Page 3 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-3
Egedal et al.
Phys. Plasmas 25, 072510 (2018)
merger between the two separate cyclotron orbits at the value of v2? corresponding to the local maximum of v2/. Below, we will refer to such mergers as orbit bifurcations.
The cut in Fig. 2(b) represents the marginal case includ- ing a point where 0 1⁄4 v2/ 1⁄4 @v2/=@q 1⁄4 @2v2/=@q2, marking the transition to the profile type in Fig. 2(c) without any local maximum. For this profile in panel (c), we have v2/ > 0 for all values of q corresponding to betatron orbits characterized by v/ < 0 for all points in their q-oscillations.
Essential for the following analysis, we will define points, ðqt ; zt Þ, where vq 1⁄4 0 as orbit turning points of the rapid q oscillations. As illustrated in panel (c), at a turning point we have v2/ðqt; zt; p/Þ 1⁄4 v2?, and it is therefore clear that v2? and the form of the q-oscillations are fully character- ized by ðqt; ztÞ.
We now consider the central interval jzj < z1, where z1 is defined in panel (d). In this interval, z 1⁄4 constant cuts of the effective potential, and v2/ ðq; z; p/ Þ include a local max- ima. We denote the radial location of the maxima as qbðzÞ, obtained as the solution to Eq. (4), and indicated by the red line in the center of panel (d). All along this line, the inner- most turning points of cyclotron orbits of the two local wells merge to form figure-eight orbits.
Points on the magenta line in panel (d) represent the outer turning points of the merging cyclotron orbits. For each value of z, these outer turning points are obtained by solving
v2/ðq; z; p/Þ 􏰂 v2/ðqbðzÞ; z; p/Þ 1⁄4 0; (5)
yielding two roots different that qb. To clarify the procedure for determining the bifurcation line, panel (a) illustrates how the points on the bifurcation line are found for z 1⁄4 0, corre- sponding to the points marked by matching red symbols in panel (d).
With our knowledge of the bifurcation line, the orbit topology of the q oscillations is now determined by the loca- tion of the turning points ðqt ; zt Þ. In panel (e), we have marked the four regions of distinct topology. Regions a and b correspond to cyclotron orbits, and region c is figure-eight orbits, while betatron orbits are found in the d regions.
The trajectories in panels (e)–(g) are representative of the orbit dynamics in the bulk of the FRC. The orbit in panel (e) with z < 0 is shown in its xy-projection in panel (f). This orbit is initialized in region c and its q-oscillation is at first of the cyclotron type, but transition into betatron motions near its turning point in the z oscillation. The orbit with z > 0 in panel (e) has a more complicated trajectory also evident in its xy-pro- jection in panel (g). This orbit is initialized as a region a cyclo- tron orbit type. In its z-motion, it encounters the bifurcation line and briefly transitions into a region c figure-eight type before reaching region d of betatron motion. In its motion back towards the midplane (z 1⁄4 0), it transitions back into a region c figure-eight orbit, and as it reaches the bifurcation line it transi- tions into a region b cyclotron type orbit.
FIG. 2. (a)–(d) Effective potential pro- file 2Veff=m 1⁄4 v2/ðq;z; p/Þ as intro- duced by Landsman et al. The z-values for the cuts in (a)–(c) are indicated in (d). (e) The topology of the q oscilla- tions can be determined by the loca- tions of the orbit turning points ðqt;ztÞ relative to the bifurcation line. Note that at these turning points vq 1⁄4 0 such that v2? 1⁄4 v2/ðqt;zt; p/Þ. (f) and (g) xy projections of the trajectories in (e).