The arithmetic geometry of resonant Rossby wave triads

TL;DR

This paper characterizes all resonant triads of Rossby and drift waves described by the Charney-Hasegawa-Mima equation, using algebraic geometry and Diophantine equations to provide a comprehensive parametrization.

Contribution

It provides a rational parametrization of the smooth points on the surface describing resonant triads, answering the fundamental question of all possible resonant wavevector configurations.

Findings

Explicit rational parametrization of resonant triads.

A fiberwise method for fixed wavevector ratios.

Connection between wave resonance and algebraic geometry.

Abstract

Linear wave solutions to the Charney-Hasegawa-Mima partial differential equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface. We give a rational parametrization of the smooth points on this surface, answering the question: What are all resonant triads? We also give a fiberwise description, yielding a procedure to answer the question: For fixed $r \in \mathbb{Q}$, what are all wavevectors $(x,y)$ that resonate with a wavevector $(a,b)$ with $a/b = r$?