7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Impact of magnetic ﬁeld on the parallel resistivity
Chao Dong,1,2 Wei Hu,3* Ding Li ,1,2† and Wenlu Zhang1,21Institute of Physics, Chinese Academy of Sciences
2
University of Chinese Academy of Sciences
3
Institute of Optoelectronic Technology, Lishui University
dli@cashq.ac.cn (speaker)

The impact of magnetic ﬁeld (MF) on the parallel
resistivity 𝜂∥ is studied for strongly magnetized plasmas
with the electron thermal gyroradius 𝜌𝑡ℎ𝑒 smaller thanthe Debye length 𝜆𝐷 but much larger than the Landaulength λL. Two previous papers [P. Ghendrih et al., Phys.
Lett. A 119, 354 (1987); S. D. Baalrud and T. Laﬂeur, Phys.
Plasmas 28, 102107 (2021)] found 𝜂∥ to increase
monotonically with MF. Unfortunately, both works usedpredetermined electron distribution functions and are thus
not self-consistent.
A self-consistent study is conducted for the impact ofMF on 𝜂∥ in this paper. Contrary to previous results, 𝜂∥is found to decrease monotonically with the MF, which isexpected to have a notable impact on various plasma
physics problems associated with 𝜂∥ , particularly on theresistive in-stabilities and transport phenomena.
The MF inﬂuence on the electron dynamics has been
considered during the collision process while its inﬂuenceon the ion dynamics is ignorable. For simplicity, theelectrons are supposed to collide only with backgroundions and not with other electrons. Since the current ismainly carried by the electrons, we can ignore the ion
response to E and take its velocity distribution function 𝑓𝑖
to be Maxwellian. Under these conditions, the evolutionof the electron distribution function 𝑓𝑒 is governed by amagnetized Fokker-Planck (FP) equation. The ﬁrst- andsecond-order magnetized FP coefﬁcients can be obtainedin the static screening approximation. The 𝑓𝑒1 can besolved self-consistently from the electron magnetized FK
equation in a Lorentz gas-like approximation. From the𝑓𝑒1 expression, we havethe two cases 𝑣̂𝑒 = 1 and 𝑣̂𝑒 = 2 . The theoreticalresults represented by the solid curves and the numericalresults by the dashed curves are in good agreement.

The parallel current 𝐽𝑧 is given by𝐽𝑧 = −𝑒 ∫ 𝑣𝑒𝑧 𝑓𝑒 (𝐯𝑒 , 𝑡) 𝑑 3 𝐯𝑒 =64√2𝜋𝜀02 (𝑘𝐵 𝑇𝑒 )3/2
𝑍𝑒 2 √𝑚𝑒 ln(𝜌𝑡ℎ𝑒 /𝑟𝐿 )𝐸from which we ﬁnally obtain the parallel resistivity as𝜂∥ =𝑍𝑒 2 √𝑚𝑒𝐸𝜌𝑡ℎ𝑒=ln ()23/2𝐽𝑧 64√2𝜋𝜀0 (𝑘𝐵 𝑇𝑒 )𝑟𝐿

It is shown that only difference from the conventional
parallel resistivity is the replacement of 𝜆𝐷 in 𝑙𝑛 Λ by𝜌𝑡ℎ𝑒 . To explain the MF dependence of 𝜂∥ , we compare
𝑓𝑒1 with MF for 𝑚𝑖 ⁄𝑚𝑒 = 3672 and 𝛼 = 0.2 , and𝑛𝑣𝑒 = 2.5𝑣𝑡ℎ𝑒 with its value 𝑓𝑒1
without MF in Fig. 3.

𝜕𝑓𝑒1 (𝐯𝑒 )𝛽𝑣𝑒4 sin 𝜃 𝑓𝑒0 (𝑣𝑒 )= 2𝜕𝜃2𝑣𝑡ℎ𝑒 [1 + 𝛼′𝑣𝑒 𝒟(𝑣𝑒 cos 𝜃)]
Figure 1 shows the contour map of 𝜕𝑓𝑒1 /𝜕𝜃 in thequasistatic state obtained numerically and the blackcontour lines from the theoretical result with 𝑚𝑖 ⁄𝑚𝑒 =3672 and 𝛼 = 0.2 . A good agreement between thenumerical and theoretical results is achieved.

The MF inﬂuence on 𝜂∥ is signiﬁcant in the white dwarf
atmospheres and the ultracold neutral plasmas with therelative corrections close to 90%. For the Alcator C-Mod
tokamak scrape-off layer plasmas, the relative correction
is about 10%. In the future, as the MF of tokamaks
increases, its inﬂuence on 𝜂∥ may become more
pronounced.
References[1] Cohen, Spitzer, and Routly, Phys. Rev. 80, 230 (1950);
Spitzer and Härm, ibid. 89, 977 (1953).
[2] J. M. Rax et al., Phys. Plasmas 26, 012303 (2019).

The change of 𝜕𝑓𝑒1 /𝜕𝜃 with 𝜃 is plotted in Fig. 2 for