The generation of a large-scale magnetic field in the kinematic regime in the absence of an α-effect is investigated by following two different approaches: the test-field method and the multiscale stability theory relying on the homogenisation technique. Our computations of the magnetic eddy diffusivity tensor of the parity-invariant flow IV of G.O. Roberts and the modified Taylor-Green flow confirm the findings of previous studies, and also explain some of their apparent contradictions. The two flows have large symmetry groups; this is used to considerably simplify the eddy diffusivity tensor. Finally, a new analytic result is presented: upon expressing the eddy diffusivity tensor in terms of solutions to auxiliary problems for the adjoint operator, we derive relations between the magnetic eddy diffusivity tensors that arise for mutually reverse small-scale flows v(x) and −v(x). Subject headings: MHD -magnetic fields -turbulence -dynamo arXiv:1501.04465v3 [physics.flu-dyn] 9 Jul 2015

Pointwise zero velocity helicity density is shown not to prevent steady flows from acting as kinematic dynamos. We present numerical evidences that such flows can generate both small-scale magnetic fields as well as, by the magnetic α-effect or negative eddy diffusivity mechanisms, largescale ones. The flows are constructed as curls of analytically defined space-periodic steady solenoidal flows, whose vorticity helicity (i.e., kinetic helicity) density is everywhere zero.

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 α-effect in such flows, the large-scale mean field experiences harmonic oscillations in time on the scale O(εt) without growth or decay. Here ε 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(ε 2 t); for the directions at which eddy diffusivity is infinite, the large-scale mean-field growth rate is finite on the scale O(ε 3/2 t). We investigate numerically this dynamo mechanism for two sample flows.

Explicit solutions for ballistic aggregation of dust-like matter, whose particles stick inelastically upon collisions, are considered. This system provides a model of large-scale structure formation in cosmology within the Zel'dovich approximation. In particular we show the equivalence of two different representations of solutions proposed in [14,5] for a flat 1D flow, extend these representations to cylindrically or spherically symmetric flows, and provide explicit counterexamples showing how exactly these representations break down in the case of non-symmetric flow.