Polarization density expressed as local Fourier expansion

7 Polarization density expressed as local Fourier expansion

Since δΦ is independent of the velocity in the particle coaordinates, the ﬁrst term (adiabatic term) in expression (224) is trivial and the velocity integration can be readily performed (assume F₀ is Maxwellian), giving

∫ δnad = q(δΦ )∂F0dv. m ∫ ∂(𝜀 ) = -q(δΦ) − m-fM dv. m T = − qδΦn , (226) T 0

which is called adiabatic response. Next, let us perform the gyro-averaging and the velocity integration of the second term in expression (224), i.e.,

∫ − -q⟨δΦ⟩α∂F0-dv. m ∂𝜀

(227)

7.1 Gyro-averaging of δΦ in guiding-center coordinates

In order to perform the gyro-averaging of δΦ, we Fourier expand δΦ in space as

∫ δΦ(x) = δΦ exp(ik ⋅x)-dk--, k (2π)3

(228)

and then express x in terms of the guiding center variables (X,v) since the gyro-averaging is taken by holding X rather than x constant. The guiding-center transformation gives

e∥(X-)- x = X + ρ(x,v ) ≈ X − v× Ω(X ).

(229)

Using expressions (228) and (229), the gyro-average of δΦ is written as

⟨ ∫ ⟩ ⟨δΦ⟩ = δΦ exp(ik⋅x )-dk-- α k (2π)3 α ⟨ ∫ ( ( e∥(X ))) dk ⟩ = δΦk exp ik⋅ X − v× -Ω(X-) (2π)3 ∫ ⟨ ( ) ⟩ α = δΦk exp(ik ⋅X ) exp − ik⋅v × e∥(X) -dk--. (230) Ω (X) α (2π)3

When doing the gyro-averaging, X is hold constant and thus e_∥(X) is also constant. Then it is straightforward to deﬁne the gyro-angle α. Let k_⊥ deﬁne one of the perpendicular direction

₁, i.e., k_⊥ = k_⊥

₁. Then another perpendicular basis vector is deﬁned by

₂ = e_∥×

₁. Then v_⊥ is written as v_⊥ = v_⊥(

₁ cosα +

₂ sinα), which deﬁnes the gyro-angle α. Then the blue expression in Eq. (230) is written as

− ik⋅v × e∥(X) = − ik⋅v⊥ (ˆe1cosα + ˆe2sin α)× e∥(X-) Ω (X) Ω(X) = − ik⋅-v⊥-(− ˆe2cosα+ ˆe1sinα) Ω(X) = − ik⊥v⊥-sin α. (231) Ω

Then the gyro-averaging in expression (230) is written as

⟨ ( ) ⟩ ⟨ ( )⟩ exp − ik ⋅v× e∥(X-) = exp − ik⊥v⊥-sin α Ω(X) α Ω α 1 ∫ 2π ( k⊥v⊥ ) = 2π- exp − i-Ω---sinα dα ( 0 ) = J0 k⊥v⊥- . (232) Ω

where use has been made of the deﬁnition of the zeroth Bessel function of the ﬁrst kind. Then ⟨δΦ⟩_α in expression (230) is written as

∫ ( ) ⟨δΦ ⟩α = δΦk exp(ik ⋅X)J0 k⊥v⊥- --dk-. Ω (2π)3

(233)

7.2 Gryo-angle integration in particle coordinates

Next, we need to perform the integration in velocity space, which is done by holding x (rather than X) constant. Therefore, it is convenient to transform back to particle coordinates. Using X = x + v × e∥(x)
Ω(x) , expression (233) is written as

∫ (k⊥v⊥ ) ( e∥) dk ⟨δΦ⟩α = δΦk exp(ik⋅x )J0 --Ω-- exp ik⋅v × Ω- (2π-)3.

(234)

Then the velocity integration is written as

∫ ⟨δΦ ⟩α ∂F0dv ∫ ∂𝜀 [∫ ( ) ] k⊥v⊥- ( e∥) ∂F0- -dk-- = δΦkexp(ik⋅x) J0 Ω exp ik ⋅v× Ω ∂𝜀 dv (2π)3. (235)

Similar to Eq. (231), except for now at x rather than X, ik ⋅ v × e∥
Ω

is written as

e∥ k⊥v⊥- ik ⋅v× Ω = i Ω sin α.

(236)

Since this is at x rather than X, k_⊥, v_⊥, and Ω are diﬀerent from those appearing in expression (231). However, since this diﬀerence is due to the variation of the equilibrium quantity e_∥∕Ω in a Larmor radius, and thus is small and is ignored in the following.

Plugging expression (236) into expression (235) and using dv = v_⊥dv_⊥dv_∥dα, we get

∫ ⟨δΦ ⟩α∂F0dv ∫ ∂ 𝜀 [∫ ( ) ( ) ] k⊥v⊥- k⊥v⊥- ∂F0- -dk-- = δΦk exp (ik ⋅x ) J0 Ω exp i Ω sinα ∂𝜀 v⊥dv⊥dv∥dα (2π)3.(237)

Note that ∂F₀∕∂𝜀 is independent of the gyro-angle α in terms of guiding-center variables. When transformed back to particle coordinates, X contained in ∂F₀∕∂𝜀 will introduce α dependence via X = x + v × e∥-
Ω

. This dependence on α is weak since the equilibrium quantities can be considered constant over a Larmor radius distance evaluated at the thermal velocity. Therefore this dependence can be ignored when performing the integration over α, i.e., in terms of particle coordinates, ∂F₀∕∂𝜀 is approximately independent of the gyro-angle α. Then the integration over α in Eq. (237) can be performed, yielding

∫ ⟨δΦ ⟩α ∂F0dv ∂𝜀 ∫ ∫ ∫ ( k⊥v⊥-)[∫ 2π ( k⊥v⊥- ) ] ∂F0 -dk-- = δΦk exp(ik⋅x ) J0 Ω 0 exp i Ω sin α dα ∂𝜀 v⊥dv⊥dv∥(2π)3 ∫ [∫ ∫ ( k v ) (k v ) ∂F ] dk = δΦk exp(ik⋅x ) J0 -⊥-⊥- 2πJ0 -⊥-⊥- --0v⊥dv⊥dv∥ ---3, (238) Ω Ω ∂𝜀 (2π )

where again use has been made of the deﬁnition of the Bessel function.

7.2.1 The remaining velocity integration can be performed analytically if F₀ is Maxwellian

In order to perform the remaining velocity integration in expression (238), we assume that F₀ is a Maxwellian distribution given by

( ) F = f = ----n0(X)-----exp − mv2- (239) 0 M (2πT(X )∕m )3∕2 2T(X) n0 ( − v2) = (2π)3∕2v3exp 2v2- , (240) t t

where v_t = ∘ ----
T∕m

, then

∂F0 m -∂𝜀-= − T-fM .

(241)

Again we will ignore the weak dependence of n₀(X) and T(X) on v introduced by X = x + v × e_∥∕Ω when transformed back to particle coordinates. (For suﬃciently large velocity, the corresponding Larmor radius will be large enough to make the equilibrium undergo substantial variation. Since the velocity integration limit is to inﬁnite, this will deﬁnitely occur. However, F₀ is exponentially decreasing with velocity, making those particles with velocity much larger than the thermal velocity negligibly few and thus can be neglected.)

Parallel integration Using Eq. (241), the expression in the square brackets of Eq. (238) is written as

∫ ∫ ( ) 2π J2 k⊥v⊥- ∂F0v dv dv 0 Ω ∂𝜀 ⊥ ⊥ ∥ m n ∫ ∫ ( k v ) 1 ( v2∥ + v⊥2) = − -----01∕2 J20 -⊥-⊥- -3exp −----2-- v⊥dv⊥dv∥ (242) T (2π) Ω vt ( 2vt) m n0 ∫ ∫ 2( k⊥v⊥ ) v2∥ + v2⊥ - - - = − T-(2π)1∕2 J0 -Ω--- exp − ---2--- v⊥d v⊥dv∥, (243)

where v_∥ = v_∥∕v_t, v_⊥ = v_⊥∕v_t. Using

∫ ∞ ( 2) √ --- exp − x- dx = 2π, −∞ 2

(244)

the integration over v_∥ in expression (243) can be performed, yielding

∫ ∞ ( ) ( -2 ) − m-n0 J20 k⊥v⊥- exp − v⊥- v⊥dv ⊥ T 0 Ω 2

(245)

Perpendicular integration Using (I veriﬁed this by using Sympy)

∫ ( ) ∞ 2 x2 2 2 0 J0(ax)exp − 2 xdx = exp(− a )I0(a ),

(246)

where I₀(a) is the zeroth modiﬁed Bessel function of the ﬁrst kind, expression (245) is written

− m-n0exp(− b)I0(b) T

(247)

where b = k_⊥²v_t²∕Ω² = k_⊥²ρ_t². Then the corresponding density (227) is written as

∫ ∫ −-q ⟨δΦ ⟩α ∂F0dv = qn0 δΦkexp(ik⋅x)exp(− b)I0(b)-dk-. m ∂𝜀 T (2π)3

(248)

7.2.2 Final form of polarization density

In Fourier space, the adiabatic term in expression (226) is written as

∫ ∫ -q(δΦ)∂F0-dv = − qn0δΦ = − qn0 δΦ exp(ik ⋅x)-dk--. m ∂𝜀 T T k (2π)3

(249)

Plugging expression (248) and (249) into expression (224), the polarization density n_p is written as

∫ np = − qn0 δΦkexp(ik ⋅x)[1 − exp(− b)I0(b)]-dk--. (250) T (2π)3

Deﬁne

Γ 0 = exp(− b)I0(b),

(251)

then Eq. (250) is written as

∫ n = − qn0 δΦ exp(ik ⋅x)[1 − Γ ]-dk-, p T k 0(2π)3

(252)

Expression (252) agrees with the result given in Yang Chen’s notes. Note that the dependence on species mass enters the formula through the Larmor radius ρ_t in Γ₀.

7.3 Pade approximation

Γ₀ deﬁned in Eq. (251) can be approximated by the Pade approximation as

-1--- Γ 0 ≈ 1+ b.

(253)

The comparison between the exact value of Γ₀ and the above Pade approximation is shown in Fig. 1.

pict

Figure 1: Comparison between the exact value of Γ₀ = exp(−(k_⊥ρ)²)I₀((k_⊥ρ)²) and the corresponding Pade approximation 1∕(1 + (k_⊥ρ)²).

Using the Pade approximation (253), the polarization density n_p in expression (252) can be written as

∫ n ≈ − qn0 δΦ exp(ik ⋅x)-k2⊥-ρ2---dk-. (254) p T k 1+ k2⊥ρ2 (2π)3

(Pade approximate is the “best” approximation of a function by a rational function of given order – under this technique, the approximant’s power series agrees with the power series of the function it is approximating.)

7.3.1 Long wavelength approximation of the polarization density

In the long wavelength limit, k_⊥ρ ≪ 1, expression (254) can be further approximated as

∫ n ≈ − qn0 δΦ exp(ik⋅x)k2ρ2 -dk-, p T k ⊥ (2π)3 = qn0ρ2∇2 δΦ. (255) T ⊥

Then the corresponding term in the Poisson equation is written as

-qn = q2n0ρ2∇2 δΦ 𝜀0 p 𝜀0T ⊥ ρ2- 2 = λ2 ∇⊥ δΦ, (256) D

where λ_D is the Debye length deﬁned by λ_D² = T𝜀₀∕(n₀q²). For typical tokamak plasmas, the thermal ion gyroradius ρ_i is much larger than λ_D. Therefore the term in expression (256) for ions is much larger than the space charge term ∇²δΦ ≡∇_⊥²δΦ + ∇_∥²δΦ ≈∇_⊥²δΦ in the Poisson equation. Therefore the space charge term can be neglected in the long wavelength limit.

Equation (256) also shows that electron polarization density is smaller than the ion polarization density by a factor of ρ_e∕ρ_i ≈ 1∕60. Note that this conclusion is drawn in the long wavelength limit. For short wavelength, the electron polarization and ion polarization density can be of similar magnitude (to be discussed later).

7.3.2 Polarization density expressed in terms of Laplacian operator

The polarization density expression (255) is for the long wavelength limit, which partially neglects FLR eﬀect. Let us go back to the more general expression (254). The Poisson equation is written

2 − 𝜀0∇⊥δΦ = qiδni + qeδne.

(257)

Write δn_i = n_pi + δn_i′, where δn_pi is the ion polarization density, then the above expression is written

− 𝜀0∇2⊥δΦ − qinpi = qiδn′i + qeδne.

(258)

Fourier transforming in space, the above equation is written

− 𝜀0k2⊥δˆΦ − qiˆnpi = qiδˆn′i + qeδˆne,

(259)

where _pi is the Fourier transformation (in space) of the polarization density n_pi and similar meanings for δ, δ_i′, and δ_e. Expression (254) implies that _pi is given by