Magnetic field dynamics in presence of Hall conductivity and thermo-diffusion

TL;DR

This work develops a rigorous MHD framework incorporating Hall conductivity and thermo-diffusion to study magnetic-field dynamics in nonuniform plasmas. By deriving and analyzing the magnetic-field evolution with Hall terms and Biermann-battery seed-field contributions, the authors reveal how thermo-diffusion and barodiffusion can generate seed fields in non-magnetized media, and how Hall currents modify field structures in various settings. The paper further applies the framework to four configurations—thrusters, neutron-star crusts, magnetized plasma cylinders, and tori—demonstrating opposing internal Hall-field directions relative to external fields and providing both analytic and non-dimensional models to guide experimental and astrophysical modeling. The resulting formalism offers a valuable tool for laboratory astrophysics and for understanding seed-field amplification across astrophysical environments.

Abstract

Anisotropy of kinetic coefficients in presence of a magnetic field is represented by Hall currents, which appear in a collisional medium due to action of the Lorentz force on the charged particles between collisions. We derive equations, describing dynamics of the magnetic field in presence of thermo-diffusion with Hall currents, using a standard electrodynamic consideration. The influence of the Hall currents, at presence of thermo-diffusion, on the magnetic field structure is considered in simple models. The equation is derived, which includes additional term for a seed magnetic field creation in the non-magnetized plasma, due to thermo-diffusion. This equation describes the action of the seed magnetic field creation by the mechanism known as "Biermann battery".

Figures (8)

• Figure 1: Conducting cylinder with Hall current $j_{Hall}$, depending on the magnitude of the radial temperature gradient, and external constant magnetic field $B_0$ along its axis. The induced magnetic field $B_1$ is determined by the Hall current. $R_1$ is the radius of the central heated region with constant temperature $T_0$. Toroidal region, colored in gray, contains Hall current and associated magnetic field, which has an opposite direction to the external field $B_0$, decreasing the resulting field along the cylinder.
• Figure 2: Torus with a initial electric current $j_0$, that produce circular magnetic field $B_0$. $B_0$ and temperature gradient $\nabla T$ create Hall electric current $j_{Hall}$. The induced magnetic field $B_1$ is determined by the Hall current $j_{Hall}$ and is oriented in the opposite direction to the $B_{0}$.
• Figure 3: Magnetic field in torus small circle induced by the Hall current, for $G=1.2\cdot 10^{-5}$, $E=0.1$, and three variants: $N_1=0.8$; $N_2=8.5$; $N_3=85.2$. These values are related to $Z =1$ and include combinations $B_{0} = 5\cdot 10^3$ G, $T_0 =2\cdot 10^{5}$ K, $\rho_0=10^{-4}$ g/cm$^{3}$ for $N_1$; $B_{0}=5\cdot 10^{2}$ G, $T_0 = 2\cdot 10^{5}$ K, $\rho_0 = 10^{-5}$ g/cm$^{3}$ for $N_2$; $B_{0}=50$ G, $T_0 = 2\cdot 10^{5}$ K, $\rho_0=10^{-6}$ g/cm$^{3}$ for $N_3$.
• Figure 4: Temperature distribution in the torus small circle for the same parameters as in Fig. 3
• Figure 5: Magnetic field in the torus small circle, induced by the Hall current, for $E = 0.1$ and three variants: $G_1= 1.2\cdot 10^{-11}$, $N_1=0.085$; $G_2=1.2\cdot 10^{-9}$, $N_2=0.8$; $G_3=1.2\cdot 10^{-7}$, $N_3=8.5$. These values are related to $Z = 1$, and include variants $T_0 = 2\cdot 10^{5}$ K, $B_{0}= 50$ G , $\rho_0=10^{-3}$ g/cm$^{3}$ for $N_1$,$G_1$; $\rho_0 = 10^{-4}$ g/cm$^{3}$ for $N_2$,$G_2$; $\rho_0 = 10^{-5}$ g/cm$^{3}$ for $N_3$$G_3$.