Abstract

The small-scale dynamo plays a substantial role in magnetizing the Universe under a large range of conditions, including subsonic turbulence at low Mach numbers, highly supersonic turbulence at high Mach numbers and a large range of magnetic Prandtl numbers Pm, i.e. the ratio of kinetic viscosity to magnetic resistivity. Low Mach numbers may, in particular, lead to the well-known, incompressible Kolmogorov turbulence, while for high Mach numbers, we are in the highly compressible regime, thus close to Burgers turbulence. In this paper, we explore whether in this large range of conditions, universal behavior can be expected. Our starting point is previous investigations in the kinematic regime. Here, analytic studies based on the Kazantsev model have shown that the behavior of the dynamo depends significantly on Pm and the type of turbulence, and numerical simulations indicate a strong dependence of the growth rate on the Mach number of the flow. Once the magnetic field saturates on the current amplification scale, backreactions occur and the growth is shifted to the next-larger scale. We employ a Fokker-Planck model to calculate the magnetic field amplification during the nonlinear regime, and find a resulting power-law growth that depends on the type of turbulence invoked. For Kolmogorov turbulence, we confirm previous results suggesting a linear growth of magnetic energy. For more general turbulent spectra, where the turbulent velocity scales with the characteristic length scale as u$_{ℓ}${\prop}{\ell}$^{ϑ}$, we find that the magnetic energy grows as (t/T$_{ed}$)$^{2ϑ/(1-ϑ)}$, with t being the time coordinate and T$_{ed}$ the eddy-turnover time on the forcing scale of turbulence. For Burgers turbulence, {\thetav} = 1/2, quadratic rather than linear growth may thus be expected, as the spectral energy increases from smaller to larger scales more rapidly. The quadratic growth is due to the initially smaller growth rates obtained for Burgers turbulence. Similarly, we show that the characteristic length scale of the magnetic field grows as t$^{1/(1-ϑ)}$ in the general case, implying t$^{3/2}$ for Kolmogorov and t$^{2}$ for Burgers turbulence. Overall, we find that high Mach numbers, as typically associated with steep spectra of turbulence, may break the previously postulated universality, and introduce a dependence on the environment also in the nonlinear regime.