Turbulent magnetic field amplification from the smallest to the largest magnetic Prandtl numbers

TL;DR

This paper uses the Kazantsev model to analyze magnetic field amplification by the small-scale dynamo across all magnetic Prandtl numbers and turbulence spectra, revealing key scaling laws and thresholds relevant for astrophysical environments.

Contribution

It provides the first comprehensive solution for the small-scale dynamo across the full range of magnetic Prandtl numbers and turbulence types, including new scaling laws and critical thresholds.

Findings

Dynamo growth rate increases rapidly near Pm ~ 1 for all turbulence types.

Steeper turbulence spectra reduce the dynamo growth rate.

Critical magnetic Reynolds number varies with turbulence spectrum, highest for Burgers.

Abstract

The small-scale dynamo provides a highly efficient mechanism for the conversion of turbulent into magnetic energy. In astrophysical environments, such turbulence often occurs at high Mach numbers, implying steep slopes in the turbulent spectra. It is thus a central question whether the small-scale dynamo can amplify magnetic fields in the interstellar or intergalactic media, where such Mach numbers occur. To address this long-standing issue, we employ the Kazantsev model for turbulent magnetic field amplification, systematically exploring the effect of different turbulent slopes, as expected for Kolmogorov, Burgers, the Larson laws and results derived from numerical simulations. With the framework employed here, we give the first solution encompassing the complete range of magnetic Prandtl numbers, including Pm << 1, Pm ~ 1 and Pm >> 1. We derive scaling laws of the growth rate as a function of hydrodynamic and magnetic Reynolds number for Pm << 1 and Pm >> 1 for all types of turbulence. A central result concerns the regime of Pm ~ 1, where the magnetic field amplification rate increases rapidly as a function of Pm. This phenomenon occurs for all types of turbulence we explored. We further find that the dynamo growth rate can be decreased by a few orders of magnitude for turbulence spectra steeper than Kolmogorov. We calculate the critical magnetic Reynolds number Rm_c for magnetic field amplification, which is highest for the Burgers case. As expected, our calculation shows a linear behaviour of the amplification rate close to the threshold proportional to Rm-Rm_c. Based on the Kazantsev model, we therefore expect the existence of the small-scale dynamo for any given value of Pm, as long as the magnetic Reynolds number is above the critical threshold.