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, $\lambda \propto \ln (\mathrm{Rm}/{\mathrm{Rm}}^{\mathrm{cr}})$, and when the magnetic Reynolds number is much larger than the threshold of the excitation of the small-scale dynamo instability, $\lambda \propto {\mathrm{Rm}}^{1/2}$, where ${\mathrm{Rm}}^{\mathrm{cr}}$ is the small-scale dynamo instability threshold in the magnetic Reynolds number $\mathrm{Rm}$. 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.

14 pages, 1 figure, revised version

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• Small-scale dynamos in simulations of stratified turbulent convection
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• Magneto-Thermal Instability In Galaxy Clusters II: Three-Dimensional Simulations
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• Small-scale turbulent dynamo for low-Prandtl number fluid: comparison of the theory with results of numerical simulations