Supplement to Resonant Rossby Wave Triads and the Swinging Spring

Abstract

The wave solutions discovered by Rossby are of fundamental importance for atmospheric dynamics. The nonlinear interactions between these waves determine the primary characteristics of the energy spectrum. These interactions take place between triplets of waves known as ?resonant triads? and, for small amplitude, they are described by the three-wave equations. These same equations also govern the dynamics of a simple mechanical system, the elastic pendulum or swinging spring. This equivalence allows us to deduce properties, not otherwise evident, of resonant triads from the behavior of the mechanical system. In particular, the characteristic stepwise precession of the swing plane, so obvious from observation of the physical spring pendulum, is also found for the Rossby triads. This phenomenon has not been previously noted and is an example of the insight coming from the mathematical equivalence of the two systems. The implications of the precession for predictability of atmospheric motions are considered. The pattern of breakdown of unstable Rossby waves is very sensitive to unobservable details of the perturbations, making accurate prediction very difficult.