Drift kinetic equation and neoclassical transport theory
Drift kinetic equation and neoclassical transport theory
Drift kinetic equation and neoclassical transport theory
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PHYS 7500 Plasma Transport Theory #6 c○Jeong-Young Ji 3<br />
The corresponding unit vectors can be written as<br />
ŝ = e 1 sin γ − e 2 cosγ (10)<br />
<strong>and</strong><br />
ˆρ = b × ŝ = e 1 cosγ + e 2 sin γ. (11)<br />
Multiplying the Lorentz force <strong>equation</strong><br />
by s, v, <strong>and</strong> ˆρ, we derive<br />
d<br />
dt (u + s) = q E − Ωsˆρ, (12)<br />
m<br />
dµ<br />
dt<br />
dU<br />
dt<br />
dγ<br />
dt<br />
= − µ B<br />
Here, in Eq. (15), we have used<br />
dB<br />
dt − mu<br />
B s · db<br />
dt + q s · E, (13)<br />
B<br />
= q(v · E + dΦ ), (14)<br />
dt<br />
= −Ω + e 1 · de 2<br />
dt − u db ˆρ ·<br />
s dt + q ˆρ · E, (15)<br />
ms<br />
ˆρ · dŝ<br />
dt = dγ<br />
dt − e 1 · de 2<br />
dt . (16)<br />
Then the gyroaverages are<br />
〈 〉 dµ<br />
= − µ ∂B<br />
dt B ∂t , (17)<br />
〈 〉 dU<br />
( ∂Φ<br />
= q<br />
dt ∂t − v ‖ · ∂A )<br />
, (18)<br />
∂t<br />
〈 〉 dγ<br />
= −Ω + 〈e 1 · ė 2 〉 − u dt<br />
2 (e 2e 1 − e 1 e 2 ) : ∇b, (19)<br />
(ab : cd = a · db · c in the Note #6)<br />
¯L = ∂ ∂t + v ‖ · ∇ − µ B<br />
∂B<br />
∂t<br />
<strong>and</strong> the lowest order drift <strong>kinetic</strong> <strong>equation</strong> for f 0 is<br />
∂<br />
( ∂Φ<br />
∂µ + q ∂t − v ‖ · ∂A ) ∂<br />
∂t ∂U , (20)<br />
∂f 0<br />
∂t + v ‖ · ∇f 0 − µ ∂B ∂f<br />
(<br />
0 ∂Φ<br />
B ∂t ∂µ + q ∂t − v ‖ · ∂A ) ∂f0<br />
∂t ∂U = ¯C(f 0 , f 0 ), (21)<br />
where e 1 · ė 2 <strong>and</strong> (e 2 e 1 − e 1 e 2 ) : ∇b terms do not appear (discuss!).<br />
Following Hazeltine 2003, the indefinite γ-integrals for ˜f 1 can be obtained<br />
by using<br />
ŝ = − ∂ˆρ ∂ŝ<br />
, ˆρ = (22)<br />
∂γ ∂γ