The Two-Fifths and One-Fifth Rules for Rossby Wave Breaking in the WKB Limit

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Abstract
A stationary Rossby wave, sinusoidal in longitude, is slowly switched on, and the meridional propagation of the resulting wave front through a shear flow is examined. Initially the flow is westerly everywhere and therefore free of critical layers. The transition from reversible to irreversible behavior as the wave amplitude is increased is described. It is shown that under slowly varying conditions in an inviscid quasi-linear model, a steady state is obtained if, and only if, the mean flow is decelerated by less than two-fifths of its initial value as a result of the passage of the wave front. If this passage causes a larger mean flow reduction, a pile-up of wave activity in the shear layer culminates in the generation of a critical layer, qualitatively as in Dunkerton's model of gravity wave–mean flow interaction. This qualitative picture is shown to be preserved in the quasi-linear model when the slowly varying assumption breaks down. Fully nonlinear calculations show that these quasi-linear results are only part of the story. Once the mean flow is decelerated by two-fifths of its initial value in the fully nonlinear model, rapid wave breaking and irreversible mixing occur in the shear layer. But more slowly developing wave breaking also occurs for wave amplitudes that are too small to produce the two-fifths deceleration. Overturning of contours can be shown to occur in the quasi-linear slowly varying model once the mean flow has been decelerated by one-fifth of its initial value, and this appears to be the critical value for wave breaking to occur in the nonlinear integrations.

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