The universal growth of magnetic energy during the nonlinear phase of subsonic and supersonic small-scale dynamos

Small-scale dynamos (SSDs) amplify magnetic fields in turbulent plasmas. Theory predicts nonlinear magnetic energy growth $E_\mathrm{mag} \propto t^{p_\mathrm{nl}}$, but this scaling has not been tested across flow regimes. Using a large ensemble of SSD simulations spanning subsonic to supersonic turbulence, we measure linear growth ($p_\mathrm{nl} = 1$) in subsonic flows and quadratic growth ($p_\mathrm{nl} = 2$) in supersonic flows. In all cases, the nonlinear dynamo converts a nearly constant fraction $\sim 1/100$ of the turbulent kinetic energy flux into magnetic energy, and the nonlinear phase has a characteristic duration $\Delta t \approx 20\,t_0$, where $t_0$ is the outer-scale turnover time. By isolating the onset of magnetic backreaction in SSDs, our statistical ensemble approach identifies a robust efficiency and duration for the nonlinear SSD that can be used to interpret more complex astrophysical and laboratory plasmas.