A vorticity wave packet breaking within a rapidly rotating vortex. Part I: The critical layer flow

Abstract

AbstractThis paper considers the interaction between a free vorticity wave packet and a rapidly rotating vortex in the slowly‐evolving regime, a long time after the initial, unsteady, and strong interaction. We study a singular, nonlinear, amplitude‐modulated, and helical mode inside a linearly stable, columnar, and axisymmetric vortex on the ‐plane. The interaction starts when the neutral mode enters resonance with the vortex on a spiraling critical surface, where the phase angular speed is equal to the rotation frequency. The singularity in the modal equation on this asymmetric surface strongly modifies the flow in its neighborhood, the three‐dimensionl helical critical layer, the region where the wave/vortex interaction occurs. This interaction generates a vertically sheared 3D mean flow of higher amplitude than the wave packet. The chosen envelope regime assumes the formation of a mean radial velocity of the same order as the wave packet amplitude, deviating the streamlines in a spiral way with respect to the rotational wind. Through matched asymptotic expansions, we find an analytical solution of the leading‐order motion equations inside the critical layer. The pattern of the latter, strongly deformed by the mean radial velocity, loses its symmetries with respect to the azimuthal and radial directions. It nevertheless conserves enough symmetries so that the leading‐order divergences of the mean radial wave fluxes are zero. The Batchelor integral condition generalized to the three‐dimensional quasi‐steady recirculating flow within the critical layer separatrices yields a leading‐order uniform axial vorticity at any vortex height.