A singular vortex Rossby wave packet within a rapidly rotating vortex

This paper describes the quasi-steady régime attained by a rapidly rotating vortex after a wave packet has interacted with it. We consider singular, nonlinear, helical, and shear asymmetric modes within a linearly stable, columnar, axisymmetric, and dry vortex in the f-plane. The normal modes enter resonance with the vortex at a certain radius rc, where the phase angular speed is equal to the rotation frequency. The related singularity in the modal equation at rc strongly modifies the flow in the 3D helical critical layer, the region where the wave/vortex interaction occurs. This interaction induces a secondary mean flow of higher amplitude than the wave packet and that diffuses at either side of the critical layer inside two spiral diffusion boundary layers. We derive the leading-order equations of the system of nonlinear coupled partial differential equations that govern the slowly evolving amplitudes of the wave packet and induced mean flow a long time after this interaction started. We show that the critical layer imposes its proper scalings and evolution equations; in particular, two slow times are involved, the faster being secular. This system leads to a more complex dynamics with respect to the previous studies on wave packets where this coupling was omitted and where, for instance, a nonlinear Schrödinger equation was derived [D. J. Benney and S. A. Maslowe, “The evolution in space and time of nonlinear waves in parallel shear flows,” Stud. Appl. Math. 54, 181 (1975)]. Matched asymptotic expansion method lets appear that the neutral modes are distorted. The main outcome is that a stronger wave/vortex interaction takes place when a wave packet is considered with respect to the case of a single mode. Numerical simulations of the leading-order inviscid Burgers-like equations of the derived system show that the wave packet rapidly breaks and that the vortex, after intensifying in the transition stage, is substantially weakened before the breaking onset. This breaking could give a dynamical explanation of the formation of an inner spiral band through the prism of the critical layer theory.