Page 8 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-8 Egedal et al.
In Figs. 9(a)–9(c), we repeat the profiles of Figs. 8(b)–8(d), now displayed as a function of ðv;nÞ. Furthermore, rather than Dv, we show profiles of sbDv as this is a quantity of special interest to the MP model being devel- oped in the subsequent sections. In the panels of Fig. 9, we also display the regions of distinct orbit topologies, with the black dashed lines marking the trapped/passing boundary corresponding to zt 1⁄4 0 and v 1⁄4 v/ in Fig. 7(b). The magenta lines with n ’ 0:4 represent the magenta bifurcation line in Fig. 7(b), while the white lines correspond to the condition Jq; bifur;max 1⁄4 ð1 􏰂 n2ÞJ0ðvÞ, with Jq; bifur;max being the maxi- mum value of Jq along the bifurcation line [marked by the white circle in Fig. 7(b)].
Additionally, the profiles in Figs. 9(d)–9(f) display vsb, as well as sbDv, computed for perpendicular and parallel compression of the equilibrium, this time considering the flux-surface closer to the edge of the configuration marked by the blue line in Fig. 1. The main difference in these pro- files is that the point of intersection between the line of mar- ginal trapping and the bifurcation line has moved outside the range of v considered.
Phys. Plasmas 25, 072510 (2018) III. STATISTICAL DESCRIPTION OF MAGNETIC
PUMPING
A. Pitch-angle-mixing operator
To obtain a kinetic formulation for MP, we first consider the hypothetical case where Jq and Jz are fully conserved adi- abatic invariants of the orbit motion. Similar to the case stud- ied by Montag et al.,18 because Jq, Jz, and p/ are all constants of motion and there are no other variables to con- sider, we must have @v=@t 1⁄4 dv=dt. Next, if f 1⁄4 f ðv; Jq; Jz;p/Þ, then it follows from Eq. (9) and the fact that Jq, Jz, and p/ are constants of motion that:
df 1⁄4 @f 􏰂 1 @Jz @f 1⁄4 0: dt @t vsb@t@v
This is a kinetic equation which describes the evolution of f in the limit where the orbit bounce time is fast compared to the evolution of the magnetic equilibrium. Furthermore, it is assumed that no pitch-angle-mixing occurs.
The effects of the important pitch-angle-mixing, caused by the breakdown of Jq as an adiabatic invariant, can be
FIG. 9. (a)–(c) The profiles of Figs. 8(b)–8(d) are displayed as functions of ðv;nÞ, with overlayed lines separating distinct orbit topologies. (d)–(f) Profiles similar to those in (a)–(c), but now calculated for p/ 1⁄4 p/0, where qW 1⁄4 p/0 corresponds to the flux sur- face marked by blue in Fig. 1, closer to the edge of the equilibrium.