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- 2012-11-19T14:59:22+01:00 (11 years ago)
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branches/2012/dev_UKMO_2012/DOC/TexFiles/Chapters/Chap_DOM.tex
r3294 r3600 502 502 time-variation of the free surface so that the transformation is time dependent: 503 503 $z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). This option can be used with full step 504 bathymetry or $s$-coordinate (hybrid eand partial step coordinates have not504 bathymetry or $s$-coordinate (hybrid and partial step coordinates have not 505 505 yet been tested in NEMO v2.3). 506 506 … … 743 743 levels are defined from the product of a depth field and either a stretching 744 744 function or its derivative, respectively: 745 745 746 \begin{equation} \label{DOM_sco_ana} 746 747 \begin{split} … … 749 750 \end{split} 750 751 \end{equation} 752 751 753 where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point 752 754 location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea 753 755 surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean 754 756 depth, since a mixed step-like and bottom-following representation of the 755 topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e). In the example provided 756 (\rou{zgr\_sco} routine, see \mdl{domzgr}) $h$ is a smooth envelope bathymetry 757 and steps are used to represent sharp bathymetric gradients. 758 759 A new flexible stretching function, modified from \citet{Song_Haidvogel_JCP94} is provided as an example: 757 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). 758 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. 759 760 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. 761 A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} stretching (\np{ln\_sco\_SH94}~=~true), is provided as an example: 762 760 763 \begin{equation} \label{DOM_sco_function} 761 764 \begin{split} 762 z &= h_c +( h-h_c)\;c s )\\765 z &= h_c +( h-h_c)\;c s \\ 763 766 c(s) &= \frac{ \left[ \tanh{ \left( \theta \, (s+b) \right)} 764 767 - \tanh{ \left( \theta \, b \right)} \right]} … … 766 769 \end{split} 767 770 \end{equation} 768 where $h_c$ is the thermocline depth and $\theta$ and $b$ are the surface and 771 772 773 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 774 \begin{figure}[!ht] \begin{center} 775 \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_sco_function.pdf} 776 \caption{ \label{Fig_sco_function} 777 Examples of the stretching function applied to a seamount; from left to right: 778 surface, surface and bottom, and bottom intensified resolutions} 779 \end{center} \end{figure} 780 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 781 782 where $h_c$ is the critical depth (\np{rn\_hc}) total depth 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 769 783 bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 770 784 $0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom 771 785 increase of the vertical resolution (Fig.~\ref{Fig_sco_function}). 772 786 773 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 774 \begin{figure}[!tb] \begin{center} 775 \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_sco_function.pdf} 776 \caption{ \label{Fig_sco_function} 777 Examples of the stretching function applied to a sea mont; from left to right: 778 surface, surface and bottom, and bottom intensified resolutions} 779 \end{center} \end{figure} 780 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 787 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: 788 789 \begin{equation} 790 z = \gamma\left(h+\zeta\right) \quad \text{ with } \quad 0 \leq \gamma \leq 1 791 \label{eq:z} 792 \end{equation} 793 794 The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate: 795 796 \begin{equation} \label{DOM_gamma_deriv} 797 \gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right) 798 \end{equation} 799 800 Where: 801 \begin{equation} \label{DOM_gamma} 802 f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} 803 \end{equation} 804 805 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: 806 807 \begin{equation} \label{DOM_zb} 808 Z_b= h a + b 809 \end{equation} 810 811 where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively. 812 813 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 814 \begin{figure}[!ht] 815 \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/FIG_DOM_compare_coordinates_surface.pdf} 816 \caption{A comparison of the \citet{Song_Haidvogel_JCP94} $S$-coordinate (solid lines), a 50 level $Z$-coordinate (contoured surfaces) and the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth. For clarity every third coordinate surface is shown.} 817 \label{fig_compare_coordinates_surface} 818 \end{figure} 819 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 820 821 This gives a smooth analytical stretching in computational space that is constrained to given specified surface and bottom grid cell depths in real space. This is not to be confused with the hybrid schemes that superimpose geopotential coordinates on terrain following coordinates thus creating a non-analytical vertical coordinate that therefore may suffer from large gradients in the vertical resolutions. This stretching is less straightforward to implement than the \citet{Song_Haidvogel_JCP94} stretching, but has the advantage of resolving diurnal processes in deep water and has generally flatter slopes. 822 823 As with the \citet{Song_Haidvogel_JCP94} stretching the stretch is only applied at depths greater than the critical depth $h_c$. In this example two options are available in depths shallower than $h_c$, with pure sigma being applied if the \np{ln\_sigcrit} is true and pure z-coordinates if it is false (the z-coordinate being equal to the depths of the stretched coordinate at $h_c$. 824 825 Minimising the horizontal slope of the vertical coordinate is important in terrain-following systems as large slopes lead to hydrostatic consistency. A hydrostatic consistency parameter diagnostic following \citet{Haney1991} has been implemented, and is output as part of the model mesh file at the start of the run. 781 826 782 827 % -------------------------------------------------------------------------------------------------------------
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