This effectively makes the rotation and gravity vectors co-linear and the normal to a local tangent plane at the Earth's surface . The Coriolis parameter will be regarded as constant and equal to
where is the latitude of the point at which the tangent plane is defined. The -plane assumption also removes the horizontal component of the Coriolis force which results from vertical motion. The configuration is similar to that of the rotating `dishpan' of laboratory experiments except that there are no lateral walls. The f-plane the momentum equation therefore becomes:
which is the Euler form of the equation of momentum with added Coriolis force.
The Continuity Equation - conservation of fluid mass
Continuity of fluid mass may be expressed as:
and, on using the vector identity
we can express eq. (3) in flux form:
.
This expression of the continuity of mass can also derived by equating the rate of change of mass within a spatially-fixed volume to the flux of mass through the bounding surface, followed by the application of Gauss’s divergence theorem i.e.
where is a vector field and integration is over the volume and surface .