Plasma fluid equations

PHYS 7500 Plasma Transport Theory #3 c○Jeong-Young Ji 9
collision operator,
A 0p,0k
ei
= n p−1
∑
e
3µ c 1 1 p−1,m
τ m!(bk−m− 2
k
− 1 1
ei θ bk−m+ 2
k
) (50)
m=0
A 01,1k
ei
A 1p,00
ei
A lp,lk
ei
−δ p1 δ k0
n e
τ ei
2
( Vei
V ei
v Te
) 2,
(51)
= − n e
b k+1/2
k
·,
τ ei v Te
(52)
= − n e
3b p+ 1 V
2 ei
p ,
τ ei v Te
(53)
= − n e
τ ei
3
4 l(l + 1) p∑
m=0
and for the ion-electron collision operator,
A lp,lk
ie
= n √
i µ
[ ( 1
)
2
τ ie θ θ − 1 λ l+1
A 10,00
ie
= n i
3
τ ie θ
c l pm (l + m − 1)!bk−m+ 1 2
k
, for l ≠ 0, (54)
p−1 δ p−1,k − 1 θ
( ) ]
l + 2p λ l p δ pk , (55)
V ei
v Te
, (56)
B 01,0k
ie
= n i µ
( 2k + 1
)
3√
− 1 b k− 1 2
k
, (57)
τ ie θ θ
B 10,1k
ie
= n i
τ ie
3
2θ bk+ 1 2
k
, (58)
B 20,2k
ie
= n i
τ ie
3
2√ µ
θ
[ 3
]
3
θ bk+ 2
k
− b k+ 1 2
k
, (59)
where V ei = V e −V i , µ = m e /m i , θ = T e /T i , and b p q = p!/q!(p−q)!. We rewrite
the formulas which are necessary for calculating Q e , Q i , R e and R i :
 01,00
ei
= 1
√
2
[
3µ(1 − 1 (
τ ei 3 θ ) − 2 Vei
) 2 ]
, (60)
v Te
 01,1k
ei
n 1k
e = √ 2 Â 10,1k
3
ei
V ei
· n 1k
e
v , (61)
Te
 10,00
ei
= − 1 √ V ei 2 , (62)
τ ei v Te
√
 10,1k
ei
= − 1 3(k + 1/2)!
τ ei (2k + 3)k!(1/2)! , (63)
 10,00
ie
= √ 2 1 n e m e v Te V ei
, (64)
n i τ ei m i v Ti v Te
√
3(k + 1/2)!
m i v Ti (2k + 3)k!(1/2)! , (65)
ˆB 10,1k
ie
= 1
τ ei
m e v Te