  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 (Bustamante and Hayat, 2013). 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 ${\mathcal{O}}(N^3)$, 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, in the form of sharp bounds on the size of the
interacting wavevectors.
