An Improved Fit to the Density Fluctuations in Supersonic Isothermal Turbulence

Abstract The density distribution of supersonic isothermal turbulence plays a critical role in many astrophysical systems. It is commonly approximated by a log-normal distribution with a variance of σ s , V 2 ≈ ln ( 1 + b 2 M V 2 ) , where s ≡ ln ρ / ρ 0 , M V is the rms volume-weighted Mach number, and b is a parameter that depends on the driving mechanism, which can be solenoidal (divergence-free), compressive (curl-free), or a mix of both. However, this fit neglects the driving correlation time, τ a , which plays a key role when compressive driving is significant. Here, we conduct turbulence simulations spanning a wide range of Mach numbers, driving mechanisms, and τ a values. In the compressive case, σ s , V 2 is not well fit by the standard expression. Instead, it scales approximately linearly with M V , and its dependence on τ a is σ s , V 2 ≈ M V [ 1 + 2 3 ( 1 + λ a ) Θ ( 1 + λ a ) ] , where λ a ≡ ln ( τ a / τ e ) , τ e is the eddy turnover time, and Θ is the Heaviside step function. Mixed driven turbulence shows a weak dependence on τ a , and for solenoidally driven turbulence, σ