Analysis of Saturation Scalings and Time-Dependent Behavior in Ion Temperature Gradient Turbulence*

Toroidal ion temperature gradient turbulence saturates by three-wave energy transfer from the instability to a stable mode through the intermediary of a zonal flow, with all three modes in the large-scale instability range. Recent theory based on this mechanism derives the dependence of saturation levels for the unstable mode, stable mode, and the zonal flow on parameters like growth rate and zonal flow damping rate. This theory is tested by comparing analytic solutions of statistical-closure energy evolution equations with numerical solutions of the primitive nonlinear two-field equations and two different approaches for numerical steady state solutions of the closure equations. To test the assumption that wavenumber dependencies are unimportant for setting levels and scalings energy evolution equations for a single triplet of wavenumbers are derived and solved. Time-dependent solutions of this system allow for the study of predator-prey oscillations involving the turbulence and zonal flows that account for the first time the true saturation energy sink. Characteristic oscillation times and phasing are investigated. Dissipation, which breaks the conjugate symmetry of unstable and stable modes, is also examined to further probe saturation physics.