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5.2 Eliminate ∂⟨δv ⋅ δA⟩α∕∂t term on the right-hand side of GK equation

Similar to the method of eliminating ∂⟨δϕ⟩α∕∂t, we define another gyro-phase independent function δh by

δh = δf −-q⟨v ⋅δA ⟩α∂F0-. m ∂𝜀
(139)

then Eq. (138) is written in terms of δh as

[ ∂ ] --+ (v∥e∥ + VD + δVD )⋅∇X δh ∂t [( ) ] + q-∂F0- ∂-+ v∥e∥ + VD + δVD ⋅∇X ⟨v ⋅δA⟩α m ∂𝜀 ∂t[( ) ]( ) q- ∂- ∂F0- + m ⟨v⋅δA ⟩α ∂t + v∥e∥ + VD + δVD ⋅∇X ∂𝜀 = − δVD ⋅∇XF0 q [ ∂⟨v ⋅δA ⟩ ] ∂F − -- − --------α − (v∥e∥ + VD + δVD ) ⋅∇X ⟨δΦ ⟩α --0, (140) m ∂t ∂𝜀
Noting that ∂F0∕∂t = 0, e∥⋅∇F0 = 0, ∇F0 ∼ O(λ1)F0, we find that the third line of the above equation is of order O(λ3) and thus can be dropped. Moving the second line to the right-hand side and noting that ⟨δL⟩α = ⟨δϕ − v ⋅ δA⟩α, the above equation is written as
[ ] ∂-+ (v∥e∥ + VD + δVD )⋅∇X δh ∂t = − δVD ⋅∇XF0 -q ∂F0- −m [(v∥e∥ +VD + δVD )⋅∇X (⟨v⋅δA − δΦ⟩α)]∂𝜀 , (141)
where two ∂⟨v ⋅ δA⟩α∕∂t terms cancel each other and no time derivatives of the perturbed fields appear on the right-hand side. Noting that δVD given by Eq. (122) is perpendicular to ∇X⟨v ⋅ δA − δΦ⟩α and thus the blue term in Eq. (141) is zero, then Eq. (141) simplifies to
[ ∂ ] -- + (v∥e∥ +VD + δVD )⋅∇X δh ∂t = − δVD ⋅∇XF0 − q-[(v e + VD )⋅∇X ⟨v ⋅δA − δΦ ⟩α]∂F0-. (142) m ∥ ∥ ∂𝜀
Using VG = v∥e∥ + VD + δVD, equation (141) can also be written as
[ ] ∂-+ V ⋅∇ δh ∂t G X q- ∂F0- = − δVD ⋅∇XF0 − m [VG ⋅∇X ⟨v⋅δA − δΦ⟩α]∂ 𝜀 . (143)
For special case δA ≈ δA∥e∥

Most gyrokinetic simulations approximate the vector potential as δA ≈ δA∥e∥. Let us simplify Eq. (142) for this case. Then ⟨v ⋅ δA⟩α is written as

⟨v ⋅δA⟩α ≈ ⟨v∥δA ∥⟩α.
(144)

Note that in terms of (X,𝜀,μ,α,σ) coordinates, v∥ is written as

∘ --------- v∥ = σ 2𝜀− 2μB0,
(145)

where B0(x) = B0(X + ρ) with ρ = ρ(X,𝜀,μ,α). Since the scale length of B0 is much larger than the thermal Larmor radius, B0(x) ≈ B0(X) and hence v∥ of thermal particles can be approximately considered to be independent of the gyro-angle α. Then v∥ can be taken out of the gyro-averaging in expression (144), yielding

⟨v ⋅δA⟩α ≈ v∥⟨δA ∥⟩α.
(146)

Using this, the term related to δA in (142) is written as

(v∥e∥ + VD )⋅∇X ⟨v ⋅δA ⟩α = (v∥e∥ + VD )⋅∇X (v∥⟨δA∥⟩α) = ⟨δA ⟩ (v e + V )⋅∇ (v )+ v (v e + V ) ⋅∇ ⟨δA ⟩(.147) ∥α ∥ ∥ D X ∥ ∥ ∥ ∥ D X ∥α
Using expression (145), (v∥e∥ + VD) ⋅∇X(v∥) is written as
(v∥e∥ + VD )⋅∇X (v∥) ≈ (v∥e∥) ⋅∇X (v∥) ( ∘ ---------) = (v∥e∥) ⋅∇X σ 2𝜀− 2μB0 (∘ ---------) = σ(v∥e∥)⋅∇X 2𝜀− 2μB0 −-2μe∥ ⋅∇XB0- = σv∥ 2√2𝜀−-2μB0- − 2μe ∥ ⋅∇XB0 = v∥-----2v------ ∥ = − μe ∥ ⋅∇XB0. (148)
(We can also obtain ∇X(v∥) = −μ(∇B0)∕v∥ by using Eq. (268)). Using the above results, equation (142) is written as
[ ∂ ] --+ (v∥e∥ + VD + δVD )⋅∇X δh ∂t = − δVD ⋅∇XF0 − q-[− (v e + VD )⋅∇X ⟨δΦ⟩α]∂F0-, m ∥ ∥ ∂𝜀 − q-[v (v e + V )⋅∇ ⟨δA ⟩ − ⟨δA ⟩ (μe ⋅∇B )]∂F0, (149) m ∥ ∥ ∥ D X ∥ α ∥ α ∥ 0 ∂𝜀
which agrees with the so-called p∥ formulation given in GEM code manual (the first line of Eq. 28), which uses p∥ = v∥ + q⟨A∥⟩α∕m as an independent variable.

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