δf form of Vlasov equation in guiding-center variables

3 δf form of Vlasov equation in guiding-center variables

3.1 Electromagnetic ﬁeld perturbation

Since the deﬁnition of the guiding-center variables (X,𝜀,μ,α) involves the equilibrium ﬁelds B₀ and E₀, to further simplify Eq. (52), we need to separate electromagnetic ﬁeld into equilibrium and perturbation parts. Writing the electromagnetic ﬁeld as

E = E0 + δE

(53)

and

B = B0 + δB,

(54)

then substituting these expressions into equation (52) and moving all terms involving the ﬁeld perturbations to the right-hand side, we obtain

∂fg ∂fg [ q ∂fg] -∂t + v ⋅∂X-+ v ⋅ [λB1 + λB2]fg − m-E0 ∂𝜀 ( ) ( ) + -q(E0 + v× B0 )× e∥ ⋅ ∂fg + q-(v × B0)⋅ eα-∂fg m ( Ω ∂X m) v⊥ ∂α -q ∂fg v⊥∂fg eα-∂fg + m E0 ⋅ v ∂𝜀 + B0 ∂μ + v⊥ ∂α = δRfg, (55)

where δR is deﬁned by

( ) q- (e∥) -∂- q- eα-∂-- δR = − m(δE + v× δB )× Ω ⋅∂X − m (v × δB) ⋅ v⊥ ∂α q ( ∂ v⊥ ∂ eα ∂ ) − mδE ⋅ v∂𝜀-+ B--∂μ-+ v-∂-α . (56) 0 ⊥

Next, let us simplify the left-hand side of Eq. (55). Note that

( ) q-(v× B )⋅ eα-∂fg m 0 v⊥ ∂α q- e∥ ×-v⊥-∂fg = m (v× B0 )⋅ v2⊥ ∂ α ∂f = − Ω--g. (57) ∂α

Note that

q (e∥) ∂fg (E0 × e∥) ∂fg ∂fg m-E0 × Ω- ⋅∂X- = c ---B0-- ⋅∂X- = vE0 ⋅ ∂X-,

(58)

where v_E0 is deﬁned by v_E0 = cE₀ × e_∥∕B₀, which is the E₀ × B₀ drift. Further note that

q-v×-B0- ( e∥) ∂fg ∂fg m c × Ω ⋅ ∂X = [(v × e∥)× e∥]⋅∂X ∂fg = [v∥e∥ − v ]⋅∂X , (59)

which can be combined with v ⋅ ∂f_g∕∂X term, yielding v_∥e_∥⋅ ∂f_g∕∂X.

Using Eqs. (58), (59), and (57), the left-hand side of equation (55) is written as

∂fg ∂fg ∂fg ∂t-+ (v∥e∥ + VE0 )⋅∂X +v ⋅[λB1 + λB2 ]fg − Ω ∂α q ( v⊥ ∂fg eα∂fg) + --E0 ⋅ ------+ ----- ≡ Lgfg, (60) m B0 ∂μ v⊥ ∂α

where L_g is often called the unperturbed Vlasov propagator in guiding-center coordinates (X,𝜀,μ,α).

[Equation (60), corresponds to Eq. (7) in Frieman-Chen’s paper[3]. In Frieman-Chen’s equation (7), there is a term

(E_mac − E₀) ⋅ v

where E_mac is a given macroscopic electric ﬁeld introduced when deﬁning the guiding-center transformation. In my derivation E_mac is chosen to be equal to the equilibrium electric ﬁeld, and thus the above term is zero.]

Using the above results, Eq. (55) is written as

Lgfg = δRfg,

(61)

i.e.

∂fg+ (v∥e∥ + VE0 )⋅ ∂fg − Ω ∂fg ∂t [ ∂X ∂α ] +v ⋅ v× -∂-(e-∥)⋅ ∂fg + ∂μ-∂fg+ ∂α∂fg ∂x Ω ∂X ∂x ∂μ ∂x ∂α q (v ⊥∂fg eα ∂fg) +m-E0 ⋅ B0--∂μ + v⊥-∂α- (e ) ( ) = −-q(δE + v× δB )× -∥ ⋅ ∂fg− q-(v × δB)⋅ eα-∂fg m ( Ω ∂X ) m v⊥ ∂α −-qδE ⋅ v ∂fg+ v-⊥∂fg + eα-∂fg . (62) m ∂𝜀 B0 ∂μ v⊥ ∂α

It is instructive to consider some special cases of the above complicated equation. Consider the case that the equilibrium magnetic ﬁeld B₀ is uniform and time-independent, E₀ = 0, and the electrostatic limit δB = 0, then equation (62) is simpliﬁed as

∂fg+ v∥e∥ ⋅ ∂fg− Ω ∂fg ∂t ∂(X ) ∂α = −-q(δE) × e∥ ⋅ ∂fg (63) m ( Ω ∂X ) −-qδE ⋅ v ∂fg+ v⊥∂fg + eα-∂fg (64) m ∂𝜀 B0 ∂μ v⊥ ∂α

If neglecting the δE perturbation, the above equation reduces to

∂f ∂f ∂f --g + v∥e∥ ⋅--g − Ω--g = 0, ∂t ∂X ∂α

(65)

which agrees with Eq. (21) discussed in Sec. 1.2.

3.2 Distribution function perturbation

Expand the distribution function f_g as

fg = Fg + δFg,

(66)

where F_g is assumed to be an equilibrium distribution function, i.e.,

LgFg = 0.

(67)

Using Eqs. (66) and (67) in Eq. (61), we obtain an equation for δF_g:

Lg δFg = δRFg + δRδFg.

(68)

3.3 Gyrokinetic ordering

To facilitate the simpliﬁcation of the Vlasov equation in the low-frequency regime, we assume the following orderings (some of which are roughly based on experiment measure of ﬂuctuations responsible for tokamak plasma transport, some of which can be invalid in some interesting cases.) These ordering are often called the standard gyrokinetic orderings.

3.3.1 Assumptions for equilibrium quantities

Deﬁne the spatial scale length L₀ of equilibrium quantities by L₀ ≈ F_g∕|∇_XF_g|. Assume that L₀ is much larger than the thermal gyro-radius ρ_i ≡ v_t∕Ω, i.e., λ ≡ ρ_i∕L₀ is a small parameter, where v_t = ∘2T--∕m is the thermal velocity. That is

1-- 1 Fgρi|∇XFg | ∼ O (λ ),

(69)

and

-1-ρ |∇ B | ∼ O(λ1). B0 i X 0

(70)

The equilibrium E₀ × B₀ ﬂow, i.e.,

vE0 = E0 × e∥∕B0 = − ∇ Φ0 × e∥∕B0,

(71)

is assumed to be weak with

|vE0|∼ O(λ1), vt

(72)

3.3.2 Assumptions for perturbations

We assume that the amplitudes of perturbations are small. Speciﬁcally, we assume

δFg- qδΦ- |δB-| 1 F0 ∼ T ∼ B0 ∼ O (λ),

(73)

where δΦ is the perturbed scalar potential deﬁned later in Eq. (79).

We consider low frequency perturbations with ω∕Ω ∼ O(λ¹), then

-1--1∂δFg-∼ O (λ1). δFgΩ ∂t

(74)

The perturbation is assumed to have a long wavelength (much longer than ρ_i) in the parallel direction

-1-|ρe ⋅∇ δF | ∼ O(λ1), δFg i∥ X g

(75)

and have a short wavelength comparable to the thermal gyro-radius in the perpendicular direction

-1-|ρi∇X⊥ δFg| ∼ O(λ0). δFg

(76)

[Combining Eq. (75) and (76), we obtain

k∥-≈ e∥ ⋅∇X-∼ O(λ), k⊥ ∇X⊥

(77)

i.e., the parallel wave number is one order smaller than the perpendicular wave-number.]

In terms of the scalar and vector potentials δΦ and δA, the perturbed electromagnetic ﬁeld is written as

δB = ∇x × δA,

(78)

and

∂δA- δE = − ∇xδΦ − ∂t .

(79)

Then

(∂-δA) δE ∥ = − ∇∥δΦ − ∂t ∥

(80)

(∂δA ) δE ⊥ = − ∇ ⊥δΦ − ---- , ∂t ⊥

(81)

Using the above orderings, it is ready see that δE_∥ is one order smaller than δE_⊥, i.e.,

δE --∥-= O (λ1). δE⊥

(82)

Most gyrokinetic simulations approximate the vector potential as δA ≈ δA_∥e_∥.

δB = ∇x × (δA ∥e∥) ≈ ∇xδA ∥ × e∥.

(83)

We will assume v_tδA_∥∼ δΦ.

3.4 Equation for macroscopic distribution function F_g

The evolution of the macroscopic quantity F_g is governed by Eq. (67), i.e.,

LgFg = 0,

(84)

where the left-hand side is written as

∂F ∂X ∂F ∂V ∂F LgFg = --g+ ---⋅---g+ --- ⋅--g- ∂t ∂t ∂X ∂t ∂V + (v∥e∥ + VE0 )⋅ ∂Fg-+ v⋅[(λB1 + λB2 )Fg ]− Ω ∂Fg ( ∂X ) ∂ α + qE0 ⋅ v⊥-∂Fg-+ eα∂Fg- m B0 ∂μ v⊥ ∂α

Expand F_g as F_g = F_g0 + F_g1 + …., where F_gi ∼ F_g0O(λⁱ). Then, the balance on order O(λ⁰) gives

∂Fg0 = 0 ∂α

(85)

i.e., F_g0 is independent of the gyro-angle α. The balance on O(λ¹) gives

( ) v e ⋅ ∂Fg0+ -qE ⋅ v⊥∂Fg0 = Ω∂Fg1. ∥ ∥ ∂X m 0 B0 ∂μ ∂α

(86)

Performing averaging over α, ∫ ₀^2π(…)dα, on the above equation and noting that F_g0 is independent of α, we obtain

(∫ ) ∫ ( ) ∫ 2π ∂Fg0 q-∂Fg0 2π v⊥- 2π ∂Fg1 0 dαv∥e∥ ⋅ ∂X + m ∂μ 0 dαE0 ⋅ B0 = 0 dαΩ ∂α

(87)

Note that a quantity A = A(x) that is independent of v will depend on v when transformed to guiding-center coordinates, i.e., A(x) = A_g(X,v). Therefore A_g depends on gyro-angle α. However, since ρ_i∕L ≪ 1 for equilibrium quantities, the gyro-angle dependence of the equilibrium quantities can be neglected. Speciﬁcally, e_∥, B₀ and Ω can be considered to be independent of α. As to v_∥, we have v_∥ = ± ∘2-(𝜀−-B-μ)-
0 . Since B₀ is considered independent of α, so does v_∥. Using these results, equation (87) is written

∫ 2π ( ) v∥e∥ ⋅ ∂Fg0+-q ∂Fg0 dαE0 ⋅ v⊥- = 0. ∂X m ∂ μ 0 B0

(88)

Using E₀ = −∇Φ₀, the above equation is written as

∫ 2π ( ) v∥e∥ ⋅ ∂Fg0 + q-∂Fg0 dα − v⊥-⋅∇-Φ0 = 0, ∂X m ∂μ 0 B0

(89)

Note that

∫ 2π dα-1-v⊥ ⋅∇X Φ0 ≈ 0, 0 B0

(90)

where the error is of O(λ²)Φ₀, and thus, accurate to O(λ), the last term of equation (89) is zero. Then equation (89) is written as

∂F v∥e∥ ⋅-g0 = 0, ∂X

(91)

which implies that F_g0 is constant along a magnetic ﬁeld line.

3.5 Equation for δF_g

Using F_g ≈ F_g0, equation (68) is written as

LgδFg = δRFg0 + δR δFg , Nonlin◟ear◝T◜erm◞∼O(λ2)

(92)

where δRδF_g is a nonlinear term which is of order O(λ²) or higher, L_gδF_g and δRF_g0 are linear terms which are of order O(λ¹) or higher. The linear term δRF_g0 is given by

q ( v × δB) (e∥ ) ∂Fg0 q ( ∂Fg0 v⊥ ∂Fg0) δRFg0 = − m- δE + ---c-- × -Ω ⋅-∂X- − m-δE ⋅ v-∂𝜀- + B0--∂μ- , ◟-------------◝◜--------------◞ ◟----------◝◜----------◞ O(λ2) O(λ1)

(93)

In obtaining (93), use has been made of ∂F_g0∕∂α = 0. Another linear term L_gδF_g is written as

LgδFg = ∂δFg-+ (v∥e∥ + VE0 )⋅ ∂δFg-+ v ⋅[λB1 + λB2]δFg − Ω ∂δFg ∂t ∂X ◟-◝∂◜α◞ O(λ1) q ( v⊥ ∂δFg eα ∂δFg) + m-E0 ⋅ B---∂μ--+ v---∂α-- , (94) 0 ⊥

where Ω∂δF_g∕∂α is of order O(λ¹) and all the other terms are of order O(λ²).

Next, to reduce the complexity of algebra, we consider the easier case in which ∂F_g0∕∂μ = 0.

3.5.1 Balance on order O(λ¹): adiabatic response

The balance between the leading terms (terms of O(λ)) in Eq. (92) requires that

∂δF q ( ∂F ) Ω----a= -δE ⋅ v--g0 , ∂α m ∂𝜀

(95)

where δF_a is a unknown distribution function to be solved from the above equation. It is ready to verify that

q ∂F δFa = --δΦ---g0, m ∂𝜀

(96)

is a solution to the above equation, accurate to O(λ). [Proof: Substitute expression (96) into the left-hand side of Eq. (95), we obtain

∂δFa ∂ ( q ∂Fg0) Ω-∂α-- = Ω ∂α- m-δΦ -∂𝜀- = -q∂Fg0 Ω-∂-(δΦ ) m ∂ 𝜀 ∂α = -q∂Fg0 Ω(∇xδΦ )⋅ ∂x (97) m ∂ 𝜀 ∂α

Using

∂x ∂ [ e∥(X)] ∂α- = ∂α- − v × Ω-(X-) = -∂[− v ]× e∥(X-) ∂α Ω(X ) = − v-⊥ (98) Ω

Eq. (97) is written as

∂ δFa q ∂Fg0 v ⊥ Ω-∂α-- = − m--∂𝜀-Ω (∇x δΦ)⋅-Ω- q∂Fg0 ( ∂δA ) = ------ δE + ---- ⋅v⊥ (99) m ∂ 𝜀 ∂t ≈ -q∂Fg0 (δE )⋅v⊥, (100) m ∂ 𝜀

where terms of O(λ²) have been dropped. Similarly, dropping the parallel electric ﬁeld term (which is of O(λ²)) on the right-hand side of Eq. (95), we ﬁnd it is identical to the right-hand side of Eq. (100)]

3.5.2 Separate δF_g into adiabatic and non-adiabatic part

As is discussed above, the terms of O(λ) can be eliminated by splitting a so-called adiabatic term form δF_g. Speciﬁcally, write δF_g as

δFg = δFa + δG,

(101)

where δF_a is given by (96), i.e.,

δF = q-δΦ∂Fg0 , a m ∂𝜀

(102)

which depends on the gyro-angle via δΦ and this term is often called adiabatic term. Plugging expression (101) into equation (92), we obtain

LgδG = δRFg0 − LgδFa+ δR δFg . ◟-----◝◜-----◞ ◟ ◝◜-◞ LinearTerms NonlinearTerms

(103)

Next, let us simplify the linear term on the right-hand side, i.e, δRF_g0 −L_gfδF_a, (which should be of O(λ²) or higher because Ω∂δF_a∕∂α cancels all the O(λ¹) terms in δRF_g0).

L_gδF_a is written

q-∂Fg0 -q ∂Fg0 LgδFa = m ∂𝜀 LgδΦ + m δΦLg ∂𝜀 q-∂Fg0 ≈ m ∂𝜀 LgδΦ, (104)

where the error is of order O(λ³). In obtaining the above expression, use has been made of e_∥⋅∂F_g0∕∂X = 0, ∂F_g0∕∂X = O(λ¹)F_g0, ∂F_g0∕∂α = 0, ∂F_g0∕∂μ = 0, and the deﬁnition of λ_B1 and λ_B2 given in expressions (39) and (40). The expression (104) involves δΦ operated by the Vlasov propagator L_g. Since δΦ takes the most simple form when expressed in particle coordinates (if in guiding-center coordinates, δΦ(x) = δΦ(X−v ×e_∥∕Ω), which depends on velocity coordinates and thus more complicated), it is convenient to use the Vlasov propagator L_g expressed in particle coordinates. Transforming L_g back to the particle coordinates, expression (104) is written

q ∂Fg0 [∂δΦ q ∂Φ ] LgδFa = m--∂𝜀- -∂t-|x,v + v⋅∇x δΦ + m-(E0 + v × B0)⋅∂v-|x [ ] = q-∂Fg0 ∂δΦ-|x,v + v⋅∇x δΦ (105) m ∂𝜀 [ ∂t ( )] q-∂Fg0 ∂δΦ- ∂δA- = m ∂𝜀 ∂t |x,v + v⋅ − δE − ∂t |x,v q ∂Fg0 [∂δΦ ∂v ⋅δA ] = m--∂𝜀- -∂t-|x,v − v⋅δE − ---∂t--|x,v . (106) [ ] = q-∂Fg0 ∂δΦ-− v ⋅δE − ∂v-⋅δA- . (107) m ∂𝜀 ∂t ∂t

Using this and expression (93), δRF_g0 − L_gδF_a is written as

q ( e ) ∂F q ( ∂F ) δRFg0 − LgδFa = − --(δE + v × δB)× -∥ ⋅ --g0− --δE ⋅ v --g0 m [ Ω ∂X ] m ∂𝜀 − -q ∂Fg0 ∂δΦ-− v ⋅δE − ∂v⋅δA-- m ∂𝜀 ∂t ∂t q- ( e∥) ∂Fg0 -q∂Fg0 [∂Φ- ∂v⋅δA-] = − m (δE + v × δB)× Ω ⋅ ∂X − m ∂ 𝜀 ∂t − ∂t ,(108)

where the two terms of O(λ¹) (the terms in blue and red) cancel each other, with the remain terms being all of O(λ²), i.e, the contribution of the adiabatic term cancels the leading order terms of O(λ¹) on the RHS of Eq. (103).

The consequence of this is that, as we will see in Sec. 3.6.1, δG is independent of the gyro-angle, accurate to order O(λ¹). Therefore, separating δF into adiabatic and non-adiabatic parts also corresponds to separating δF into gyro-angle dependent and gyro-angle independent parts.

3.5.3 Linear term expressed in terms of δΦ and δA

Let us rewrite the linear term (108) in terms of δΦ and δA. The δE + v ×δB term in expression (108) is written as

δE+ v × δB = − ∇ δΦ − ∂δA-+ v × ∇ ×δA. x ∂t x

(109)

Note that this term needs to be accurate to only O(λ). Then

δE + v× δB ≈ − ∇xδΦ + v× ∇x × δA,

(110)

where the error is of O(λ²). Using the vector identity v ×∇_x ×δA = (∇δA) ⋅v − (v ⋅∇)δA and noting v is constant for ∇_x operator, the above equation is written

δE + v × δB = − ∇x δΦ+ ∇x (δA ⋅v )− (v ⋅∇x )δA

(111)

Note that Eq. (41) indicates that ∇_xδΦ ≈∇_XδΦ, where the error is of O(λ²), then the above equation is written

δE+ v × δB = − ∇X δΦ + ∇X (δA ⋅v)− (v⋅∇X )δA

(112)

Further note that the parallel gradients in the above equation are of O(λ²) and thus can be dropped. Then expression (112) is written

δE + v × δB = − ∇X ⊥δΦ + ∇X ⊥(δA ⋅v)− (v⊥ ⋅∇X⊥ )δA. = − ∇X ⊥δL − v⊥ ⋅∇X⊥ δA, (113)

where δL is deﬁned by

δL = δΦ − v ⋅δA.

(114)

Using expression (113), equation (108) is written

q [ e ] ∂F q∂δL ∂F δRF0 − LgδFa = − -- (− ∇X ⊥δL − v⊥ ⋅∇X ⊥δA) ×-∥ ⋅---g0− --------g0, m Ω ∂X m ∂t ∂𝜀

(115)

where all terms are of O(λ²).

3.6 Equation for the non-adiabatic part δG

Plugging expression (115) into Eq. (103), we obtain

[ ] LgδG = − q- (− ∇X ⊥ δL− v⊥ ⋅∇X ⊥δA )× e∥ ⋅ ∂Fg0− q-∂δL∂Fg0 + δRδFg, m Ω ∂X m ∂t ∂ 𝜀

(116)

where L_g is given by Eq. (94), i.e.,

∂ ∂ ∂ Lg = ∂t + (v∥e∥ + VE0) ⋅∂X-+ v ⋅[λB1 + λB2]− Ω ∂α q ( v⊥ ∂ eα ∂ ) + --E0 ⋅ ------+ ----- , (117) m B0 ∂μ v⊥∂ α

3.6.1 Expansion of δG

Expand δG as

δG = δG₀ + δG₁ + …,

where δG_i ∼ O(λⁱ⁺¹)F_g0, and note that the right-hand side of Eq. (116) is of O(λ²), then, the balance on order O(λ¹) requires

∂ δG ----0 = 0, ∂ α

(118)

i.e., δG₀ is gyro-phase independent.

The balance on order O(λ²) requires (for the special case of E₀ = 0):

∂δG0-+ v∥e∥ ⋅ ∂δG0-+v ⋅[λB1 + λB2 ]δG0 ∂t[ ∂X ] = − q- (− ∇X ⊥δL − v⊥ ⋅∇X δA)× e∥ ⋅ ∂Fg0 − q-∂δL-∂Fg0 + δR δFg. (119) m Ω ∂X m ∂t ∂𝜀

3.6.2 Gyro-averaging

Deﬁne the gyro-average operator ⟨…⟩_α by

∫ 2π ⟨h⟩ = (2π)− 1 hdα, α 0

(120)

where h = h(X,α,𝜀,μ) is an arbitrary function of guiding-center variables. The gyro-averaging is an integration in the velocity space. [For a ﬁeld quantity, which is independent of the velocity in particle coordinates, i.e., h = h(x), it is ready to see that the above averaging is a spatial averaging over a gyro-ring.]

Gyro-averaging Eq. (119), we obtain

∂δG0 ⟨ ∂δG0⟩ -∂t--+ v∥e∥ ⋅-∂X-- + ⟨v ⋅[λB1 + λB2]δG0 ⟩α [ ] = − q- − ∇X ⊥⟨δL⟩α × e∥ ⋅ ∂Fg0 −-q∂⟨δL⟩α ∂Fg0+ ⟨δRδFg⟩α, (121) m Ω ∂X m ∂t ∂𝜀

where use has been made of ⟨(v_⊥⋅∇_X)δA⟩_α ≈ 0, where the error is of order higher than O(λ²). Note that v_∥ = ± ∘ ----------
2(𝜀− B0μ )

. Since B₀ is approximately independent of α, so does v_∥. Using this, the ﬁrst gyro-averaging on the left-hand side of the above equation is written

⟨ ∂δG ⟩ ∂δG ∂δG v∥e∥ ⋅---0 = ⟨v∥e∥⟩⋅ ---0-= v∥e∥ ⋅---0 ∂X α ∂X ∂X

(122)

The second gyro-averaging on the left-hand side of Eq. (121) can be written as

⟨v ⋅[λB1 + λB2]δG0⟩α = VD ⋅∇X δG0,

(123)

where V_D is the magnetic curvature and gradient drift (Eq. (123) is derived in Appendix xx, to do later). Then Eq. (121) is written

[ ∂ ] ∂t + (v∥e∥ +VD ) ⋅∇X δG0 [ ] = − q- − ∇X ⊥ ⟨δL ⟩α × e∥ ⋅ ∂Fg0−-q ∂⟨δL-⟩α-∂Fg0 +⟨δR δFg ⟩α. (124) m Ω ∂X m ∂t ∂𝜀

3.6.3 Simpliﬁcation of the nonlinear term

Next, we try to simplify the nonlinear term ⟨δRδF_g⟩_α appearing in Eq. (124), which is written as

⟨ ( q ∂F )⟩ ⟨δR δFg⟩α = δR -δΦ --g0 + δG0 ⟨ m( ∂𝜀 ) ⟩ α = -qδR δΦ ∂Fg0 +⟨δR δG0 ⟩α (125) m ∂𝜀 α

First, let us focus on the ﬁrst term, which can be written as

( ) ( ) ( ) ( ) ( ) δR δΦ∂Fg0 ≈ − q-∂Fg0 δE + v-×-δB- × e∥ ⋅ ∂δΦ-−-q ∂Fg0 v×-δB- ⋅ eα-∂δΦ- ∂ 𝜀 m ∂𝜀 c Ω ∂X m ∂𝜀 c v⊥ ∂α q ∂Fg0 ( ∂ δΦ v⊥ ∂δΦ eα∂ δΦ ) q ∂2Fg0 − m--∂𝜀-δE ⋅ v-∂𝜀-+ B0-∂μ--+ v⊥-∂α- + m-δΦδE ⋅v-∂𝜀2- ( ) 2 = − q- δE+ v×-δB- ⋅∇vδΦ + q-δΦδE ⋅v∂-Fg20 m c m ∂𝜀 = qδΦ δE ⋅v ∂2Fg0 (126) m ∂𝜀2

Using the above results, the nonlinear term ⟨δRδF⟩_α is written as

⟨ 2 ⟩ ⟨δRδF⟩α = q- δΦδE ⋅v∂-Fg0 + ⟨δRδG0⟩α m ∂𝜀2 α

(127)

Accurate to O(λ²),the ﬁrst term on the right-hand side of the above is zero. [Proof:

⟨ 2 ⟩ ⟨ 2 ⟩ δΦδE ⋅v∂-Fg20 = ∂-Fg20δΦ∇ δΦ ⋅v ∂ 𝜀 α ∂𝜀 α ∂2Fg0 2 = ∂𝜀2 ⟨v ⋅∇ (δΦ )⟩α ∂2Fg0 2 ≈ ∂𝜀2 ⟨v ⊥ ⋅∇ (δΦ )⟩α ≈ 0, (128)

where use has been made of ⟨v_⊥⋅∇_XδΦ⟩_α ≈ 0, where the error is of O(λ²). Using the above results, expression (127) is written as

⟨δRδFg⟩α = ⟨δR δG0⟩α.

(129)

Using the expression of δR given by Eq. (56), the above expression is written as

⟨( ) ⟩ q- v×-δB- (e∥) ∂δG0- ⟨δR δG0⟩α = − m δE + c × Ω α ⋅ ∂X q ∂δG q ∂δG ⟨ v ⟩ − ------0⟨δE ⋅v⟩α − -----0- δE ⋅-⊥- (130) m ∂𝜀 m ∂μ B0 α

where use has been made of ∂δG₀∕∂α = 0. Using Eq. (113), we obtain

q-⟨ ( e∥)⟩ -q e∥ − m (δE +v × δB) × Ω α = m ∇X ⊥⟨δL⟩α × Ω .

(131)

The other two terms in Eq. (130) can be proved to be zero. [Proof:

− q-∂δG0-⟨δE ⋅v⟩α = q-∂δG0⟨v ⋅∇xΦ ⟩α m ∂𝜀 m ∂𝜀 ≈ q-∂δG0⟨v⊥ ⋅∇xΦ ⟩α m ∂𝜀 ≈ q-∂δG0⟨v ⋅∇ Φ⟩ m ∂𝜀 ⊥ X α ≈ 0 (132)

⟨ ⟩ ⟨ ⟩ − q-∂δG0- δE ⋅ v⊥ = q-∂δG0- 1-v⊥ ⋅∇x Φ m ∂μ B0 α m ∂μ ⟨ B0 ⟩α q-∂δG0- 1-- ≈ m ∂μ B0v⊥ ⋅∇X Φ α ≈ 0 (133)

] Using the above results, the nonlinear term is ﬁnally written as

⟨δRδG ⟩ = -q[∇ ⟨δL⟩ × e∥]⋅∇ δG . 0α m X⊥ α Ω X 0

(134)

Using this in Eq. (129), we obtain

q [ e∥] ⟨δRδFg⟩α = m- ∇X ⊥⟨δL⟩α × Ω- ⋅∇X δG0,

(135)

which is of O(λ²).

3.6.4 Final equation for the non-adiabatic part of perturbed distribution function

Using the above results, the gyro-averaged kinetic equation for δG₀ is ﬁnally written as

( ) ∂δG0 | q e∥| -∂t--+ |(v∥e∥ + VD − m-∇X ⟨δL ⟩α × Ω-|) ⋅∇X δG0 ◟---non◝l◜inear---◞ ( e ) = q-∇X ⟨δL ⟩α × -∥ ⋅∇XFg0 − q-∂⟨δL⟩α ∂Fg0 . (136) ◟-m---------◝◜Ω---------◞ m◟---∂t◝◜---∂𝜀◞ spatial− drive velocit− space− damp

where V_D is the equilibrium guiding-center drift velocity, ⟨…⟩_α is the gyro-phase averaging operator, δL = δΦ − v ⋅δA, and δG₀ = δG₀(X,𝜀,μ,t) is gyro-angle independent and is related to the perturbed distribution function δF_g by

δF = -qδΦ ∂Fg0+ δG , g m ∂𝜀 0

(137)

where the ﬁrst term is called “adiabatic term”, which depends on the gyro-phase α via δΦ. Equation (136) is the special case (∂F_g0∕∂μ|_𝜀 = 0) of the Frieman-Chen nonlinear gyrokinetic equation given in Ref. [3]. Note that the nonlinear terms only appear on the left-hand side of Eq. (136) and all the terms on the right-hand side are linear. The term

− q-∇X ⟨δL ⟩α × e∥ ≡ δVD, m Ω

(138)

consists of the δE × B₀ drift and magnetic ﬂuttering term (refer to expression (328) in Sec. C.3). For notaiton ease, this term is denoted by δV_D.

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