Accurate determination of a balanced axisymmetric vortex in a compressible atmosphere

We give a simple method to calculate without approximation the balanced density field of an axisymmetric vortex in a compressible atmosphere in various coordinate systems given the tangential wind speed as a function of radius and height and the vertical density profile at large radius. The method is generally applicable, but the example considered is relevant to tropical cyclones. The exact solution is used to investigate the accuracy of making the anelastic approximation in a tropical cyclone, i.e. the neglect of the radial variation of density when calculating the gradient wind. We show that the core of a baroclinic vortex with tangential wind speed decreasing with height is positively buoyant in terms of density differences compared at constant height, but at some levels may be interpreted as cold-cored or warm-cored depending on the surfaces along which the temperature deviation is measured. However, it is everywhere warn-cored if the potential temperature deviation is considered. In contrast, a barotropic vortex in a stably-stratified atmosphere is cold-cored at all levels when viewed in terms of the temperature deviation at constant height or constant Q, but warm-cored when viewed in terms of the potential temperature deviation along these surfaces. The calculations provide a possible explanation for the observed reduction in surface air temperature in the inner core of tropical cyclones.