In Part I of this paper with the same title [Stewartson (1978)] the evolution of a Rossby wave of amplitudeO(ε) forced on a uniform shear was studied. Timestwere considered such thatt=O(1) and ε½t= O(1) and it was shown that as ε½t→ ∞ the vorticity in the neighbourhood of the critical layer does not tend to a limit though the velocity jump across it tends to zero. The discussion concentrated on inviscid flow so that λ= R−1ε−3/2was zero. In the present paper the unsteady investigation is extended to values of λ ≪ 1, and it is shown that the vorticity in the critical layer diffuses outwards until it has an effect on the imposed shear which is larger by a factor ε−½than that due of the wave disturbance. This is corroborated by a closer examination of the Benney-Bergeron theory for λ ≪ 1 in which the correctness of the conjecture of Haberman that the vorticity must be continuous across the critical layer is demonstrated. Thus in this limit also there is anO(ε½) modification to the imposed shear.