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Nonlocal wave turbulence in the Charney-Hasegawa-Mima equation: a short review

Rossby wave turbulence, as modelled by the Charney-Hasegawa-Mima (CHM) equation, is nonlocal in scale. As a result, the formal stationary Kolmogorov-Zakharov solutions of the Rossby wave kinetic equation, which describe local cascades, are not valid. Rather the solution of the kinetic equation is dominated by interactions between the large and small scales. This suggests an alternative analytic approach based on an expansion of the collision integral in a small parameter obtained from scale separation. This expansion approximates the integral collision operator in the kinetic equation by anisotropic diffusion operators in wavenumber space as first shown in a series of papers by Balk, Nazarenko and Zakharov in the early 1990's. In this note we summarize the foundations of this theory and provide the technical details which were absent from the original papers.