  Nuclear burning plays a key role in a wide range of astrophysical stellar
transients, including thermonuclear, pair instability, and core collapse
supernovae, as well as kilonovae and collapsars. Turbulence is now understood
to also play a key role in these astrophysical transients. Here we demonstrate
that turbulent nuclear burning may lead to large enhancements above the uniform
background burning rate, since turbulent dissipation gives rise to temperature
fluctuations, and in general the nuclear burning rates are highly sensitive to
temperature. We derive results for the turbulent enhancement of the nuclear
burning rate under the influence of strong turbulence in the distributed
burning regime in homogeneous isotropic turbulence, using probability
distribution function (PDF) methods. We demonstrate that the turbulent
enhancement obeys a universal scaling law in the limit of weak turbulence. We
further demonstrate that, for a wide range of key nuclear reactions, such as
C$^{12}$(O$^{16}$, $\alpha$)Mg$^{24}$ and triple-$\alpha$, even relatively
modest temperature fluctuations, of the order ten percent, can lead to
enhancements of 1 - 3 orders of magnitude in the turbulent nuclear burning
rate. We verify the predicted turbulent enhancement directly against numerical
simulations, and find very good agreement. We also present an estimation for
the onset of turbulent detonation initiation, and discuss implications of our
results for the modeling of stellar transients.
