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![Page 1: Statistical Properties (PS, PDF) of Density Fields in Isothermal Hydrodynamic Turbulent Flows Jongsoo Kim Korea Astronomy and Space Science Institute Collaborators:](https://reader034.fdocuments.net/reader034/viewer/2022051200/5a4d1b447f8b9ab0599a2b79/html5/thumbnails/1.jpg)
Statistical Properties (PS, PDF) of Density Fields in Isothermal Hydrodynamic Turbulent
Flows
Jongsoo KimKorea Astronomy and Space Science Institute
Collaborators: Dongsu Ryu, Enrique Vazquez-Semadeni
Kim, & Ryu 2005, ApJL (PS)Kim, VS, Passot, & Ryu 2006, in preparation (PDF)
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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
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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
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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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;0 vt
δvvvv
2at
•Isothermal Hydrodynamic equations
•Isothermal TVD Code (Kim, et al. 1998)
;121rmsrms
avM
•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 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
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PC cluster in KASI• 128 Intel Xeon processors
(64 nodes) • Gigabit Ethernet
interconnect• 128GB memory • 6TB disk space
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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.
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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.
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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.
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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.
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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.
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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.
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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.
ln
2)ln(lnexp
2
1ln)(ln 2
20
2ddP
for a mass-conserving system2
ln2
0
• However, the density PDFs of large-scale driven turbulent flows with high Mrms numbers (for example, in molecular clouds) were not explored.
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2D isothermal HD (VS 94)
Mrms=0.58
Need to explore flows with higher Mach numbers.
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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
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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.
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2D driven experiments
color-coded density movie density PDFMrms ~8; 1<k<2; resolution 10242
When the large-sclae dense filaments and voids form, the density PDFquite significantly deviate from the log-nomal distribution.
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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.
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2D driven experiments
1<k<2 Mrms~8time-averaged density PDF; resolution 10242
As the Mrms and the driving wavelength increase, the density PDFsdeviate from the log-normal distribution.
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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.
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3D decay experiments
1<k<2|vk||2~ k-4
time-evolution of density PDF; resolution 5123
As the turbulent flows decay, their density PDFs are converged to the log-normal distribution.
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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.