Numerical Analysis of Magnetic Field Amplification by Turbulence

We apply a Fourier spectral numerical method to 3D incompressible MHD turbulence with a magnetic Prandtl number $Pr \geq 1$. We examine the processes by which an initially weak, large-scale seed magnetic field and an initially weak, small-scale, impulse-like seed magnetic field are amplified. We find that in both cases the magnetic energy spectrum grows at all scales. The growth rates at different amplification stages are analyzed. For a large-scale seed magnetic field, the magnetic energy density grows as $\sim t^2$ for the first few turbulence eddy turnover times, followed by an exponential growth stage, of which the growth rate is largely determined by the turbulence eddy turnover time. For a seed magnetic field that is initially set up at a small scale in the turbulence, during the kinematic development stage, the growth rate of magnetic energy is $\propto 1/\tau_{max}$, where $\tau_{max}$ is the eddy turnover time of the smallest eddies of the turbulence. The kinematic growth stage is followed by a dynamic growth stage, during which the growth rate of total magnetic energy is determined by both the magnetic energy amplification within the turbulence inertial range and that within the turbulence dissipation range.