Split-weight scheme for electrons in GEM code

E Split-weight scheme for electrons in GEM code

The perturbed distribution function is decomposed as given by Eq. (203), i.e.,

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

(335)

where the term in blue is the so-called adiabatic response, which depends on the gyro-angle in guiding-center coordinates. 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 𝜖_g. Speciﬁcally, δh in the above is split as

δh = δhs + 𝜖g q-⟨δΦ⟩α∂F0. m ∂𝜀

(336)

(If 𝜀_g = 1, then the two ⟨δΦ⟩_α terms in Eq. (335) and (336) cancel each other.) Substituting this expression into Eq. (), we obtain the following equation for δh_s:

[ ∂ ] ∂t + (v∥e∥ + VD + δVD )⋅∇X δhs [ ] +𝜖g q-∂F0 ∂- +(v∥e∥ + VD + δVD ) ⋅∇X ⟨δΦ⟩α m ∂𝜀 ∂[t ] q- -∂ ∂F0- +𝜖gm ⟨δΦ⟩α ∂t + (v∥e∥ + VD + δVD )⋅∇X ∂𝜀 = − δVD ⋅∇XF0 q ∂F0 −m-[(v∥e∥ +VD + δVD )⋅∇X (⟨v ⋅δA − δΦ⟩α)]∂𝜀-. (337)

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 + V + δV )⋅∇ δh ∂t ∥ ∥ D D X s = − δVD ⋅∇XF0 q { [ ∂⟨δΦ ⟩α ]} ∂F0 −m- (v∥e∥ + VD + δVD )⋅∇X [⟨v ⋅δA ⟩α − ⟨δΦ ⟩α]+ 𝜖g --∂t--+ VG ⋅∇X ⟨δΦ ⟩α (∂3𝜀38.)

E.0.1 special case of 𝜖_g = 1

For the special case of 𝜖_g = 1 (the default and most used case in GEM code, Yang Chen said 𝜖_g < 1 cases are sometimes not accurate, so he gave up using it since 2009), equation (338) can be simpliﬁed as:

[ ∂ ] ∂t + (v∥e∥ + VD + δVD )⋅∇X δhs = − δ[VD ⋅∇XF0 ] − q- VG ⋅∇X ⟨v ⋅δA⟩α + ∂⟨δΦ⟩α ∂F0, (339) m ∂t ∂𝜀

where two V_G ⋅⟨δΦ⟩_α terms cancel each other. Because the v_∥E_∥ term is one of the factors that make kinetic electron simulations diﬃcult, eliminating V_G ⋅⟨δΦ⟩_α term may be beneﬁcial for obtaining stable algorithms.

For 𝜖_g = 1, δF is written as

q ∂F q ∂F q ∂F δF = δhs + 𝜀g-⟨δΦ⟩α--g0 + --(δΦ − ⟨δΦ⟩α)--0-+ --⟨v ⋅δA ⟩α --0- m ∂𝜀 m ∂ 𝜀 m ∂𝜀 = δhs +-qδΦ ∂Fg0+ -q⟨v ⋅δA⟩α∂F0-, (340) m ∂𝜀 m ∂𝜀

where 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 (339) 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 (339) can also be obtained from the original Frieman-Chen equation (136) by writing δG₀ as

δG = δh + -q⟨v ⋅δA ⟩ ∂F0-, 0 s m α ∂𝜀

(341)

In this case, δF is written as

δF = δh + -qδΦ ∂Fg0+ -q⟨v ⋅δA⟩ ∂F0-, s m ∂𝜀 m α ∂𝜀

(342)

Substituting expression (341) into equation (136), we obtain the following equation for δh_s:

[ ] ∂-+ (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-, (343) m ∂t ∂𝜀

[ ∂ ] ∂t + (v∥e∥ + VD + δVD )⋅∇X δhs = − δ[VD ⋅∇XF0 ] − q- ∂-⟨δΦ-⟩α + (v∥e∥ + VD + δVD )⋅∇X ⟨v ⋅δA ⟩α ∂F0-, (344) m ∂t ∂𝜀

which agrees with Eq. (339).

In GEM, the split weight method is used only for electrons, and ∂⟨δΦ⟩_α∕∂t is approximated by ∂δΦ∕∂t, which is obtained from the vorticity equation (rather than from time-diﬀerence scheme).

When using the split weight scheme, a ∂δϕ∕∂t terms appear in the right-hand-side of the weight evolution equation. GEM makes use of the vorticity equation (time derivative of the Poissson equation) to evaluate ∂δϕ∕∂t.

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E Split-weight scheme for electrons in GEM code

E.0.1 special case of 𝜖g = 1

E.0.1 special case of 𝜖_g = 1