The nature of mean-field generation in three classes of optimal dynamos

Abstract

In recent years, several optimal dynamos have been discovered. They minimize the magnetic energy dissipation or, equivalently, maximize the growth rate at a fixed magnetic Reynolds number. In the optimal dynamo of Willis (2012, Phys. Rev. Lett. 109, 251101), we find mean-field dynamo action for planar averages. One component of the magnetic field grows exponentially while the other decays in an oscillatory fashion near onset. This behavior is different from that of an α 2 dynamo, where the two non-vanishing components of the planar averages are coupled and have the same growth rate. For the Willis dynamo, we find that the mean field is excited by a negative turbulent magnetic diffusivity, which has a non-uniform spatial profile near onset. The temporal oscillations in the decaying component are caused by the corresponding component of the diffusivity tensor being complex when the mean field is decaying and, in this way, time-dependent. The growing mean field can be modeled by a negative magnetic diffusivity combined with a positive magnetic hyperdiffusivity. In two other classes of optimal dynamos of Chen et al. (2015, J. Fluid Mech. 783, 23), we find, to some extent, similar mean-field dynamo actions. When the magnetic boundary conditions are mixed, the two components of the planar averaged field grow at different rates when the dynamo is 15% supercritical. When the mean magnetic field satisfies homogeneous boundary conditions (where the magnetic field is tangential to the boundary), mean-field dynamo action is found for one-dimensional averages, but not for planar averages. Despite having different spatial profiles, both dynamos show negative turbulent magnetic diffusivities. Our finding suggests negative turbulent magnetic diffusivities may support a broader class of dynamos than previously thought, including these three optimal dynamos. † Email address for correspondence: brandenb@nordita.org

Cross-helicity effect onα-type dynamo in non-equilibrium turbulence

“…When is not small, higher-order derivatives in space and time may play important roles in (3.6). See § 7.2 of Brandenburg & Chen (2020) for an example. See also Appendix A.4 of Yokoi (2013) regarding the assumptions and limitations of the TSDIA.…”

Compressible Test-field Method and Its Application to Shear Dynamos

“…For Pr M = 5, the LSD grows the slowest, and because its SSD saturation strength was lower, 10 growth is seen also in B x . Its growth rate, however, is different from that of B y , which is somewhat atypical of "standard" dynamos and could be taken to be indicative of eigenmodes that consist of only one component; see Rheinhardt et al (2014) and Brandenburg & Chen (2020) for examples. However, for an LSD based on the coherent effects alone, this can be ruled out here.…”

Compressible test-field method and its application to shear dynamos

“…For Pr M = 5, the LSD grows the slowest, and, because its SSD's saturation strength was lower 4 , growth is seen also in B x . Its growth rate, however, is different from that of B y , which is somewhat atypical of "standard" dynamos and could be taken indicative of eigenmodes that consist of only one component; see Rheinhardt et al (2014); Brandenburg & Chen (2020) for examples. However, for an LSD based on the coherent effects alone this can be ruled out here.…”