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In recent years, several optimal dynamos have been discovered. They minimizethe magnetic energy dissipation or, equivalently, maximize the growth rate at afixed 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 decaysin an oscillatory fashion near onset. This behavior is different from that ofan alpha^2 dynamo, where the two non-vanishing components of the planaraverages are coupled and have the same growth rate. For the Willis dynamo, wefind that the mean field is excited by a negative turbulent magneticdiffusivity, which has a non-uniform spatial profile near onset. The temporaloscillations in the decaying component are caused by the correspondingcomponent of the diffusivity tensor being complex when the mean field isdecaying and, in this way, time-dependent. The growing mean field can bemodeled by a negative magnetic diffusivity combined with a positive magnetichyperdiffusivity. 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 dynamoactions. When the magnetic boundary conditions are mixed, the two components ofthe planar averaged field grow at different rates when the dynamo is 15%supercritical. When the mean magnetic field satisfies homogeneous boundaryconditions (where the magnetic field is tangential to the boundary), mean-fielddynamo action is found for one-dimensional averages, but not for planaraverages. Despite having different spatial profiles, both dynamos show negativeturbulent magnetic diffusivities. Our finding suggests negative turbulentmagnetic diffusivities may support a broader class of dynamos than previouslythought, including these three optimal dynamos.

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