Constraints on Parametric Hurricane Wind Profiles

Fundamental fluid dynamics impose physical consistency restraints on parametric tropical cyclone (TC) wind profiles used for idealized theoretical models or full-physics model initialization. A typical TC wind profile exhibits a wind maximum 10-50 km from the storm center surrounded by an initially steep, then gradual, decay with distance away from the center. All profiles share a broad cyclonic vorticity maximum around the center, but differ at large radius. Here, we focus on three types of vortex: asymptotically bounded, finitely bounded, and unbounded.

The circulations of asymptotically-bounded vortices approach zero at large radius whereas the circulation of finitely-bounded vortices become identically zero beyond some finite radius. They both contain cyclonic vorticity near the center surrounded by an equal amount of anticyclonic vorticity. This condition implies the existence of an outer waveguide in which the reversed vorticity gradient can support downstream-propagating vortex Rossby waves (VRWs).

Unbounded (e.g., Rankine) vortices, have no vorticity anywhere but in the core. This characteristic violates Stokes Theorem on a sphere (or indeed, on any closed manifold) since there must be equal and opposite vorticity inside and outside of any closed circulation contour. Moreover, unbounded vortices lack outer waveguides and have infinite kinetic energy and angular momentum. The combination of these physical inconsistencies make for an unrealistic representation of TCs; at least in the context of large scale dynamics and vortex motion. If one were focused solely on inner core dynamics, then Rankine-like vortices might be reasonable. To better understand the argument for use of bounded vortices, imagine a transparent sphere with no vorticity except for a lone cyclonic patch at some arbitrary location. From the perspective of an observer at the antipode, the circulation would appear to be anticyclonic but with no enclosed vorticity.

In a three-dimensional atmosphere with no rigid lid, vortex tubes must either terminate on the surface or reconnect. Free-slip and no-slip surface boundary conditions yield similar scenarios. In the former, vortex tubes rise upward from the surface in the vortex core, spread at tropopause level, then return to the surface where they end. In the latter, the boundary layer wind is exactly zero at the surface. The resulting strong shear between the surface and top of boundary layer contains horizontal vortex tubes that follow the same pattern as free-slip except that they reconnect in the boundary layer as opposed to terminating at the surface. The region where the vortex tubes rise contains cyclonic vertical vorticity near the center; anticyclonic vertical vorticity exists where the vortex tubes reconnect with the horizontal boundary layer tubes at the vortex periphery. This configuration is consistent with the bounded vortices described earlier.

These vortices have implications for idealized modeling TC motion. Asymptotically-bounded and Rankine vortices in a Barotropic Nondivergent vortex-following model on a beta plane yield northwestward storm motion two to three times faster than the observed beta drift speed. Finitely-bounded vortices with narrow profiles exhibit somewhat slower motion since the vortex must be smaller because beyond some finite radius from the center, because the area-integrated vorticity becomes exactly zero. Nevertheless it is important to note that diffusion, vorticity filamentation at a VRW critical radius, and nonlinearity play significant roles in controlling translation speed. Nonlinearity dominates since wave-wave interactions force wavenumber-one gyres of opposite polarity to the linearly-forced beta gyres, such that the flow between them is opposite to the beta-induced relative flow. These anti-beta gyres counteract the normal beta gyres' northwest flow to limit the overall storm motion.