  We study large-scale kinematic dynamo action of steady mirror-antisymmetric
flows of incompressible fluid, that involve small spatial scales only, by
asymptotic methods of the multiscale stability theory. It turns out that, due
to the magnetic $\alpha$-effect in such flows, the large-scale mean field
experiences harmonic oscillations in time on the scale O($\varepsilon t$)
without growth or decay. Here $\varepsilon$ is the spatial scale ratio and $t$
is the fast time of the order of the flow turnover time. The interaction of the
accompanying fluctuating magnetic field with the flow gives rise to an
anisotropic magnetic eddy diffusivity, whose dependence on the direction of the
large-scale wave vector generically exhibits a singular behaviour, and thus to
negative eddy diffusivity for whichever molecular magnetic diffusivity.
Consequently, such flows always act as kinematic dynamos on the time scale
O($\varepsilon^2t$); for the directions at which eddy diffusivity is infinite,
the large-scale mean-field growth rate is finite on the scale
O($\varepsilon^{3/2}t$). We investigate numerically this dynamo mechanism for
two sample flows.
