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

TL;DR

This paper models the small-scale dynamo process, analyzing how turbulence characteristics and physical parameters like Reynolds and Prandtl numbers influence magnetic field amplification, providing critical thresholds and growth rate dependencies.

Contribution

It introduces a turbulence-dependent model for the small-scale dynamo using the Kazantsev equation and WKB-approximation, highlighting the impact of turbulence spectra on dynamo thresholds and growth rates.

Findings

Critical magnetic Reynolds number is ~110 for Kolmogorov turbulence.

Critical magnetic Reynolds number is ~2700 for Burgers turbulence.

Growth rate scales with Re as Re^((1-theta)/(1+theta)) at high magnetic Prandtl numbers.

Abstract

The small-scale dynamo is a process by which turbulent kinetic energy is converted into magnetic energy, and thus is expected to depend crucially on the nature of turbulence. In this work, we present a model for the small-scale dynamo that takes into account the slope of the turbulent velocity spectrum v(l) ~ l^theta, where l and v(l) are the size of a turbulent fluctuation and the typical velocity on that scale. The time evolution of the fluctuation component of the magnetic field, i.e., the small-scale field, is described by the Kazantsev equation. We solve this linear differential equation for its eigenvalues with the quantum-mechanical WKB-approximation. The validity of this method is estimated as a function of the magnetic Prandtl number Pm. We calculate the minimal magnetic Reynolds number for dynamo action, Rm_crit, using our model of the turbulent velocity correlation function. For Kolmogorov turbulence (theta=1/3), we find that the critical magnetic Reynolds number is approximately 110 and for Burgers turbulence (theta=1/2) approximately 2700. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is Gamma ~ Re^((1-theta)/(1+theta)) in the limit of infinite magnetic Prandtl numbers. For decreasing magnetic Prandtl number (down to Pm approximately larger than 10), the growth rate of the small-scale dynamo decreases. The details of this drop depend on the WKB-approximation, which becomes invalid for a magnetic Prandtl number of about unity.