Barotropic instability in the inner core of tropical cyclones

Abstract

The theory of barotropic stability of a vortex is presented including Rayleigh's condition, Fjertoft's condition, Ripa's Theorem and Arnold's Theorem. The probable profile of potential vorticity (PV) in a tropical cyclone is discussed. It is likely that this profile has at least one reversal of the radial gradient of PV in the inner region of the storm. This reversal of PV gradient is a necessary condition for barotropic instability. Linear normal mode analysis of many tangential wind profiles from the data set of Gray and Shea (1976) indicate that barotropic instability may be a common feature of mature tropical cyclones. The modified Rankine profile, the Holland (1980) profile and two profiles developed in this paper are also analyzed. Results indicate that a single reversal of vorticity gradient over the entire radial extent of the storm may produce low wavenumber instability while more localized reversals tend toward higher wavenumber instability. These instabilities have e-folding times on the order of a few hours and are generally located in the vicinity of the PV gradient reversal which is typically just within the radius of maximum winds. These results lead us to conclude that barotropic instability may be the primary cause of the polygonal eye walls which are observed in many tropical cyclones.