TY - JOUR

T1 - Magnetic field amplification by small-scale dynamo action

T2 - Dependence on turbulence models and Reynolds and Prandtl numbers

AU - Schober, Jennifer

AU - Schleicher, Dominik

AU - Federrath, Christoph

AU - Klessen, Ralf

AU - Banerjee, Robi

PY - 2012/2/3

Y1 - 2012/2/3

N2 - The small-scale dynamo is a process by which turbulent kinetic energy is converted into magnetic energy, and thus it is expected to depend crucially on the nature of the turbulence. In this paper, we present a model for the small-scale dynamo that takes into account the slope of the turbulent velocity spectrum v(), where and v() 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 (=1/3), we find that the critical magnetic Reynolds number is RmcritK110 and for Burgers turbulence (=1/2) RmcritB2700. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is ΓRe (1 -) /(1 +) in the limit of infinite magnetic Prandtl number. For decreasing magnetic Prandtl number (down to Pm10), 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.

AB - The small-scale dynamo is a process by which turbulent kinetic energy is converted into magnetic energy, and thus it is expected to depend crucially on the nature of the turbulence. In this paper, we present a model for the small-scale dynamo that takes into account the slope of the turbulent velocity spectrum v(), where and v() 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 (=1/3), we find that the critical magnetic Reynolds number is RmcritK110 and for Burgers turbulence (=1/2) RmcritB2700. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is ΓRe (1 -) /(1 +) in the limit of infinite magnetic Prandtl number. For decreasing magnetic Prandtl number (down to Pm10), 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.

UR - http://www.scopus.com/inward/record.url?scp=84857597311&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.85.026303

DO - 10.1103/PhysRevE.85.026303

M3 - Article

SN - 2470-0045

VL - 85

JO - Physical Review E

JF - Physical Review E

IS - 2

M1 - 026303

ER -