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( ) ∝ ϑ , 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, Rmcrit, using our model of the turbulent velocity correlation function. For Kolmogorov turbulence (ϑ = 1/3), we find that the critical magnetic Reynolds number is Rm K crit ≈ 110 and for Burgers turbulence (ϑ = 1/2) Rm B crit ≈ 2700. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is Γ ∝ Rein the limit of infinite magnetic Prandtl numbers. 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.

A new interpretation of the far-infrared – radio correlation and the expected breakdown at high redshift

“…Most relevantly, their study demonstrated that the dynamo still exists at high Mach numbers, even though the saturation level is somewhat reduced. A similar behavior has been suggested based on analytical calculations by Schober et al (2012c), which was extended into the non-linear regime where backreactions occur by Schleicher et al (2013). A series of simulations by Balsara et al (2001Balsara et al ( , 2004; Balsara & Kim (2005) further demonstrated that the dynamo is efficient in astrophysical environments when driven by supernovae.…”

The implications of dust for high-redshift protogalaxies and the formation of binary disks

“…In the absence of rotation, the small-scale dynamo is expected to produce tangled magnetic fields, close to equipartition with turbulent energy (Arshakian et al 2009;Schleicher et al 2010a). This process was shown to be efficient for a large range of turbulence models and Mach numbers (Federrath et al 2011a;Schober et al 2012). The presence of rotating disks, on the other hand, gives rise to large-scale dynamo effects amplifying the mean magnetic field (Brandenburg & Subramanian 2005).…”

Magnetic fields in primordial accretion disks

“…Turk et al (2012) and Latif et al (2013d) performed cosmological magneto-hydrodynamical simulations and confirmed that the small-scale dynamo is operational during the formation of protogalaxies, and the operation of the smallscale dynamo has been confirmed even for turbulence driven by supernova explosions (Balsara et al 2004;Balsara & Kim 2005). Furthermore, substantial progress has been made in the theoretical understanding of the dynamo, including the regime at high Mach numbers, different types of turbulence, and a large range of magnetic Prandtl numbers (Federrath et al 2011a(Federrath et al , 2014Schober et al 2012b;Bovino et al 2013;Schleicher et al 2013a).…”

“…The conditions for the dynamo to operate require turbulence and a critical magnetic Reynolds number Rm crit = v /η, where v and are the characteristic turbulent velocity and length scales and η the turbulent diffusivity. The minimum magnetic Reynolds number for the dynamo to operate is about Rm crit ∼ 100; for details see Brandenburg & Subramanian (2005) and Schober et al (2012b). Both the Ohmic and ambipolar resistivity during primordial star formation are found to be sufficiently low to stop efficient magnetic field amplification (Schober et al 2012a).…”

“…So, both the Weibel instability and plasma fluctuations can produce a magnetic field on small scales. The small-scale dynamo (Kazantsev 1968) can enhance the magnetic field strength via turbulence (Brandenburg & Subramanian 2005;Schober et al 2012b) and lead to the rapid production of a strong, tangled magnetic field. After saturation on small scales, the dynamo enters the non-linear regime, and larger scales are magnetised on their respective eddy-turnover timescales (Schekochihin et al 2002;Schleicher et al 2013b).…”