Understanding Density Fluctuations in Supersonic, Isothermal Turbulence

Supersonic turbulence occurs in many environments, particularly in astrophysics. In the crucial case of isothermal turbulence, the probability density function (PDF) of the logarithmic density, $s$, is well measured, but a theoretical understanding of the processes leading to this distribution remains elusive. We investigate these processes using Lagrangian tracer particles to track $s$ and $\frac{ds}{dt}$ in direct numerical simulations, and we show that their evolution can be modeled as a stochastic differential process with time-correlated noise. The temporal correlation functions of $s$ and $\frac{ds}{dt}$ decay exponentially, as predicted by the model, and the decay timescale is $\approx$ 1/6 the eddy turnover time. The behavior of the conditional averages of $\frac{ds}{dt}$ and $\frac{d^2s}{dt^2}$ is also well explained by the model, which shows that the density PDF arises from a balance between stochastic compressions/expansions, which tend to broaden the PDF, and the acceleration/deceleration of shocks by density gradients, which tends to narrow it.