Steady multipolar planar vortices with nonlinear critical layers

摘要

This article considers a family of steady multipolar planar vortices, which are the superposition of an axisymmetric mean flow, and an azimuthal disturbance in the context of inviscid, incompressible flow. This configuration leads to strongly nonlinear critical layers where the angular velocities of the mean flow and the disturbance are comparable. The poles located on the same critical radius possess the same uniform vorticity, whose weak amplitude is of the same order as the azimuthal disturbance. This problem is examined through a perturbation expansion in which relevant nonlinear terms are retained in the critical layer equations, while viscosity is neglected. In particular, the associated singularity at the meeting point of the separatrices is treated by employing appropriate re-scaled variables. Matched asymptotic expansions are then used to obtain a complete analytical description of these vortices.