Density Power Spectrum of Compressible Hydrodynamic Turbulent Flows

Turbulent flows are ubiquitous in astrophysical environments, and understanding density structures and their statistics in turbulent media is of great importance in astrophysics. In this Letter, we study the density power spectra P ρ of transonic and supersonic turbulent flows through one- and three-dimensional simulations of driven, isothermal hydrodynamic turbulence with rms Mach numbers in the range 1 ≲ M rms ≲ 10. From one-dimensional experiments we find that the slope of the density power spectra becomes gradually shallower as the rms Mach number increases. This is because the density distribution is transformed from the profile with discontinuities having P ρ ∝ k -2 for M rms ~ 1 to the profile with peaks having P ρ ∝ k 0 for M rms ≫ 1. We also find that the same trend is carried to three dimensions; that is, the density power spectrum flattens as the Mach number increases. But the density power spectrum of the flow with M rms ~ 1 has a Kolmogorov slope. The flattening is a consequence of the dominant density structures of filaments and sheets. Observations have claimed different slopes of density power spectra for electron density and cold H I gas in the interstellar medium. We argue that while the Kolmogorov spectrum for electron density reflects the transonic turbulence of M rms ~ 1 in the warm ionized medium, the shallower spectrum of cold H I gas reflects the supersonic turbulence of M rms ~ a few in the cold neutral medium.