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The small-scale dynamo provides a highly efficient mechanism for theconversion of turbulent into magnetic energy. In astrophysical environments,such turbulence often occurs at high Mach numbers, implying steep slopes in theturbulent spectra. It is thus a central question whether the small-scale dynamocan amplify magnetic fields in the interstellar or intergalactic media, wheresuch Mach numbers occur. To address this long-standing issue, we employ theKazantsev model for turbulent magnetic field amplification, systematicallyexploring the effect of different turbulent slopes, as expected for Kolmogorov,Burgers, the Larson laws and results derived from numerical simulations. Withthe framework employed here, we give the first solution encompassing thecomplete 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 andmagnetic Reynolds number for Pm << 1 and Pm >> 1 for all types of turbulence. Acentral result concerns the regime of Pm ~ 1, where the magnetic fieldamplification rate increases rapidly as a function of Pm. This phenomenonoccurs for all types of turbulence we explored. We further find that the dynamogrowth rate can be decreased by a few orders of magnitude for turbulencespectra steeper than Kolmogorov. We calculate the critical magnetic Reynoldsnumber Rm_c for magnetic field amplification, which is highest for the Burgerscase. As expected, our calculation shows a linear behaviour of theamplification rate close to the threshold proportional to Rm-Rm_c. Based on theKazantsev model, we therefore expect the existence of the small-scale dynamofor any given value of Pm, as long as the magnetic Reynolds number is above thecritical threshold.

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