The model uses
orthogonal curvilinear coordinates in the horizontal and the sigma
coordinate system in the vertical. Let 
, 
be the horizontal
coordinates and z the vertical coordinate of a right handed
orthogonal curvilinear coordinate system. We will write down the
equations in the z system before transforming to the 
coordinate
system in the vertical. Let 
and 
be the velocities in the 
and 
directions and w the velocity in the vertical direction.
(Note that for curvilinear formulation this notation is more
convenient and logical than U and V. Just remember
).
The continuity equation is
where 
, 
are the metric
coefficients satisfying the identity
where ds is the length
of a segment in 
, 
, z space.
The governing equations are simplified by assuming that the fluid is incompressible and essentially of constant density. Thus variations in density are ignored except when the density is multiplied by gravitational acceleration, thus retaining the important stratification (i.e. buoyancy) effects. This is the so-called Boussinesq approximation. All terms except the gravitational and pressure gradient terms are ignored in the momentum equation in the vertical, equivalent to a hydrostatic balance, even though the fluid is in motion.
