Linear stability analysis of steady state tropical cyclones with single or double eyewalls

The structure of an eyewall within an intense tropical cyclone is known to strongly influence its intensity. Furthermore, intense tropical cyclones often undergo an eyewall replacement cycle, a process in which an outer eyewall forms, contracts, and replaces an inner eyewall. During this process, there will exist two concentric eyewalls. To examine the effects of such structures, analytical solutions of the transverse circulation equation associated with a balanced vortex model have been proposed in the literature. For a single eyewall, the analytical solution is defined over three regions, the eye, eyewall, and far-field, while in the case of concentric or double eyewalls, one subdivides the domain into five regions, the eye, inner eyewall, moat, outer eyewall, and far-field. The goal of the present paper is to present a stability analysis for steady state vortex solutions corresponding to tropical cyclones with either single or double eyewalls. We first obtain general steady state solutions and a general approach for the stability or instability of such states involving a perturbation analysis (which makes use of small perturbations to the vortex structure in each of the three or five regions). We then make several simplifying assumptions on the form of the stationary solutions so as to make the stability analysis tractable while maintaining physical relevance. The obtained stability criteria depend on the model parameters as well as on the structure of the perturbations the steady state solutions undergo.Making use of parameter values calibrated from the literature, we then provide bifurcation diagrams giving regions of stability or instability. Such analytical stability results for steady state vortex solutions of the transverse circulation equation complement existing numerical stability results in the literature.