A steady nonlinear and singular vortex Rossby wave within a rapidly rotating vortex. Part I : theory

The goal of this paper is to predict the evolution of a rapidly rotating vortex when neutral waves are propagating therein. We consider a singular and nonlinear vortex Rossby helical wave within a linearly stable, barotropic, axisymmetric, and dry vortex in the -plane. The wave enters resonance with the vortex at a certain radius, where the wave phase angular speed is equal to the rotation frequency. The singularity in the modal equation at this radius strongly modifies the flow in the critical layer (CL), the region where the wave/vortical flow interaction occurs. We resolve the singularity by reintroducing nonlinearity in the inner-flow equations in the steady-régime assumption. In this régime, the neutral mode is weakly singular. We indeed assume that the CL forms at the outer edge of the vortex where the basic axial vorticity has become small compared to the inertial frequency, owing to vortex erosion. Through matched asymptotic expansions, we determine the critical-layer induced distorted flow characterized by a secondary mean flow, whose amplitude is larger than the wave amplitude, which diffuses in an asymmetric way at either side of the CL that winds up around the vortex axis. The analysis shows that, after intensifying in the transition stage, the vortex is weakening in the steady stage, but in a negligible way with the slow viscous homogenization time scale. The CL extent is nevertheless evolving with this same time all the more so since the ratio vertical wavenumber over azimuthal one is large.