Page 6 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-6 Egedal et al. Phys. Plasmas 25, 072510 (2018)
FIG. 6. (a) Example of an orbit in its (z, q)-projection, with turning points following one particular contour of constant Jq (red line). Orbit transitions are observed where the Jq contour intersects the magenta bifurcation line. The orbit is reflected in its z-motion where the Jq contour intersects the blue lines denot- ing v2/ ðq; zÞ 1⁄4 v2 . For this particular orbit, such intersections are observed for z ’ 140 and z ’ 280, confining the orbit to this limited range of the device. In the trajectory projections shown above, the transitions in the orbit are highlighted by changing colors both in (a) and (b). (c) and (d) Example of an orbit for which the Jq contour does not intersect the magenta bifurcation line. Thus, this orbit is at all locations of the figure-8 type. The orbit is again reflected in the z- direction where Jq contour intersects the blue line where again v2/ ðq; zÞ 1⁄4 v2 . (e) and (f) Example of an orbit for which the Jq contour does intersect the magenta bifurcation line. However, here the trajectory is trapped outside the bifurcation line and the orbits are again at all locations of the figure-8 type.
Fig. 7(b) contains color contours of v? evaluated in the ðzt ; Jq Þ-plane. Again, the blue and red lines correspond to Jq and v, respectively, of the orbit in Figs. 6(c) and 6(d), while the magenta line marks the locations of orbit bifurcations.
By considering cuts of v2?ðzt; JqÞ as a function of zt for
fixed Jq, we can analyze the z-oscillation in a manner similar
to that applied for the q-oscillations in Sec. IIB. In this
particular profile, we notice that for fixed Jq 1⁄4 Jq0 with pffiffiffiffiffiffi 2
Jq0 􏰐 14, the profiles of v?ðzt; Jq 1⁄4 Jq0; p/Þ have local maxima at zt 1⁄4 0, allowing ions to be confined to one half of the device, as was explored above with the orbits in Figs. 6(a), 6(b), 6(e), and 6(f).
Ions with total energies above the local maximum of v2? will pass freely across the midplane. For example, in Fig.
7(c), the green line is v? evaluated along the Jq 1⁄4 15 cut pffiffiffiffi
(red line) in Fig. 7(b). Again, this value of Jq 1⁄4 15 corre- sponds to the orbit in Figs. 6(c) and 6(d) with the speed v1⁄47.8 indicated by the blue line in Fig. 7(c). The orbit is confined to the interval in z for which v? < v. Within this interval, we find for the z velocity vz 1⁄4 6ðv2 􏰂 v2?Þ1=2, repre- sented by the black line in Fig. 7(c). The calculation is vali- dated by the magenta line, which is vz of the full orbit considered in Figs. 6(c) and 6(d).
F. The Jz action integral and related quantities
The area encircled bÞy the vzðzÞ curve in Fig. 7(c) is the Jz action integral Jz 1⁄4 vzdz. Thus, using the profile of
v? ðzt ; Jq Þ, we can readily calculate Jz for any combination of
ðv; Jq Þ. For the value of p/ considered above the result of
this calculation is shown in Fig. 8(a). Consistent with Eq. pffiffiffiffi
(A5) in the Appendix, we find Jq ’ 2v?, where v? is a characteristic speed of the perpendicular (xy)-orbit motion.
Thus, in Fig. 8(a), the y-axis in Fig. 8(a) is approximately 2v?. Naturally, no values of Jz are displayed in the unphysi- cal region corresponding to v? > v.
The Jz action integral is useful as the orbit bounce time sb can be obtained as the speed differential of Jz
@Jz
vsb 1⁄4 @v : (8)
Þ
This expression follows simply because @Jz=@v 1⁄4 vdz= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ
v2 􏰂v2? 1⁄4v dz=vz 1⁄4vsb. The profile of vsb in Fig. 8(b) is obtained using Eq. (8). In this profile, we notice the line where vsb is enhanced. This line coincides with orbits for which vz ’ 0 for zt 1⁄4 0. These orbits have a stagnation point
at zt 1⁄4 0, yielding a logarithmic singularity in the sb. Orbits 􏰍 pffiffiffiffi􏰄
with v; Jq falling to the left of this singularity do not
reach zt 1⁄4 0 in their orbit motion and, as discussed above, are thus confined to one half of the device.
The ion speeds will change during modification of the equilibrium, which can be quantified through the changes imposed on Jz
pffiffiffiffi