The evolution of the critical layer of a Rossby wave

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Abstract
The evolution of a Rossby wave forced on a uniform shear is considered, the strength of the wave being characterised by a parameter ∊ ≪ 1. In the first stage of the process, when t ∼ 1, a steady flow is established everywhere outside a layer near the line y = 0, known as the critical layer, in which the vorticity oscillates with increasing amplitude. The velocity jump across the layer is however steady and of the same wavelength as the forcing oscillation. At larger times (∊ 1/2 t ∼ 1) the velocity jump oscillates slowly and higher harmonics are generated. It is shown by considering a model equation that as ∊1/2 t → ∞ the velocity jump tends to zero but that violent oscillations in the velocity normal to the layer are induced. The relation of this theory with the Benney-Bergeron-Davis theory is discussed and the effect of viscosity briefly considered.

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