An alternative view on the role of the β-effect in the Rossby wave propagation mechanism

The role of the β-effect in the Rossby wave propagation mechanism is examined in the linearised shallow water equations directly in momentum-height variables, without recourse to potential vorticity (PV). Rigorous asymptotic expansion of the equations, with respect to the small non-dimensionalised β parameter, reveals in detail how the Coriolis force acting on the small ageostrophic terms translates the geostrophic leading-order solution to propagate westward in concert. This information cannot be obtained directly from the conventional PV perspective on the propagation mechanism. Furthermore, a comparison between the β-effect in planetary Rossby waves and the sloping-bottom effect in promoting topographic Rossby waves shows that the ageostrophic terms play different roles in the two cases. This is despite the fact that from the PV viewpoint whether the advection of mean PV gradient is set up by changes in planetary vorticity or by mean depth is inconsequential.