Analytical solutions of vortex Rossby waves in a discrete barotropic model

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 solutionof 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 distributionof 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 solutionof 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 cangrow 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,meteorological society of japan.