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Changeset 3618 for branches/2012/dev_UKMO_2012/DOC/TexFiles/Chapters/Chap_DOM.tex – NEMO

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Timestamp:
2012-11-20T19:20:57+01:00 (10 years ago)
Author:
johnsiddorn
Message:

updates to reflect reviewers comments on coding standards and documentation

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1 edited

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  • branches/2012/dev_UKMO_2012/DOC/TexFiles/Chapters/Chap_DOM.tex

    r3600 r3618  
    759759 
    760760Options 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 
     762The 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}: 
     763 
     764\begin{equation} 
     765  z = s_{min}+C\left(s\right)\left(H-s_{min}\right) 
     766  \label{eq:SH94_1} 
     767\end{equation} 
     768 
     769where $s_{min}$ 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). 
     770 
     771\begin{equation} 
     772  s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1 
     773  \label{eq:s} 
     774\end{equation} 
    762775 
    763776\begin{equation} \label{DOM_sco_function} 
    764777\begin{split} 
    765 z  &= h_c +( h-h_c)\;c s   \\ 
    766 c(s)  &=  \frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)}  
     778C(s)  &=  \frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)}  
    767779               - \tanh{ \left(  \theta \, b      \right)}  \right]} 
    768780            {2\;\sinh \left( \theta \right)} 
     
    770782\end{equation} 
    771783 
     784A 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} 
    772791 
    773792%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    780799%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    781800 
    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  
     801where $H_c$ 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  
    783802bottom control parameters such that $0\leqslant \theta \leqslant 20$, and  
    784803$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom  
     
    788807 
    789808\begin{equation} 
    790 z = \gamma\left(h+\zeta\right) \quad \text{ with } \quad 0 \leq \gamma \leq 1 
     809z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1 
    791810\label{eq:z} 
    792811\end{equation} 
     
    800819Where: 
    801820\begin{equation} \label{DOM_gamma} 
    802 f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} 
     821f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1}  
    803822\end{equation} 
    804823 
     
    819838%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    820839 
    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. 
     840This 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 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. 
    822841 
    823842As 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 $h_c$. 
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