Barotropic Instability of Axisymmetric Double-Ring Vortices

A tropical cyclone containing concentric eyewalls can be approximated as an axisymmetric double-ring vortex using a five-region vorticity model involving the following five piecewise-constant vorticity regions: an inner-core region (the eye), an inner-ring region (the inner eyewall), an inter-ring region (the moat), an outer-ring region (the outer eyewall), and a far-field region of zero vorticity. This work utilizes the nondivergent barotropic model to examine the barotropic instability of a number of different families of such axisymmetric double-ring vortices. A linear instability analysis involving a simple eigenvalue/eigenvector problem is used to determine the most unstable mode and growth rate for each vortex within each of these families, and an objective method based on the energetics of the linear dynamics is developed and used to identify the type (location) of each instability observed. For such five-region vortices, there are three primary types of instability that can occur: (1) instability across the outer ring of enhanced vorticity, (2) instability across the low vorticity moat, and (3) instability across the inner ring of enhanced vorticity. For several select unstable vortices, the nonlinear evolution into a more stable structure is simulated using a nondivergent barotropic model and smoothed versions of the associated five-region vortices. This work especially focuses on investigating which double-ring vortices exhibit mode 2 instabilities across the moat, a phenomenon that we believe (1) has been observed in a number of tropical cyclones during eyewall replacement cycles, including Hurricane Maria (2017), and (2) plays a role in the rearrangement and mixing of the inner-ring vorticity that must occur during such cycles as the inner eyewall vanishes.