The nonlinear dynamics of zonal flows (ZFs) is investigated when geodesic acoustic modes (GAMs) have substantial influence on plasma states. Simultaneous existence of multiple GAMs with different radial phase velocities gives additional nonlinear mode couplings, and asymmetry of the turbulence spectrum induces energy exchanges between GAMs and a ZF. A set of model equations is derived to describe the nonlinear dynamics of a ZF, GAMs and ambient turbulence. The model includes the mechanism of ZF generation by a pair of GAMs, and the back interaction from the turbulence is solved self-consistently. Two stationary solutions are obtained; one is the known solution that the GAM propagates as a travelling wave with no ZF excitation, and the other is a new solution that a pair of GAMs forms a standing wave, which induces a ZF. The accessibility to the steady states, and the threshold for the transition between them are obtained. The GAM and ZF formation in the new state affects the background plasmas. The effects on the turbulent heat diffusivity and the ion heating rate are discussed.
Modulation behaviour and possible existence criterion of geodesic acoustic modes in tokamak devices