Terrain-Following Coordinate Transformation

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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:

(1)

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:

(2)

where

(3)

As a trade-off for this geometric simplification, the dynamic equations become somewhat more complicated. The resulting dynamic equations, after dropping the carats, are:

(4)


(5)


(6)


(7)


(8)


(9)

where

(10)
(11)

The vertical velocity in coordinates is

(12)

and

(13)

Vertical Boundary Conditions

In the stretched coordinate system, the vertical boundary conditions become:

top ():

(14)

and bottom ():

(15)

Note the simplification of the boundary conditions on vertical velocity that arises from the coordinate transformation.