The α-effect in 2D fast dynamos

Abstract

We look at the large-scale dynamo properties of spatially periodic, time dependent, helical 2D flows of the form u(x, t) = (∂ _y  ψ (x, y, t), −∂ _x  ψ (x, y, t), −ψ (x, y, t). These flows act as kinematic fast dynamos and are able to generate a mean magnetic field uniform and constant in the xy-plane but whose direction varies periodically along z with wavenumber k. Using Mean Field Electrodynamics, the generation mechanism can be understood in terms of a k-dependent α-effect, which depends on the magnetic Reynolds number, R _m . We calculate this effect for different motions and investigate how its limit as k → 0 depends on R _m and on the properties of the flows such as their spatial structure or correlation time. This work generalises earlier studies based on 2D steady flows to motions with time dependence.

Keywords:

• Magnetic fields
• Fast dynamos
• Alpha-effect

Acknowledgements

The author is deeply grateful to D.W. Hughes and S.M. Tobias for the support provided whilst carrying out this research and for numerous useful comments on the manuscript, to A.D. Gilbert, F. Plunian and A.M. Rucklidge for fruitful discussions and to the anonymous referees for helpful comments. This project was carried out with the financial support of the EPSRC and the University of Leeds and completed under PPARC grant PP/D00179X/1.

Notes

^†We identified the symmetry of these modes using timeseries of the mean field (see figure 2).

^†This might however depend on the value of ε. As ε → 0, the chaotic regions shrink and the α-effect might be affected. Indeed for ε = 0, we recover the Roberts flow for which α → 0 as R _m  → ∞. The results presented here are however limited to ε = 0.5, 0.75 and 1.0 for which the chaotic regions are substantial (see figure 1).