[prev] [prev-tail] [tail] [up]

5.8 Pade approximation

Γ0 defined in Eq. (209) can be approximated by the Pade approximation as

1 Γ 0 ≈----. 1+ b
(211)

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

pict

Fig. 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 (211), the polarization density np in expression (210) can be written as

∫ 2 2 np ≈ − qn0 δΦkexp(ik⋅x)--k⊥ρ2---dk--. (212) T 1 + k⊥ρ2(2π)3
(Padé 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.)
Long wavelength approximation of the polarization density

In the long wavelength limit, k⊥ρ ≪ 1, expression (212) can be further approximated as

∫ n ≈ − qn0 δΦ exp(ik ⋅x)k2ρ2-dk--, p T k ⊥ (2π)3 = qn0 ρ2∇2 δΦ. (213) T ⊥
Then the corresponding term in the Poisson equation is written as
q-n = q2n0-ρ2∇2δΦ 𝜀0 p 𝜀0T ⊥ ρ2- 2 = λ2∇ ⊥δΦ, (214) D
where λD is the Debye length defined by λD2 = T𝜀0∕(n0q2). For typical tokamak plasmas, the thermal ion gyroradius ρi is much larger than λD. Therefore the term in expression (214) 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 (214) also shows that electron polarization density is smaller than the ion polarization density by a factor of ρe∕ρi ≈ 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).

Polarization density expressed in terms of Laplacian operator

The polarization density expression (213) is for the long wavelength limit, which partially neglects FLR effect. Let us go back to the more general expression (212). The Poisson equation is written

− 𝜀0∇2⊥ δΦ = qiδni + qeδne.
(215)

Write δni = npi + δni′, where δnpi is the ion polarization density, then the above expression is written

− 𝜀0∇2⊥δΦ − qinpi = qiδn′i + qeδne.
(216)

Fourier transforming in space, the above equation is written

− 𝜀0k2⊥δˆΦ − qiˆnpi = qiδˆn′i + qeδˆne,
(217)

where ˆnpi is the Fourier transformation (in space) of the polarization density npi and similar meanings for δˆΦ , δnˆi′, and δˆne. Expression (212) implies that ˆnpi is given by

qini0- --k2⊥ρ2i-- ˆnpi = − Ti δˆΦ1 + k2ρ2. ⊥ i
(218)

Using this, equation (217) is written

( ) 2 ˆ qini0--k2⊥-ρ2i-- ˆ ′ − 𝜀0k⊥ δΦ− qi − Ti 1+ k2⊥ρ2i δΦ = qiδˆni + qeδnˆe,
(219)

Multiplying both sides by (1 + k⊥2ρi2)∕𝜀0, the above equation is written

2 2 2 qi( qini0 2 2 ) 1 2 2 ′ − (1 + k⊥ρi)k⊥δˆΦ− 𝜀0- − -Ti-(k⊥ρi)δˆΦ = 𝜀0(1+ k⊥ρi)(qiδˆni + qeδˆne).
(220)

Next, transforming the above equation back to the real space, we obtain

qi( qini0 ) 1 − (1− ρ2i∇2⊥ )∇2⊥δΦ − 𝜀- -T--ρ2i∇2⊥δΦ = 𝜀-(1− ρ2i∇2⊥)(qiδn′i + qeδne). 0 i 0
(221)

Neglecting the Debye shielding term, the above equation is written

( ) ρ2i-- 2 -1 2 2 ′ − λ2Di∇ ⊥δΦ = 𝜀0(1− ρi∇ ⊥)(qiδni + qeδne),
(222)

which is the equation actually solved in many gyrokinetic codes, where λDi2 = 𝜀0Ti∕(qi2ni0).

[prev] [prev-tail] [front] [up]