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A detailed study of the Charney-Hasegawa-Mima model and its extensions ispresented. These simple nonlinear partial differential equations suggested forboth Rossby waves in the atmosphere and also drift waves in amagnetically-confined plasma exhibit some remarkable and nontrivial properties,which in their qualitative form survive in more realistic and complicatedmodels, and as such form a conceptual basis for understanding the turbulenceand zonal flow dynamics in real plasma and geophysical systems. Two idealisedscenarios of generation of zonal flows by small-scale turbulence are explored:a modulational instability and turbulent cascades.  A detailed study of the generation of zonal flows by the modulationalinstability reveals that the dynamics of this zonal flow generation mechanismdiffer widely depending on the initial degree of nonlinearity. A numericalproof is provided for the extra invariant in Rossby and drift wave turbulence-zonostrophy and the invariant cascades are shown to be characterised by thezonostrophy pushing the energy to the zonal scales.  A small scale instability forcing applied to the model demonstrates thewell-known drift wave - zonal flow feedback loop in which the turbulence whichinitially leads to the zonal flow creation, is completely suppressed and thezonal flows saturate. The turbulence spectrum is shown to diffuse in a mannerwhich has been mathematically predicted.  The insights gained from this simple model could provide a basis forequivalent studies in more sophisticated plasma and geophysical fluid dynamicsmodels in an effort to fully understand the zonal flow generation, theturbulent transport suppression and the zonal flow saturation processes in boththe plasma and geophysical contexts as well as other wave and turbulencesystems where order evolves from chaos.

Rossby and Drift Wave Turbulence and Zonal Flows: the Charney-Hasegawa-Mima model and its extensions