Plasma fluid equations

PHYS 7500 Plasma Transport Theory #3 c○Jeong-Young Ji 11
Appendix: how to write the moment equations
(0,1) equation:
√ √
√3
σ1 1 = 5 4 , ˆΞ 0 10 = − 2 , Û0 c10 = − 2
3 , ˆΨ 0+
11 = 5
6 , ˆΦ 0+
11 = √ 15
2 √ , 2 Û0+
10 = − 2 √
3
∞∑
ˆD 1kn 0 0k = d t }{{}
n 01 +(d t lnT)(ˆΞ 0 11 }{{}
n 01 +ˆΞ 0 10n 00 ) + ∇ · V(Û0 c11 }{{}
n 01 +Û0 c10n 00 )
0
0
0
k=0
∑
k
+(∇V) · Û0 l1k n 0k + (∇V) ·
}{{}
Û0 r1k
}{{}
0
0
3
= −√
2 n 1 √ √
2 2
T d tT −
3 n∇ · V = − 1
3 T (3 2 nd tT + nT ∇ · V)
ˆD 0+
1k · n1k = v T ∇ · (ˆΨ 0+
11 n11 +
∑
k
∑
k
=
=
ˆΨ
0+
10 n10
}{{}
0
+v −1
T [ q m (E + V × B) − d tV] ·
ˆD 0++
1k
: n 2k = ∇V : Û0+ 10 n20 = − 2 √
3
∇V :
ˆD 0−
1k n−1k = 0,
(1,0) equation:
σ0 2 = 1 1−
2
, ˆΘ 00 = −√ 2,
∑
k
k
n 0k
) + v T ∇ lnT · (ˆΦ 0+
ˆΘ
0+
10 n10 }{{}
0
11 n11 +
ˆΦ
0+
10 n10
√ √ √
5 15
5
6 v T ∇ · n 11 +
2 √ 2 v T ∇ lnT · n 11 =
6 v T
√ √
5 1
2
6 vT
2 ∇ · (vT 3 1
n11 ) = −
3 T ∇ · q
√ √
2 2
2T π = − 1
3 T ∇V : π
∑
ˆD 0−−
1k
n −2k = 0
1− ˆΨ 00 = √ 1 1−
2
, ˆΦ 00 = √ 1 2
,
ˆΨ
1+
00
= 1, ˆΦ
1+
00 = 1
∞∑
ˆD 0k 1 n1k = d t }{{}
n 10 +(d t lnT)ˆΞ 1 00 }{{}
n10 +∇ · VÛ1 c00 }{{}
n10
k=0
0
0
0
+(∇V) · Û1 l00 }{{}
n10 +(∇V) · Û1 r00 }{{}
n10 = 0
0
0
ˆD 1+
0k · n2k 1+
= v T ∇ · ˆΨ 00 n20 1+
+ v T ∇ lnT · ˆΦ 00 n20 + v −1
= v T ∇ · n 20 + v T
T
}{{}
0
(∇ · n 11 + 3 ∇ lnT · n11)
2
T [ q m (E + V × B) − d tV] ·
∇T · n20 a = v √
T
2
T ∇ · (Tn20 ) = ∇ · π
mv T
)
ˆΘ
1+
0k
}{{}
0
n 2k