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

TL;DR

This paper investigates the growth rate scaling of small-scale dynamo instability at low magnetic Prandtl numbers and examines the existence and observational challenges of the Golitsyn magnetic fluctuation spectrum.

Contribution

It derives the asymptotic behaviors of the dynamo growth rate near threshold and at high magnetic Reynolds numbers, and analyzes the conditions for the Golitsyn spectrum's existence.

Findings

Growth rate scales as ln(Rm/Rm^{cr}) near threshold

Growth rate scales as Rm^{1/2} at high Rm

Golitsyn spectrum requires finite velocity correlation time

Abstract

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 scaling for the growth rate of small-scale dynamo instability in the vicinity of the dynamo threshold; (ii) the existence of the Golitsyn spectrum of magnetic fluctuations in small-scale dynamos. 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, $\lambda \propto \ln({\rm Rm}/ {\rm Rm}^{\rm cr})$, and when the magnetic Reynolds number is much larger than the threshold of the excitation of the small-scale dynamo instability, $\lambda \propto {\rm Rm}^{1/2}$, where ${\rm Rm}^{\rm cr}$ is the small-scale dynamo instability threshold in the magnetic Reynolds number ${\rm Rm}$. We 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 it is so difficult to observe this spectrum in direct numerical simulations for the small-scale dynamo with low magnetic Prandtl numbers.