Complete classification of discrete resonant Rossby/drift wave triads on periodic domains

Abstract

We consider the set of Diophantine equations that arise in the context of the partial differential equation called "barotropic vorticity equation" on periodic domains, when nonlinear wave interactions are studied to leading order in the amplitudes. The solutions to this set of Diophantine equations are of interest in atmosphere (Rossby waves) and Tokamak plasmas (drift waves), because they provide the values of the spectral wavevectors that interact resonantly via three-wave interactions. These wavenumbers come in "triads", i.e., groups of three wavevectors.We provide the full solution to the Diophantine equations in the physically sensible limit when the Rossby deformation radius is infinite. The method is completely new, and relies on mapping the unknown variables via rational transformations, first to rational points on elliptic curves and surfaces, and from there to rational points on quadratic forms of "Minkowski" type (such as the familiar space-time in special relativity). Classical methods invented centuries ago by Fermat, Euler, Lagrange, Minkowski, are used to classify all solutions to our original Diophantine equations, thus providing a computational method to generate numerically all the resonant triads in the system. Computationally speaking, our method has a clear advantage over brute-force numerical search: on a 10000 2 grid, the brute-force search would take 15 years using optimised C ++ codes on a cluster, whereas our method takes about 40 minutes using a laptop.Moreover, the method is extended to generate so-called quasi-resonant triads, which are defined by relaxing the resonant condition on the frequencies, allowing for a small mismatch. Quasi-resonant triads' distribution in wavevector space is robust with respect to physical perturbations, unlike resonant triads' distribution. Therefore, the extended method is really valuable in practical terms. We show that the set of quasi-resonant triads form an intricate network of connected triads, forming clusters whose structure depends on the value of the allowed mismatch. It is believed that understanding this network is absolutely relevant to understanding turbulence. We provide some quantitative comparison between the clusters' structure and the onset of fully nonlinear turbulent regime in the barotropic vorticity equation, and we provide perspectives for new research.

The Arithmetic Geometry of Resonant Rossby Wave Triads

“…Several authors framed the problem of enumerating resonant triads as a number-theoretic question [3,15]. For convenience, change notation and set (k 1 , 1 , k 3 , 3 ) = (a, b, x, y).…”

“…Bustamante and Hayat [3] classify integer solutions to (1.15) by mapping them bijectively to representations of zero by a certain quadratic form, yielding an algorithm for enumerating all resonant triads (see further discussion in subsection 1.8). Kishimoto and Yoneda [15] show that there are no solutions where b = 0.…”

“…As Bustamante and Hayat [3] stress in their introduction, information about highfrequency resonances is needed to describe small-scale turbulence in numerical simulations. Moreover, mesoscale waves observed in Jupiter's atmosphere have wavelengths of about 20 km, and would have wavenumbers of approximately 20000 if "extended to cover a significant part of Jupiter's surface" [3]. 1 The set of resonant triads may be described as the set of integer solutions to a Diophantine equation, as explained in the next section.…”

“…The CHME is 2 and β > 0 and F ≥ 0 are constants. 3 The following are the physical meanings of the parameters: (x, y) are spatial coordinates, t is a temporal coordinate, and ψ is the streamfunction of an incompressible flow.…”

“…Mathematically, Rossby waves are modeled as linear wave solutions to a nonlinear differential equation-the Charney-Hasegawa-Mima equation (CHME) [3]-expressing conservation of potential vorticity (see [23], especially section 3.16). Rossby waves exist in the β-plane model, in which the Coriolis force varies as a linear function of the latitude (see [23, sec-tion 3.17 and Chapter 6]).…”