The evolution of the critical layer of a rossby wave. Part II

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Abstract
In Part I of this paper with the same title [Stewartson (1978)] the evolution of a Rossby wave of amplitude O(∊) forced on a uniform shear was studied. Times t were considered such that t = O (1) and ∊½ t= O (1) and it was shown that as ∊½ t→ ∞ 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 −1 ∊ −3/2 was 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 ∊−½ than 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 an O (∊½) modification to the imposed shear.

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