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Just before making landfall in Puerto Rico, Hurricane Maria (2017) underwent a concentric eyewall cycle in which the outer convective ring appeared robust while the inner ring first distorted into an ellipse and then disintegrated. The present work offers further support for the simple interpretation of this event in terms of the non-divergent barotropic model, which serves as the basis for a linear stability analysis and for non-linear numerical simulations. For the linear stability analysis the model’s axisymmetric basic state vorticity distribution is piece-wise uniform in five regions: the eye, the inner eyewall, the moat, the outer eyewall, and the far field. The stability of such structures is investigated by solving a simple eigenvalue/eigenvector problem and, in the case of instability, the non-linear evolution into a more stable structure is simulated using the non-linear barotropic model. Three types of instability and vorticity rearrangement are identified: (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. The first and third types of instability occur when the rings of enhanced vorticity are sufficiently narrow, with non-linear mixing resulting in broader and weaker vorticity rings. The second type of instability, most relevant to Hurricane Maria, occurs when the radial extent of the moat is sufficiently narrow that unstable interactions occur between the outer edge of the primary eyewall and the inner edge of the secondary eyewall. The non-linear dynamics of this type of instability distort the inner eyewall into an ellipse that splits and later recombines, resulting in a vorticity tripole. This type of instability may occur near the end of a concentric eyewall cycle. 

The problem of tropical cyclone rapid intensification is reduced to a potential vorticity (PV) equation and a second order, inhomogeneous, partial differential equation for the azimuthal wind. The latter equation has the form of a Klein-Gordon equation, the right-hand side of which involves the radial derivative of the evolving PV field. When the PV field evolves rapidly, inertia-gravity waves are excited at the edges of the evolving PV structure. In contrast, when the PV field evolves slowly, the second order time derivative term in the Klein-Gordon equation is negligible, inertia-gravity waves are not excited, and the equation reduces to an invertibility principle for the PV. The above concepts are presented in the context of an axisymmetric shallow water model, in both its linear and nonlinear forms. The nonlinear results show a remarkable sensitivity of vortex intensification to the percentage of mass that is diabatically removed from the region inside a given absolute angular momentum surface. 

Solutions of the secondary (transverse) circulation equation for an axisymmetric, gradient balanced vortex are used to better understand the distribution of subsidence in the eye of a tropical cyclone. This secondary circulation equation is derived using both the physical radius coordinate r and the potential radius coordinate R . In the R -coordinate version, baroclinic effects are implicit in the coordinate transformation and are recovered in the final step of transforming the solution for the streamfunction Ψ back from R -space to r -space. Two types of elliptic problems for Ψ are formulated: 1) the full secondary circulation problem, which is formulated on 0 ≤ R < ∞ , with the diabatic forcing due to eyewall convection appearing on the right-hand side of the elliptic equation; 2) the restricted secondary circulation problem, which is formulated on 0 ≤ R ≤ R ew , where the constant R ew is the potential radius of the inside edge of the eyewall, with no diabatic forcing but with the streamfunction specified along R = R ew . The restricted secondary circulation problem can be solved semi-analytically for the case of vertically sheared, Rankine vortex cores. The solutions identify the conditions under which large values of radial and vertical advection of θ are located in the lower troposphere at the outer edge of the eye, thereby producing a warm-ring thermal structure.