Fluctuation dynamos in supersonic turbulence at ${\rm Pm} \gtrsim 1$

Fluctuation dynamos provide a robust mechanism for amplifying weak seed magnetic fields in turbulent astrophysical plasmas. However, their behaviour in the highly compressible regimes characteristic of the interstellar medium remains incompletely understood. Using high-resolution 3D magnetohydrodynamic simulations of supersonic turbulence with rms Mach number $\mathcal{M}_{\rm rms} \approx 11$, we explore fluctuation dynamos across magnetic Prandtl numbers ${\rm Pm} = 1-10$. At ${\rm Pm}=1$, dynamo growth is slower and saturates at lower magnetic-to-kinetic energy ratios, with amplification in the kinematic phase dominated by compression rather than line stretching. In contrast, at ${\rm Pm}=10$, vortical stretching emerges as the dominant mechanism, yielding faster growth, higher saturation levels, and stronger suppression of density--magnetic field correlations by magnetic pressure. This transition is reflected in the correlation coefficient between density and magnetic field strength, which is strongly positive at ${\rm Pm}=1$ but decreases significantly at higher ${\rm Pm}$. Across all runs, the ratio of velocity-to-magnetic integral scales is $\sim 3.4$, in the saturated phase, independent of ${\rm Pm}$, while the ratio of viscous to resistive dissipation scales increases with the increase in ${\rm Pm}$. Synthetic Faraday rotation measures reveal coherence lengths of $\sim$one-fourth to one-third of the forcing scale across the range of ${\rm Pm}$ explored. Using these coherence scales, we discuss the potential contribution of fluctuation dynamos to Faraday rotation expected from turbulent, gas-rich young disk galaxies.