On the convergence of bootstrap current to the Shaing–Callen limit in stellarators

The bootstrap current in stellarators can be presented as a sum of a collisionless value given by the Shaing–Callen asymptotic formula and an off-set current, which non-trivially depends on plasma collisionality and radial electric field. Using NEO-2 modeling, analytical estimates and semi-analytical studies with the help of a propagator method, it is shown that the off-set current in the $1/\nu$ regime does not converge with decreasing collisionality $\nu _\ast$ but rather shows oscillations over $\log \nu _\ast$ with an amplitude of the order of the bootstrap current in an equivalent tokamak. The convergence to the Shaing–Callen limit appears in regimes with significant orbit precession, in particular, due to a finite radial electric field, where the off-set current decreases as $\nu _\ast ^{3/5}$ . The off-set current strongly increases in case of nearly aligned magnetic field maxima on the field line where it diverges as $\nu _\ast ^{-1/2}$ in the $1/\nu$ regime and saturates due to the precession at a level exceeding the equivalent tokamak value by ${v_E^\ast }^{-1/2}$ , where $v_E^\ast$ is the perpendicular Mach number. The latter off-set, however, can be minimized by further aligning the local magnetic field maxima and by fulfilling an extra integral condition of “equivalent ripples” for the magnetic field. A criterion for the accuracy of this alignment and of ripple equivalence is derived. In addition, the possibility of the bootstrap effect at the magnetic axis caused by the above off-set is also discussed.