 The Power Spectra and Point Distribution Functions of

1.  The Power Spectra and Point Distribution Functions of Density Fields in Isothermal, HD Turbulent Flows Korea Astronomy and Space Science Institute Jongsoo Kim Collaborators: Dongsu Ryu Enrique Vazquez-Semadeni Thierry Passot Kim, & Ryu 2005, ApJL (PS) Kim, VS, Passot, & Ryu 2006, in preparation (PDF)

2. PC AU Armstrong et al. 1995 ApJ, Nature 1981 11/3(5/3)=3.66(1.66) : the 3D (1D) slope of Komogorov PS • 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

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

4. 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. -2.4 -2.75 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.

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

6. 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.

7. Time evolution of velocity and density fields: (II) Mrms=6.0 • Resolution: 8196 • 1D 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.

8. 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.

9. Mrms=12 Mrms=1.2 Comparison of sliced density images from 3D simulations • Large-scale driving with a wavenumber ranges 1<k<2 • Resolution: 5123 • Filaments and sheets with high density are formed in a flow with Mrms=12.

10. 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: 5123 • 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.

11. 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. mass conservation • However, the density PDFs of large-scale driven turbulent flows with high Mrms numbers (for example, in molecular clouds) were not explored.

12. 2D isothermal HD (VS 94) Mrms=0.58 Need to explore flows with higher Mach numbers.

13. 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 |vk |2~ k-4

14. 1D driven experiments with flat velocity spectra time-averaged density PDF; resolution 8196 Driving with a flat spectrum over the wavenumber range, 1<k<19 Large-scale driving in the wavenumber range, 1<k<2 The density PDFs of large-scale driven flows significantly deviate from the log-normal distribution.

15. 2D driven experiments Mrms ~8; 1<k<2; resolution 10242 color-coded density image density PDF As the large-sclae dense filaments and voids form, the density PDF quite significantly deviate from the log-nomal distribution.

16. 2D driven experiments Mrms ~1; 15<k<16; resolution 10242 density PDF color-coded density image Density PDFs of the low Mach number flow driven at small scales almost perfectly follow the log-nomal distribution.

17. 2D driven experiments time-averaged density PDF; resolution 10242 1<k<2 Mrms~8 As the Mrms and the driving wavelength increase, the density PDFs deviate from the log-normal distribution.

18. 3D driven experiments density PDFs with different Mrms; resolution 5123 |vk|2~ k-4 1<k<2 A density PDF of a large-scale driven flow with Mrms=7 quite significantly deviates from the log-normal distribution.

19. 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.