Abstract
We study the kinematics of nonlinear resonance broadening of interacting Rossby waves as modelled by the Charney–Hasegawa–Mima equation on a biperiodic domain. We focus on the set of wave modes which can interact quasi-resonantly at a particular level of resonance broadening and aim to characterize how the structure of this set changes as the level of resonance broadening is varied. The commonly held view that resonance broadening can be thought of as a thickening of the resonant manifold is misleading. We show that in fact the set of modes corresponding to a single quasi-resonant triad has a non-trivial structure and that its area in fact diverges for a finite degree of broadening. We also study the connectivity of the network of modes which is generated when quasi-resonant triads share common modes. This network has been argued to form the backbone for energy transfer in Rossby wave turbulence. We show that this network undergoes a percolation transition when the level of resonance broadening exceeds a critical value. Below this critical value, the largest connected component of the quasi-resonant network contains a negligible fraction of the total number of modes in the system whereas above this critical value a finite fraction of the total number of modes in the system are contained in the largest connected component. We argue that this percolation transition should correspond to the transition to turbulence in the system.
Highlights
Exchange is generally quite slow and occurs most efficiently between groups of modes which are in resonance
We show that a percolation transition occurs at a critical value, δ∗, of the resonance broadening as δ is increased
We have presented a kinematic analysis of the properties of quasi-resonant triads in the CHM equation
Summary
It is less surprising that the boundary of the quasi-resonant set can diverge for finite δ once we appreciate that the exact resonant manifold of a zonal mode is unbounded. One might counter this with the observation that it is only in the weakly nonlinear regime that it makes sense to be discussing quasi-resonant interactions in the first place and in this regime, the broadening is necessarily small. Equation (15) tells us, that no matter how small the broadening, there are always modes close to the zonal axis, whose quasi-resonant set diverges
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