-Coordinate System
Coastal ocean models usually use a terrain-following -coordinate
system to handle the effects of bottom topography. Here the water column is
divided into the same number of grid cells independence of depth . Let
(
denote Cartesian coordinates and (
sigma
coordinates. In most sigma coordinate ocean models the relationship between
the two coordinate systems are:
where z and increase vertically upward such that 
at
the surface and z=-H and 
at the bottom. H=H(x,y) is the bottom topography.
A problem has long been recognized in computing the horizontal pressure
gradient in the -coordinate system (e.g., Gary, 1973; Haney, 1991; and
Mellor et al., 1994, McCalpin, 1994): the horizontal pressure gradient becomes
a difference between two terms, which leads to a large truncation error at a
steep topography.