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

Axel Brandenburg,Long Chen

doi:10.1017/s0022377820000082

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 (Phys. Rev. Lett., vol. 109, 2012, 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 behaviour is different from that of an$\unicode[STIX]{x1D6FC}^{2}$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 modelled by a negative magnetic diffusivity combined with a positive magnetic hyperdiffusivity. In two other classes of optimal dynamos of Chenet al.(J. Fluid Mech., vol. 783, 2015, pp. 23–45), 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 that negative turbulent magnetic diffusivities may support a broader class of dynamos than previously thought, including these three optimal dynamos.

Highlights

Since the works of Varley, Wheatstone and Siemens of around 1867, we know that electromagnetic dynamos can be self-excited, i.e. they work without permanent magnets to turn kinetic energy into electromagnetic energy
For the TTT case, mean-field dynamo action has only been obtained for column averaged fields
Applying the test-field method (TFM) for xy averages, we find that the only nonvanishing components of the eight transport coefficients are ηxx and ηyy

Summary

Introduction

Since the works of Varley, Wheatstone and Siemens of around 1867, we know that electromagnetic dynamos can be self-excited, i.e. they work without permanent magnets to turn kinetic energy into electromagnetic energy. Unlike those technical dynamos with wires, homogeneous dynamos work in uniformly conducting media (Larmor 1919). Moss (1990) found that the critical magnetic Reynolds number, i.e. the ratio of inertial to resistive electromagnetic forces, is rather large for the dynamo of Gailitis (1970) to be excited This means that the typical scale and velocity can be very large, so an experimental verification is difficult for that flow. For the Ponomarenko dynamo, by contrast, the critical magnetic Reynolds number is sufficiently low so that an experimental verification was successful (Gailitis et al 2000)

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Results

Conclusion