Changeset 6289 for trunk/DOC/TexFiles/Chapters/Chap_DOM.tex
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- 2016-02-05T00:47:05+01:00 (8 years ago)
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trunk/DOC/TexFiles/Chapters/Chap_DOM.tex (modified) (12 diffs)
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trunk/DOC/TexFiles/Chapters/Chap_DOM.tex
r6140 r6289 138 138 and $f$-points, and its divergence defined at $t$-points: 139 139 \begin{eqnarray} \label{Eq_DOM_curl} 140 \nabla \times {\rm 140 \nabla \times {\rm{\bf A}}\equiv & 141 141 \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right) &\ \mathbf{i} \\ 142 142 +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1 \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right) &\ \mathbf{j} \\ 143 143 +& \frac{1}{e_{1f} \,e_{2f} } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2 \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right) &\ \mathbf{k} 144 144 \end{eqnarray} 145 \begin{equation} \label{Eq_DOM_div} 146 \nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}\,e_{3t}}\left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 147 +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 148 \end{equation} 149 150 In the special case of a pure $z$-coordinate system, \eqref{Eq_DOM_lap} and 151 \eqref{Eq_DOM_div} can be simplified. In this case, the vertical scale factor 152 becomes a function of the single variable $k$ and thus does not depend on the 153 horizontal location of a grid point. For example \eqref{Eq_DOM_div} reduces to: 154 \begin{equation*} 155 \nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}} \left( \delta_i \left[e_{2u}\,a_1 \right] 156 +\delta_j \left[e_{1v}\, a_2 \right] \right) 157 +\frac{1}{e_{3t}} \delta_k \left[ a_3 \right] 158 \end{equation*} 145 \begin{eqnarray} \label{Eq_DOM_div} 146 \nabla \cdot \rm{\bf A} \equiv 147 \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 148 +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 149 \end{eqnarray} 159 150 160 151 The vertical average over the whole water column denoted by an overbar becomes … … 183 174 Let $a$ and $b$ be two fields defined on the mesh, with value zero inside 184 175 continental area. Using integration by parts it can be shown that the differencing 185 operators ($\delta_i$, $\delta_j$ and $\delta_k$) are anti-symmetric linear186 operators,and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$,176 operators ($\delta_i$, $\delta_j$ and $\delta_k$) are skew-symmetric linear operators, 177 and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$, 187 178 $\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear 188 179 operators, $i.e.$ … … 425 416 426 417 The choice of the grid must be consistent with the boundary conditions specified 427 by the parameter \np{jperio}(see {\S\ref{LBC}).418 by \np{jperio}, a parameter found in \ngn{namcfg} namelist (see {\S\ref{LBC}). 428 419 429 420 % ------------------------------------------------------------------------------------------------------------- … … 505 496 If \np{ln\_isfcav}~=~true, an extra file input file describing the ice shelf draft 506 497 (in meters) (\ifile{isf\_draft\_meter}) is needed and all the location where the isf cavity thinnest 507 than \np{rn\_isfhmin} meters are grounded ( iemasked).498 than \np{rn\_isfhmin} meters are grounded ($i.e.$ masked). 508 499 509 500 After reading the bathymetry, the algorithm for vertical grid definition differs … … 540 531 541 532 Three options are possible for defining the bathymetry, according to the 542 namelist variable \np{nn\_bathy} :533 namelist variable \np{nn\_bathy} (found in \ngn{namdom} namelist): 543 534 \begin{description} 544 535 \item[\np{nn\_bathy} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ … … 721 712 usually 10\%, of the default thickness $e_{3t}(jk)$). 722 713 723 \colorbox{yellow}{Add a figure here of pstep especially at last ocean level}714 \gmcomment{ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level } } 724 715 725 716 % ------------------------------------------------------------------------------------------------------------- … … 749 740 depth, since a mixed step-like and bottom-following representation of the 750 741 topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e) or an envelop bathymetry can be defined (Fig.~\ref{Fig_z_zps_s_sps}f). 751 The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate. The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} as the minimum and maximum depths at which the terrain-following vertical coordinate is calculated. 752 753 Options for stretching the coordinate are provided as examples, but care must be taken to ensure that the vertical stretch used is appropriate for the application. 754 755 The original default NEMO s-coordinate stretching is available if neither of the other options are specified as true (\np{ln\_sco\_SH94}~=~false and \np{ln\_sco\_SF12}~=~false.) This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}: 742 The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects 743 the sea bed and becomes a pseudo z-coordinate. 744 The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} 745 as the minimum and maximum depths at which the terrain-following vertical coordinate is calculated. 746 747 Options for stretching the coordinate are provided as examples, but care must be taken to ensure 748 that the vertical stretch used is appropriate for the application. 749 750 The original default NEMO s-coordinate stretching is available if neither of the other options 751 are specified as true (\np{ln\_s\_SH94}~=~false and \np{ln\_s\_SF12}~=~false). 752 This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}: 756 753 757 754 \begin{equation} … … 760 757 \end{equation} 761 758 762 where $s_{min}$ is the depth at which the s-coordinate stretching starts and allows a z-coordinate to placed on top of the stretched coordinate, and z is the depth (negative down from the asea surface). 759 where $s_{min}$ is the depth at which the s-coordinate stretching starts and 760 allows a z-coordinate to placed on top of the stretched coordinate, 761 and z is the depth (negative down from the asea surface). 763 762 764 763 \begin{equation} … … 775 774 \end{equation} 776 775 777 A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} stretching (\np{ln\_sco\_SH94}~=~true), is also available and is more commonly used for shelf seas modelling: 776 A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} 777 stretching (\np{ln\_s\_SH94}~=~true), is also available and is more commonly used for shelf seas modelling: 778 778 779 779 \begin{equation} … … 792 792 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 793 793 794 where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from pure $\sigma$ to the stretched coordinate, and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) are the surface and 795 bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 794 where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from 795 pure $\sigma$ to the stretched coordinate, and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) 796 are the surface and bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 796 797 $0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom 797 798 increase of the vertical resolution (Fig.~\ref{Fig_sco_function}). 798 799 799 Another example has been provided at version 3.5 (\np{ln\_sco\_SF12}) that allows a fixed surface resolution in an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}. In this case the a stretching function $\gamma$ is defined such that: 800 Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows 801 a fixed surface resolution in an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}. 802 In this case the a stretching function $\gamma$ is defined such that: 800 803 801 804 \begin{equation} … … 815 818 \end{equation} 816 819 817 This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and user prescribed surface (\np{rn\_zs}) and bottom depths. The bottom cell depth in this example is given as a function of water depth: 820 This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of 821 the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards 822 the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and user prescribed surface (\np{rn\_zs}) 823 and bottom depths. The bottom cell depth in this example is given as a function of water depth: 818 824 819 825 \begin{equation} \label{DOM_zb} … … 843 849 \label{DOM_zgr_star} 844 850 845 This option is described in the Report by Levier \textit{et al.} (2007), available on 846 the \NEMO web site. 851 This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO web site. 847 852 848 853 %gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances
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