H 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 (347) is written as

H.1 Linear case

Neglecting the nonlinear terms, drift-kinetic equation (392) is written

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 (393) by qv_∥ and then integrating over velocity space, we obtain

Equation (394) 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 (393) into variable (X,μ,v_∥).

H.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

(395)

(396)

and

(397)

Use this, we have

and

Then, in terms of variable (X′,μ,v_∥), equation (393) is written

where ∇≡ ∂∕∂X′|_μ,v∥.

H.3 Parallel momentum equation

Multiplying the linear drift kinetic equation (400) by qv_∥ and then integrating over velocity space, we obtain

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. (401) are all zero. Among the rest terms, only the following term

(402)

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

Using these results, the parallel momentum equation (401) is written

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 (404) 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

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

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. (406) is written

where δp_∥ = ∫ mv_∥²δfdv. Similarly, the term −q ∫ dvv_∥⋅∇_XF₀ is written as

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

(409)

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

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

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

Define

(412)

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

(413)

then Eq. (411) is written as

H.4 Special case in uniform magnetic field

In the case of uniform magnetic field, the parallel momentum equation (411) is written as

(415)

H.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. F or Appendix in Yang Chen’s paper[1])

(416)

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

Drift kinetic equation

Drift kinetic equation is written

(417)

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.

Parallel momentum equation

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

(418)

which can be written as

(419)

Using dv = B⁻¹2πmdv_∥dμ, the last term on the RHS of the above equation is written

where use has been made of b ⋅∇p_∥0 = 0.

(422)

Using Eq. () in Eq. (), we obtain

(423)

————–

(424)

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