Mean-field dynamo due to fluctuating turbulent diffusivity

In systems where the standard $\alpha$ effect is inoperative, one often explains the existence of mean magnetic fields by invoking the `incoherent $\alpha$ effect', which appeals to fluctuations of the mean kinetic helicity. Previous studies, while considering fluctuations in the mean kinetic helicity, treated the mean turbulent kinetic energy as a constant, despite the fact that both these quantities involve second-order velocity correlations. The mean turbulent kinetic energy causes both turbulent diffusion and diamagnetic pumping of the mean magnetic field. In this work, we use a double-averaging procedure to analytically show that fluctuations of the mean turbulent kinetic energy (giving rise to $\eta$-fluctuations, where $\eta$ is the turbulent diffusivity) can lead to the growth of a large-scale magnetic field even when the kinetic helicity is zero pointwise. Constraints on the operation of such a dynamo are expressed in terms of dynamo numbers that depend on the correlation length, correlation time, and strength of these fluctuations. In the white-noise limit, we find that these fluctuations reduce the overall turbulent diffusion, while also contributing a drift term which does not affect the growth of the field. We also study the effects of nonzero correlation time and anisotropy. Diamagnetic pumping, which arises due to inhomogeneities in the turbulent kinetic energy, leads to growing mean field solutions even when the $\eta$-fluctuations are isotropic. Our results suggest that fluctuations of the turbulent kinetic energy may be relevant in astrophysical contexts.