Finite amplitude disturbances in a rotating fluid

A liquid is rotating with angular velocity vector along the z-axis. A uniform gravity field acts along the negative z-axis and the liquid is stratified with arbitrary density gradient. A steady disturbance exists with all properties independent of one of the horizontal coordinates, say, y. It is then shown that the primitive equations can be integrated once to yield a second-order vorticity equation. This equation is generally non-linear and complicated but greatly simplifies when the fluid is homogeneous. In one special and interesting case the problem becomes identical to two-dimensional flow of a certain stratified fluid in a uniform gravity field.