Potential Vorticity Inversion for Tropical Cyclones Using the Asymmetric Balance Theory

Abstract

A three-dimensional model is developed, based upon the recently derived asymmetric balance (AB) formulation of Shapiro and Montgomery, to study the evolution of rapidly rotating vortices, including hurricanes. A particular advantage of the AB theory, unlike other balanced models, is its ability to incorporate divergence of the same order as the vorticity. The main assumption of the AB theory is that the squared local Rossby number ?1, where the squared local Rossby number is defined by the ratio of the orbital frequency squared to the inertial stability. The AB theory leads to a set of prognostic equations that are manipulated so that the first- and second-order local time tendencies can be evaluated diagnostically at a given time. Using the diagnostic version of the AB equations the potential vorticity (PV) distribution from a primitive equation (PE) model is inverted to obtain the corresponding balanced height and wind fields. As far as the authors are aware, this is the first time that the AB equations have been solved in three dimensions. A calculation is described in which the PE model is initialized with an axisymmetric barotropic vortex in a vertical shear flow. Vertical shear leads to a wavenumber 1 asymmetry in the PV distribution. Associated with this asymmetry is a component of flow across the vortex center, which has an influence on the vortex motion. In this calculation the PE model provides not only the PV distribution but also the data to test the accuracy of the newly derived AB theory. The wavenumber 1 distributions of the radial, tangential, and vertical velocity fields diagnosed using the AB theory are compared with the results of the PE model. The agreement in amplitude and orientation is found to be good. The relative error between the amplitude maxima of the velocities in the PE calculations and the diagnostically derived AB fields is comparable with the maximum size of the squared local Rossby number. Although the main assumption of the AB theory is not strictly satisfied in these calculations, meaningful comparisons can be made between the PE results and the AB solutions. Presenting the results of the velocity fields in the moving coordinate system and use of the piecewise inversion makes it possible to isolate the influence of the upper-level PV anomaly on the lower-level part of the vortex and the influence of the lower-level PV anomaly on the upper-level part of the vortex. In a further calculation a vortex is initialized in a horizontal shear flow and diabatic heating and friction are included. The prescribed heating is related to the boundary layer convergence. The heating produces strong vertical gradients in the tangential wind so that the PV of the symmetric vortex becomes negative after 24 h. As in the nonlinear balance equations, the AB formulation requires the PV to be positive in order to be able to find a solution. A comparison between the velocity fields of the PE model and the diagnostically derived AB solutions after 12 h shows a good agreement in amplitude and orientation at lower levels but significant differences in amplitude at upper levels. At upper levels a vortex has not developed after 12 h and the standard Rossby number is the appropriate measure of the validity and accuracy as in the quasigeostrophic approximation. As in the case with no heating the agreement between the velocity components of the AB and PE model depends on the magnitude of the squared local Rossby number or standard Rossby number.