Terrain-Following Coordinate Transformation

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Vertical Terrain-Following Coordinates

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.