Transcript No Slide Title

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)
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
PC
AU
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
Deshpande et al. 2000
-2.4
-2.75
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.
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

   v   0;
t
 v

   v  v    a 2  δv
 t

 
•Driving method (Mac Low 99)
δv is a Gaussian random perturbation field with either a power
spectrum | v |2  k 4 or a flat power spectrum with a predefined
wavenumber ranges.
We drive the flow in such a way that root-mean-square
Mach number, Mrms, has a certain value.
vrms
M rms 
 1  12;
a
•Initial Condition: uniform density
•Periodic Boundary Condition
•Isothermal TVD Code (Kim, et al. 1998)
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 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: 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 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 lognormal distribution.
 (ln   ln 0 ) 2 
P(ln  )d ln  
exp
 d ln 
2
2
2
2


2
ln  0  
mass conservation
1
2
• 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.
1D Driven isothermal HD
(Passot & VS 98)
3D decaying isothermal MHD
(Ostriker et al. 01)
Drive with a flat velocity PS
initial PS |vk |2~ k-4
over the wavenumber range 1<k<19
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.
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.
2D driven experiments
Mrms ~1; 15<k<16; resolution 10242
color-coded density image
density PDF
Density PDFs of the low Mach number flow driven at small scales
almost perfectly follow the log-nomal distribution.
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.
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.
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.