  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 at
a mesoscale. Most previous studies, while considering fluctuations in the mean
kinetic helicity, treated the mean turbulent kinetic energy at the mesoscale as
a constant, despite the fact that both these quantities involve second-order
velocity correlations. The mean turbulent kinetic energy affects the mean
magnetic field through both turbulent diffusion and turbulent diamagnetism. In
this work, we use a double-averaging procedure to analytically show that
fluctuations of the mean turbulent kinetic energy at the mesoscale (giving rise
to $\eta$-fluctuations at the mesoscale, where the scalar $\eta$ 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. Turbulent diamagnetism, which arises due to inhomogeneities in
the turbulent kinetic energy, leads to growing mean field solutions even when
the $\eta$-fluctuations are statistically isotropic.
