Abstract
We study drift/Rossby wave turbulence described by the large-scale limit of the Charney–Hasegawa–Mima equation. We define the zonal and meridional regions as and respectively, where is in a plane perpendicular to the magnetic field such that kx is along the isopycnals and ky is along the plasma density gradient. We prove that the only types of resonant triads allowed are and . Therefore, if the spectrum of weak large-scale drift/Rossby turbulence is initially in Z it will remain in Z indefinitely. We present a generalised Fjørtoft’s argument to find transfer directions for the quadratic invariants in the two-dimensional -space. Using direct numerical simulations, we test and confirm our theoretical predictions for weak large-scale drift/Rossby turbulence, and establish qualitative differences with cases when turbulence is strong. We demonstrate that the qualitative features of the large-scale limit survive when the typical turbulent scale is only moderately greater than the Larmor/Rossby radius.
Highlights
Drift waves in plasmas and Rossby waves in the ocean and planetary atmospheres, though unrelated at the first sight, have common features in their dynamics and statistics, which at a basic level can be described by the same model—the Charney–Hasegawa–Mima (CHM) equation (2.1)
Prompted by the fact of its conservation, we have proposed and proven that the following resonant triads are prohibited, M « M + M, M « Z + Z, Z « M + Z and Z « M + M, where Z and M are the zonal and meridional sets of modes defined in proposition 1
This proposition has a drastic consequence for the weakly nonlinear dynamics of the large-scale CHM systems: spectrum initially fully concentrated in the zonal sector Z cannot ever leave this sector
Summary
Drift waves in plasmas and Rossby waves in the ocean and planetary atmospheres, though unrelated at the first sight, have common features in their dynamics and statistics, which at a basic level can be described by the same model—the Charney–Hasegawa–Mima (CHM) equation (2.1). Much of the research using this model has concentrated on its small-scale limit, r ¥, where ρ is the ion Larmor radius in plasmas or the Rossby radius of deformation in oceans and atmospheres. Defining the zonal region of wave vectors Z ≔ {k : ∣ky∣ > 3 kx} and the meridional region M ≔ {k : ∣ky∣ < 3 kx}, we prove that the only allowed resonant triad interactions are M « M + Z and Z « Z + Z This property has profound effects on the nonlinear evolution. It leads to the claim that the spectrum of weak large-scale drift/Rossby turbulence which is initially in Z will remain in Z indefinitely We will use this property and the semi-action invariant for revising Fjørtoft’s argument of Balk et al (1991) aimed at predicting directions of the turbulent cascades in the two-dimensional (2D) scale space. We study robustness of our results for systems where the large-scale limit is not well satisfied, and find that most qualitative features survive even for flows with typical scales exceeding ρ only by a factor of two
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