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 (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 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.

Image Encryption Using Elliptic Curves and Rossby/Drift Wave Triads

“…A Rossby wave solution is given explicitly by the parameterized function , where is an arbitrary constant, is the so-called dispersion relation, and is called the wave vector. For simplicity, we take and in what follows [ 32 , 33 ].…”

“…New parameterization: In [ 40 ], Kopp parameterized the resonant triads and in terms of parameters u and t it follows by [ 40 ] (Equation (1.22)) that: In 2019, Hayat et al [ 33 ] found a new parameterisation of and D in terms of auxiliary parameters and hence and are given by: …”

“…The proposed sequence is cryptographically a good source of pseudo-randomness because triads are highly sensitive to the auxiliary parameters [ 33 ] and inverse detuning level . It is shown in [ 32 ] that the intricate structure of clusters formed by triads depends on the chosen , and the size of the clusters increases as the inverse detuning level increases.…”

“…Based on the above discussion, we propose an improved image encryption algorithm, based on quasi-resonant Rossby/drift wave triads [ 32 , 33 ] (triads, for short) and Mordell elliptic curves (MECs). The triads are utilized in the generation of pseudo-random numbers and MECs are employed to create dynamic S-boxes.…”