The nonlinear dynamics of the modulational instability of drift waves and the associated zonal flows

The linear and nonlinear dynamics of zonal flows and their interactions with drift wave turbulence is considered in the simple but illuminating generalized Charney-Hasegawa-Mima model due to Smolyakov et. al. (A.I. Smolyakov, P.H. Diamond and V.I. Shevchenko, Phys. Plasmas, 7, 1349 (2000)). Two positive definite, exact, integral invariants associated with the full generalized Charney-Hasegawa-Mima system are derived. These invariants are the generalizations of the well-known energy and enstrophy integrals of the original Charney-Hasegawa-Mima equation. Taking the initial pump amplitude as fixed (but small), it is shown that the system experiences a classic 'modulational instability'. This is characterized by the growth of a specified, infinitesimal amplitude zonal flow and side-bands of the pump generated by it beating with the zonal flow. The threshold for this growth is determined and found to be readily satisfied under typical conditions on the pump amplitude and zonal flow perturbation wave number. The pump is then allowed to evolve according to a simple 'four-wave', nonlinear model. Two positive definite invariants associated with the four-wave evolution are identified. The generalized Charney-Hasegawa-Mima equations are solved numerically