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5.4 Generalized split-weight scheme for electrons

The perturbed distribution function is decomposed as given by Eq. (151), i.e.,

δF = δh + q-(δΦ− ⟨δΦ⟩α)∂F0-+ q-⟨v⋅δA ⟩α ∂F0, m ∂𝜀 m ∂𝜀
(168)

where the term in blue is the so-called adiabatic response, which depends on the gyro-angle. Recall that the red term ⟨δΦ⟩α, which is independent of the gyro-angle, is introduced in order to eliminate the time derivative ∂⟨δΦ⟩α∕∂t term on the right-hand side of the original Frieman-Chen gyrokinetic equation.

The so-called generalized split-weight scheme corresponds to going back to the original Frieman-Chen gyrokinetic equation by introducing another ⟨δΦ⟩α term with a free small parameter 𝜖g. Specifically, δh in the above is split as

q ∂F δh = δhs + 𝜖g-⟨δΦ⟩α--0. m ∂ 𝜀
(169)

(If 𝜀g = 1, then the two ⟨δΦ⟩α terms in Eq. (168) and (169) cancel each other.) Substituting this expression into Eq. (143), we obtain the following equation for δhs:

[ ] -∂ + (v∥e∥ + VD + δVD )⋅∇X δhs ∂t [ ] q-∂F0- ∂- + 𝜖gm ∂𝜀 ∂t + (v∥e∥ + VD + δVD )⋅∇X ⟨δΦ ⟩α q [ ∂ ] ∂F0 + 𝜖gm-⟨δΦ⟩α ∂t + (v∥e∥ + VD + δVD )⋅∇X -∂𝜀- = − δVD ⋅∇XF0 − q-[(v∥e∥ + VD + δVD ) ⋅∇X (⟨v ⋅δA − δΦ ⟩α )]∂F0-. (170) m ∂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, the above equation is written as
[ ] ∂-+ (v∥e∥ + VD + δVD )⋅∇X δhs ∂t = − δVD ⋅∇XF0 -q{ [∂-⟨δΦ⟩α ]} ∂F0- − m (v∥e∥ + VD + δVD )⋅∇X [⟨v ⋅δA⟩α − ⟨δΦ ⟩α]+ 𝜖g ∂t + VG ⋅∇X ⟨δΦ⟩α ∂(𝜀1 .71)
special case of 𝜖g = 1

For the special case of 𝜖g = 1 (the default and most used case in GEM code, Yang Chen said 𝜖g < 1 cases are sometimes not accurate, so he gave up using it since 2009), equation (171) can be simplified as:

[ ∂ ] ∂t + (v∥e∥ + VD + δVD )⋅∇X δhs = − δV[D ⋅∇XF0 ] − q- VG ⋅∇X ⟨v ⋅δA ⟩α + ∂-⟨δΦ-⟩α ∂F0-, (172) m ∂t ∂𝜀
where two VG ⋅⟨δΦ⟩α terms cancel each other. Because the v∥E∥ term is one of the factors that make kinetic electron simulations difficult, eliminating VG ⋅⟨δΦ⟩α term may be beneficial for obtaining stable algorithms.

For 𝜖g = 1, δF is written as

δF = δhs + 𝜀g q-⟨δΦ ⟩α ∂Fg0+-q(δΦ − ⟨δΦ ⟩α)∂F0-+-q⟨v ⋅δA ⟩α∂F0- m ∂𝜀 m ∂𝜀 m ∂𝜀 = δhs + q-δΦ∂Fg0 + q-⟨v⋅δA ⟩α∂F0, (173) m ∂𝜀 m ∂𝜀
where the polarization density term is partially canceled, yielding only the pure adiabatic response. The adiabatic term will be moved to the left-hand side of the Poisson’s equation. The descretization of this term is much easier than the polarization density. This term is already in GEM.

Equation (172) actually goes back to the original Frieman-Chen equation. The only difference is that q- m⟨v ⋅δA⟩α∂F0 ∂𝜀 is further split from the perturbed distribution function. Considering this, equation (172) can also be obtained from the original Frieman-Chen equation (133) by writing δG0 as

δG = δh + -q⟨v ⋅δA ⟩ ∂F0-, 0 s m α ∂𝜀
(174)

In this case, δF is written as

q ∂F q ∂F δF = δhs +--δΦ --g0+ --⟨v ⋅δA⟩α---0, m ∂𝜀 m ∂𝜀
(175)

Substituting expression (174) into equation (133), we obtain the following equation for δhs:

[ ] ∂-+ (v∥e∥ + VD + δVD )⋅∇X δhs ∂t [ ] -q∂F0- -∂ + m ∂𝜀 ∂t + (v∥e∥ + VD + δVD )⋅∇X ⟨v ⋅δA ⟩α q [ ∂ ] ∂F0 + m-⟨v⋅δA ⟩α ∂t + (v∥e∥ +VD + δVD )⋅∇X -∂𝜀- = − δVD ⋅∇XF0 − q-∂⟨δΦ-−-v⋅δA-⟩α-∂F0, (176) 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, the above equation is written as
[ ∂ ] ∂t + (v∥e∥ + VD + δVD )⋅∇X δhs = − δ[VD ⋅∇XF0 ] − -q ∂⟨δΦ-⟩α-+ [(v∥e∥ + VD + δVD )⋅∇X ]⟨v ⋅δA⟩α ∂F0, (177) m ∂t ∂𝜀
which agrees with Eq. (172).

In GEM, the split weight method is used only for electrons and the ∂⟨δΦ⟩α∕∂t is approximated by ∂δΦ∕∂t and this term is obtained from the vorticity equation (rather than from an implicit iteration).

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