Quasilinear saturation of the kinetic ion mixing mode

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Abstract
The linear and quasilinear theories of the ion mixing mode are discussed in the electrostatic limit. The source of free energy for the instability is the ion temperature gradient. The parameter relevant to this calculation is ηi=(d ln Ti/dr)/(d ln n/dr). Generally, this mode has a phase velocity comparable to the ion thermal velocity and there exist values of ηi that yield maximum growth. By using a time‐asymptotic formalism, we are able to discuss the quasilinear saturation of this mode. Short wavelength modes saturate at small amplitudes, ‖eφ1/Ti‖2 ≤(ηi−ηc)/ηc, where ηc is the value of ηi at marginal stability, and the saturated state is stable against further perturbations. Long wavelength modes also saturate, though the saturation is of the ‘‘hard’’ type: the saturated state itself is unstable. When saturation occurs, it is caused by reduced inverse ion Landau damping brought about both by the nonlinear frequency shift and by quasilinear modifications to the ion distribution function.