Tearing Instability of Isolated Compressible Vortices

Abstract

The present paper addresses the instability of a compressible vortex flow. We consider a family of so-called isolated vortices – the circular planar vortices that have zero circulation (net vorticity). The term “isolation” implies the presence of a shield in the vorticity field – a region of vorticity of common sign surrounding the central domain where the vorticity is oppositely signed. To model such a vorticity field, a generalized Taylor-type profile for the swirl velocity in the radial direction is employed, which involves two parameters: intensity μ (proportional to the maximal velocity) and steepness β (characterizing the scale of the shield zone). Vortices are assumed to be compressible and homentropic. The linear-stability analysis is carried out, which shows that isolated vortices can exhibit both stable and unstable behavior depending on the model parameters μ and β. By numerical simulations of the non-linear stage, the unstable normal modes are shown to evolve towards tearing of the basic vortex with formation of smaller secondary vortical structures.