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

TL;DR

This paper investigates the kinematic phase of the supersonic small-scale dynamo in astrophysical turbulence, revealing how compressibility and shocks influence magnetic field amplification and structure across different regimes.

Contribution

It introduces new methods to measure dissipation and magnetic scales, and uncovers universal and regime-dependent behaviors of magnetic energy concentration in compressible turbulence.

Findings

Universal relation $\, ext{ell}_ u/ ext{ell}_ exteta \, extasciitilde \, ext{Pm}^{1/2}$ for $ ext{Pm} \, extgreater \,1$

Incompressible SSDs concentrate magnetic energy at $ ext{ell}_ ext{p} \, extasciitilde \, ext{ell}_ exteta$

Compressible SSDs with shocks concentrate magnetic energy at larger scales $ ext{ell}_ ext{p} \, extgreater \, ext{ell}_ exteta$

Abstract

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.