Analyses of observational data on hurricanes in the tropical atmosphere indicate the existence of spiral rainbands which propagate outward from the eye and affect the structure and intensity of the hurricane. These disturbances may be described as vortex Rossby waves. This paper describes the evolution of barotropic vortex Rossby waves in a cyclonic vortex in a two-dimensional configuration where the variation of the Coriolis force with latitude is ignored. The waves are forced by a constant-amplitude boundary condition at a fixed radius from the center of the vortex and propagate outward. The mean flow angular velocity profile is taken to be a quadratic function of the radial distance from the center of the vortex and there is a critical radius at which it is equal to the phase speed of the waves. For the case of waves with steady amplitude, an exact solution is derived for the steady linearized equations in terms of hypergeometric functions; this solution is valid in the outer region away from the critical radius. For the case of waves with time-dependent amplitude, asymptotic solutions of the linearized equations, valid for late time, are obtained in the outer and inner regions. It is found that there are strong qualitative similarities between the conclusions on the evolution of the vortex waves in this configuration and those obtained in the case of Rossby waves in a rectangular configuration where the latitudinal gradient of the Coriolis parameter is taken into account. In particular, the amplitude of the steady-state outer solution is greatly attenuated and there is a phase change of -π across the critical radius, and in the linear time-dependent configuration, the outer solution approaches a steady state in the limit of infinite time, while the amplitude of the inner solution grows on a logarithmic time scale and the width of the critical layer approaches zero.
Studies in Applied Mathematics
Carleton University

Nikitina, L.V. (L. V.), & Campbell, L. (2015). Dynamics of Vortex Rossby Waves in Tropical Cyclones, Part 1: Linear Time-Dependent Evolution on an f-Plane. Studies in Applied Mathematics, 135(4), 377–421. doi:10.1111/sapm.12094