  We derive scaling relations based on free-fall 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 and
found to hold in the saturation phase, but exhibit discrepancies during the
mixing and decays phases, suggesting a need to relax either the free-fall 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
time scale 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.
