Fundamental scales in the kinematic phase of the turbulent dynamo

Abstract

The turbulent dynamo is a powerful mechanism that converts turbulent kinetic energy to magnetic energy. A key question regarding the magnetic field amplification by turbulence, is, on what scale, $k_{\mathrm{p}}$, do magnetic fields become most concentrated? There has been some disagreement about whether $k_{\mathrm{p}}$ is controlled by the viscous scale, $k_{\hat{I}}½$ (where turbulent kinetic energy dissipates), or the resistive scale, $k_{\hat{I}}\cdot$ (where magnetic fields dissipate). Here we use direct numerical simulations of magnetohydrodynamic turbulence to measure characteristic scales in the kinematic phase of the turbulent dynamo. We run $104$-simulations with hydrodynamic Reynolds numbers of $10\le \mathrm{Re}\le 3600$, and magnetic Reynolds numbers of $270\le \mathrm{Rm}\le 4000$, to explore the dependence of $k_{\mathrm{p}}$ on $k_{\hat{I}}½$ and $k_{\hat{I}}\cdot$. Using physically motivated models for the kinetic and magnetic energy spectra, we measure $k_{\hat{I}}½$, $k_{\hat{I}}\cdot$ and $k_{\mathrm{p}}$, making sure that the obtained scales are numerically converged. We determine the overall dissipation scale relations $k_{\hat{I}}½=(0.025_{-0.006}^{+0.005})k_{\mathrm{turb}}{\mathrm{Re}}^{3/4}$ and $k_{\hat{I}}\cdot =(0.88_{-0.23}^{+0.21})k_{\hat{I}}½{\mathrm{Pm}}^{1/2}$, where $k_{\mathrm{turb}}$ is the turbulence driving wavenumber and $\mathrm{Pm}=\mathrm{Rm}/\mathrm{Re}$ is the magnetic Prandtl number. We demonstrate that the principle dependence of $k_{\mathrm{p}}$ is on $k_{\hat{I}}\cdot$. For plasmas where $\mathrm{Re}\gtrsim 100$, we find that $k_{\mathrm{p}}=(1.2_{-0.2}^{+0.2})k_{\hat{I}}\cdot$, with the proportionality constant related to the power-law `Kazantsev' exponent of the magnetic power spectrum. Throughout this study, we find a dichotomy in the fundamental properties of the dynamo where $\mathrm{Re}>100$, compared to $\mathrm{Re}<100$. We report a minimum critical hydrodynamic Reynolds number, ${\mathrm{Re}}_{\mathrm{crit}}=100$ for bonafide turbulent dynamo action.

15 pages, 10 figures; accepted for publication in MNRAS, 2022 April 4