The role of the sonic scale in the growth of magnetic field in compressible turbulence

Abstract

ABSTRACT We study the growth of small fluctuations of magnetic field in supersonic turbulence, the small-scale dynamo. The growth is due to the smallest and fastest turbulent eddies above the resistive scale. We observe that for supersonic turbulence these eddies are localized below the sonic scale ls, defined as the scale where the typical velocity of the turbulent eddies equals the speed of sound, and are therefore effectively incompressible. All previous studies have ignored the existence of the sonic scale and consequently treated the entire inertial range as made up of compressible eddies. However, at large Mach numbers ls is much smaller than the integral scale of the turbulence so the fastest growing mode of the magnetic field belongs to small-scale incompressible turbulence. We determine this mode and the associated growth rate numerically with the aid of a white noise in time model of turbulence whose approximate validity for the description of the Navier–Stokes turbulence is explained. For that purpose, we introduce a new non-dimensional number Rsm that we name the magnetosonic Reynolds number that describes the division of the magnetic field amplification range between small-scale incompressible eddies and large-scale supersonic ones. We show that indeed, as Rsm grows (which means that the incompressible eddies occupy a larger portion of the magnetic field amplification range) the growth rate of the fastest growing mode increases, while the spatial distribution of the growing magnetic field shifts to smaller scales. Our result implies the existence of small-scale dynamo for compressible homogeneous turbulence.