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branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_Model_Basics.tex
r2349 r2376 6 6 \label{PE} 7 7 \minitoc 8 9 8 10 9 \newpage … … 114 113 115 114 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 116 \begin{figure}[!ht] \label{Fig_ocean_bc}\begin{center}115 \begin{figure}[!ht] \begin{center} 117 116 \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_I_ocean_bc.pdf} 118 \caption{The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, where $H$ 119 is the depth of the sea floor and $\eta$ the height of the sea surface. Both $H$ and $\eta $ 120 are referenced to $z=0$.} 117 \caption{ \label{Fig_ocean_bc} 118 The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,t)$, where $H$ 119 is the depth of the sea floor and $\eta$ the height of the sea surface. 120 Both $H$ and $\eta$ are referenced to $z=0$.} 121 121 \end{center} \end{figure} 122 122 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 167 167 168 168 169 \newpage170 $\ $\newline % force a new ligne169 %\newpage 170 %$\ $\newline % force a new ligne 171 171 172 172 % ================================================================ … … 371 371 372 372 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 373 \begin{figure}[!tb] \label{Fig_referential}\begin{center}373 \begin{figure}[!tb] \begin{center} 374 374 \includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_I_earth_referential.pdf} 375 \caption{the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear 375 \caption{ \label{Fig_referential} 376 the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear 376 377 coordinate system (\textbf{i},\textbf{j},\textbf{k}). } 377 378 \end{center} \end{figure} … … 703 704 \label{PE_gco} 704 705 705 %\gmcomment{706 706 The ocean domain presents a huge diversity of situation in the vertical. First the ocean surface is a time dependent surface (moving surface). Second the ocean floor depends on the geographical position, varying from more than 6,000 meters in abyssal trenches to zero at the coast. Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing. 707 707 Therefore, in order to represent the ocean with respect to the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height $e.g.$ an $z$*-coordinate; for the second point, a space variation to fit the change of bottom topography $e.g.$ a terrain-following or $\sigma$-coordinate; and for the third point, one will be tempted to use a space and time dependent coordinate that follows the isopycnal surfaces, $e.g.$ an isopycnic coordinate. … … 717 717 The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces. 718 718 719 A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient... 720 721 the generalized vertical coordinates used in ocean modelling are not orthogonal, which contrasts with many other applications in mathematical physics. Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter. 722 723 the horizontal velocity in ocean models measures motions in the horizontal plane, perpendicular to the local gravitational field. That is, horizontal velocity is mathematically the same regardless the vertical coordinate, be it geopotential, isopycnal, pressure, or terrain following. The key motivation for maintaining the same horizontal velocity component is that the hydrostatic and geostrophic balances are dominant in the large-scale ocean. Use of an alternative quasi-horizontal velocity, for example one oriented parallel to the generalized surface, would lead to unacceptable numerical errors. Correspondingly, the vertical direction is anti-parallel to the gravitational force in all of the coordinate systems. We do not choose the alternative of a quasi-vertical direction oriented normal to the surface of a constant generalized vertical coordinate. 724 725 It is the method used to measure transport across the generalized vertical coordinate surfaces which differs between the vertical coordinate choices. That is, computation of the dia-surface velocity component represents the fundamental distinction between the various coordinates. In some models, such as geopotential, pressure, 726 and terrain following, this transport is typically diagnosed from volume or mass conservation. In other models, such as isopycnal layered models, this transport is prescribed based on assumptions about the physical processes producing a flux across the layer interfaces. 727 728 729 In this section we first establish the PE in the generalised vertical $s$-coordinate, then we discuss the particular cases available in \NEMO, namely $z$, $z$*, $s$, and $\tilde z$. 719 %\gmcomment{ 720 721 %A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient... 722 723 the generalized vertical coordinates used in ocean modelling are not orthogonal, 724 which contrasts with many other applications in mathematical physics. 725 Hence, it is useful to keep in mind the following properties that may seem 726 odd on initial encounter. 727 728 The horizontal velocity in ocean models measures motions in the horizontal plane, 729 perpendicular to the local gravitational field. That is, horizontal velocity is mathematically 730 the same regardless the vertical coordinate, be it geopotential, isopycnal, pressure, 731 or terrain following. The key motivation for maintaining the same horizontal velocity 732 component is that the hydrostatic and geostrophic balances are dominant in the large-scale ocean. 733 Use of an alternative quasi-horizontal velocity, for example one oriented parallel 734 to the generalized surface, would lead to unacceptable numerical errors. 735 Correspondingly, the vertical direction is anti-parallel to the gravitational force in all 736 of the coordinate systems. We do not choose the alternative of a quasi-vertical 737 direction oriented normal to the surface of a constant generalized vertical coordinate. 738 739 It is the method used to measure transport across the generalized vertical coordinate 740 surfaces which differs between the vertical coordinate choices. That is, computation 741 of the dia-surface velocity component represents the fundamental distinction between 742 the various coordinates. In some models, such as geopotential, pressure, and 743 terrain following, this transport is typically diagnosed from volume or mass conservation. 744 In other models, such as isopycnal layered models, this transport is prescribed based 745 on assumptions about the physical processes producing a flux across the layer interfaces. 746 747 748 In this section we first establish the PE in the generalised vertical $s$-coordinate, 749 then we discuss the particular cases available in \NEMO, namely $z$, $z$*, $s$, and $\tilde z$. 730 750 %} 731 751 … … 821 841 822 842 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 823 \begin{figure}[!b] \label{Fig_z_zstar}\begin{center}843 \begin{figure}[!b] \begin{center} 824 844 \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zstar.pdf} 825 \caption{(a) $z$-coordinate in linear free-surface case ; (b) $z-$coordinate in non-linear 826 free surface case (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate 845 \caption{ \label{Fig_z_zstar} 846 (a) $z$-coordinate in linear free-surface case ; 847 (b) $z-$coordinate in non-linear free surface case ; 848 (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate 827 849 \citep{Adcroft_Campin_OM04} ).} 828 850 \end{center} \end{figure}
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