Nonlinear Finite Beta Saturation of the Drift Cyclotron Loss Cone Instability

The nonlinear saturation of a single Drift Cyclotron Loss Cone mode is considered in the case of a mirror confined finite (beta) plasma. The instability is modeled in slab geometry with the local approximation made. Using a variation of the Bogoliubov technique, analytic expressions are derived for the time asymptotic distribution function, electrostatic potential, and vector potential. The saturation amplitude and frequency shift are determined as functions of the fractional increase of the density above its threshold value. Freedom is retained in the equilibrium ion velocity space distribution function so that the effect of a partially filled loss cone may be investigated. A term neglected in a previous low (beta) application of this technique to the DCLC is found to affect the saturation results. In a collisionless, lossless model, the effect of a partially filled loss cone on the saturation amplitude and density, density gradient space stability boundary is demonstrated. The nonlinear frequency shift is indeterminate. Stable, small amplitude saturated states are found in some regions of parameter space for a partially filled Subtracted Maxwellians loss cone distributions, though amplitudes are smaller than those required in the aforementioned low (beta) work. A delta function equilibrium velocity space distribution is found to have only large amplitude saturated states. The nonlinear shift in the electrostatic potential is found to be negligibly small in this model, and the nonlinear shift in the vector potential is found to be just that needed to restore particle-field pressure balance. A second model is employed which takes into account plasma losses through a simple Krook type collision term in the Boltzman equation. Ion losses are assumed to be dominated by electron drag into the loss cone and the electron losses are modeled by the classical expression derived by Pastukhov. Threshold conditions are only slightly modified by including losses. Saturation results are found to be substantially affected. The ambipolar potential rises in the presence of instability due to enhanced electron losses. Stable saturation amplitudes are found which vary linearly with electron temperature. Finite (beta) is shown to have a stabilizing effect. Comparison with 2XIIB experimental results shows some agreement using a Subtracted Maxwellians ion distribution, through saturation amplitudes are larger than observed experimentally. Improvement is obtained by more closely modeling the observed ion distribution. Predicted nonlinear frequency shifts, however, are far too large.