Fluctuations of Electrical Conductivity: A New Source for Astrophysical Magnetic Fields

We consider the generation of a magnetic field by the flow of a fluid for which the electrical conductivity is nonuniform. A new amplification mechanism is found which leads to dynamo action for flows much simpler than those considered so far. In particular, the fluctuations of the electrical conductivity provide a way to bypass antidynamo theorems. For astrophysical objects, we show through three-dimensional global numerical simulations that the temperature-driven fluctuations of the electrical conductivity can amplify an otherwise decaying large scale equatorial dipolar field. This effect could play a role for the generation of the unusually tilted magnetic field of the iced giants Neptune and Uranus.

Figure 1

Sketch of the different steps involved in the amplification mechanism ${\alpha }^{\sigma }$ for a typical geophysical flow. Top: Two adjacent convective cells (gray cylinders) with axial vorticity $\omega$ are subject to a transverse azimuthal magnetic field $B$ (red). Middle: Both upward and downward axial currents $J\propto (\mathbf{v}\times \mathbf{B})$ (blue) are induced between the convective cells. Bottom: In the presence of conductivity gradients correlated to the vorticity (maximum gradient represented by pink dashed lines), large (respectively, low) conductivity increases (respectively, decreases) the induced current: the resulting net upward current $J^{\prime }$ is parallel to the vorticity.

Figure 2

The growth rates for the 2D flow considered in the text as a function of $\mathbf{K}$, for $\mathrm{Rm}=1/6$ and three different $\delta \eta$. Symbols correspond to numerically evaluated growth rates and straight lines to the analytical prediction.

Figure 3

Growth rate of the magnetic energy in the kinematic phase as a function of the (temperature-driven) electrical conductivity modulation in a dynamo simulation, for $E=6\times {10}^{-4}$, $\mathrm{Pm}=7.9$, and $\mathrm{Ra}/{\mathrm{Ra}}_c=2.2$, for axial (black) and equatorial (red) dipole modes. Note that the axial dipole obtained at $\delta \eta /{\eta }_0=0$ is replaced by a transverse dipole in presence of conductivity modulation.

Figure 4

Structure of the saturated equatorial dipole generated for $E=10^{-3}$, $\mathrm{Pm}=7$, $\mathrm{Ra}/{\mathrm{Ra}}_c=2$, and $\delta \eta /{\eta }_0=0.4$. The colored sphere indicates amplitude of the radial magnetic field at the surface of the core-mantle boundary and magnetic field lines in the insulating mantle are shown.

Figure 5

Equatorial cut of the purely hydrodynamical state obtained for $E=6\times {10}^{-4}$ and $\mathrm{Ra}/{\mathrm{Ra}}_c=2.2$. The color plot displays the amplitude of the azimuthal temperature gradient ${\partial }_{\varphi }T$, whereas black lines are isocontours of the axial component of the vorticity $(\nabla \times \mathbf{u})·{\mathbf{e}}_{\mathbf{z}}$. Note the strong correlation between the two quantities.