Saturation of collisionless ion temperature gradient turbulence via symmetric dynamics. I. The zonal flow

Collisionless zonal flow (ZF) saturation is analytically investigated using a reduced two-field fluid model for ion temperature gradient-driven turbulence subject to a three-wavevector, five-mode truncation that includes the interaction of an unstable mode, the ZF and a stable mode. Using weak-turbulence closure theory, a wave kinetic equation (WKE) is derived describing the temporal evolution of the ZF energy spectrum for this system according to linear and nonlinear dynamics. The terms in the nonlinear time evolution operator, which describe resonant energy transfer among triads of fluctuations, are expressed as matrices whose elements are arithmetic combinations of the linear eigenvalues of the individual modes participating in a given interaction. In the collisionless limit, the matrices characterising ZF drive and damping become highly symmetric, and conditions are found for spectral saturation in this regime. A set of stationary solutions to the WKE are found that describe a state of turbulence in which the nonlinearly driven stable modes reach identical levels as the linearly driven unstable modes, producing identical rates of up- and down-gradient thermal energy transport at each fluctuation length scale. Generalisation to the non-truncated system is briefly discussed.