Magnetic field amplification by small-scale dynamo action: Dependence on turbulence models and Reynolds and Prandtl numbers

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, Rm₂ₑ₈ₓ, 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.