Page 16 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-16 Egedal et al.
these two types of orbit motions, leading to non-adiabatic changes in Jq-action variable. Because the ion energy is con- served during these Jq changes, we denote this process as pitch-angle-mixing characterized by mixing operator Lmix.
The pitch-angle-mixing built-in to the ion dynamics in FRC plasma turns out to be highly beneficial to the effective- ness of MP for heating the plasma. The ion pressure anisot- ropy that builds through modification of the magnetic equilibrium is isotropized with a characteristic 1=e decay time of about ten orbit bounce times. By applying periodic perturbations of the equilibrium at this frequency, the pres- sure anisotropy remains finite and shifted in phase from the equilibrium perturbations. As a result when averaged over time, the perturbations are associated with net mechanical work against the ion pressure components and are thus asso- ciated with direct heating of the ion population.
The details of this heating process are captured by the kinetic framework derived here. While the kinetic frame- work is suitable for comprehensive modeling of the pump- ing process, we also present a reduced model which allows for more direct estimates of the heating rates that can be expected by MP. In contrast to collisional pumping, the col- lisionless pitch-angle-mixing in FRC plasma causes the absolute heating efficiency of MP to increase with the plasma temperature. Applying 1% linear compression of the equilibrium, it is found that for each pump cycle the fast ion energy in the FRC is likely to increase by a factor of 2:5 􏰃 10􏰂5 , with an optimal pump frequencies on the order of 10–30 kHz. The heating rates scale with square of the perturbation amplitude, and our numerical example shows that MP can deliver megawatts of heating power directly to the ions. By combining the full framework for MP with the output of MHD models predicting the pump perturbations, the level of ion heating can be computed for any particular experimental configurations. Thus, the result of the present paper will be useful for a detailed study and optimization of MP for heating of FRC plasmas. Our pre- liminary results suggest that MP could very well help boosting the ion temperatures in FRC reactors to the levels needed for thermonuclear fusion.
APPENDIX: FRC CONSTANTS OF MOTION VARIABLES A. Jacobian of phase space volumes
Above, we used f ðx; vÞ 1⁄4 f ðU; Jq; p/; tÞ for characteriz- ing the electron phase space distributions. For our manipula- tions, it is important to know the relationship between the phase space volumes ðDxÞ3ðDvÞ3 and DUDJqDp/. For this, we introduce the Jacobian J defined as
and
Phys. Plasmas 25, 072510 (2018) ðx;vÞ ! ðU;l;s;l;/t;/gÞ; (A2)
ðU;l;s;l;/t;/gÞ ! ðU;l;p/;l;/t;/gÞ: (A3)
Here, l is the length along the poloidal (q; Z) projection of the orbit, while s is the spatial coordinate perpendicular to the orbit in the poloidal plane. To eliminate ambiguities, we define the unit vectors at ^s 1⁄4 rp/=jrp/j, and ^l 1⁄4 /^ 􏰃 ^s, where rp/ is evaluated for constant U and l.
To find the Jacobian j1 for the transformation in Eq. (A2) we basically consider the transformation v!ðU;l; /gÞ where /g is the gyrophase. We note
@U1⁄4mv; @l1⁄4mv?; @/g 1⁄4b􏰃v; @v @v B @v v2?
that for xi 1⁄4 ðvk; v?1; v?2Þ and Xi 1⁄4 ðU; l; /gÞ the 2mv mv mv 3
such
Jacobian matrix becomes
@Xj6k ?1 ?27 @x 1⁄44 0 mv?1=B mv?2=B5; i 0 􏰂 v ? 2 = v 2? v ? 1 = v 2?
and it is readily seen that its determinant reduces to j1 1⁄4 m2 jvk j=B. To obtain the Jacobian for the transformation in Eq. (A3), we use xi 1⁄4 ðU;l;sÞ and Xi 1⁄4 ðU;l;p/Þ to get
21 0dp=dU3 @Xj 6 /7
@x1⁄440 1 dp/=dl 5; i0 0 jrp/j
and it is clear that only the terms in the diagonal are impor- tant such that j2 1⁄4 jrp/ j.
We can calculate ðDxÞ3ðDvÞ3 by integrating over the ignorable coordinates /t ; /g , and l. Integration over /t and /g each yields a factor of 2p such that
332þ1
ðDxÞ ðDvÞ 1⁄4 DUDlDp/ 4p 1⁄4 DU Dl Dp/ 4p2
1⁄4DUDlDp4p dl1; / m2 vqz
4p2 1⁄4DUDlDp/ m2 sb:
dlj j ; þ12
dl B
2 þ m2jvkjjrp/j
;
(A4)
ðDxÞ3 ðDvÞ3 J􏰄DUDJDp :
To calculate J , we first assume that the orbits follow standard guiding center theory and are characterized by ðU;l;p/Þ. Using manipulations similar to those in Refs. 18 and 20, we split the coordinate transformation into two main steps
Here, we have used that the poloidal projection of the guid- ing center velocity can be expressed as vqz 1⁄4 jrp/jjvkj=B.
With Eq. (A4), we have used standard guiding center theory to determine that ðDxÞ3 ðDvÞ3 / sb , where sb is the orbit bounce time. We note that in regions where the orbits are well-magnetized, we have
(A1)
q/
qþ2p
l1⁄44p
q þ/g1⁄42p v?rLd/g 1⁄44p2 vqdq:
/g1⁄40
0
To relate this expression to Jq, it is important to recall that in our definition of Jq in Eq. (7) we integrate over all values of