Critical magnetic Reynolds number of the weakly-collisional turbulent dynamo in supersonic plasmas

Weakly-collisional plasmas are found in several turbulent astrophysical environments, such as the Intra-galaxy-cluster medium (ICM) and the solar wind, where magnetic fields can be amplified by the turbulent dynamo. The onset of dynamo action is determined by the critical magnetic Reynolds number, ${\mathrm{R}\mathrm{m}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$, above which turbulent motions amplify magnetic energy. Although ${\mathrm{R}\mathrm{m}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ has been studied extensively in collisional magnetohydrodynamic (MHD) turbulence, its value in supersonic weakly-collisional plasmas remains unknown. Here we determine ${\mathrm{R}\mathrm{m}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ using three-dimensional hybrid particle-in-cell (HPIC) simulations of solenoidally driven turbulence with a sonic Mach number $\approx 2$. We vary the magnetic Reynolds number over $\mathrm{R}\mathrm{m}\approx 31$-$1000$ and consider two initial plasma magnetisations, characterised by initial Larmor ratios $r_{\mathrm{L}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{o}\mathrm{r},0}\mathrm{/}{L}_{\mathrm{b}\mathrm{o}\mathrm{x}}=1$ and $100$. We find that the magnetic energy grows exponentially at sufficiently large $\mathrm{R}\mathrm{m}$ and decays when magnetic diffusion dominates at low $\mathrm{R}\mathrm{m}$. By fitting the magnetic-energy growth rate as a function of $\mathrm{R}\mathrm{m}$, we measure ${\mathrm{R}\mathrm{m}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=190_{-17}^{+21}$ and $198_{-18}^{+16}$ for the initially magnetised and unmagnetised plasma, respectively. These values are the same within their uncertainties, indicating that the initial plasma magnetisation does not significantly affect ${\mathrm{R}\mathrm{m}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ in the supersonic regime. The measured values are higher than those previously obtained for the subsonic weakly-collisional dynamo, showing that increasing compressibility makes weakly-collisional turbulent dynamo action more difficult to excite. The measured values are also approximately 1.5 times higher than the ${\mathrm{R}\mathrm{m}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ value reported for supersonic MHD turbulence.