The Arithmetic Geometry of Resonant Rossby Wave Triads

Linear wave solutions to the Charney--Hasegawa--Mima 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 $X$. The set of all resonant triads was found by Bustamante and Hayat (2013) via mapping to quadratic forms. Our work independently finds all resonant triads via a rational parametrization of $X$. We provide a fiberwise description of $X$ as a rational singular elliptic surface, yielding many new results about the set of wavevectors belonging to resonant triads. In particular, we show there is an infinite number of resonant triads (with relatively prime wavevectors) containing a wavevector $(a,b)$ with $a/b=r$, where $r$ is any given rational, and we provide a method to find these triads. This is applied to find all resonant Rossby wave packets that are stationary in the east-west direction.