  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 characterise 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 nontrivial
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.
