Drift-kinetic limit

I Drift-kinetic limit

In the drift-kinetic limit, ⟨v_⊥⋅δE⟩_α = 0, ⟨δB_∥v_⊥⟩_α = 0, and ⟨δh⟩_α = δh, where δh is an arbitrary ﬁeld quantity. Using these, gyrokinetic equation (334) is written as

[ ( ) ] -∂ + v∥e∥ + vD + vE + v∥δB⊥ ⋅∇X δf ∂t ( ) B0 [( ) ] = − v + v δB-⊥ ⋅∇ F − q- v e + v + v δB-⊥ ⋅δE ∂F0. (384) E ∥ B0 X 0 m ∥ ∥ D ∥ B0 ∂ 𝜀

I.1 Linear case

Neglecting the nonlinear terms, drift-kinetic equation (384) is written

[ ] ∂- ∂t +(v∥e∥ + vD )⋅∇X δf ( δB ) q ∂F = − vE + v∥--⊥- ⋅∇XF0 − --[(v∥e∥ +vD )⋅δE]---0. (385) B0 m ∂𝜀

Next let us derive the parallel momentum equation from the linear drift kinetic equation (this is needed in my simulation). Multiplying the linear drift kinetic equation (385) by qv_∥ and then integrating over velocity space, we obtain

∫ ∂δj∥ = − q dvv (ve + v )⋅∇ δf ∂t ∥( ∥∥ D ) X ∫ δB⊥- -q ∫ ∂F0- − q dvv ∥ vE + v∥ B0 ⋅∇XF0 − m q dvv∥[(v∥e∥ + vD )⋅δE]∂𝜀 . (386)

Equation (386) involve ∇_Xδf and this should be avoided in particle methods whose goal is to avoid directly evaluating the derivatives of δf over phase-space coordinates. On the other hand, the partial derivatives of velocity moment of δf are allowed. Therefore, we would like to make the velocity integration of δf appear. Note that ∇_Xδf here is taken by holding (𝜀,μ) constant and thus v_∥ is not a constant and thus can not be moved inside ∇_X. Next, to facilitate performing the integration over v_∥, we transform the linear drift kinetic equation (385) into variable (X,μ,v_∥).

I.2 Transform from (X,μ,𝜀) to (X,μ,v_∥) coordinates

The kinetic equation given above is written in terms of variable (X,μ,𝜀). Next, we transform it into coordinates (X′,μ′,v_∥) which is deﬁned by

′ X (X,μ,𝜀) = X,

(387)

′ μ (X,μ,𝜀) = μ,

(388)

and

v(X, μ,𝜀) = ∘2-(𝜀-−-μB-(X-)). ∥ 0

(389)

Use this, we have

′ ′ ∂δG0-|μ,𝜀 = ∂X--∂δG0-+ ∂μ-∂δG0-+ ∂v∥ ∂δG0- ∂X ∂X ∂X ′ ∂X ∂μ′ ∂X ∂v∥ ∂δG0- ∂δG0- μ-∂B0-∂δG0- = ∂X′ |μ,v∥ + 0 ∂μ′ − v∥ dX ∂v∥ , (390)

and

∂F0 ∂F0 ∂μ′ ∂F0 ∂v∥ -∂𝜀- = ∂μ-′∂𝜀-+ -∂v--∂𝜀 ∥ = 0∂F0-+ ∂F0-∂v∥ ∂μ ′ ∂v∥ ∂𝜀 ∂F0-1- = ∂v∥ v∥ (391)

Then, in terms of variable (X′,μ,v_∥), equation (385) is written

∂δf-+ (v e + v )⋅∇ δf − e ⋅μ∇B ∂δf- ∂t ∥ ∥ D ∥ ∂v∥ ( δB⊥ ) ( vE δB ⊥) ∂F0 q [( vD) ] ∂F0 = − vE + v∥B--- ⋅∇F0 + v--+ -B-- ⋅μ∇B ∂v-− m- e∥ + v-- ⋅δE ∂v(3,92) 0 ∥ 0 ∥ ∥ ∥

where ∇≡ ∂∕∂X′|_{μ,v_∥}.

I.3 Parallel momentum equation

Multiplying the linear drift kinetic equation (392) by qv_∥ and then integrating over velocity space, we obtain

∫ ∫ ∂δj∥ + q dvv (ve + v )⋅∇ δf − q dvv e ⋅μ ∇B ∂δf- ∂t ∥ ∥∥ D X ∥ ∥ ∂v∥ ∫ ( δB ⊥) ∫ ( vE δB ⊥) ∂F0 = − q dvv∥ vE +v∥-B0- ⋅∇XF0 + q dvv∥ v--+ -B0- ⋅μ∇B ∂v-- ∫ [( ) ] ∥ ∥ − q-q dvv∥ e∥ + vD ⋅δE ∂F0. (393) m v∥ ∂v∥

Consider the simple case that F₀ does not carry current, i.e., F₀(X,μ,v_∥) is an even function about v_∥. Then it is obvious that the integration of the terms in red in Eq. (393) are all zero. Among the rest terms, only the following term

∫ − q-q dvv∥[(v∥e∥)⋅δE ]∂F0-1- m ∂v∥v∥

(394)

explicitly depends on δE. Using dv = 2πBdv_∥dμ, the integration in the above expression can be analytically performed, giving

∫ − q-q dvv∥[(v∥e∥)⋅δE]∂F0-1- m ∂v∥ v∥ q2∫ ∂F0 = − m- 2πBdv ∥dμv∥δE∥∂v∥- 2∫ ∫ = − q- 2πBd μδE∥ v∥∂F0dv∥ m ( ∂v∥ ) q2∫ ∫ = − m 2πBd μδE∥ 0− F0dv∥ q2 = --δE ∥n0. (395) m

Using these results, the parallel momentum equation (393) is written

∂δj∥ q2 ∫ ∫ ∂δf- ∂t = m δE ∥n0 − q dvv∥(v∥e∥ + vD )⋅∇X δf + q dvv ∥e∥ ⋅μ∇B ∂v∥ ∫ ( ) ∫ ( ) − q dvv∥ v∥δB⊥- ⋅∇F0 +q dvv∥ δB⊥- ⋅μ∇B ∂F0-, (396) B0 B0 ∂v∥

where the explicit dependence on δE is via the ﬁrst term q²n₀δE_∥∕m, with all the the other terms being explicitly independent of δE (δf and δB implicitly depend on δE).

Equation (396) involve derivatives of δf with respect to space and v_∥ and these should be avoided in the particle method whose goal is to avoid directly evaluating these derivatives. Using integration by parts, the terms involving ∂∕∂v_∥ can be simpliﬁed, yielding

∫ ∫ ∂δj∥ = q2δE n − q dvv (ve + v )⋅∇ δf − q(e ⋅∇B ) μδfdv ∂t m ∥ 0 ∥ ∥ ∥ D X ∥ 0 ∫ ( δB-⊥) ( δB⊥-) ∫ − q dvv∥ v∥ B0 ⋅∇F0 − q B0 ⋅(∇B0 ) μF0dv, (397)

Deﬁne p_⊥0 = ∫ mv_⊥²F₀∕2dv and δp_⊥ = ∫ mv_⊥²δf∕2dv, then the above equation is written

∫ ∂δj∥ q2- -δp⊥- ∂t = m δE ∥n0 − q dvv∥(v∥e∥ + vD )⋅∇X δf − q(e∥ ⋅∇B0 )mB0 ∫ ( δB⊥ ) ( δB⊥ ) p⊥0 − q dvv∥ v∥-B-- ⋅∇F0 − q B--- ⋅(∇B0 )mB--, (398) 0 0 0

Next, we try to eliminate the spatial gradient of δf by changing the order of integration. The second term on the right-hand side of Eq. (398) is written

∫ − q dvv∥(v∥e∥)⋅∇X δf, ∫ 2 = − q 2πB0dv∥dμv∥e∥ ⋅∇X δf ∫ = − q2πB0e ∥ ⋅∇X v2∥δfdv∥dμ ( ∫ ) = − qB0e ∥ ⋅∇X -1-- mv2∥δfdv ( mB0 ) δp∥- = − qB0e ∥ ⋅∇X mB0 , (399)

where δp_∥ = ∫ mv_∥²δfdv. Similarly, the term −q ∫ dvv_∥ ( )
v∥δB⊥-
B0

⋅∇_XF₀ is written as

∫ ( ) − q dvv v δB⊥- ⋅∇ F ∥ ∥ B0 X 0 ∫ ( 2δB-⊥) = − q 2πB0dv∥dμ v∥ B0 ⋅∇XF0 ( δB ) ∫ = − q --⊥- ⋅B0 ∇X (v2∥F02 πdv∥dμ ) ( B0 ) [ ∫ ] = − q δB⊥- ⋅B ∇ -1 (mv2F dv) B0 0 X B ∥ 0 ( p∥0-) = − qδB⊥ ⋅∇X mB0 (400)

where p_∥0 = ∫ mv_∥²F₀dv. Similarly, the term −q ∫ dvv_∥v_D ⋅∇_Xδf can be written as the gradient of moments of δf. Let us work on this. The drift v_D is given by

B0v∥∇ × b μ vD = --ΩB-⋆---v∥ + ΩB-⋆B0 ×∇B0. ∥ ∥

(401)

where B_∥^⋆ = B₀ ( v∥- )
1 + Ωb ⋅∇ × b (refer to my another notes). Using b ⋅∇× b ≈ 0, we obtain B_∥^⋆ ≈ B. Then v_D is written

v_D =

∇× b +

b ×∇B₀.

Using this and dv = 2πB₀dv_∥dμ, the term −q ∫ dvv_∥v_D ⋅∇_Xδf is written as

( ) ∫ ∫ v2∥ μ- − q dvv∥vD ⋅∇X δf = − q 2πB0dv∥dμv∥ Ω ∇ × b + Ωb × ∇B0 ⋅∇X δf ∫ ∫ = − q2πB0 1-(∇ × b )⋅∇X v3∥δfdv∥dμ− q2πB0 1-(b × ∇B0 )⋅∇X v∥μδfdv∥dμ Ω ( ∫ ) Ω ( ∫ ) = − qB 1-(∇ × b) ⋅∇ -1- v3δfdv − qB 1(b × ∇B )⋅∇ 1-- v μδfdv , 0Ω X B0 ∥ 0Ω 0 X B0 ∥ ( 1 ∫ 3 ) ( 1 ∫ ) = − m (∇ × b)⋅∇X B-- v∥δfdv − m (b × ∇B0 )⋅∇X B-- v∥μδfdv , (402) 0 0

which are the third order moments of δf and may be neglect-able (a guess, not veriﬁed). Using the above results, the linear parallel momentum equation is ﬁnally written

∂δj∥ e2ne0- ( δp∥-) δp⊥- ∂t = m δE ∥ + eB0b ⋅∇X mB0 + e(b ⋅∇B0 )mB0 ( p ) ( δB ) p +e δB⊥ ⋅∇X --∥0- + e --⊥- ⋅(∇B0 )-⊥0-. mB0( ∫ B0 ) mB0 ( ∫ ) − m (∇ × b) ⋅∇X 1-- v3δfdv − m (b× ∇B0 )⋅∇X -1- v μδfdv (403) B0 ∥ B0 ∥

Deﬁne

( p ) ∇B p D0 = ∇ --∥0 + ---0 -⊥0-, mB0 B0 mB0

(404)

which, for the isotropic case (p_∥0 = p_⊥0 = p₀), is simpliﬁed to

∇p D0 = ---0. mB0

(405)

then Eq. (403) is written as

∂δj∥ e2n0 -∂t- = -m--δE∥ + eδB ⊥ ⋅D0 ( δp∥ ) δp + eB0b ⋅∇X ---- + e(b ⋅∇B0 )--⊥- mB0( ∫ ) mB0 ( ∫ ) − m (∇ × b) ⋅∇X -1- v3δfdv − m (b× ∇B0 )⋅∇X -1- v∥μδfdv .(406) B0 ∥ B0

I.4 Special case in uniform magnetic ﬁeld

In the case of uniform magnetic ﬁeld, the parallel momentum equation (403) is written as

∂δj∥ q- δB⊥- ∂t = m qE∥ne0 − qe∥ ⋅∇X (δp∥)− q B0 ⋅∇Xp ∥0.

(407)

I.5 Electron perpendicular ﬂow

Using the gyrokinetic theory and taking the drift-kinetic limit, the perturbed perpendicular electron ﬂow, δV_e⊥, is written (see Sec. G or Appendix in Yang Chen’s paper[2])

ne0 1 ne0δVe ⊥ = B0-δE ×b − eB0b × ∇ δp⊥e ◟--◝◜---◞ ◟----◝◜-----◞ E×B ﬂow diamagneticﬂow

(408)

where n_e0 is the equilibrium electron number density, δp_e⊥ is the perturbed perpendicular pressure of electrons.

I.5.1 Drift kinetic equation

Drift kinetic equation is written

( ) ∂f- ˜ δE-×-b ( e- ˜ ) ∂f- ∂t + v∥b + vD + B0 ⋅∇f + − m δE∥ − μb ⋅∇B ∂v∥ = 0,

(409)

where f = f(x,μ,v_∥,t), μ = mv_⊥²∕B₀ is the magnetic moment, = b + δB_⊥∕B₀, b = B₀∕B₀ is the unit vector along the equilibrium magnetic ﬁeld, v_D = v_D(x,μ,v_∥) is the guiding-center drift in the equilibrium magnetic ﬁeld. δE and δB are the perturbed electric ﬁeld and magnetic ﬁeld, respectively.