Terrain-Following Coordinate Transformation
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 Vertical S-coordinate for the form of used here. Also, see Shchepetkin and McWilliams, 2005 for a discussion about the nature of this form of and how it
differs from that used in SCRUM.
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, after dropping the
carats, are:
where
The vertical velocity in coordinates is
and
Vertical Boundary Conditions
In the stretched coordinate system, the vertical boundary conditions
become:
top ():
and bottom ():
Note the simplification of the boundary conditions on vertical
velocity that arises from the coordinate transformation.