The stability of a Rossby wave

Abstract

Abstract The stability of a finite-amplitude Rossby wave on a β-plane with respect to a small amplitude perturbation is examined. Normal modes are defined, without further approximation, by a third-order Floquet system. The parametric instability exhibited by the system is examined analytically and numerically. When the amplitude of the Rossby wave tends to zero this instability is shown to be analogous to the resonant interaction of the Rossby wave with components of the perturbation. The stability of a Rossby wave is seen to depend upon two parameters, M and ξ. The parameter M is proportional to the amplitude of the Rossby wave and ξ is its orientation on the β-plane. Curves of marginal stability in the (M, ξ) space are found numerically.