Fundamental scales in the kinematic phase of the turbulent dynamo

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, $k_{\rm p}$, do magnetic fields become most concentrated? There has been some disagreement about whether $k_{\rm p}$ is controlled by the viscous scale, $k_\nu$ (where turbulent kinetic energy dissipates), or the resistive scale, $k_\eta$ (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 $10 \leq {\rm Re} \leq 3600$, and magnetic Reynolds numbers of $270 \leq {\rm Rm} \leq 4000$, to explore the dependence of $k_{\rm p}$ on $k_\nu$ and $k_\eta$. Using physically motivated models for the kinetic and magnetic energy spectra, we measure $k_\nu$, $k_\eta$ and $k_{\rm p}$, making sure that the obtained scales are numerically converged. We determine the overall dissipation scale relations $k_\nu = (0.025^{+0.005}_{-0.006})\, k_{\rm turb}\, {\rm Re}^{3/4}$ and $k_\eta = (0.88^{+0.21}_{-0.23})\, k_\nu\, {\rm Pm}^{1/2}$, where $k_{\rm turb}$ is the turbulence driving wavenumber and ${\rm Pm}={\rm Rm}/{\rm Re}$ is the magnetic Prandtl number. We demonstrate that the principle dependence of $k_{\rm p}$ is on $k_\eta$. For plasmas where ${\rm Re} \gtrsim 100$, we find that $k_{\rm p} = (1.2_{-0.2}^{+0.2})\, k_\eta$, 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 ${\rm Re} > 100$, compared to ${\rm Re} < 100$. We report a minimum critical hydrodynamic Reynolds number, ${\rm Re}_{\rm crit} = 100$ for bonafide turbulent dynamo action.