The 3-D velocity field is split into its baroclinic part
and
its depth integrated part according to:

where

Depth integrating the moment equations and the continuity equation gives, neglecting for a moment the atmospheric pressure terms and the vertical viscosity terms, gives:

and

where

The non-linear and the internal pressure terms are represented through and . The vertical integration is exact except for the horizontal viscosity terms. These terms take mainly care of small scale oscillations, and the 2-D field is computed according to (11) where is replaced by or chosen to be constant in space and time.

The equations for the baroclinic fields and become after subtracting (53) from (38) and (54) from (39) :

and