Nonlinear aspects of the vorticity equation for nondivergent, planetary circulations of the atmosphere may be explored by representation of the solution in spherical surface harmonics. A systematic analytic investigation is presented here of the truncated spectral vorticity equation thus obtained. The principal results pertain to systems with three spherical-harmonic components (which may have as many as six degrees of freedom): the spectral equations for all three-component systems are solvable by quadratures; and in those cases in which the components exchange energy, the quadratures lead either to elliptic or to circular functions of the time. The solutions, and in particular the energy exchange therefore are periodic in all three-component systems.
Extensions to nontrivial multicomponent systems are made in a few cases for which complete solutions are possible. The simplest configuration is one with an arbitrary zonal flow and a single tesseral component. No energy exchange occurs; but the phase of the tesseral component is propagated with a speed determined by the distributions of angular velocity and vorticity gradient in the zonal flow. With an arbitrary zonal flow and two tesseral components of the same rank (equal wave number), there is a periodic exchange of energy governed by solutions expressible in terms of elliptic functions. The corresponding linearized equations may be regarded as providing a means of approximating conventional perturbation analysis of stability of an arbitrary (inviscid) zonal flow, and are found to yield an explicit necessary and sufficient condition for stability.