Statistical Properties of Density Fields in Isothermal
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Transcript Statistical Properties of Density Fields in Isothermal
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. 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
2
v v a δv
v 0;
t
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 adjust the amplitude of the velocity field 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)
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: 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.
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: 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
for a mass-conserving system
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 movie
density PDF
When 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.
3D decay experiments
time-evolution of density PDF; resolution 5123
|vk||2~ k-4
1<k<2
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.