On the Equilibration in Nonlinear Barotropic Instability

Abstract

This paper investigate the equilibration of finite amplitude waves in a forced, dissipative, high resolution, barotropic beta-plane model. The external forcing is prescribed in the form of an easterly Bickley jet and the dissipation is due to Rayleigh friction. Under moderate supercriticality, the system evolves towards a steady wave state consisting of one dominant wave and its higher zonal harmonics after a complex transient stage. The wavelength of the dominant wave is longer for a wider and stronger jet, a weaker viscosity, and/or a smaller beta effect. The dominant wave has a symmetric meridianal structure with respect to the jet and its next higher harmonic an antisymmetric structure. The latter is energetically sustained by the primary wave. All wave in a steady state are phase-locked as indicated by a common zonal propagation speed. The zonal flow is modified by the nonlinear feedback until the equilibrated waves become dynamically neutral. This system is also demonstrated to have the property of hysteresis. These results complement nicely with the laboratory results of Niino and Misawa (1984). When the forcing is sufficiently large, a state of vacillation of all the waves would emerge with one subset of the waves fluctuates out of phase with respect to the complementary subset of waves. The wave-wave interaction plays a central role in such a state. A detailed energetics analysis is also presented. When the forcing is further increased, the equilibrated state is chaos. The most favorable conditions for the vacillation states to prevail are intermediate forcing and damping as in a baroclinic system (Mak, 1985). The intensity of the jet required for vacillation is considerably stronger than that of typica; jets in the earth's atmosphere, implying that barotropic dynamics alone are likely to give rise to steady waves rather than vacillation in the atmosphere.