Finite Amplitude Stability of a Rossby Wave in a Channel.

The stability characteristics of a barotropic inviscid Rossby wave in a mid latitude channel are examined in the absence and then in the presence of weak nonlinearities. Two cases are investigated: one in which the disturbance is a pure wave field and a second in which the disturbance contains a zonal flow. In the first case, the linear stability analysis resolves the general wavelike disturbance into two decoupled sets of modes. The stability characteristics of each set are defined in terms of a critical amplitude of the basic state (Rossby) wave, A(,c). Curves of marginal stability are obtained for each set. The relative importance of each set to the instability is then determined as a function of the zonal space scales of both the basic wave and the disturbance field. A finite amplitude version of the linear stability analysis is performed by introducing weak nonlinearities to balance an assumed initial weak instability. A long time scale determined by the slow linear growth and an asymptotic expansion of the disturbance field are used to determine the evolution of the system. When the nonlinearities are stabilizing the system exhibits a vacillatory exchange of energy between the basic state wave and the disturbance field. In the first case investigated, the mean field is not modified at any order. For a large region of wavenumber space, especially the shorter scales, the system is nonlinearly unstable. The nonlinear destabilization is attributed to the severity of the truncation of the disturbance field. In the second case investigated, the disturbance field contains a zonal (mean) flow among its spectral components. Both the linear and nonlinear analyses require the basic state wave to be confined to shorter zonal scales. In the allowable region, the nonlinearities are always stabilizing and provide a means of reinforcing zonal flows.