Small-scale dynamo saturation across magnetic Prandtl numbers using the EDQNM closure

Small-scale dynamos (SSDs) are believed to be the primary source of magnetic fields in all turbulent astrophysical systems, especially those with weak rotation such as elliptical galaxies and galaxy clusters. The initial kinematic phase of these dynamos is relatively well understood. Here we demonstrate analytically and numerically that, in an appropriate limit, the eddy-damped quasi-normal Markovian (EDQNM) closure for incompressible magnetohydrodynamic turbulence is strictly equivalent to the earlier models of kinematic dynamos. Moreover, it allows the extension of the kinematic dynamo framework to multi-scale turbulent flows and into the nonlinear regime. The EDQNM closure also enables us to explore a wide parameter range which is inaccessible to direct numerical simulations of the SSD. Using nonhelical EDQNM simulations, we identify several asymptotic regimes of nonlinear dynamo action when the system is highly turbulent with fluid Reynolds number $Re \gtrsim 10^6$ for magnetic Prandtl number $Pm > 1$ and magnetic Reynolds number $Rm \gtrsim 10^6$ for $Pm < 1$: 1) the kinematic growth rate approaches a value independent of $Pm$, 2) the saturated magnetic to kinetic energy ratio similarly converges to $\simeq 0.55$ across $Pm$, while the ratio of magnetic to kinetic integral wavenumbers asymptotes to $\simeq 3$. For all $Pm$, we further find strong feedback between magnetic field and velocity field largely via Alfv\'{e}nisation leading to a saturated kinetic and magnetic spectra with almost the same inertial range with a slope of $-3/2$. These findings could provide guidance for future global simulations and for modeling the nonlinear regime of astrophysical systems living in these extreme limits.