Changeset 3680 for branches/2012/dev_MERGE_2012/DOC/TexFiles/Chapters/Chap_DOM.tex

Timestamp:

2012-11-27T15:42:24+01:00 (11 years ago)

Author:

rblod

Message:

First commit of the final branch for 2012 (future nemo_3_5), see ticket #1028

File:

• branches/2012/dev_MERGE_2012/DOC/TexFiles/Chapters/Chap_DOM.tex (modified) (3 diffs)

branches/2012/dev_MERGE_2012/DOC/TexFiles/Chapters/Chap_DOM.tex

@@ -502 +502 @@
 time-variation of the free surface so that the transformation is time dependent: 
 $z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). This option can be used with full step 
-bathymetry or $s$-coordinate (hybride and partial step coordinates have not 
+bathymetry or $s$-coordinate (hybrid and partial step coordinates have not 
 yet been tested in NEMO v2.3). 
 
@@ -743 +743 @@
 levels are defined from the product of a depth field and either a stretching 
 function or its derivative, respectively:
+
 \begin{equation} \label{DOM_sco_ana}
 \begin{split}
@@ -749 +750 @@
 \end{split}
 \end{equation}
+
 where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point 
 location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea 
 surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean 
 depth, since a mixed step-like and bottom-following representation of the 
-topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e). In the example provided 
-(\rou{zgr\_sco} routine, see \mdl{domzgr}) $h$ is a smooth envelope bathymetry 
-and steps are used to represent sharp bathymetric gradients.
-
-A new flexible stretching function, modified from \citet{Song_Haidvogel_JCP94} is provided as an example:
+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).
+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.
+
+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.
+
+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)}
 \end{split}
 \end{equation}
-where $h_c$ is the thermocline depth and $\theta$ and $b$ are the surface and 
+
+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}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht]    \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_sco_function.pdf}
+\caption{  \label{Fig_sco_function}   
+Examples of the stretching function applied to a seamount; from left to right: 
+surface, surface and bottom, and bottom intensified resolutions}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+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 
 increase of the vertical resolution (Fig.~\ref{Fig_sco_function}).
 
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!tb]    \begin{center}
-\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_sco_function.pdf}
-\caption{  \label{Fig_sco_function}   
-Examples of the stretching function applied to a sea mont; from left to right: 
-surface, surface and bottom, and bottom intensified resolutions}
-\end{center}   \end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+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:
+
+\begin{equation}
+z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1
+\label{eq:z}
+\end{equation}
+
+The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate:
+
+\begin{equation} \label{DOM_gamma_deriv}
+\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)
+\end{equation}
+
+Where:
+\begin{equation} \label{DOM_gamma}
+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}
+
+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:
+
+\begin{equation} \label{DOM_zb}
+Z_b= h a + b
+\end{equation}
+
+where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht]
+   \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/FIG_DOM_compare_coordinates_surface.pdf}
+        \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.}
+    \label{fig_compare_coordinates_surface}
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+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 $h_c$.
+
+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.
 
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