Density Power Spectrum of Compressible Hydrodynamic Turbulent Flows

Turbulent flows are ubiquitous in astrophysical environments, andunderstanding density structures and their statistics in turbulent media is ofgreat importance in astrophysics. In this paper, we study the density powerspectra, $P_{\rho}$, of transonic and supersonic turbulent flows through oneand three-dimensional simulations of driven, isothermal hydrodynamic turbulencewith root-mean-square Mach number in the range of $1 \la M_{\rm rms} \la 10$. From one-dimensional experiments we find that the slope of the density powerspectra becomes gradually shallower as the rms Mach number increases. It isbecause the density distribution transforms from the profile with {\itdiscontinuities} having $P_{\rho} \propto k^{-2}$ for $M_{\rm rms} \sim 1$ tothe profile with {\it peaks} having $P_{\rho} \propto k^0$ for $M_{\rm rms} \gg1$. We also find that the same trend is carried to three-dimension; that is,the density power spectrum flattens as the Mach number increases. But thedensity power spectrum of the flow with $M_{\rm rms} \sim 1$ has the Kolmogorovslope. The flattening is the consequence of the dominant density structures of{\it filaments} and {\it sheets}. Observations have claimed different slopes ofdensity power spectra for electron density and cold H I gas in the interstellarmedium. We argue that while the Kolmogorov spectrum for electron densityreflects the {\it transonic} turbulence of $M_{\rm rms} \sim 1$ in the warmionized medium, the shallower spectrum of cold H I gas reflects the {\itsupersonic} turbulence of $M_{\rm rms} \sim$ a few in the cold neutral medium.