Second order nonlinear interactions among Rossby -waves

Abstract

In the weakly nonlinear interaction theory of mid-latitude Rossby waves (RWs), a perturbative solution for the quasi-geostrophic (QG) stream function is sought in the form = ⁽⁰⁾ + ε⁽¹⁾ + ε²⁽²⁾ + ..., where ε is the β-Rossby number. If the leading order solution ⁽⁰⁾ is any superposition of RWs, the second order perturbation equation for ⁽²⁾ always has resonant forcing. This is in contrast to first order nonlinear interactions where resonant triads form a very restricted set of all possible interactions. An example is worked out in a laterally unbounded ocean and taking ⁽⁰⁾ as the superposition of two arbitrary RWs. At second order, multiple time scales lead to an O(ε²) Doppler shift of the frequency of each wave, proportional to the amplitude squared of the other one. Using realistic wave parameters for wave-1 and the amplitude of wave-2, the frequency shift is not negligible in regions of wave number space of wave-2 near resonance at first order. It is thus conceivable to haw a field of weakly interacting RWs, such that at O(ε) there are no resonant interactions; however the Doppler shift in their frequencies, albeit small, will always take place due to resonance at O(ε²).