Formally, we transform the equations from a
coordinate system to the
system, where
, , and .
Let **f** be a dependent
variable that denotes any quantity representative of an ocean
property. Then the derivatives of this quantity are related in the
two coordinate systems thus:

where ,
the total depth
of the water column. Note that as defined in
equation (67), goes
from 0 at the free surface to **-1** at the bottom **z=-H**.
(It is
also possible to define such that it goes from **+1** to **-1**). Using
equations (68) and dropping overbars for
greater clarity,
continuity equation (52), the momentum equations
(54) to (56),
conservation equations (60) and (61)
and the turbulence equations (65) and (66)
can be written as:

and

where , , , and denote horizontal diffusion terms for momentum and scalars , and respectively, and and denote the horizontal gradients of pressure. Notice that the overbar has been dropped from the in equations (69)--(75); which is a pseudo-vertical velocity in the new coordinate system given by