Kinetic equilibria of very high-$\beta $ plasmas

Plasma equilibria with many large ion orbits, such as an advanced beam-driven field-reversed configuration, are neither static (Grad-Shafranov) nor describable as a flowing, multi-fluid. A fully-kinetic treatment of the ions is essential for such high-$\beta $ plasmas. A kinetic equilibrium is needed to properly support realistic stability and transport analyses, both of which are strongly affected by large-orbit ions. A hybrid equilibrium model has been developed with a fully-kinetic treatment of both thermal ions and a rapidly-rotating ``beam-ion'' component, such as produced by neutral beam injection, relevant to the C-2U experiments at TAE. It employs analytic Vlasov solutions in that the distribution depends only on the two constants of motion, the Hamiltonian ($H)$ and the canonical angular momentum ($P_{\theta })$. Electrons are treated as a pressure-bearing fluid. Since realistic forms of $f(H$,$P_{\theta })$ are affected by collisions, $f$ is limited to solutions of a simplified Fokker-Planck equation. Importantly, a kinetic end-loss condition applies to unconfined ions, using a particle sink at a rate consistent with Monte-Carlo-like simulations of end loss accounting for a strong end mirror.