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 most established SSD theories have only considered incompressible flows. To address this gap, we perform viscoresistive SSD simulations across a range of sonic Mach numbers (M), hydrodynamic Reynolds numbers (Re), and magnetic Prandtl numbers (Pm), focusing on the exponential growth phase. From these simulations, we develop robust measurements of the kinetic and magnetic energy dissipation scales (l_ν and l_η, respectively), and show that l_ν/l_η ∼ Pm¹/² is a universal feature of turbulent (Re ≥ Re_crit ≈ 100), Pm ≥ 1 SSDs, regardless of M. We also measure the scale of maximum magnetic field strength (l_p), where we confirm that incompressible SSDs (where either M ≤ 1 or Re < Re_crit) concentrate magnetic energy at l_p ∼ l_η with inversely correlated field strength and curvature. By contrast, for compressible SSDs (where M > 1 and Re ≥ Re_crit), shocks concentrate magnetic energy in large, overdense, coherent structures with l_p ∼ (l_turb/l_shock)¹/³l_η ≫ l_η, where l_shock is the characteristic shock width, and l_turb is the outer scale of the turbulent field. When Pm < Re²/³, the shift of l_p (from the incompressible to compressible flow regime) is large enough to move the peak magnetic energy scale out of the subviscous range, and the plasma converges on a hierarchy of scales: l_turb > l_p > l_shock > l_ν > l_η. In the compressible flow regime, more broadly, we also find that magnetic field-line curvature becomes nearly independent of the field strength, not because the field geometry has changed, but instead the field becomes locally amplified through flux-frozen compression by shocks. These results have implications for various astrophysical plasma environments in the early Universe, and cosmic ray transport models in the interstellar medium.