A singular vorticity wave packet within a rapidly rotating vortex: spiralling versus oscillating motions

Abstract

This paper considers a free vorticity wave packet propagating within a rapidly rotating vortex in the quasi-steady regime, a long time after the wave packet strongly and unsteadily interacted with the vortex. We study a singular, nonlinear, helical and asymmetric shear mode inside a linearly stable, columnar and axisymmetric vortex on the $f$-plane. The amplitude-modulated mode enters resonance with the vortex at a certain radius $r_{c}$, where the phase angular speed is equal to the rotation frequency. The singularity in the modal equation at $r_{c}$ strongly modifies the flow in the three-dimensional helical critical layer, the region around $r_{c}$ where the wave/vortex interaction occurs. This interaction generates a vertically sheared three-dimensional 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, leading to the streamlines exhibiting a spiral motion in the neighbourhood of the critical layer. Radar images frequently show such spiral bands in tropical cyclones or tornadoes. Through matched asymptotic expansions, we find an analytical solution of the leading-order equations inside the critical layer. The generalized Batchelor integral condition applied to the quasi-steady, three-dimensional motion inside the separatrices yields a leading-order, non-uniform three-dimensional vorticity. The critical-layer pattern, strongly deformed by the mean radial velocity, loses its symmetries with respect to the azimuthal and radial directions, which makes the leading-order mean radial wave fluxes non-zero. Finally, a stronger wave/vortex interaction occurs with respect to previous studies where a steady neutral vortical mode or an envelope of larger extent was involved.