Role of stable modes in zonal flow regulated turbulence

Abstract

Zonal flows are shown to regulate ion temperature gradient turbulence by enabling efficient energy transfer from the instability to a damped eigenmode in the unstable wavenumber range. The damped mode also saturates turbulence when zonal flows are not active in saturation dynamics, for example, in electron temperature gradient turbulence, but the transfer from unstable to stable mode is less efficient and requires a larger amplitude to balance the instability drive. From numerical solutions of a fluid model with a single damped eigenmode, an eigenmode decomposition of the nonlinear evolution shows that the dominant energy transfer involves the triplet correlation of the unstable mode, the zonal flow, and the stable mode at three wavenumbers satisfying k=k′+k″. In this triplet, nearly all of the energy from the instability goes to the damped mode. The very small fraction going to the zonal flow is balanced by small zonal flow damping. This combination of unstable mode, zonal flow, and stable mode minimizes the nonlinear frequency mismatch and avails itself of large coupling strengths associated with the zonal flow.