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| Such "<math>s\!\,</math>" coordinate systems have long been used, with slight | | Such "<math>s\!\,</math>" coordinate systems have long been used, with slight |
| appropriate modification, in both meteorology and oceanography | | appropriate modification, in both meteorology and oceanography |
| (e.g., Phillips \cite{Phil} and Freeman et al.\ \cite{FHD}).
| | [e.g., Phillips (1957) and Freeman et al. (1972)]. |
| To proceed, we make the coordinate transformation: | | To proceed, we make the coordinate transformation: |
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| |
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Revision as of 19:22, 7 November 2006
From the point of view of the computational model, it is highly
convenient to introduce a stretched vertical coordinate system which
essentially "flattens out" the variable bottom at .
Such "" coordinate systems have long been used, with slight
appropriate modification, in both meteorology and oceanography
[e.g., Phillips (1957) and Freeman et al. (1972)].
To proceed, we make the coordinate transformation:
See S-coordinate for the form of used here.
In the stretched system, the vertical coordinate spans the
range ; we are therefore left with
level upper () and lower () bounding
surfaces. The chain rules for this transformation are:
where
As a trade-off for this geometric
simplification, the dynamic equations become somewhat more
complicated. The resulting dynamic equations are, after dropping the
carats:
where
The vertical velocity in coordinates is
and
Vertical Boundary Conditions
In the stretched coordinate system, the vertical boundary conditions
become:
top ():
and bottom ():
& \left(\frac{K_m}{H_z}\right) \frac{\partial v}{\partial s} = \tau^y_b (x,y,t)</math>
& \left(\frac{K_T}{H_z}\right) \frac{\partial T}{\partial s} = 0</math>
Note the simplification of the boundary conditions on vertical
velocity that arises from the coordinate transformation.