Nonlinear Hall effect in the stationary cylinder with a radial heat flux

TL;DR

The paper examines how a radial temperature gradient in a magnetized cylinder can induce an azimuthal Hall current that generates an axial magnetic component $B_1$, which can counteract an externally supplied field $B_0$ in stationary state. Using tensor transport theory in the Lorentz gas approximation, it derives the heat- and charge-flux relations and couples them to Maxwell equations in a cylindrical geometry, yielding a nonlinear two-equation system for the Hall field and temperature. Numerical results for neutron-star-crust-like parameters and laboratory conditions show that the Hall-induced field suppresses the external field, with the effect growing with the Hall parameter and thermomagnetic coefficients. These findings inform models of magneto-thermal evolution in compact objects and offer guidance for laboratory experiments modeling Hall-current feedback on magnetic fields, while highlighting the need to include degeneracy and relativistic corrections in more realistic settings.

Abstract

A conducting cylinder with a uniform magnetic field along its axis and radial temperature gradient is considered at the stationary state. At large temperature gradients the azimuthal Hall electrical current creates the axial magnetic field which strength may be comparable with the original one. It is shown, that the magnetic field, generated by the azimuthal Hall current, leads to the decrease of magnetic field originated by external sources, and this suppression increases with increase of the electromotive force, connected with a thermodiffusion. Obtained results can help to investigate influence of the Hall current on the coupled magneto-thermal evolution of magnetic and electric fields in neutron stars, white dwarfs, and, possibly, in a laboratory facilities.

Figures (14)

• 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, coloured 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: The same cylinder as in Fig.\ref{['cylinder']}, with opposite direction of the constant magnetic field $B_0$. We see, that the magnetic field $B_1$, induced by Hall currents $j_{Hall}$ is again opposite to the direction of $B_0$. Therefore the resulting magnetic field decreases, for any direction of the magnetic field $B_0$.
• Figure 3: Magnetic field in the cylinder, induced by the Hall current, for $G = 5.2\cdot 10^{-5}$, $E = 0.012$, and three values of $N$: $N_1 = 6.0\cdot 10^{-2}$, $N_2 = 0.6$, $N_3 = 6.0$ . These values are related to $Z = 26$, and include combinations $B_{0} = 10^{14}\ G , \quad T_0 = 10^{9} \ K,\quad \rho_0 = 10^{9}$ g/cm$^{3}$ for $N_1$$B_{0} = 10^{13}\ G , \quad T_0 = 10^{9} \ K,\quad \rho_0 = 10^{8}$ g/cm$^{3}$ for $N_2$; $B_{0} = 10^{12}\ G , \quad T_0 = 10^{9} \ K,\quad \rho_0 = 10^{7}$ g/cm$^{3}$ for $N_3$; .
• Figure 4: Temperature distribution in the cylinder for the same parameters as in Fig.\ref{['figureGconst']}.
• Figure 5: Magnetic field in the cylinder, induced by the Hall current, for $N = 6.0$ and three variants: $G_1 = 2.2\cdot 10^{-5}, E_1 = 0.018$; $G_2 = 3.1\cdot 10^{-4}, E_2 = 0.012$; $G_3 = 5.2\cdot 10^{-5}, E_3 = 0.012$. These values are related to $Z = 26$, and include combinations $B_{0} = 10^{13}\ G , \quad T_0 = 3.5\cdot 10^{9} \ K, \quad \rho_0 = 10^{9}$ g/cm$^{3}$ for $G_1$,$E_1$; $B_{0} =10^{13} \ G , \quad T_0 = 1.8\cdot 10^{9} \ K, \quad \rho_0 = 10^{8}$ g/cm$^{3}$ for $G_2$,$E_2$; $B_{0} = 10^{12} \ G ,\quad T_0 = 10^{9} \ K,\quad \rho_0 = 10^{7}$ g/cm$^{3}$ for $G_3$,$E_3$.