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. [Phys. Plasmas 7, 1349 (2000)]. Two positive definite, exact, integral invariants associated with the full generalized Charney–Hasegawa–Mima system are derived. For an initial monochromatic drift wave pump with small but finite amplitude, a modulational instability can occur, characterized by growing zonal flow and sideband perturbations (i.e., a four-wave interaction). The pump threshold for instability is readily satisfied, depending on the zonal flow wave number. The fully nonlinear Charney–Hasegawa–Mima equations are solved with a numerical scheme which is validated by demonstrating the conservation of the two exact invariants. The simulations show that the validity of the four-wave model is limited to approximately three instability growth times. The radial structure of the zonal flow can be “jet-like” or highly oscillatory depending upon the ratio of the system size to the density scale length and initial conditions. It is found that zonal flows can be dramatically reduced if the most unstable zonal flow wave number does not fit into the system.