Impact of magnetic field on the parallel resistivity

The impact of magnetic field (MF) on the parallel resistivity ${\eta }_{\parallel }$ is studied for strongly magnetized plasmas with the electron thermal gyroradius ${\rho }_{\text{the}}$ smaller than the Debye length ${\lambda }_D$ but much larger than the Landau length ${\lambda }_L$. 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 ${\eta }_{\parallel }$ to increase monotonically with MF. Unfortunately, both works used predetermined electron distribution functions and are thus not self-consistent. In this paper, we analyze the MF dependence of ${\eta }_{\parallel }$ self-consistently by solving the electron magnetized kinetic equation in a Lorentz gaslike approximation. It is found ${\eta }_{\parallel }$ decreases monotonically with MF, with ${\lambda }_D$ in the usual Coulomb logarithm $ln\mathrm{\Lambda }=ln({\lambda }_D/{\lambda }_L)$ being replaced by ${\rho }_{\text{the}}$. The underlying physics is that the electrons affected only by the collisions with impact parameters between ${\lambda }_L$ and ${\rho }_{\text{the}}$ carry almost all the parallel current.

Figure 1

The variation of ${\mathcal{R}}_f\equiv \langle \mathrm{\Delta }{V}_{ez}\rangle /\langle \mathrm{\Delta }{v}_{ez}\rangle$ with $|v_{ez}|$ for $T_i=T_e, v_{e\perp }=v_{\text{the}}$, and (a) $m_i/m_e=1836, \alpha =0.1$, (b) $m_i/m_e=1836, \alpha =0.2$, (c) $m_i/m_e=3672, \alpha =0.1$, and (d) $m_i/m_e=3672, \alpha =0.2$. The inset shows ${\mathcal{R}}_f$ in the region $11<|v_{ez}|/v_{\text{thi}}<30$.

Figure 2

The variation of ${\mathcal{R}}_d\equiv \langle \mathrm{\Delta }{V}_{ez}\mathrm{\Delta }{V}_{ez}\rangle /\langle \mathrm{\Delta }{v}_{ez}\mathrm{\Delta }{v}_{ez}\rangle$ with $|v_{ez}|$ with the same parameter settings as in Fig. 1 for the four cases. The inset shows ${\mathcal{R}}_d$ in the region $4<|v_{ez}|/v_{\text{thi}}<15$.

Figure 3

Contour map of $\partial {\widehat{f}}_{e1}/\partial \theta$ in the quasisteady state obtained numerically and the black contour lines from the theoretical result in Eq. (13) with $m_i/m_e=3672$ and $\alpha =0.2$.

Figure 4

Change of $\partial {\widehat{f}}_{e1}/\partial \theta$ with $\theta$ for the two cases ${\widehat{v}}_e=1$ and ${\widehat{v}}_e=2$ with $m_i/m_e=3672$ and $\alpha =0.2$. The solid curves represent the theoretical results and the dashed curves the numerical results. The results without MF multiplied by $1/(1-\alpha )$ are shown for comparison by the dotted curves. The dashed black vertical line indicates the position of $\theta =\pi /2$.

Figure 5

Comparison of $f_{e1}$ in Eq. (12) for $m_i/m_e=3672, \alpha =0.2$, and $v_e=2.5v_{\text{the}}$ with its value $f_{e1}^n$ without MF. The red curve represents ${\widehat{f}}_{e1}$, the green curve ${\widehat{f}}_{e1}^n$, and the blue curve ${\widehat{f}}_{e1}^n/(1-\alpha )$.