Four motional invariants in axisymmetric tori equilibria

Abstract

In addition to the standard set (ε,μ,pφ) of three invariants in axisymmetric tori, there exists a fourth independent radial drift invariant Ir. For confined particles, the net radial drift has to be zero, whereby the drift orbit average Ir=⟨r¯0⟩ of the gyro center radial Clebsch coordinate is constant. To lowest order in the banana width, the radial invariant is the gyro center radial coordinate r¯0(x,v), and to this order the gyro center moves on a magnetic flux surface. The gyro center orbit projected on the (r,z) plane determines the radial invariant and first order banana width corrections to Ir are calculated. The radial drift invariant exists for trapped as well as passing particles. The new invariant is applied to construct Vlasov equilibria, where the magnetic field satisfies a generalized Grad-Shafranov equation with a poloidal plasma current and a bridge to ideal magnetohydrodynamic equilibria is found. For equilibria with sufficiently small banana widths and radial drift excursions, the approximation Ir≈r¯0(x,v) can be used for the equilibrium state.