Page 14 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-14 Egedal et al.
Phys. Plasmas 25, 072510 (2018)
The effectiveness of MP is largely determined by the profiles of A􏰁0;jðvÞ representing the effective drive for particu- lar perturbations Plj. Note that for calculating A􏰁0;j, we use 􏰆ð@Jz=@tÞ=v 1⁄4 xsbDv, where the relevant profiles of sbDv are shown in Figs. 9(e) and 9(f). The resulting A􏰁0;j for paral- lel compression is shown in Fig. 14(c). Because C0;0 1⁄4 0 we have that F0 1⁄4 0, such that A􏰁0;0 becomes unimportant (and is therefore not shown). In general, we note that the A􏰁0;j-terms represent a measure of the coupling of the sbDv profile in Fig. 9(f) to the basis functions Plj in Fig. 13. Because of the steady increase in sbDv as a function of n, there is a strong coupling to Pl1 and the A􏰁0;1 term dominates in Fig. 14(c). When evaluating the sum in Eq. (30), we then only need to consider j 1⁄4 1.
Still for the case of parallel compression, the profiles of B􏰁0;j are shown by the full lines in Fig. 14(d), while the dashed lines correspond to Eq. (35). In general, the B􏰁0;j- terms are not particularly well-approximated by Eq. (35). This indicates that our averaging over the n dependency in sb did indeed cause Eq. (33) not to be fully consistent with par- ticle conservation. Nevertheless, Eq. (35) for j 1⁄4 1 yields val- ues within a factor of 2 for v<4 and is quite accurate for v > 4, further providing evidence that the pumping process is well approximated by the particle conserving form Eq. (34).
To summarize the results for the particular case of 1% parallel compression, we have found T0;0 ’ 720=v1=2; J0 ’ 15v3=2; D/B ’ 0:06=v0:8; A0;1 ’ 2, such that A1;0 ’ 4A0;1 ’ 8, C1;1 1⁄4 􏰆4D/B lj1 ðlj1 þ 1Þ ’ 0:69=v0:8. Because A0;1 􏰉 A0;j for j>1, we only include the j1⁄41 term in Eq. (34) to get
FIG. 15. Considering pumping by a 1% parallel compression, the efficiency of pumping, characterized by sb0Dv=ðvxÞ, is shown as a function of x=xb0. Here, xb0 is the average angular bounce frequency for v 1⁄4 1. The blue line is evaluated with v 1⁄4 1, while the red line is for v 1⁄4 10.
is thus possible to couple the heating power preferentially to the more energetic part of the fast ion population.
For perpendicular compression, the profiles of A􏰁0;j and
B􏰁0;j in Figs. 14(e) and 14(f) show that higher orders Plj terms
are excited. This is consistent with the more structured form
of sbDv in Fig. 9(e), strongly peaked along the trapped/pass-
ing boundary, showing that perpendicular compression inter-
actsstronglywithstagnationorbits.ThevariationsinA􏰁0;j
and B􏰁0;j as functions of v are related to the decrease in n for
the trapped/passing boundary as the velocity is increased,
impacting the coupling to the basis functions, Plj ðnÞ. This
explains the strong variation of A􏰁0;j and B􏰁0;j as v is increased.
We note that because Fj / A􏰁0;j, each term in the sum of Eq.
(34) scales with A􏰁2 . The negative signs of A􏰁0;j are therefore 0;j
not associated with a reduction in heating efficiency. Because the combined amplitude for the A􏰁0;j-terms is similar to the A􏰁0;1 for the parallel compression case, we con- clude that perpendicular and parallel compressions will have similar efficiencies. However, the larger values of j excited for perpendicular compression yield larger values of Cj;j / ljðlj þ 1Þ. In turn, the larger values of j are associated with higher effective scattering rates Cj;j =T0;0 . Compared to Dv=x in Fig. 15, we therefore expect a broader profile with the range of effective frequencies reaching a factor of l5ðl5 þ1Þ=ðl1ðl1 þ 1ÞÞ ’ 10 higher values. On the other hand, as mentioned above, the perpendicular compression most strongly affects the near stagnation orbits for which sb 􏰉 hsbin 1⁄4 T0;0. It is therefore possible that the reduced model considered here is not appropriate for the perpendicu- lar compression case. In particular, it is likely that T0;0 in Eq. (34) should be replaced by a larger effective bounce time for the participating ions, reducing the efficiency of heating by these perpendicular compressions. This issue will be the sub-
ject of future investigations.
C. Global heating rate
To determine the overall efficiency of MP, we apply Eq. (36) to compute the relative increase in energy for the ion population for each pump cycle
@f0s x2 @A0;1vJ0 A1;0􏰃eff @f0s @t1⁄42T0;0vJ0@v T0;0 x2þ􏰃2 @v;
eff
1@2 @f0s v x2􏰃eff
’v2@vvDv@v; Dv1⁄465000x2þ􏰃2 ; (36) eff
where we have introduced the effective scattering frequency 􏰃eff 􏰈 􏰆C1;1 =T0;0 ’ 10􏰆3 =v0:3 . Note that A0;1 and A1;0 are both proportional to the magnitude of the compression DB=B, such that the effectiveness of the pumping scales like ðDB=BÞ2 .
In Eq. (36), the function Dv is the diffusion coefficient in velocity space induced by pumping, and it provides a direct measure of the pumping efficiency as a function of the angu- lar pump frequency x. For a practical implementation of magnetic pumping, at fixed DB=B the input power in heating the system is likely to scale proportionally to x. Then, the efficiency of the system can be optimized by selecting x such as to maximize Dv=x. For convenience, we introduce the nominal bounce time sb0 1⁄4 720 characteristic for ions with v ’ 1. Likewise, we introduce the nominal bounce angular frequency xb0 1⁄4 2p=sb0 for normalizing the drive frequency. In Fig. 15, we observe how Dv=x is optimized for x=xb0 1⁄4 sb0 􏰃eff =ð2pÞ. For v 1⁄4 1, we have sb0 􏰃eff =ð2pÞ ’ 0:11, while for v 1⁄4 10 we find sb0 􏰃eff =ð2pÞ ’ 0:055. This shift to lower frequencies at higher velocities is caused by the v dependency of D/B / v􏰆0:8, stronger than the falloff of sb / v􏰆0:5. By choosing a relatively low pump frequency, it