie\ $|u|=cst$) then the diffusive flux is 150 150 \begin{equation} 151 151 \label{eq:tra_ldf_lap} … … 166 166 \end{split} 167 167 \end{equation} 168 if the velocity is uniform (\ie $|u|=cst$) and168 if the velocity is uniform (\ie\ $|u|=cst$) and 169 169 choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right]$ 170 170 … … 218 218 not $2\rdt$ as it can be found sometimes in literature. 219 219 The leap-Frog time stepping is a second order centered scheme. 220 As such it respects the quadratic invariant in integral forms, \ie the following continuous property,220 As such it respects the quadratic invariant in integral forms, \ie\ the following continuous property, 221 221 \[ 222 222 % \label{eq:Energy} … … 256 256 257 257 Let try to define a scheme that get its inspiration from the \citet{griffies.gnanadesikan.ea_JPO98} scheme, 258 but is formulated within the \NEMO framework259 (\ie using scale factors rather than grid-size and having a position of $T$-points that258 but is formulated within the \NEMO\ framework 259 (\ie\ using scale factors rather than grid-size and having a position of $T$-points that 260 260 is not necessary in the middle of vertical velocity points, see \autoref{fig:zgr_e3}). 261 261 … … 271 271 (see \autoref{chap:LDF}). 272 272 Nevertheless, this technique works fine for $T$ and $S$ as they are active tracers 273 (\ie they enter the computation of density), but it does not work for a passive tracer.273 (\ie\ they enter the computation of density), but it does not work for a passive tracer. 274 274 \citep{griffies.gnanadesikan.ea_JPO98} introduce a different way to discretise the off-diagonal terms that 275 275 nicely solve the problem. … … 386 386 \item[$\bullet$ implicit treatment in the vertical] 387 387 In the diagonal term associated with the vertical divergence of the iso-neutral fluxes 388 \ie the term associated with a second order vertical derivative)388 \ie\ the term associated with a second order vertical derivative) 389 389 appears only tracer values associated with a single water column. 390 390 This is of paramount importance since it means that … … 431 431 It is a key property for a diffusion term. 432 432 It means that the operator is also a dissipation term, 433 \ie it is a sink term for the square of the quantity on which it is applied.433 \ie\ it is a sink term for the square of the quantity on which it is applied. 434 434 It therfore ensures that, when the diffusivity coefficient is large enough, 435 435 the field on which it is applied become free of grid-point noise. … … 457 457 the formulation of which depends on the slopes of iso-neutral surfaces. 458 458 Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 459 \ie \autoref{eq:ldfslp_geo} is used in $z$-coordinate,459 \ie\ \autoref{eq:ldfslp_geo} is used in $z$-coordinate, 460 460 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates. 461 461 … … 578 578 Nevertheless this property can be used to choose a discret form of \autoref{eq:eiv_skew_continuous} which 579 579 is consistent with the iso-neutral operator \autoref{eq:Gf_operator}. 580 Using the slopes \autoref{eq:Gf_slopes} and defining $A_e$ at $T$-point(\ie as $A$,580 Using the slopes \autoref{eq:Gf_slopes} and defining $A_e$ at $T$-point(\ie\ as $A$, 581 581 the eddy diffusivity coefficient), the resulting discret form is given by: 582 582 \begin{equation} … … 600 600 it uses the same definition for the slopes. 601 601 It also ensures the conservation of the tracer variance (see Appendix \autoref{apdx:eiv_skew}), 602 \ie it does not include a diffusive component but is a "pure" advection term.602 \ie\ it does not include a diffusive component but is a "pure" advection term. 603 603 604 604 $\ $\newpage %force an empty line … … 840 840 Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the $i$ direction. 841 841 Therefore the sum over the domain is zero, 842 \ie the variance of the tracer is preserved by the discretisation of the skew fluxes.842 \ie\ the variance of the tracer is preserved by the discretisation of the skew fluxes. 843 843 844 844 \biblio
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