The Power Spectra and Point Distribution Functions of

Density Fields in Isothermal, HD Turbulent Flows

Korea Astronomy and Space Science InstituteJongsoo Kim

Collaborators: Dongsu Ryu Enrique Vazquez-Semadeni Thierry Passot

Kim, & Ryu 2005, ApJL (PS)Kim, VS, Passot, & Ryu 2006, in preparation (PDF)

Armstrong et al. 1995 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.

-2.4

-2.75

Deshpande et al. 2000

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.

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•Isothermal Hydrodynamic equations

•Isothermal TVD Code (Kim, et al. 1998)

;121rmsrms

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vM

•Periodic Boundary Condition

is a Gaussian random perturbation field with either a power spectrum or a flat power spectrum with a predefined wavenumber ranges.

δv42|| kv

We drive the flow in such a way that root-mean-squareMach number, Mrms, has a certain value.

•Driving method (Mac Low 99)

•Initial Condition: uniform density

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: 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.

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 spectraare still nearly -2.•Slopes of the spectra with higherMach 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: 5123

•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: 5123

•Spectral slopes are obtained withleast-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.

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mass conservation2

ln2

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• 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 PSover the wavenumber range 1<k<19

initial PS |vk |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 image density PDF

Mrms ~8; 1<k<2; resolution 10242

As the large-sclae dense filaments and voids form, the density PDFquite significantly deviate from the log-nomal distribution.

2D driven experiments

color-coded density image density PDF

Mrms ~1; 15<k<16; resolution 10242

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 10242

As the Mrms and the driving wavelength increase, the density PDFsdeviate from the log-normal distribution.

1<k<2|vk|2~ k-4

3D driven experimentsdensity PDFs with different Mrms; resolution 5123

A density PDF of a large-scale driven flow with Mrms=7 quite significantly deviates from 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.