Dynamics of Atmospheres and Oceans 35 (2002) 153–177 Vortex Rossby waves on smooth circular vortices Part I. Theory Gilbert Brunet a,∗ , Michael T. Montgomery b a RPN/SMC, Canada, 2121 Route Trans-Canadienne, Dorval, Que., Canada H9P 1J3 b Colorado State University, Fort Collins, CO, 80523, USA Received 18 January 2001; accepted 29 October 2001 Abstract A complete theory of the linear initial-value problem for Rossby waves on a class of smooth circular vortices in both f-plane and polar-region geometries is presented in the limit of small and large Rossby deformation radius. Although restricted to the interior region of barotropically stable circular vortices possessing a single extrema in tangential wind, the theory covers all azimuthal wavenumbers. The non-dimensional evolution equation for perturbation potential vorticity is shown to depend on only one parameter, G, involving the azimuthal wavenumber, the basic state radial potential vorticity gradient, the interior deformation radius, and the interior Rossby number. In Hankel transform space the problem admits a Schrödinger’s equation formulation which per- mits a qualitative and quantitative discussion of the interaction between vortex Rossby wave distur- bances and the mean vortex. New conservation laws are developed which give exact time-evolving bounds for disturbance kinetic energy. Using results from the theory of Lie groups a nontrivial sepa- ration of variables can be achieved to obtain an exact solution for asymmetric balanced disturbances covering a wide range of geophysical vortex applications including tropical cyclone, polar vortex, and cyclone/anticyclone interiors in barotropic dynamics. The expansion for square summable po- tential vorticity comprises a discrete basis of radially propagating sheared vortex Rossby wave packets with nontrivial transient behavior. The solution representation is new, and for this class of swirling flows gives deeper physical insight into the dynamics of perturbed vortex interiors than the more traditional approach of Laplace transform or continuous-spectrum normal-mode repre- sentations. In general, initial disturbances are shown to excite two regions of wave activity. At the extrema of these barotropically stable vortices and for a certain range of wavenumbers, the Rossby wave dynamics are shown to become nonlinear for all initial conditions. Extensions of the theory are proposed. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Continuous-spectrum; Cylindrical coordinates; Initial-value problem; Vortex Rossby waves ∗ Corresponding author. Tel.: +1-514-421-4617; fax: +1-514-421-2106. E-mail address: gilbert.brunet@ec.gc.ca (G. Brunet). 0377-0265/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S0377-0265(01)00087-2