Non-linear saturation of the trapped-ion mode by mode coupling in two dimensions

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Abstract
A study of the possible non-linear saturation by mode coupling of the dissipative trappedion mode is presented in which both radial and poloidal variations are considered. The saturation mechanism consists of, the non-linear coupling via convection of energy from linearly unstable modes to stable modes. Stabilization is provided at short poloidal wavelengths by Landau damping from trapped and circulating ions for ηi ≡ d ln(Ti)/d ln(n0) < 2/3, at short radial wavelengths by effects associated with the finite ion banana excursions and at long wavelengths by ion collisions. The authors' analysis applies only to the case ηi < 2/3. A one-dimensional, non-linear partial differential equation for the electrostatic potential derived in earlier work is extended to two dimensions and to third order in amplitude. Included approximately in a systematic way are kinetic effects, e.g. Landau damping and its spatial dependence due to magnetic shear. The stability and accessibility of equilibria are considered for cases far from as well as close to marginal stability. In the first case, three-wave interactions are found to be important when the spectrum of unstable modes is sufficiently narrow. In the latter case, it is found that, for a single unstable mode, a four-wave self-interaction can provide a saturation mechanism. Accessibility of equilibria is determined in both cases. In the case where three-wave interactions are dominant, the stability analysis is inconclusive. However, plasma parameters are found for which equilibria established by means of the four-wave self-interaction are stable to linear perturbation. Cross-field transport is calculated, and the scaling of results is considered for tokamak parameters.