Dynamics of vorticity defects in shear

Abstract

Matched asymptotic expansions are used to obtain a reduced description of the nonlinear and viscous evolution of small, localized vorticity defects embedded in a Couette flow. This vorticity defect approximation is similar to the Vlasov equation, and to other reduced descriptions used to study forced Rossby wave critical layers and their secondary instabilities. The linear stability theory of the vorticity defect approximation is developed in a concise and complete form. The dispersion relations for the normal modes of both inviscid and viscous defects are obtained explicitly. The Nyquist method is used to obtain necessary and sufficient conditions for instability, and to understand qualitatively how changes in the basic state alter the stability properties. The linear initial value problem is solved explicitly with Laplace transforms; the resulting solutions exhibit the transient growth and eventual decay of the streamfunction associated with the continuous spectrum. The expansion scheme can be generalized to handle vorticity defects in non-Couette, but monotonic, velocity profiles.