The initial value problem of vortex Rossby waves (VRWs) is analytically solved in a linearized barotropic system on an f plane. The basic axisymmetric vorticity q is assumed to be piecewise uniform in the radial direction so that the radial gradient dq/dr and the disturbance vorticity q are expressed in terms of Dirac delta functions. After Fourier transformation in the azimuthal direction with the wavenumber m, the linearized vorticity equation becomes a system of ordinary differential equations with respect to time; these can be analytically solved to give a closed-form solution with a prescribed initial value.
For a monopolar q, the solution of q starting from the innermost radius exhibits the outward propagation of VRWs. As the outer disturbances are generated, the inner disturbance is diminished. On the other hand, in the case of a solution forced at the innermost radius, the inner disturbance is not diminished, and the outward propagation of VRWs forms a distribution of spiral-shaped disturbance vorticity.
For a basic vorticity q with a moat, and if the radial distribution of q satisfies a certain additional condition, the solution of q with |m| ≠ 1 grows exponentially or linearly in time as a result of the interaction of counterpropagating VRWs near the moat. Although the solution of q with |m| = 1 cannot grow exponentially for any q, it can grow as a linear function of time. This linear growth may be regarded as a result of resonance between two internal modes of the system.
2013 by Meteorological Society of Japan