Enhanced dynamo growth in nonhomogeneous conducting fluids

We address magnetic-field generation by dynamo action in systems with inhomogeneous electrical conductivity and magnetic permeability. More specifically, we first show that the Taylor-Couette kinematic dynamo undergoes a drastic reduction of its stability threshold when a (zero-mean) modulation of the fluid's electrical conductivity or magnetic permeability is introduced. These results are obtained outside the mean-field regime, for which this effect was initially proposed. Beyond this illustrative example, we extend a duality argument put forward by Favier and Proctor (2013) to show that swapping the distributions of conductivity and permeability and changing $u\to -u$ leaves the dynamo threshold unchanged. This allows one to make connections between a priori unrelated dynamo studies. Finally, we discuss the possibility of observing such an effect both in laboratory and astrophysical settings.

Figure 1

From left to right: Axial slices of (a) the azimuthal magnetic field in the fluid domain (color map), with streamline pattern superimposed ($\mathrm{\Psi }$ isocontours); (b) magnetic energy density (averaged in the azimuthal direction); (c) electrical energy density (again, averaged in the azimuthal direction); all of them corresponding to the reference case $\text{Rm}=100$, $\lambda =0$. (d) Isocontours of the eletrical resistivity modulation ${\rho }^{\prime }/{\rho }_0$ as defined in (9). (e),(f) Isosurfaces of magnetic energy (drawn for $10\%$ of the maximum value): (e) $\lambda =0$, $\text{Rm}=100$; (f) $\lambda =0.9$, $\text{Rm}=80$. The inner cylinder boundary is shown for reference.

Figure 2

Critical magnetic Reynolds number for kinematic dynamo ${\text{Rm}}_c$ as a function of the amplitude of the resistivity modulation $\lambda$. Note that the prefactor in (9) ensures that, for example, $\lambda =0.5$ corresponds to a maximum resistivity variation of $50\%$ around its mean value. Inset: Asymptotic kinematic growth rate for $\text{Rm}=70$ and $\lambda =0.25$, with different values of the phase shift $\mathrm{\Phi }$ in (10): $\mathrm{\Phi }=\pi (▿)$; $\pi /2(\square )$; $\pi /4(▹)$; $\pi /6(\circ )$ and $0\left(▵\right)$. The symbol $*$ corresponds to a different modulation wavelength with ${\rho }^{\prime }\propto cos(\pi z+\pi /2)$, and the dashed line marks the value of the reference growth rate ($\lambda =0$).