Vertical S-coordinate

S-coordinate

Following Song and Haidvogel (1994), the vertical coordinate has been chosen to be:

$z=\zeta +\left(1+{\zeta \over h}\right)\left[h_{c}s+(h-h_{c})C(s)\right],\qquad \qquad -1\leq s\leq 0$

where $h_{c}\!\,$ is either the minimum depth or a shallower depth above which we wish to have more resolution. $C(s)\!\,$ is defined as:

$C(s)=(1-b){\sinh(\theta s) \over \sinh \theta }+b{\tanh[\theta (s+{1 \over 2})]-\tanh({1 \over 2}\theta ) \over 2\tanh({1 \over 2}\theta )}$

where $\theta \!\,$ and $b\!\,$ are surface and bottom control parameters. Their ranges are $0<\theta \leq 20\!\,$ and $0\leq b\leq 1\!\,$, respectively. The first equation leads to $z=\zeta \!\,$ for $s=0\!\,$ and $z=h\!\,$ for $s=-1\!\,$.

Some features of this coordinate system:

• It is a generalization of the $\sigma \!\,$-coordinate system. Letting $\theta \!\,$ go to zero and using L'Hopital's rule, we get:

$z=(\zeta +h)(1+s)-h\!\,$

which is the $\sigma \!\,$-coordinate.

• It has a linear dependence on $\zeta \!\,$ and is infinitely differentiable in $s\!\,$.
• The larger the value of $\theta \!\,$, the more resolution is kept above $h_{c}\!\,$.
• For $b=0\!\,$, the resolution all goes to the surface as $\theta \!\,$ is increased.
• For $b=1\!\,$, the resolution goes to both the surface and the bottom equally as $\theta \!\,$ is increased.
• For $\theta \neq 0\!\,$ there is a subtle mismatch in the discretization of the model equations, for instance in the horizontal viscosity term. We recommend that you stick with "reasonable" values of $\theta \!\,$, say $\theta \leq 5\!\,$.
• Some problems turn out to be sensitive to the value of $\theta \!\,$ used.

The following figure shows the $s\!\,$-surfaces for several values of $\theta \!\,$ and $b\!\,$ for one of our domains. It was produced by a Matlab tool written by Hernan Arango which is available from our web site.

[figure]

\caption{The $s\!\,$-surfaces for the North Atlantic with (a) $\theta =0.0001\!\,$ and $b=0\!\,$, (b) $\theta =8\!\,$ and $b=0\!\,$, (c) $\theta =8\!\,$ and $b=1\!\,$. (d) The actual values used in this domain were $\theta =5\!\,$ and $b=0.4\!\,$.}

We find it convenient to define:

$H_{z}\equiv {\partial z \over \partial s}$

The derivative of $C(s)\!\,$ can be computed analytically:

${\partial C(s) \over \partial s}=(1-b){\cosh(\theta s) \over \sinh \theta }\theta +b{\coth({1 \over 2}\theta ) \over 2\cosh ^{2}[\theta (s+{1 \over 2})]}\theta$

However, we choose to compute $H_{z}\!\,$ discretely as $\Delta z/\Delta s\!\,$ since this leads to the vertical sum of $H_{z}\!\,$ being exactly the total water depth $D\!\,$.