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

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

2011-09-21

arxiv.org/abs/1109.4571v2

Astronomy and Astrophysics

Astrophysics

Cosmology and Extragalactic Astrophysics

13 pages, 8 figures; published in Phys. Rev. E 2012

Scientific paper

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.

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