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The arithmetic geometry of resonant Rossby wave triads. (English) Zbl 1374.14021

Summary: Linear wave solutions to the Charney-Hasegaw-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 M. D. Bustamante and U. Hayat [Commun. Nonlinear Sci. Numer. Simul. 18, No. 9, 2402–2419 (2013; Zbl 1310.35058)] 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.

MSC:

14G05 Rational points
11D41 Higher degree equations; Fermat’s equation
76B65 Rossby waves (MSC2010)
86A10 Meteorology and atmospheric physics
11D45 Counting solutions of Diophantine equations
11G05 Elliptic curves over global fields
11G35 Varieties over global fields
14M20 Rational and unirational varieties
35Q35 PDEs in connection with fluid mechanics
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations

Citations:

Zbl 1310.35058

Software:

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References:

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