Finite-amplitude stability of a Rossby wave in a channel

The linear stability characteristics and life cycle, in the presence of weak nonlinearities, of an inviscid Rossby wave are examined in a mid-latitude channel. Two cases are investigated: one in which the superimposed disturbance is purely wavelike and a second in which it includes a zonal flow component.

The linear analysis reveals that in the first case, the disturbance is resolved into two decoupled sets of modes. The most unstable set as a function of wavenumber is then determined by obtaining the curves of marginal stability for each set. The second case is characterized by a single unstable set.

The finite amplitude asymptotic analysis of the first case reveals that, when the nonlinearities are stabilizing, the system exhibits an amplitude vacillation on a slow time scale. It also reveals that the most severe meridional truncation is valid for the longer zonal scales and does not affect the mean field at any order. If an additional meridional mode is included, the resulting truncation becomes valid for the shorter zonal scales and alters the mean field at second order. In the second case, the nonlinearities are always stabilizing.