Fundamental MHD scales -- II: the kinematic phase of the supersonic small-scale dynamo

Many astrophysical small-scale dynamos (SSDs) amplify weak magnetic fields via highly compressible, supersonic turbulence, but established SSD theories have overlooked these compressible effects. To address this, we perform visco-resistive SSD simulations across a range of sonic Mach numbers ($\mathcal{M}$), hydrodynamic Reynolds numbers ($\mathrm{Re}$), and magnetic Prandtl numbers ($\mathrm{Pm}$). We develop robust methods to measure kinetic and magnetic energy dissipation scales ($\ell_\nu$ and $\ell_\eta$) and the scale of strongest magnetic fields ($\ell_\mathrm{p}$) during the kinematic phase. We demonstrate that $\ell_\nu/\ell_\eta \sim \mathrm{Pm}^{1/2}$ is a universal feature for $\mathrm{Pm} \geq 1$ SSDs, regardless of $\mathcal{M}$ or $\mathrm{Re}$. Incompressible SSDs (either $\mathcal{M} \leq 1$ or $\mathrm{Re} < \mathrm{Re}\mathrm{crit} \approx 100$) concentrate magnetic energy at $\ell_\mathrm{p} \sim \ell_\eta$ with inversely correlated field strength and curvature. However, for compressible SSDs ($\mathcal{M} > 1$ and $\mathrm{Re} > \mathrm{Re}\mathrm{crit}$), shocks concentrate magnetic energy in large structures with $\ell_\mathrm{p} \sim (\ell_\mathrm{turb} / \ell_\mathrm{shock})^{1/3} \ell_\eta \gg \ell_\eta$, where $\ell_\mathrm{shock}$ is the characteristic shock width, and $\ell_\mathrm{turb}$ is the outer scale of the turbulent field. In this regime, magnetic field-line curvature becomes nearly independent of field strength. These results have implications for galaxy mergers and cosmic ray transport models in the interstellar medium.