  A detailed study of the Charney-Hasegawa-Mima model and its extensions is
presented. These simple nonlinear partial differential equations suggested for
both Rossby waves in the atmosphere and also drift waves in a
magnetically-confined plasma exhibit some remarkable and nontrivial properties,
which in their qualitative form survive in more realistic and complicated
models, and as such form a conceptual basis for understanding the turbulence
and zonal flow dynamics in real plasma and geophysical systems. Two idealised
scenarios 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 modulational
instability reveals that the dynamics of this zonal flow generation mechanism
differ widely depending on the initial degree of nonlinearity. A numerical
proof is provided for the extra invariant in Rossby and drift wave turbulence
-zonostrophy and the invariant cascades are shown to be characterised by the
zonostrophy pushing the energy to the zonal scales.
  A small scale instability forcing applied to the model demonstrates the
well-known drift wave - zonal flow feedback loop in which the turbulence which
initially leads to the zonal flow creation, is completely suppressed and the
zonal flows saturate. The turbulence spectrum is shown to diffuse in a manner
which has been mathematically predicted.
  The insights gained from this simple model could provide a basis for
equivalent studies in more sophisticated plasma and geophysical fluid dynamics
models in an effort to fully understand the zonal flow generation, the
turbulent transport suppression and the zonal flow saturation processes in both
the plasma and geophysical contexts as well as other wave and turbulence
systems where order evolves from chaos.
