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 have 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 Rmc 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 − Rmc). On the basis of the Kazantsev model, we therefore expect the existence of the small-scale dynamo for a given value of Pm as long as the magnetic Reynolds number is above the critical threshold.
Highlights
The conversion of kinetic energy into magnetic energy, the so-called dynamo action, plays an important role in a wide range of astrophysical applications
For a narrow range of Pm we found a strong increase of the growth rate, in particular for the fastest growing mode, which depends on the fact that, for 5 ≤ Pm ≤
The presence of the higher growing modes is important since it gives an additional contribution to the magnetic field amplification that becomes more marked for Pm ≫ 1, where the Γvalues are 6 orders of magnitude larger than for Pm ≪ 1
Summary
The conversion of kinetic energy into magnetic energy, the so-called dynamo action, plays an important role in a wide range of astrophysical applications. The Kraichnan model [17, 18] considers the advection of a passive scalar field, which may for instance represent some chemical species, while the Kazantsev model [16] describes the turbulent diffusion of a passive vector field Both models employ the same assumptions, in particular a velocity field based on a zero mean Gaussian random process as well as a δ-correlation in time. The classical studies of the Kazantsev model typically focused only on Kolmogorov turbulence As discussed above, the latter is often not applicable in astrophysical environments, and steeper slopes for the turbulent spectra are frequently found both in numerical simulations and observational data sets. We show the different growing modes for Kolmogorov turbulence, discuss the dependence on different types of turbulence, analyze the behavior close to the threshold and discuss the implications of our results
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