Page 11 - Theory of ion dynamics and heating by magnetic pumping in FRC plasma

072510-11 Egedal et al.
Phys. Plasmas 25, 072510 (2018)
well characterized by nc 1⁄4 0:4. Furthermore, D/B ðv; p/Þ is obtained by averaging hðD/BÞ2i over n for n > nc.
Using the substitution x 1⁄4 2n2 􏰂 1, the scattering opera- tor in Eq. (16) can then be written as
8< 0 ; 0 􏰆 n 􏰆 n c sbLmix1⁄4:D @ð1􏰂x2Þ@; n 􏰆n􏰆1: (18)
We recognize Eq. (18) as the Legendre operator for which the equation LmixPl þ kPl 1⁄4 0 is solved by the Legendre functions, Pl. For the typical case where 􏰂1 􏰆 x 􏰆 1, the index l is an integer. However, in our case, the operator only applies in the interval 2n2c 􏰂 1 􏰆 x 􏰆 1, and the appropriate non-integer order lj of the Legendre functions Plj must be determined by a boundary condition at xc 1⁄4 n2c 􏰂 1.
To obtain the boundary condition for Pl, we note that in the following we will solve the full pumping equation in Eq. (12) through the use of the expansion:
/B@x @xc
fðv;p/;nÞ1⁄4Xfjðv;p/ÞPljð2n2 􏰂1Þ: j
(19)
@􏰌􏰌􏰌 @􏰌􏰌􏰌 @n􏰌􏰌􏰌@􏰌􏰌􏰌 @v􏰌 1⁄4@v􏰌þ@v􏰌@n􏰌;
Using that sb Lmix Plj 1⁄4 􏰂4D/B lj ðlj þ 1ÞPlj , we
now
eigenfunctions
X 1@Jz
FIG.
􏰂1Þ 1⁄4 0. The eigenvalues are kj 1⁄4 ljðlj þ 1Þ, where the non-integer values of lj are determined from Eq. (20), here for nc 1⁄4 0:4.
where
13. Eigenfunctions
of the
equation
jk jk
Lmix Plj ð2n2 􏰂 1Þ þ kj Plj ð2n2
Because n is conserved for n < nc, there can be no diffusion across the x 1⁄4 xc boundary. The appropriate boundary condi- tion for the basis functions is then
Jq 􏰌n Jq v 􏰌 1⁄4 @ 􏰌􏰌 þ 1 􏰂 n2 dJ0 @ 􏰌􏰌 :
@ 􏰌􏰌􏰌 @x Plj ðxÞ􏰌
@v􏰌n 2nJ0 dv @n􏰌v After multiplying Eq. (12) by sb, we find
1⁄4 0:
(20)
XX2! @fj 1@Jz @fj 1􏰂n dJ0 @Plj
x1⁄4xc
sbPlj@t􏰂v@t Plj@vþ2nJ0 dvfj@n jj
This equation is readily solved numerically for lj. For exam- ple, Table I provides the first eigenvalues and normalization factors for nc 1⁄4 0:4, such that xc 1⁄4 􏰂0:68. The correspond- ing Plj ð2n2 􏰂 1Þ solutions are shown in Fig. 13. These eigen- functions form a complete set, allowing any distribution function f ðv; p/ ; nÞ to be expanded in the form given by Eq. (19). Furthermore, the functions are orthogonal on the inter- val defined through
(21)
(22)
can simplify the scattering term in the kinetic equation in Eq. (12). However, as a subtlety, given we are now using v and n as the independent variables, the @=@vjJq term in Eq. (12)
1⁄4 􏰂4D/B ljðlj þ 1Þ X Plj fj; (23) j
ð1 nc
where again Plj 1⁄4 Plj ð2n2 􏰂 1Þ.
We next expand the factors in front of fj in terms of
the basis functions. Because p/ is fully conserved, we can do this separately for each p/. The kinetic equation then reads
nPlj ð2n2 􏰂 1Þ Plk ð2n2 􏰂 1Þ dn 1⁄4 ajdj;k; where aj is the normalization factor
ð1
aj 1⁄4 nPljð2n2 􏰂1ÞPljð2n2 􏰂1Þdn:
nc
XXT P @fjþXXA P @fj k;j lk @t k;j lk @v
þXXBk;jPlkfj 1⁄4 XXCk;jPlkfj; (24) jk jk
XTk;jPlk 1⁄4sbPlj; k
Ak;jPlk 1⁄4􏰂v@tPlj;
XBk;jPlk 1⁄4􏰂1@Jz1􏰂n2dJ0@Plj ;
Xk v@t2nJ0dv@n
C P 1⁄4􏰂4D lðl þ1ÞP : (25)
k
Given the orthogonality of the Plj set expressed in Eq. (21), the functions Tk;j; Ak;j, Bk;j, and Ck;j are readily isolated
must be evaluated appropriately
TABLE I. Eigenvalues and normalization factors of the Plj for nc 1⁄4 0:4.
k
j012345
lj 0 1.27 2.59 3.94 5.28 6.63 aj 0.42 0.11 0.061 0.042 0.032 0.026
k;j lk /B j j lj