Page 9 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma
P. 9
072510-9 Egedal et al.
included in the model through a mixing operator Lmix such
that the full kinetic equation reads
@f1@Jz@f1⁄4Lmixf: (12)
To derive an expression for Lmix, we consider a phase-space- volume DV 1⁄4 DU DJq Dp/. The number of particles in this volume is given by fJDV, where J is the Jacobian in Eq. (A6) derived in the Appendix. During a bounce time, the par- ticles will all go once around the orbit, such that for each point along the orbit we can define the flux of particles (per DV) as C 1⁄4 J f =sb. The non-adiabatic changes in Jq occur over non-localized regions, but for deriving the mixing oper- ator we can assume that the changes occur in a vanishingly small volume localized somewhere along the orbit. The flux of particles going into the mixing volume is simply the flux of particles going around the orbit
Phys. Plasmas 25, 072510 (2018)
further numerical calculations. The operator also fulfills the condition that it must conserve particles, which is readily seen by considering the integral
ð ð1
ðLmixfÞJ dJq 1⁄4 ðLmixfÞJ @Jq dn 1⁄4 0:
@t vsb@t@v
C ðnÞ1⁄4DN1⁄4Jf: in Dtsb
(13)
B. Rates of pitch-angle-mixing
For orbits with no bifurcation points, we find that n is a well-conserved adiabatic invariant, such that hðD/BÞ2i 1⁄4 0. For orbits which do include bifurcation points, the rate of pitch-angle-mixing hðD/BÞ2i is a function of ðv; n; p/Þ and can be determined numerically by tracing large numbers of orbit sections across the bifurcation line, allowing for an eval- uation of hðD/BÞ2i. For a particular value of n and v, the pitch-angle-mixing is highly sensitive to the value of /q at the point where the orbit reaches the bifurcation line. Here, /q is the phase angle of the q-oscillation. An efficient numeri- cal scheme to calculate hðD/BÞ2i must therefore apply trajec- tories where /q is controlled and distributed uniformly.
0 @n
also note that in the limit where hðD/B Þ2 i is constant,
We
the form of Lmix is identical to an approximation derived by Cordey19 for the drift orbit averaged Lorentz scattering oper- ator applied to trapped fast ions in tokamaks.
We assume that the changes DJq are uncorrelated and as particles bounce repeatedly they undergo a “random walk” in Jq. This is thus a diffusion process which will obey Fick’s law such that the outgoing flux of particles is given by
@@ Cout 1⁄4 1þ hðDJqÞ2i Cin:
Here, hðDJqÞ2i is a function of (U;Jq;p/) and can be deter- mined numerically by tracing a large number of orbits through the magnetic equilibrium while evaluating the changes in Jq.
The change in the number of particles in DV can now be computed as J Df 1⁄4 DtðCout Cin Þ such that
@f 1⁄4 Df 1⁄4 1 ðCout CinÞ;
1@2@J 1⁄4J@J hðDJqÞi@J s f;
In Fig. 10, we consider a case where we evaluate hðD/ Þ2 i for v 1⁄4 6 and n 1⁄4 0:93 (corresponding to
@Jq @Jq
(14)
pffiffiffiffi B pffiffiffiffi
J0 1⁄4 14:50; Jq 1⁄4 5:38, and /B 1⁄4 0:38). The red con-
@tmix DtJ
tours in Fig. 10(a) represent the considered value of Jq, and the intersection point of this Jq contour and the magenta bifurcation line are used as the starting point for an orbit with v1⁄46, which is tracked a short distance back in time. The second turning point of this orbit reaching the initial Jq contour is identified and represents a 2p shift in /q compared to the orbits starting point. The 10 black crosses between the turning points considered are then applied as initial condition for orbits (uniformly distributed in /q and tracked forward in time). Figure 10(b) shows the orbits that are followed until they exit and return to the region encircled by the bifurcation line. Within this region, a turning point ðqt ; zt Þ is identified such that by using Eq. (7) the new values of Jq and /B can be recorded. In turn, this procedure allows hðD/B Þ2 i to be computed for a set of orbits uniformly distributed over 2p in /q . The values of hðD/B Þ2 i indicated in Fig. 10(c) were obtained based by this method, applying a total of 300 uni- formly distributed orbit sections. The changes in /B are observed to be particularly sensitive to /q when a small change causes a transition in the trajectory to go above or below the magnetic center of the device.
Using the method outlined with Fig. 10, we can calculate hðD/BÞ2i as a function of ðn; vÞ. The results of such a calcula- tion are shown in Fig. 11, for a value of p/ corresponding to orbits around the green flux-surface close to the center of the equilibrium highlighted in Fig. 1. In Fig. 11, we have marked the regions in the ðv;nÞ-plane for which the corresponding orbits have no bifurcation points, and in those regions we have hðD/B Þ2 i 1⁄4 0. For orbits which do include bifurcation points, we observe typical values of hðD/B Þ2 i ’ 0:015.
qqb 1@2@
1⁄4s@JhðDJqÞi@Jf; (15) bqq
where in the last step we used that J =sb is independent of Jq. It is useful to introduce the representative pitch-angle /B 1⁄4 cos1ðnÞ. The level of pitch-angle-mixing can then be
described by hðD/BÞ2i per bounce motion. We then have
@J 2 @n 2
hðDJqÞ2i 1⁄4 q hðD/BÞ2i;
@n @/B
where @Jq=@n 1⁄4 2nJ0 and ð@n=@/BÞ2 1⁄4 1 n2. Using these results and the chain-rule for differentiation, Eq. (15) can now be written as @f =@tjmix 1⁄4 Lmixf , where
Lmix 1⁄4 1 @ nð1 n2ÞhðD/BÞ2i @ : nsb @n @n
(16)
The operator has the desired form including differentials with respect to the variable n, which later will be useful for
