Second order nonlinear interactions among Rossby -waves



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 O2) 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 O2).


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