In this presentation I will discuss analytical and numerical solutions of an initial-value problem that describes vortex Rossby waves in an atmospheric fluid flow. Vortex Rossby waves occur in the atmosphere in the form of outward-propagating disturbances within cyclonic vortices such as hurricanes. They result from the force due to the radial gradient of the vorticity of the cyclone. The rotation of the Earth contributes to the beta-gyre effect, or beta drift, where there is a secondary dipole circulation and a resulting displacement of the vortex. An understanding of the beta-gyre effect is important for accurate forecasting of the tracks of tropical cyclones.