A New Interpretation of the Beta Term in the Vorticity Equation

The β term, usually defined as the northward gradient of the vertical planetary vorticity, is here defined and interpreted as the northward planetary vorticity. The β term in the vorticity equation, relative to a rotating reference frame, is put into an exact correspondence with the planetary vorticity tilting term, which has a meaning independent of the coordinate system. For barotropic, isochoric, horizontal flow there is an exact balance between the rate of change of vertical vorticity and the planetary vorticity tilting term due to the northward components of velocity and planetary vorticity. This interpretation of the β term in the vorticity equation seems to be simpler than the interpretation based on the northward gradient of the vertical planetary vorticity because it only involves the components, not the derivatives, of the planetary vorticity, and because it is independent of the coordinate system. It is also shown, as a consequence of this new interpretation, that the Sverdrup relation is practically equivalent to the vertical component of the Taylor–Proudman constraint.