Page 10 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-10
Egedal et al.
Phys. Plasmas 25, 072510 (2018)
FIG. 10. Outline of numerical scheme for calculating hðD/B Þ2 i, here for /B 1⁄4 0:38 (or n 1⁄4 0:93) and v1⁄46. In (a), an orbit is tracked back in time, and its first turning points on its Jq-contour are used for generating starting points for trajectories uniformly distributed in phase angle /q of the fast q oscillations. Ten trajectories evenly distributed in /q are given in (b). (c) Values of /B obtained for a set of 300 trajectories, with the red line indicat- ing the initial value of /B 1⁄4 0:38. The resulting value of hðD/BÞ2i is displayed.
The ions with no bifurcations along their orbits experi- ence no pitch-angle-mixing. Because heating by MP requires pitch-angle-mixing, these ions are not subject to this heating mechanism. The boundaries in the ðv;nÞ-plane for the “Passing, No bifurcation” orbits [see Figs. 6(c) and 6(d)] are highly sensitive to the value of p/ considered. For flux surfa- ces closer to the edge of the equilibrium, higher velocities are required for reaching this regime. To illustrate this, we have included Fig. 12 providing a colormap of hðD/BÞ2i, here calculated for a value of p/ corresponding to the flux- surface highlighted in blue in Fig. 1. For the range of v con- sidered, the region of passing ions with no bifurcations is now eliminated. Meanwhile, the region of trapped orbits with no bifurcations is still characterized by the range of pitch-angles where 0 􏰆 n 􏰆 0:4.
C. Expansion of pitch-angle dependencies
In this section, we will develop further the model for magnetic pumping. As a first step, we need a simplified
hðD/BÞ2i ’
between orbits with and without bifurcations. The curve for
FIG. 11. Rates of pitch-angle-mixing hðD/BÞ2i as a function of v and n. The values are obtained using the method outlined in Fig. 10, with orbits cen- tered on the green contour of W in Fig. 1 near the center of the equilibrium. In regions of the parameter for which the orbits have no bifurcations, we have hðD/BÞ2i 1⁄4 0. The region marked “Passing, No bifurcation” is charac- terized by the orbit type of Figs. 6(c) and 6(d), while the region marked “Trapped, No bifurcations” is characterized by the orbit type of Figs. 6(e) and 6(f).
pitch-angle-mixing operator, Lmix. We will first consider the case in Fig. 12 where the mixing rates are well approximated by a function of the form
c
( 0; D/Bðv;p/Þ;
0 􏰆 n 􏰆 n nc 􏰆 n 􏰆 1:
c (17) Here, n is the critical pitch-angle, marking the transition
nc in the ðv; nÞ-plane can be deduced from the relationship between Jq and v/ along the magenta transition line in Fig. 7(b). The marginal cases where the z-reflections occur on the transition line are then identified by imposing v 1⁄4 v/. As seen in the panels of Figs. 9, 11, and 12, the transition line is
FIG. 12. Rates of pitch-angle-mixing hðD/BÞ2i as a function of v and n for a value of p/ corresponding to the blue contour of W in Fig. 1.