A mean field dynamo from negative eddy diffusivity

Abstract

Using direct numerical simulations, we verify that "flow IV" of Roberts (1972) exhibits dynamo action dominated by horizontally averaged large-scale magnetic field. With the test-field method we compute the turbulent magnetic diffusivity and find that it is negative and overcomes the molecular diffusivity, thus explaining quantitatively the large-scale dynamo for magnetic Reynolds numbers above ≈ 8. As expected for a dynamo of this type, but contrary to α-effect dynamos, the two horizontal field components grow independently of each other and have arbitrary amplitude ratios and phase differences. Small length scales of the mean magnetic field are shown to be stabilized by the turbulent magnetic diffusivity becoming positive at larger wavenumbers. Oscillatory decaying or growing solutions have also been found in certain wavenumber intervals and sufficiently large values of the magnetic Reynolds number. For magnetic Reynolds numbers below ≈ 0.5 the turbulent magnetic diffusivity is confirmed to be positive, as expected for all incompressible flows. Earlier claims of a dynamo driven by a modified Taylor-Green flow through negative eddy diffusivity could not be confirmed.

Negative Magnetic Eddy Diffusivities From the Test-Field Method and Multiscale Stability Theory

“…1 of molecular diffusivities over the critical value for the onset of generation of the small-scale magnetic field. The form (38) of the operator of eddy diffusivity corroborates the conclusions of Devlen et al (2013) that the eddy diffusivity tensor for R-IV is diagonal and has a double eigenvalue, i.e., its action on fields depending on the vertical slow variable (which was the object of the studies of G.O. Roberts 1972 andDevlen et al 2013) is homogeneous.…”

“…We follow Devlen et al (2013) in investigating largescale generation by R-IV. In the spatial variables introduced by Tilgner (2004)…”

“…To the best of our knowledge, it was first proposed by Schrinner et al (2005Schrinner et al ( , 2007. Perhaps, the most detailed description of the TFM procedure applied by Devlen et al (2013) is found in Brandenburg et al (2008a). The recipe is to solve equation (2) for zero-mean magnetic perturbation b , where b is a test field.…”

Advances in mean-field dynamo theory and applications to astrophysical turbulence

“…where η t (k) was found to be sufficiently negative for k < ∼ 0.8 k 0 , but positive (corresponding to decay) for larger values of k (Devlen et al 2013). Thus, on small length scales, the solution is always stable.…”

Robustness of oscillatoryα2dynamos in spherical wedges

“…One possibility is to study the effect of decreasing the (microphysical) magnetic diffusivity even further. Another possibility is to study the memory effect, which has recently been identified as a means to facilitate oscillatory behaviour (Rheinhardt & Brandenburg 2012), although so far only decaying solutions have been found to be modified in that way (Devlen et al 2013). However, under suitable conditions such solutions can indeed become oscillatory (Rheinhardt et al 2014) and may present a possible solution to the problem where equatorward motion obtained via varying the α profile with a = (0, 0, 1), for example, is limited to certain latitudes.…”