title: Magnetic field amplification by small-scale dynamo action: dependence on turbulence models and Reynolds and Prandtl numbers.
abstract: 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(ℓ)proportional ℓ([symbol see text])V}, 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 ([symbol see text] = 1/3), we find that the critical magnetic Reynolds number is Rm(crit) (K) ≈ 110 and for Burgers turbulence ([symbol see text] = 1/2) Rm(crit)(B) ≈ 2700. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is Γ proportional Re((1-[symbol see text])/(1+[symbol see text])) in the limit of infinite magnetic Prandtl number. For decreasing magnetic Prandtl number (down to Pm >/~ 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.
authors.affiliation: Institut für Theoretische Astrophysik, Zentrum für Astronomie der Universität Heidelberg, Albert-Überle-Strasse 2, D-69120 Heidelberg, Germany. schober@stud.uni-heidelberg.de
authors.forename: Jennifer
authors.initials: J
authors.lastname: Schober
authors.forename: Dominik
authors.initials: D
authors.lastname: Schleicher
authors.forename: Christoph
authors.initials: C
authors.lastname: Federrath
authors.forename: Ralf
authors.initials: R
authors.lastname: Klessen
authors.forename: Robi
authors.initials: R
authors.lastname: Banerjee