Changeset 10442 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex
r10414 r10442 10 10 \minitoc 11 11 12 %%% Appendix put in gmcomment as it has not been updated for z*and s coordinate12 %%% Appendix put in gmcomment as it has not been updated for \zstar and s coordinate 13 13 %I'm writting this appendix. It will be available in a forthcoming release of the documentation 14 14 … … 39 39 $dv=e_1\,e_2\,e_3 \,di\,dj\,dk$ is the volume element, with only $e_3$ that depends on time. 40 40 $D$ and $S$ are the ocean domain volume and surface, respectively. 41 No wetting/drying is allow ( $i.e.$$\frac{\partial S}{\partial t} = 0$).41 No wetting/drying is allow (\ie $\frac{\partial S}{\partial t} = 0$). 42 42 Let $k_s$ and $k_b$ be the ocean surface and bottom, resp. 43 ( $i.e.$$s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth).43 (\ie $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth). 44 44 \begin{flalign*} 45 45 z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s} e_3(\tilde{k}) \;d\tilde{k} … … 99 99 \label{sec:C.1} 100 100 101 The discretization of pimitive equation in $s$-coordinate ( $i.e.$time and space varying vertical coordinate)101 The discretization of pimitive equation in $s$-coordinate (\ie time and space varying vertical coordinate) 102 102 must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy. 103 103 104 104 Let us first establish those constraint in the continuous world. 105 The total energy ( $i.e.$kinetic plus potential energies) is conserved:105 The total energy (\ie kinetic plus potential energies) is conserved: 106 106 \begin{flalign} 107 107 \label{eq:Tot_Energy} … … 487 487 + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } 488 488 \] 489 Indeed, using successively \autoref{eq:DOM_di_adj} ( $i.e.$the skew symmetry property of the $\delta$ operator)489 Indeed, using successively \autoref{eq:DOM_di_adj} (\ie the skew symmetry property of the $\delta$ operator) 490 490 and the continuity equation, then \autoref{eq:DOM_di_adj} again, 491 491 then the commutativity of operators $\overline {\,\cdot \,}$ and $\delta$, and finally \autoref{eq:DOM_mi_adj} 492 ( $i.e.$the symmetry property of the $\overline {\,\cdot \,}$ operator)492 (\ie the symmetry property of the $\overline {\,\cdot \,}$ operator) 493 493 applied in the horizontal and vertical directions, it becomes: 494 494 \begin{flalign*} … … 599 599 600 600 When the equation of state is linear 601 ( $i.e.$when an advection-diffusion equation for density can be derived from those of temperature and salinity)601 (\ie when an advection-diffusion equation for density can be derived from those of temperature and salinity) 602 602 the change of KE due to the work of pressure forces is balanced by 603 603 the change of potential energy due to buoyancy forces: … … 621 621 % 622 622 \allowdisplaybreaks 623 \intertext{Using successively \autoref{eq:DOM_di_adj}, $i.e.$the skew symmetry property of623 \intertext{Using successively \autoref{eq:DOM_di_adj}, \ie the skew symmetry property of 624 624 the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{eq:dynhpg_sco}, 625 625 the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, … … 811 811 812 812 Let us first consider the first term of the scalar product 813 ( $i.e.$just the the terms associated with the i-component of the advection):813 (\ie just the the terms associated with the i-component of the advection): 814 814 \begin{flalign*} 815 815 & - \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv \\ … … 867 867 When the UBS scheme is used to evaluate the flux form momentum advection, 868 868 the discrete operator does not contribute to the global budget of linear momentum (flux form). 869 The horizontal kinetic energy is not conserved, but forced to decay ( $i.e.$the scheme is diffusive).869 The horizontal kinetic energy is not conserved, but forced to decay (\ie the scheme is diffusive). 870 870 871 871 % ================================================================ … … 893 893 894 894 The scheme does not allow but the conservation of the total kinetic energy but the conservation of $q^2$, 895 the potential enstrophy for a horizontally non-divergent flow ( $i.e.$when $\chi$=$0$).895 the potential enstrophy for a horizontally non-divergent flow (\ie when $\chi$=$0$). 896 896 Indeed, using the symmetry or skew symmetry properties of the operators 897 897 ( \autoref{eq:DOM_mi_adj} and \autoref{eq:DOM_di_adj}), … … 942 942 } 943 943 \end{flalign*} 944 The later equality is obtain only when the flow is horizontally non-divergent, $i.e.$$\chi$=$0$.944 The later equality is obtain only when the flow is horizontally non-divergent, \ie $\chi$=$0$. 945 945 946 946 % ------------------------------------------------------------------------------------------------------------- … … 971 971 \end{equation} 972 972 973 This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ( $i.e.$$\chi=0$).973 This formulation does conserve the potential enstrophy for a horizontally non-divergent flow (\ie $\chi=0$). 974 974 975 975 Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, … … 1026 1026 the internal dynamics and physics (equations in flux form). 1027 1027 For advection, 1028 only the CEN2 scheme ( $i.e.$$2^{nd}$ order finite different scheme) conserves the global variance of tracer.1028 only the CEN2 scheme (\ie $2^{nd}$ order finite different scheme) conserves the global variance of tracer. 1029 1029 Nevertheless the other schemes ensure that the global variance decreases 1030 ( $i.e.$they are at least slightly diffusive).1030 (\ie they are at least slightly diffusive). 1031 1031 For diffusion, all the schemes ensure the decrease of the total tracer variance, except the iso-neutral operator. 1032 1032 There is generally no strict conservation of mass, … … 1072 1072 1073 1073 The conservation of the variance of tracer due to the advection tendency can be achieved only with the CEN2 scheme, 1074 $i.e.$when $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$.1074 \ie when $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$. 1075 1075 It can be demonstarted as follows: 1076 1076 \begin{flalign*} … … 1108 1108 the conservation of potential vorticity and the horizontal divergence, 1109 1109 and the dissipation of the square of these quantities 1110 ( $i.e.$enstrophy and the variance of the horizontal divergence) as well as1110 (\ie enstrophy and the variance of the horizontal divergence) as well as 1111 1111 the dissipation of the horizontal kinetic energy. 1112 1112 In particular, when the eddy coefficients are horizontally uniform, … … 1346 1346 \end{flalign*} 1347 1347 1348 If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, $i.e.$1348 If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, \ie 1349 1349 \begin{flalign*} 1350 1350 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times … … 1396 1396 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&& 1397 1397 \end{flalign*} 1398 and the square of the horizontal divergence decreases ( $i.e.$the horizontal divergence is dissipated) if1398 and the square of the horizontal divergence decreases (\ie the horizontal divergence is dissipated) if 1399 1399 the vertical diffusion coefficient is uniform over the whole domain: 1400 1400 … … 1463 1463 the heat and salt contents are conserved (equations in flux form). 1464 1464 Since a flux form is used to compute the temperature and salinity, 1465 the quadratic form of these quantities ( $i.e.$their variance) globally tends to diminish.1465 the quadratic form of these quantities (\ie their variance) globally tends to diminish. 1466 1466 As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. 1467 1467 … … 1530 1530 \biblio 1531 1531 1532 \pindex 1533 1532 1534 \end{document}
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