Page 4 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-4 Egedal et al.
C. The Jq action integral
As discussed above, because of the rapid q-oscilla- tions the vz component can be considered constant during each individual oscillation cycle, permitting the above analysis based on the effective potential, v2/ðq; z; p/Þ. Meanwhile, from the ðq;zÞ-profile of the effective poten- tial, we can also deduce information about the orbit dynamics in the z-direction. As an example, consider the trajectory in Fig. 3, initialized in the midplane with v2? 1⁄4 60 and v2 1⁄4 120. Near the midplane, the turning points fall on the v2/ 1⁄4 60 contour. For larger values of z, energy from v2z is transferred to v2?, and the orbit reaches its extreme values of z as the turning points in the q-oscil- lations reach the v2/ 1⁄4 120 contour.
The two green lines in Fig. 3 represent the loci of the orbit turning points where vq 1⁄4 0. By evaluating v2/ðq;z;p/Þ along such loci, we determine the perpendicular orbit energy v2?1⁄4v2qþv2/1⁄4v2/asafunctionofz.Inturn,giventhatvis constant, we also find v2z ðzÞ 1⁄4 v2 􏰂 v2?ðzÞ.
The above observations motivate the development of methods by which the loci of the turning points for the q- oscillations can be determined. This, in turn, will provide an effective potential v2?ðzÞ for the particle dynamics in the z- direction. The key to this analysis is again the separation of time scales between the oscillations in q and the oscillations in z,16,17 from which it follows that the action integrals Jq 1⁄4 Þ vqdq and Jz 1⁄4 Þ vzdz are adiabatic invariants in the limit where 􏰋 1⁄4 qs=zs ! 0.
For realistic scenarios where 􏰋 is finite, Jq and Jz may not be strictly invariant over the full orbit motion. The break- down of Jq and Jz as invariants can occur when an ion reaches the surface where Bz is changing sign (Bz 1⁄4 0) with vq 􏰁 v/. In this case, the ion will experience the force of v/Bq for a relatively long period of time, yielding a non- adiabatic transfer of energy between v2z and v2? . The described dynamics associated with Bz changing sign and vq 􏰁 v/ are directly related to the orbit bifurcations dis- cussed above. The related pitch-angle-mixing will be charac- terized in subsequent sections, but for now we will explore how ion orbits in the FRC are characterized by the variables ðv; Jq ; p/ Þ. Thus, we first consider Jq to be a fully conserved
FIG. 3. Example of a trajectory initialized with v2? 1⁄4 60 and v 1⁄4 120. The value of v2? as a function of z can be determined as the value of v2/ along the loci of the turning points, marked by the green lines.
Phys. Plasmas 25, 072510 (2018)
adiabatic invariant, and only later will we add back the effects of Jq not being fully conserved.
We find it useful to establish the relationships between ðqt; ztÞ and Jq. Again, for turning points, we have vqðqt; ztÞ 1⁄4 0, and it follows that v2? 1⁄4 v2/ðqt; ztÞ. Furthermore, given that v? is a slowly varying function of z and that z does not change significantly over a q-oscillation, we may consider v? a constant at this short time scale. This assumption is sim- ilar to that made in regular guiding center theory, where var- iations in v? vanish to second order in ql=RB, where ql is the Larmor radius and RB is the magnetic field radius of curva- ture. Conveniently, these observations allow us to obtain an expression for v2q as a function of q for a q-oscillation with a turning point at ðqt ; zt Þ
v2qðq; qt; ztÞ 1⁄4 v2/ðqt; ztÞ 􏰂 v2/ðq; ztÞ: (6)
Figure 4(a) shows vq ðq; qt ; zt Þ of Eq. (6), calculated using qt 1⁄4 47 and applying the value of p/ corresponding to the orbit in Fig. 1.
FIG. 4. (a) Profile of vqðq;qt;ztÞ given in Eq. (6) evaluated with radial turn- ing points at qt 1⁄4 47. (b) Profile of log10ðJqÞ defined in Eq. (7) as a function of the turning points positions ðqt;ztÞ. The magenta line indicates where orbit bifurcations occur. (c) Zoomed-in view of log10 ðJq Þ, overlayed with the orbits of Fig. 2(e). The loci of turning points for these orbits are observed to coincide with contours of constant Jq,.