(Color online) Analogy between the 4-wave interaction model [Chen et al. 26] and the disparate scale shell model.
(Color online) The cartoon of the ZF mediated 2D “cascade.” Here, while the scales close to energy injection (and larger) remain anisotropic, small scales are assumed to be isotropic. The region between the zonal flows and small scales are left out, since its description is the subject of quasi-linear theory. The spectrum in that region is known to be very anisotropic, thus it makes no sense to plot it in 1-D.
(Color online) Schematic description of energy injection, enstrophy dissipation, and predator-prey dynamics between meso-scale flows and the drift-wave spectrum.
(Color online) An exaggerated cartoon of the spectral features of predator prey oscillations. Initially, the ZF amplitude is low (and so is the forward enstrophy transfer rate). Therefore, the fluctuations are trapped near the most unstable mode during the initial growth of the fluctuation level. As the fluctuation level goes up, the zonal flow amplitude also goes up, which enhances the forward enstrophy transfer (via ZF shearing). Once the zonal flow amplitude becomes large enough, the enstrophy near the injection scale is transferred rapidly to large k, where it is efficiently damped. This leads to a reduction of the overall fluctuation level, and the zonal flow gets damped (due to νF ). Once the zonal flow level is reduced, the turbulence can recover again and the cycle repeats itself.
(Color online) Starting behavior in unstable regime with numerical result (left) and the approximate solution from Eq. (13) (right) for following values of parameters: .
(Color online) Predator-prey oscillations between large scale structure (zonal flow in black) and micro-turbulence (drift-waves in grey) for νF = 0.4.
Limit cycle in phase space for νF = 0.4.
(Color online) Strange attractor in 3D space phase for chaotic regime with νF = 0.1.
(Color online) (a) One dimensional map of for νF = 0.15 to νF = 1.5. (b) Frequency map of to νF = 1.5. (c) Averaged dynamics of drift wave and zonal flow.
One dimensional map of from γ = 0.01 to γ = 0.5 with νF = 0.5 and νk 2 g 2 = 0.8.
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