Dynamo action in sliding plates of anisotropic electrical conductivity

With materials of anisotropic electrical conductivity, it is possible to generate a dynamo with a simple velocity field, of the type precluded by Cowling's theorems with isotropic materials. Following a previous study by Ruderman and Ruzmaikin [M. S. Ruderman and A. A. Ruzmaikin, Magnetic field generation in an anisotropically conducting fluid, Geophys. Astrophys. Fluid Dyn. 28, 77 (1984)], who considered the dynamo effect induced by a uniform shear flow, we determine the conditions for the dynamo threshold when a solid plate is sliding over another one, both with anisotropic electrical conductivity. We obtain numerical solutions for a general class of anisotropy and obtain the conditions for the lowest magnetic Reynolds number, using a collocation Chebyshev method. In a particular geometry of anisotropy and wave number, we also derive an analytical solution, where the eigenvectors are just combinations of four exponential functions. An explicit analytical expression is obtained for the critical magnetic Reynolds number. Above the critical magnetic Reynolds number, we have also derived an analytical expression for the growth rate showing that this is a “very fast” dynamo, extrapolating on the “slow” and “fast” terminology introduced by Vainshtein and Zeldovich [S. I. Vainshtein and Y. B. Zeldovich, Reviews of topical problems: Origin of magnetic fields in astrophysics (turbulent “dynamo” mechanisms), Sov. Phys. Usp. 15, 159 (1972)].

Figure 1

Two plates of finite and equal thickness $H$ slide on each other with relative velocity $\pm U$. Each plate has a uniform anisotropic electrical conductivity, characterized by a lower value in the direction $\mathbf{q}$. The thin lines drawn on each cross section of the plates correspond to the directions of large electrical conductivity.

Figure 2

Angles $\alpha$ and $\beta$ define the orientation of the anisotropy. The angle between the $z$ axis and $\mathbf{q}$ is denoted $\alpha$ while $\beta$ denotes the angle between the $x$ axis and the projection of $\mathbf{q}$ in the $(x,y)$ plane: $q_x=cos\beta sin\alpha , q_y=sin\beta sin\alpha$, and $q_z=cos\alpha$.

Figure 3

Critical magnetic Reynolds number, for a uniform anisotropy ($\alpha =0.5$ rad, $\beta =0, {\eta }_1=1000$) in both plates, as a function of $k_x$ and $k_y$.

Figure 4

Configuration of uniform anisotropy, with $\beta =0$, so that the direction of large electrical resistivity lies in the $(x,z)$ plane.

Figure 5

Analytical critical magnetic Reynolds number, from Eq. (46), for a uniform anisotropy ($\alpha =0.5$ rad, $\beta =0$) in both plates, as a function of $k_y$, for four values of anisotropy: ${\eta }_1=2$, 10, 100, and 1000. The limiting case of infinite ${\eta }_1$ is also shown.

Figure 6

Analytical minimal critical magnetic Reynolds number $R_{mc}^{\text{min}}$, from Eq. (46), for a uniform anisotropy ($\beta =0$) in both plates, as a function of $\alpha$, for four values of anisotropy: ${\eta }_1=2$, 10, 100, and 1000. The limiting case of infinite ${\eta }_1$ is also plotted. The corresponding wave number $k_{yc}^{\text{min}}$ is plotted below.

Figure 7

(a) Analytical eigenvector of both $P$ and $T$ components, for ${\eta }_1=1000, \alpha =0.5$, and the minimal $R_{mc}\simeq 3.6347$, attained at $k_{yc}\simeq 0.6225$. (b) Isovalues of the eigenvector in the ($y,z)$ plane of both $T$ and $P$ components for the same parameters.

Figure 8

(a) Same configuration as in Fig. 4, but only one platelet of high electrical conductivity is shown in each plate. (b) Schematic electric current circuits (thick solid lines) in the $(y,z)$ plane are labeled $I_x$ in the $x$ direction and $\mathbf{j}$ in the $(y,z)$ plane. The corresponding schematic magnetic field is shown (thick dashed lines), $\mathbf{B}$ in the $(y,z)$ plane and $b_x$ in the $x$ direction. The pattern is periodic in the $y$ direction and invariant in the $x$ direction.

Figure 9

Optimal growth rate and corresponding wave number $k_y$, for a uniform anisotropy ($\alpha =0.5$ rad, $\beta =0$) in both plates, as a function of $R_m$, for five values of anisotropy: ${\eta }_1=2$, 10, 100, 1000, and $\infty$. Note that the growth rate is divided by $R_m$, which corresponds to the dimensional growth rate divided by $U/H$.