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

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.