On resonant Rossby-Haurwitz triads

The dynamics of non-divergent flow on a rotating sphere are described by the conservation of absolute vorticity. The
analytical study of the non-linear barotropic vorticity equation is greatly facilitated by the expansion of the solution in
spherical harmonics and truncation at low order. The normal modes are the well-known Rossby–Haurwitz (RH) waves,
which represent the natural oscillations of the system. Triads of RH waves, which satisfy conditions for resonance, are
of critical importance for the distribution of energy in the atmosphere.
We show how non-linear interactions of resonant RH triads may result in dynamic instability of large-scale components.
We also demonstrate a mathematical equivalence between the equations for an orographically forced triad
and a simple mechanical system, the forced-damped swinging spring. This equivalence yields insight concerning the
bounded response to a constant forcing in the absence of damping. An examination of triad interactions in atmospheric
reanalysis data would be of great interest.