2017
DOI: 10.1137/16m1077593
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The Arithmetic Geometry of Resonant Rossby Wave Triads

Abstract: Abstract. 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 … Show more

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Cited by 4 publications
(6 citation statements)
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“…It is interesting that elliptic curves are recognized in images of the solar corona and cromosphere in the ultraviolet and extreme ultraviolet wavelengths (on data provided by Solar Dynamics Observatory SDO), see [2]. An interesting occurrence of elliptic curves in the natural sciences is also described in G. S. Kopp's paper [4], which states that 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.…”
Section: Introductionmentioning
confidence: 93%
“…It is interesting that elliptic curves are recognized in images of the solar corona and cromosphere in the ultraviolet and extreme ultraviolet wavelengths (on data provided by Solar Dynamics Observatory SDO), see [2]. An interesting occurrence of elliptic curves in the natural sciences is also described in G. S. Kopp's paper [4], which states that 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.…”
Section: Introductionmentioning
confidence: 93%
“…Equations (12) and (13) summarise our explicit parameterisation of the resonant triads' wavevector ratios (k 1 /k 3 , l 3 /k 3 , l 1 /k 3 ) ∈ Q 3 in terms of the new parameters (A, B) ∈ Q 2 . In this context it is worth referring the interested reader to the work by Kopp (2017), who found another parameterisation independently.…”
Section: An Explicit New Parameterisation For Resonant Triadsmentioning
confidence: 99%
“…Solutions exist only if (0 ≤)F ∈ Q. If we set F = 0 (the so-called small-scale limit) then a mapping to an elliptic surface (and a subsequent mapping to quadratic forms) allows us to parameterise the solutions, as shown for the first time by Bustamante and Hayat (2013) and further developed by Kopp (2017). See also a particular case discussed independently by Kishimoto and Yoneda (2017).…”
Section: Governing Equationsmentioning
confidence: 99%
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“…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: …”
Section: Preliminariesmentioning
confidence: 99%