Drift kinetic equation and neoclassical transport theory

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PHYS 7500 Plasma Transport Theory #6 c○Jeong-Young Ji 2 Then the kinetic equation can be written as (L − C)f = δ −1 Ω ∂f ∂γ (3) where γ is the azimuthal angle of the velocity variable along b = B/B, and δ is introduced explicitly for the perturbation expansion. Note that Ω = qB/m is the largest frequency scale in the theory. Now we expand the distribution function as The δ −1 order equation yields f = f 0 + δf 1 + δ 2 f 2 + · · · . (4) Ω ∂f 0 ∂γ = 0. (5) This means that f 0 is gyrophase independent, f 0 = ¯f 0 ≡ 〈f 0 〉, where 〈A(v)〉 ≡ 1 2π ∫ 2π 0 A(v)dγ (6) denotes gyroaveraging A. Here it should be emphasized that the average is taken at a fixed position x, which is different from when taken at a fixed guiding center in the gyrokinetic equation. The δ 0 -order equation is Lf 0 − C(f 0 , f 0 ) = Ω ∂f 1 ∂γ . (7) Its gyroaverage, ¯Lf 0 − ¯C(f 0 , f 0 ) = 0, (8) provides the equation for the lowest order distribution f 0 . Then the gyrophasedependent part of f 1 can be found from (7) (with other variables suppressed): ˜f 1 (γ) = 1 Ω ∫ γ dγ [Lf 0 − C(f 0 , f 0 )] = 1 Ω ∫ γ dγ [( L − ¯L ) f 0 ] [use Eq. (8)] ≡ L H f 0 . At this point let us investigate L to calculate ¯L. For convenience, we use u = v ‖ and s = v ⊥ , and define the gyroradius vector ρ by ρ = b × v Ω = ρˆρ, ρ = ms |q|B , ˆρ = s ˆρ. (9) |Ω|
PHYS 7500 Plasma Transport Theory #6 c○Jeong-Young Ji 3 The corresponding unit vectors can be written as ŝ = e 1 sin γ − e 2 cosγ (10) and ˆρ = b × ŝ = e 1 cosγ + e 2 sin γ. (11) Multiplying the Lorentz force equation by s, v, and ˆρ, we derive d dt (u + s) = q E − Ωsˆρ, (12) m dµ dt dU dt dγ dt = − µ B Here, in Eq. (15), we have used dB dt − mu B s · db dt + q s · E, (13) B = q(v · E + dΦ ), (14) dt = −Ω + e 1 · de 2 dt − u db ˆρ · s dt + q ˆρ · E, (15) ms ˆρ · dŝ dt = dγ dt − e 1 · de 2 dt . (16) Then the gyroaverages are 〈〉 dµ = − µ ∂B dt B ∂t , (17) 〈〉 dU ( ∂Φ = q dt ∂t − v ‖ · ∂A ) , (18) ∂t 〈〉 dγ = −Ω + 〈e 1 · ė 2 〉 − u dt 2 (e 2e 1 − e 1 e 2 ) : ∇b, (19) (ab : cd = a · db · c in the Note #6) ¯L = ∂ ∂t + v ‖ · ∇ − µ B ∂B ∂t and the lowest order drift kinetic equation for f 0 is ∂ ( ∂Φ ∂µ + q ∂t − v ‖ · ∂A ) ∂ ∂t ∂U , (20) ∂f 0 ∂t + v ‖ · ∇f 0 − µ ∂B ∂f ( 0 ∂Φ B ∂t ∂µ + q ∂t − v ‖ · ∂A ) ∂f0 ∂t ∂U = ¯C(f 0 , f 0 ), (21) where e 1 · ė 2 and (e 2 e 1 − e 1 e 2 ) : ∇b terms do not appear (discuss!). Following Hazeltine 2003, the indefinite γ-integrals for ˜f 1 can be obtained by using ŝ = − ∂ˆρ ∂ŝ , ˆρ = (22) ∂γ ∂γ
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Drift kinetic equation and neoclassical transport theory

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