Magnetic Field Amplification by Small-Scale Dynamo Action: Dependence on Turbulence Models and Reynolds and Prandtl Numbers

摘要

The small-scale dynamo is a process by which turbulent kinetic energy isconverted into magnetic energy, and thus is expected to depend crucially on thenature of turbulence. In this work, we present a model for the small-scaledynamo that takes into account the slope of the turbulent velocity spectrumv(l) ~ l^theta, where l and v(l) are the size of a turbulent fluctuation andthe typical velocity on that scale. The time evolution of the fluctuationcomponent of the magnetic field, i.e., the small-scale field, is described bythe Kazantsev equation. We solve this linear differential equation for itseigenvalues with the quantum-mechanical WKB-approximation. The validity of thismethod is estimated as a function of the magnetic Prandtl number Pm. Wecalculate the minimal magnetic Reynolds number for dynamo action, Rm_crit,using our model of the turbulent velocity correlation function. For Kolmogorovturbulence (theta=1/3), we find that the critical magnetic Reynolds number isapproximately 110 and for Burgers turbulence (theta=1/2) approximately 2700.Furthermore, we derive that the growth rate of the small-scale magnetic fieldfor a general type of turbulence is Gamma ~ Re^((1-theta)/(1+theta)) in thelimit of infinite magnetic Prandtl numbers. For decreasing magnetic Prandtlnumber (down to Pm approximately larger than 10), the growth rate of thesmall-scale dynamo decreases. The details of this drop depend on theWKB-approximation, which becomes invalid for a magnetic Prandtl number of aboutunity.