The two-fifths and one-fifth rules for Rossby wave breaking in the WKB limit

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 behaviour as the wave amplitude is increased is described. 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. 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. -from Authors