Index: branches/2012/dev_UKMO_2012/DOC/TexFiles/Chapters/Chap_DOM.tex
===================================================================
--- branches/2012/dev_UKMO_2012/DOC/TexFiles/Chapters/Chap_DOM.tex (revision 3602)
+++ branches/2012/dev_UKMO_2012/DOC/TexFiles/Chapters/Chap_DOM.tex (revision 3618)
@@ -759,10 +759,22 @@
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.
-A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} stretching (\np{ln\_sco\_SH94}~=~true), is provided as an example:
+
+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}:
+
+\begin{equation}
+ z = s_{min}+C\left(s\right)\left(H-s_{min}\right)
+ \label{eq:SH94_1}
+\end{equation}
+
+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).
+
+\begin{equation}
+ s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1
+ \label{eq:s}
+\end{equation}
\begin{equation} \label{DOM_sco_function}
\begin{split}
-z &= h_c +( h-h_c)\;c s \\
-c(s) &= \frac{ \left[ \tanh{ \left( \theta \, (s+b) \right)}
+C(s) &= \frac{ \left[ \tanh{ \left( \theta \, (s+b) \right)}
- \tanh{ \left( \theta \, b \right)} \right]}
{2\;\sinh \left( \theta \right)}
@@ -770,4 +782,11 @@
\end{equation}
+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:
+
+\begin{equation}
+ C\left(s\right) = \left(1 - b \right)\frac{ \sinh\left( \theta s\right)}{\sinh\left(\theta\right)} + \\
+ 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)}
+ \label{eq:SH94_2}
+\end{equation}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -780,5 +799,5 @@
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-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
+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
bottom control parameters such that $0\leqslant \theta \leqslant 20$, and
$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom
@@ -788,5 +807,5 @@
\begin{equation}
-z = \gamma\left(h+\zeta\right) \quad \text{ with } \quad 0 \leq \gamma \leq 1
+z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1
\label{eq:z}
\end{equation}
@@ -800,5 +819,5 @@
Where:
\begin{equation} \label{DOM_gamma}
-f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2}
+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}{n-1}
\end{equation}
@@ -819,5 +838,5 @@
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-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.
+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 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.
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 $h_c$.