POTENTIAL VORTICITY INVERSION FOR TROPICAL CYCLONES USING THE ASYMMETRIC BALANCE THEORY

A three-dimensional model is developed, based upon the recently derive d asymmetric balance (AB) formulation of Shapiro and Montgomery, to st udy 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 vor ticity. The main assumption of the AB theory is that the squared local Rossby number much less than 1, where the squared local Rossby number is defined by the ratio of the orbital frequency squared to the inert ial stability. The AB theory leads to a set of prognostic equations th at are manipulated so that the first-and second-order local time tende ncies can be evaluated diagnostically at a given time. Using the diagn ostic version of the AB equations the potential vorticity (PV) distrib ution from a primitive equation (PE) model is inverted to obtain the c orresponding balanced height and wind fields. As far as the authors ar e 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 s hear flow. Vertical shear leads to a wavenumber 1 asymmetry in the PV distribution. Associated with this asymmetry is a component of flow ac ross the vortex center, which has an influence on the vortex motion. I n this calculation the PE model provides not only the PV distribution but also the data to rest the accuracy of the newly derived AB theory. The wavenumber 1 distributions of the radial, tangential, and vertica l 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 A B fields is comparable with the maximum size of the squared local Ross by number. Although the main assumption of the AB theory is not strict ly satisfied in these calculations, meaningful comparisons can be made between the PE results and the AB solutions. Presenting the results o f the velocity fields in the moving coordinate system and use of the p iecewise inversion makes it possible to isolate the influence of the u pper-level PV anomaly on the lower-level part of the vortex and the in fluence of the lower-level PV anomaly on the upper-level part of the v ortex. In a further calculation a vortex is initialized in a horizonta l shear flow and diabatic heating and friction are included. The presc ribed heating is related to the boundary layer convergence. The heatin g produces strong vertical gradients in the tangential wind so that th e PV of the symmetric vortex becomes negative after 24 h. As in the no nlinear balance equations, the AB formulation requires the PV to be po sitive in order to be able to find a solution. A comparison between th e velocity fields of the PE model and the diagnostically derived AB so lutions after 12 h shows a good agreement in amplitude and orientation at lower levels but significant differences in amplitude at upper lev els. At upper levels a vortex has not developed after 12 h and the sta ndard Rossby number is the appropriate measure of the validity and acc uracy as in the quasigeostrophic approximation. As in the case with no heating the agreement between the velocity components of the AB and P E model depends on the magnitude of the squared local Rossby number or standard Rossby number.