Statistical Properties (PS, PDF) of Density Fields in Isothermal Hydrodynamic Turbulent Flows Jongsoo Kim Korea Astronomy and Space Science Institute Collaborators: Dongsu Ryu, Enrique Vazquez-Semadeni Kim, & Ryu 2005, ApJL (PS) Kim, VS, Passot, & Ryu 2006, in preparation (PDF)

Armstrong et al ApJ, Nature 1981 PC AU Electron density PS (M~1) Composite PS from observations of ISM velocity, RM, DM, ISS fluctuations, etc. A dotted line represents the Komogorov PS A dash-dotted line does the PS with a -4 slope 11/3(5/3)=3.66(1.66) : the 3D (1D) slope of Komogorov PS

HI optical depth image CAS A VLA obs. angular resol.: 7 arcsec sampling interval: 1.6 arcsec velocity reol.: 0.6km/sec Deshpande et al. 2000

Density PS of cold HI gas (M~2-3 from Heilies and Troland 03) -A dash line represents a dirty PS obtained after averaging the PW of 11 channels. -A solid line represents a true PS obtained after CLEANing Deshpande et al Why is the spectral slope of HI PS shallower than that of electron PS?  We would like to answer this question in terms of Mrms.

Isothermal Hydrodynamic equations Isothermal TVD Code (Kim, et al. 1998) Periodic Boundary Condition is a Gaussian random perturbation field with either a power spectrum or a flat power spectrum with a predefined wavenumber ranges. - We adjust the amplitude of the velocity field in such a way that root-mean-square Mach number, Mrms, has a certain value. Driving method (Mac Low 99) Initial Condition: uniform density

PC cluster in KASI 128 Intel Xeon processors (64 nodes) Gigabit Ethernet interconnect 128GB memory 6TB disk space

Time evolution of velocity and density fields: (I) Mrms=1.0 Resolution: 8196 cells 1D isothermal HD simulation driven a flat spectrum with a wavenumber range 1<k<2 (Step function-like) Discontinuities in both velocity and density fields develop on top of sinusoidal perturbations with long-wavelengths FT of the step function gives -2 spectral slope.

Time evolution of velocity and density fields: (II) Mrms=6.0 Resolution: D isothermal HD simulation driven a flat spectrum with a wavenumber range 1<k<2 Step function-like (spectrum with a slope -2) velocity discontinuities are from by shock interactions. Interactions of strong shocks make density peaks, whose functional shape is similar to a delta function FT of a delta function gives a flat spectrum.

Velocity power spectra from 1D HD simulations Large scale driving with a wavenumber ranges 1<k<2 Resolution: 8196 Because of 1D, there are only sound waves (no eddy motions). Slopes of the spectra are nearly equal to -2, irrespective of Mrms numbers.

Density power spectra from 1D HD simulations Large scale driving with a wavenumber ranges 1<k<2 Resolution: 8196 For subsonic (Mrms=0.8) or mildly supersonic (Mrms=1.7) cases, the slopes of the spectra are still nearly -2. Slopes of the spectra with higher Mach numbers becomes flat especially in the low wavenumber region. Flat density spectra are not related to B-fields and dimensionality.

Comparison of sliced density images from 3D simulations Mrms=1.2 Mrms=12 Large-scale driving with a wavenumber ranges 1<k<2 Resolution: Filaments and sheets with high density are formed in a flow with Mrms=12.

Density power spectra from 3D HD simulations Statistical error bars of time-averaged density PS Large scale driving with a wavenumber ranges 1<k<2 Resolution: Spectral slopes are obtained with least-square fits over the ranges 4<k<14 As Mrms increases, the slope becomes flat in the inertial range.

Density PDF Previous numerical studies (for example, VS94, PN97, PN99, Passot and VS 98, E. Ostriker et al. 01) showed that density PDFs of isothermal (gamma=1), turbulent flows follow a log- normal distribution. for a mass-conserving system However, the density PDFs of large-scale driven turbulent flows with high Mrms numbers (for example, in molecular clouds) were not explored.

2D isothermal HD (VS 94) Mrms=0.58 Need to explore flows with higher Mach numbers.

3D decaying isothermal MHD (Ostriker et al. 01) 1D Driven isothermal HD (Passot & VS 98) Drive with a flat velocity PS over the wavenumber range 1<k<19 initial PS |v k | 2 ~ k -4

1D driven experiments with flat velocity spectra Driving with a flat spectrum over the wavenumber range, 1<k<19 Large-scale driving in the wavenumber range, 1<k<2 time-averaged density PDF; resolution 8196 The density PDFs of large-scale driven flows significantly deviate from the log-normal distribution.

2D driven experiments color-coded density movie density PDF Mrms ~8; 1<k<2; resolution When the large-sclae dense filaments and voids form, the density PDF quite significantly deviate from the log-nomal distribution.

2D driven experiments color-coded density image density PDF Mrms ~1; 15<k<16; resolution Density PDFs of the low Mach number flow driven at small scales almost perfectly follow the log-nomal distribution.

2D driven experiments 1<k<2 Mrms~8 time-averaged density PDF; resolution As the Mrms and the driving wavelength increase, the density PDFs deviate from the log-normal distribution.

1<k<2 |v k | 2 ~ k -4 3D driven experiments density PDFs with different Mrms; resolution A density PDF of a large-scale driven flow with Mrms=7 quite significantly deviates from the log-normal distribution.

3D decay experiments 1<k<2 |v k| | 2 ~ k -4 time-evolution of density PDF; resolution As the turbulent flows decay, their density PDFs are converged to the log-normal distribution.

Conclusions As the Mrms of compressible turbulent flow increases, the density power spectrum becomes flat. This is due to density peaks (filaments and sheets) formed by shock interactions. The Kolmogorov slope of the electron-density PS is explained by the fact that the WIM has a transonic Mach number; while the shallower slope of a patch of cold HI gas is due to the fact that it has a Mach number of a few. Density PDFs of isothermal HD, turbulent flows deviates significantly from the log-normal distirbution as the Mrms and the driving scale increase.