Wednesday, June 12, 2013

1306.2495 (S. Sridhar et al.)

Dynamo action due to alpha fluctuations in a shear flow: mean--field theory    [PDF]

S. Sridhar, Nishant K. Singh
We present an analytical theory of the growth of a large-scale mean magnetic field in a linear shear flow with fluctuations in time of the alpha parameter (equivalently, kinetic helicity). Using shearing coordinates and Fourier variables we derive a set of coupled integro-differential equations, governing the dynamics of the mean magnetic field, that are non perturbative in the rate of shear. When the alpha fluctuations are of white-noise form, the mean electromotive force (EMF) is identical to the negative diffusive form derived by Kraichnan for the case of no shear; the physical reason is that shear takes time to act, and white-noise fluctuations have zero correlation time. We demonstrate that the white-noise case does not allow for large-scale dynamo action. We then allow for a small but non zero correlation time and show that, for a slowly varying mean magnetic field, the mean EMF has additional terms that depend on a combination of shear and alpha fluctuations; the mean-field equations now reduce to a set of coupled partial differential equations. A dispersion relation for modes is derived and studied in detail for growing solutions. Our salient results are: (i) a necessary condition for dynamo action giving the minimum value of shear required; (ii) two types of dynamos depending on the different forms taken by the growth rate as a function of wavenumber; (iii) explicit expressions for the growth rate and wavenumber of the fastest growing mode; these are not only consistent with the scalings with shear seen in numerical simulations, but also provide an estimate of the strength of alpha fluctuations.
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