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branches/2012/dev_UKMO_2012/DOC/TexFiles/Chapters/Chap_DOM.tex
r3600 r3618 759 759 760 760 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: 761 762 The original default NEMO scoordinate 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}: 763 764 \begin{equation} 765 z = s_{min}+C\left(s\right)\left(Hs_{min}\right) 766 \label{eq:SH94_1} 767 \end{equation} 768 769 where $s_{min}$ is the depth at which the scoordinate stretching starts and allows a zcoordinate to placed on top of the stretched coordinate, and z is the depth (negative down from the asea surface). 770 771 \begin{equation} 772 s = \frac{k}{n1} \quad \text{ and } \quad 0 \leq k \leq n1 773 \label{eq:s} 774 \end{equation} 762 775 763 776 \begin{equation} \label{DOM_sco_function} 764 777 \begin{split} 765 z &= h_c +( hh_c)\;c s \\ 766 c(s) &= \frac{ \left[ \tanh{ \left( \theta \, (s+b) \right)} 778 C(s) &= \frac{ \left[ \tanh{ \left( \theta \, (s+b) \right)} 767 779  \tanh{ \left( \theta \, b \right)} \right]} 768 780 {2\;\sinh \left( \theta \right)} … … 770 782 \end{equation} 771 783 784 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: 785 786 \begin{equation} 787 C\left(s\right) = \left(1  b \right)\frac{ \sinh\left( \theta s\right)}{\sinh\left(\theta\right)} + \\ 788 b\frac{ \tanh \left[ \theta \left(s + \frac{1}{2} \right)\right]  \tanh\left(\frac{\theta}{2}\right)}{ 2\tanh\left (\frac{\theta}{2}\right)} 789 \label{eq:SH94_2} 790 \end{equation} 772 791 773 792 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 780 799 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 781 800 782 where $ h_c$ is the critical depth (\np{rn\_hc}) total depthat which the coordinate transitions from pure $\sigma$ to the stretched coordinate, and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) are the surface and801 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 783 802 bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 784 803 $0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom … … 788 807 789 808 \begin{equation} 790 z = \gamma\left(h+\zeta\right)\quad \text{ with } \quad 0 \leq \gamma \leq 1809 z = \gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1 791 810 \label{eq:z} 792 811 \end{equation} … … 800 819 Where: 801 820 \begin{equation} \label{DOM_gamma} 802 f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}\left(\alpha+1\right)\sigma^{\alpha+2} 821 f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n1} 803 822 \end{equation} 804 823 … … 819 838 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 820 839 821 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 nonanalytical 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.840 This gives a smooth analytical stretching in computational space that is constrained to given specified surface and bottom grid cell thicknesses in real space. This is not to be confused with the hybrid schemes that superimpose geopotential coordinates on terrain following coordinates thus creating a nonanalytical 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. 822 841 823 842 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 zcoordinates if it is false (the zcoordinate being equal to the depths of the stretched coordinate at $h_c$.
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