Continuous Mode in the Theory of the Barotropic Instability of Zonal Flows

Evolution of the two-dimensional small disturbance superimposed on a viscous parallel shear flow is investigated under the beta-plane approximation of revolution. Formal solution is obtained for an arbitrarily localized initial distribution of vorticity. The eigenfunctions of the normal mode are proved to compose a complete set. In the region where the basic flow is uniform the eigenfunction of the continuous mode behaves as the Rossby wave. The interaction between the shear and the Rossby wave is found to be an inherent character of the eigenfunction of the continuous mode. The properties of the critical latitude of the Rossby wave are attributable to those of the friction layer in the eigenfunction.