Large ${\rm Pm}$ small-scale kinematic dynamo in protoneutron stars

Abstract: Magnetars are young, isolated neutron stars that possess an exceptionally strong magnetic field, with surface dipolar strengths on the order of $10<sup>{15}$ G. One of the plausible scenarios for generating such a strong field is an exponential amplification by a turbulent convective dynamo during the protoneutron star phase. However, the short expected duration of the convection ($\sim 10$ s) imposes a stringent constraint on the dynamo growth rate. We perform an extensive set of 82 three-dimensional convective dynamo simulations in the anelastic approximation and investigate the kinematic phase to quantify the dynamo growth rate $γ$. We find that $γ$ increases with both the magnetic Prandtl number ${\rm Pm}$ and the Rayleigh number ${\rm Ra}$, with the most unstable mode becoming highly non-axisymmetric and multipolar. We further observe a gradual transition from large-scale to small-scale dynamo as the magnetic Reynolds number ${\rm Rm}$ increases, resulting in a magnetic field that is predominantly concentrated at small scales. The trend remains unchanged when the outer magnetic boundary condition is varied. Since resolving the increasingly small scales becomes numerically impractical, we employ the theoretical small-scale Kazantsev dynamo model to explore the large ${\rm Pm}$ regime characteristic of protoneutron stars. The model qualitatively captures the growth rate behaviour observed in simulations and, upon extrapolation to the large ${\rm Pm}$ limit, indicates that $γ$ is only weakly dependent on the resistivity in a PNS. Under conditions relevant to the PNS, this model predicts a magnetic energy growth rate of the order of $\sim 1$ ms$<sup>{-1}$.