Page 12 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-12 Egedal et al.
1 ð1
Phys. Plasmas 25, 072510 (2018) important for the evolution of the slowly evolving fs compo-
Tk;jðp/; vÞ 1⁄4
ak
Ak;jðp/; v; tÞ 1⁄4 􏰄
j
n sbPlj Plk dn; cs
n
1 ð1 n @J
nents. In fact, by taking a time average of Eq. (24) over a pump cycle, we can evaluate directly the impact on fj
z Plj Plk dn; ak ðnc v @t
* !+ X@fs X􏰄􏰏ixt􏰂@f~
11n@J1􏰄n2dJ@Pl (26) z0j
Tjþ <ixAe<j k;j @t k;j @v
Bk;jðp/;v;tÞ 1⁄4 􏰄ak n v @t 2nJ0 dv @n Plkdn;
j*j+t
ðc
11
Ck;jðp/; vÞ 1⁄4 􏰄 n 4D/B ljðlj þ 1ÞPlj Plk dn;
X􏰄 􏰂􏰋􏰆
þ < ixB􏰏 eixt < f~ k;j j
1⁄4C fs: (29) k;k k
When carrying out the time averaging, we encounter terms of the form
akn jt c
1⁄4 􏰄4D/B lkðlk þ 1Þdk;j:
The kinetic equation in Eq. (24) provides a set of cou- pled differential equations for the functions fj ðv; tÞ. The form of Eq. (24) is similar to that studied by Lichko et al.11 in the context of solar wind energization. However, one difference is the appearance of the tensor Tk;j accounting for effects related to sb’s dependency on n.
Following the approach of Ref. 11, we consider periodic perturbations of the equilibrium providing a periodic drive
ixt 􏰏
through the terms Ak;jðv; tÞ 1⁄4 ixe Ak;jðvÞ and Bk;jðv; tÞ
1⁄4 ixeixtB􏰏k;jðvÞ. We further assume that each fj is comprised of a slowly varying component and a rapidly varying compo- nent, denoted hereafter as
f 1⁄4fsðv;tÞþf~ðvÞeixt: (27) jjj
The Ck;j terms account for the reduction in anisotropy due to pitch-angle-mixing. Through the approximation intro- duced by averaging out the n dependency of hðD/BÞ2i ’ D/B ðp/ ; vÞ, there is no cross-coupling introduced through the mixing process between the Plj terms, and only the diag- onal terms Ck;k are finite. While the analysis can readily be generalized, it is not expected that this assumption will affect the accuracy of the pumping model. The Ck;k functions include the factors lkðlk þ 1Þ corresponding to a more rapid isotropization of smaller n-wavelength perturbations. Mathematically, it is also clear that the grPowing lkðlk þ 1Þ- factors ensures that the expansion for f 1⁄4 j fj Plj converges rapidly.
With the assumption that f1 􏰃 f0, which is generally jus- tified for pumping with DB=B < 0:1, we only consider the action of pumping directly on the background distribution f0. Because B􏰏k;0 1⁄4 0, when inserting fj in Eq. (27) into Eq. (24) the main drive terms are those including A􏰏k;0, yielding per- turbations proportional to Plk eixt. Using the orthogonality of the Plk set and Eq. (24), we can obtain a separate equation for each value of k
h<ðieixtÞ <ðf~ eixtÞi 1⁄4 h<ðeixtÞ <ð􏰄if~eixtÞi jt jt
ix
Tk;jf j þ ixAk;0 @v 1⁄4 Ck;kf k:
(28)
Equation (28) is linear in f~ and is readily inverted for sj
x2T0;0 þ Cj;j @v Furthermore, we then find that
1⁄4
Equation (29) then becomes
X @fjs X1 @
T þ xA􏰏
k;j@t 2 k;j@vj
jf~j
j cos ðargð􏰄if~ÞÞ:
2j
jf~j cos ðargð􏰄if~ÞÞ j
jj
þ X 1 xB􏰏 jf~j cos ðargð􏰄if~ÞÞ 1⁄4 C
j 2 k;j j j k;kk
f s:
(30)
X~􏰏@f0s ~
Together, Eqs. (28) and (30) provide an infinite set of coupled second order partial differential equations, describ- ing a complete pumping model for FRC plasmas including pitch-angle-mixing due to non-adiabatic particle motion. In the sections above, we have shown how all quantities in model are numerically accessible using the framework of the ðv; p/ ; Jq Þ constant of motion variables.
IV. APPLICATION OF THE PUMPING MODEL A. A reduced pumping model
The model described by Eqs. (28) and (30) is suitable for numerical evaluation at particular pumping frequencies. However, averaging over the n dependency of sb, the model can be simplified significantly, providing improved physical insight. From the profiles in Figs. 9(a) and 9(d), we observed the strongest n-variations of vsb along the trapped/passing boundary. In fact, orbits at this boundary (reaching the mid- plane z 1⁄4 0 with vz ! 0) are stagnation orbits characterized by a logarithmic singularity in their bounce time, sb ! 1. Thus, the averaging of sb over n mostly impacts the accuracy by which the small subset of near-stagnation-orbits is repre- sented in the model. With the simplification of replacing sb by T0;0 􏰆 hsbin, it is clear from Eq. (26) that Tk;j 1⁄4 hsbindk;j, permitting a closed form solution of Eq. (28)
~ ðiCj;j􏰄xT0;0ÞxA􏰏j;0 @f0s
fj1⁄4 2 2 : (31)
j
given x and f0 . It is also clear from Eq. (28) that because
Ck;k is a factor of i (90􏰊) out of phase compared to the drive
A , the solutions f~ will be offset in phase from A . These k;0 j k;0
phase shifts are important because when inserting the real part <ðf~P eixtÞ back into the real part of Eq. (24), they yield
non-vanishing time averages of the terms involving f~, j
jf~j cos ðargð􏰄if~ÞÞ 1⁄4 xF ; jjj
j lj
with