Higher-order corrections for Rossby waves in a zonal channel on the β-plane

A formal analytic perturbation expansion in the β term is carried out for the Rossby wave solution of the shallow-water equations in a zonal channel on the β-plane. Apart from a quantization of the meridional wave number, the presence of zonal boundaries alters, to first order, both the velocity and the geopotential structures of the wave but does not alter the phase speed of the wave. The ageostrophic component of the velocity field is identical in first order with that of the unbounded β-plane and is therefore not related to the presence of boundaries. In contrast, the first-order correction to the geostrophic velocity component is inherently related to the presence of walls as it ensures the vanishing of the total meridional velocity on the boundaries. This first-order correction to the geostrophic field yields only a third-order correction in the Rossby phase speed, as can be expected from symmetry considerations.