Discrete resonant Rossby/drift wave triads: Explicit parameterisations and a fast direct numerical search algorithm

Abstract

We report results on the explicit parameterisation of discrete Rossby-wave resonant triads of the Charney-Hasegawa-Mima equation in the small-scale limit (i.e. large Rossby deformation radius), following up from our previous solution in terms of elliptic curves [6]. We find an explicit parameterisation of the discrete resonant wavevectors in terms of two rational variables. We show that these new variables are restricted to a bounded region and find this region explicitly. We argue that this can be used to reduce the complexity of a direct numerical search for discrete triad resonances. Also, we introduce a new direct numerical method to search for discrete resonances. This numerical method has complexity O(N3) (up to logarithmic factors), where N is the largest wavenumber in the search. We apply this new method to find all discrete irreducible resonant triads in the wavevector box of size 5000, in a calculation that took about 10.5 days on a 16-core machine. Finally, based on our method of mapping to elliptic curves, we discuss some dynamical implications regarding the spread of quadratic invariants across scales via resonant triad interactions, through the analysis of sharp bounds on the relative sizes of the resonantly interacting wavevectors.