Scaling of Small-scale Dynamo Properties in the Rayleigh-Taylor Instability

We derive scaling relations based on freefall and isotropy assumptions for the kinematic small-scale dynamo growth rate and amplification factor over the course of the mixing, saturation, and decay phases of the Rayleigh-Taylor instability (RTI) in a fully ionized plasma. The scaling relations are tested using sets of three-dimensional, visco-resistive MHD simulations of the RTI. They are found to hold in the saturation phase, but exhibit discrepancies during the mixing and decay phases, suggesting a need to relax either the freefall or isotropy assumptions. Application of the scaling relations allows for quantitative prediction of the net amplification of magnetic energy in the kinematic dynamo phase and therefore a determination of whether the magnetic energy either remains sub-equipartition at all velocity scales or reaches equipartition with at least some scales of the turbulent kinetic energy in laboratory and astrophysical scenarios. As an example, we consider the dynamo in RTI-unstable regions of the outer envelope of a binary neutron star merger, and predict that the kinematic regime of the small-scale dynamo ends on the timescale of nanoseconds and then reaches saturation on a timescale of microseconds, which are both fast compared to the millisecond relaxation time of the post-merger.