Resonant Rossby Wave Triads and the Swinging Spring
Peter Lynch, Met Éireann, Glasnevin Hill, Dublin 9, Ireland
Submitted to Bull. Amer. Met. Soc.
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.