Dynamic Nonlinearity in Large‐Scale Dynamos with Shear

Abstract

We supplement the mean field dynamo growth equation with the total magnetic helicity evolution equation. This provides an explicitly time dependent model for alpha quenching in dynamo theory. For dynamos without shear, this approach accounts for the observed large scale field growth and saturation in numerical simulations. After a significant kinematic phase, the dynamo is resistively quenched, i.e. the saturation time depends on the microscopic resistivity. This is independent of whether or not the turbulent diffusivity is resistively quenched. We find that the approach is also successful for dynamos that include shear and exhibit migratory waves (cycles). In this case however, whether or not the cycle period remains of the order of the dynamical time scale at large magnetic Reynolds numbers does depend how on how the turbulent magnetic diffusivity quenches. Since this is unconstrained by magnetic helicity conservation, the diffusivity is presently an input parameter. Comparison to current numerical experiments suggests a turbulent diffusivity that depends only weakly on the magnetic Reynolds number, but higher resolution simulations are needed.

Analytic solution of an oscillatory migratoryα2stellar dynamo

“…A117, page 4 of 7 A. Brandenburg: Analytic solution of an oscillatory migratory α 2 stellar dynamo we find m ≈ 0.883315, which is close to the value m = 1 for α 2 dynamos in periodic domains (Blackman & Brandenburg 2002).…”

“…(25) of Blackman & Brandenburg (2002). For α 2 dynamos in periodic domains, one finds k m /k 1 = 1, but here we obtain k m /k 1 ≈ 2.253027.…”

“…Interestingly, this is also the value of the magnetic Taylor microscale wavenumber of the mean field, k T , defined through k 2 T = |Ĵ | 2 dz |B | 2 dz, i.e., k T = k m . Finally, for the fractional current helicity of the mean field (Blackman & Brandenburg 2002),…”

Cycles and cycle modulations

“…This value is larger than the one expected from the theory where this ratio should be equal to the ratio of the respective values of k f (Blackman & Brandenburg 2002), namely 2.2/1.5 ≈ 1.5.…”

Magnetic Helicity and Large Scale Magnetic Fields: A Primer

“…Flow Dominated: For this subclass, the input energy is kinetic energy dominated and the EMF can be sustained, for example, by kinetic helicity, as in the classic Parker-type solar dynamo and its extensions (e.g. Moffatt 1978;Parker 1979;Krause & Rädler 1980;Pouquet et al 1976;Blackman & Brandenburg 2002). Alternatively, there is emerging agreement that a combination of shear and fluctuating kinetic helicity can conspire to produce an LSD (Vishniac & Brandenburg 1997;Brandenburg 1995;Brandenburg 2005;Yousef et al 2008;Heinemann et al 2011;Mitra & Brandenburg 2012;Sridhar & Singh 2013).…”