Page 17 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-17 Egedal et al.
Phys. Plasmas 25, 072510 (2018)
q for which v2qðq; qt; ztÞ > 0. For values of zt in the FRC where l is a good invariant, there will be two separate cyclotron motions included in the integral in Eq. (7). Thus, for the regions where l is applicable, it follows that Jq 1⁄4 2 Þ/g1⁄4p vqdq such that:
l 1⁄4 qJq=ð2pÞ: (A5) To summarize, together with Eq. (A4), we have now found
that the Jacobian of the full transformation is given by
ðDxÞ3 ðDvÞ3 8p3
J 􏰄DUDJ Dp 1⁄4m2qsb: (A6)
q/
Again, Eq. (A6) was derived based on standard guiding center theory where l is a good adiabatic invariant. We next present a proof that this expression also holds in regions of non-standard orbits. For this, we split the orbit in Fig. 18 into three sections. The blue and the red sections with particle transit times t1 and t3 are characterized by standard gyromo- tion well capture by guiding center theory. Meanwhile, for the green section with transit time t2 is not described by guiding center theory. The total bounce time is sb 1⁄4 t1 þ t2 þt3, and the total Jacobian we denote as Jtot 1⁄4J1 þJ2 þJ3, where J1 1⁄4 ðt1=sbÞJ and J3 1⁄4 ðt3=sbÞJ, with J given by Eq. (A6). For the non-standard orbit section, we write J 2 1⁄4 ða=sbÞJ , where a is an unknown constant which needs to be determined.
We now add a source S 1⁄4 N/sb, injecting particles at the beginning of orbit section 1. The source is turned on at t 1⁄4 0 and starts filling section 1 uniformly with the front of par- ticles propagating away from the source reaching the end of section 1 at t 1⁄4 t1. At this time, a total of N1 1⁄4 t1N=sb par- ticles has been injected, and given S was constant in time, f must be uniformly distributed over the orbit section with f 1⁄4N1=J1 1⁄4N=J.
The source continues to inject particles at the given rate, and the front of elevated f continues to propagate around the orbit sections. At t 1⁄4 sb, this front has reached the end of the
/g 1⁄40
FIG. 18. Example of a typical FRC orbit. The blue and red sections (with particle transit times t1 and t3) follow guiding center theory, while the green orbit section (with transit time t2) is a nonstandard orbit not described by guiding center theory. Despite these differences, we find that all three orbit sections are characterized by the Jacobian J in Eq. (A6).
FIG. 19. J0 is the maximum value of Jq shown as a function of v. The green and blue curves correspond to the similarly colored flux surfaces in Fig. 1. The red dashed line is J0 ’ 15v1:5.
full orbit, and a total of N particles has now been ejected. Because f is constant along the full orbit (follows from df/dt 1⁄4 0), all orbit-sections must be characterized by the same value of f 1⁄4 N=J determined above for orbit section 1. Particle conservation then requires that N 1⁄4 f ðJ 1 þ J 2 þJ 3Þ. Given the results above, we can solve this equation to find a 1⁄4 t2. This is consistent with Eq. (A6), which we hereby have proven to hold also for non-standard orbits.
Because J is a local quantity, Eq. (A6) must be general and not contingent on the non-standard orbit section being con- nected to orbit sections characterised by guiding center orbits. This last conclusion can be proven by considering a general case of a fully non-standard orbit. For any such case, we can “cut” the equilibrium at z 1⁄4 0 and add a new equilibrium sec- tion that joins the previous equilibrium smoothly. For any non- standard orbit, we can always find a new equilibrium section for which the orbit will undergo bifurcation to the standard orbit type, and the proof above can then be applied.
Finally, the primary variables used in the pumping model are v and n. Here, n is defined through Jq 1⁄4 J0ð1 􏰂n2Þ, where J0 is the maximum value of Jq for a given v. From Fig. 19, it is clear that J0 ’ 15v1:5 nearly independent of the value of p/ considered. Because dU=dv 1⁄4 mv and dJq=dn 1⁄4 􏰂2nJ0, the Jacobian in terms of the primary varia- bles becomes
ðDxÞ3 ðDvÞ3 16p3
J v;n 􏰄 DvDnDp 1⁄4 mq nvJ0 sb: (A7)
/
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