Breakdown of adiabatic invariance in spherical tokamaks

Thermal ions in spherical tokamaks have two adiabatic invariants: the magnetic moment and the longitudinal invariant. For hot ions, variations in magnetic-field strength over a gyro period can become sufficiently large to cause breakdown of the adiabatic invariance. The magnetic moment is more sensitive to perturbations than the longitudinal invariant and there exists an intermediate regime, super-adiabaticity, where the longitudinal invariant remains adiabatic, but the magnetic moment does not. The motion of super-adiabatic ions remains integrable and confinement is thus preserved. However, above a threshold energy, the longitudinal invariant becomes non-adiabatic too, and confinement is lost as the motion becomes chaotic. We predict beam ions in present-day spherical tokamaks to be super-adiabatic but fusion alphas in proposed burning-plasma spherical tokamaks to be non-adiabatic. Consider a charged particle, with mass m and charge q, gyrating with speed v in a magnetic field, with field strength B. Assume axi-symmetric toroidal geometry with coordinates (R, φ, Z). The motion preserves the energy E (because the Lorentz force performs no work) and canonical toroidal momentum Pφ (because the toroidal coordinate is ignorable). There is no third exact constant of motion, but the normalized magnetic moment