1112.3926 (N. Kleeorin et al.)

Growth rate of small-scale dynamo at low magnetic Prandtl numbers    [PDF]

N. Kleeorin, I. Rogachevskii

In this study we discuss two key issues related to a small-scale dynamo instability at low magnetic Prandtl numbers and large magnetic Reynolds numbers, namely: (i) the validity of the scaling, $\lambda \propto Rm^{1/2}$, for the growth rate of small-scale dynamo instability in the vicinity of the threshold; (ii) the existence of the Golitsyn spectrum of magnetic fluctuations in small-scale dynamos. Our analysis shows that there are two different asymptotics for the small-scale dynamo growth rate: in the vicinity of the threshold of the excitation of the small-scale dynamo instability, $\lambda \propto \ln(Rm / Rm^{cr})$, and when the magnetic Reynolds number is much larger than the threshold of the excitation of the small-scale dynamo instability, $\lambda \propto Rm^{1/2}$, where $Rm^{cr}$ is the small-scale dynamo instability threshold in the magnetic Reynolds number. We also demonstrated that the existence of the Golitsyn spectrum of magnetic fluctuations requires a finite correlation time of the random velocity field. On the other hand, the influence of the Golitsyn spectrum on the small-scale dynamo instability is minor. This is the reason why so difficult to observe this spectrum in DNS for the small-scale dynamo with low magnetic Prandtl numbers.

View original: http://arxiv.org/abs/1112.3926