Introduction : Correlations : Bispectrum :
Conclusions
"Space is big. Really big. You just won't belive how vastly hugely
mindbogglingly big it is. I mean you may think it's a long way down the road
to the chemist, but that's just peanuts to space." -Douglas Adams, The
Hitch Hiker's Guide to the Galaxy
Statistical Studies of
Magnetohydrodynamical Turbulence of the Interstellar Medium
Introduction
The interstellar medium (ISM) refers to the matter that exists between stars
within a galaxy. The ISM is highly complex and consists of a mixture of
ionized and neutral hydrogen, helium, and metals. The ISM is extremely
diffuse and collisions between atoms can offen take up to seconds (a very long
time on the atomic scale!). The ISM is important for astrophysics in that it
is the birthplace of stars. In return, the stars provide energy imput and
matter through photons, stellar winds, and supernovae.
In the photo above is shown approximatly one
square degree of nebula IC 4603 near the star Antares. Large amounts of interstellar dust form the
ISM. The dust begins life in the atmosphere of young stars and is disperesed
as the star dies in the form of a supernova Image source
The Interstellar Medium is highly turbulent and thus extremely difficult to
study numerically. This turbulence is believed by many to be a driving factor
in molecular cloud dynamics and star formation. Thus because of its
intermediate role between stellar and galactic scales, the ISM is a crucial
component in determining the lifespan of active star formation. Most of the
ISM is made of plasmas in different states. Because of this, most of the ISM
is imbedded in a vast magnetic field. Magnetohydrodynamical (MHD) turbulence of the
interstellar medium is the topic of my summer research and is currently a very
mysterious project in spite of many attempts to study it.
Over the course of the summer I applied many different statistical methods to
modeled density perturbations produced by MHD turbulence. These statistics
included skewness, correlations, and the bispectrum. The data I
worked with was modeled MHD turbulence using a variety of sonic and Alfvenic
mach numbers. We used a set of MHD numerical simulations, obtained for
different Sonic
and Alfvenic Mach number to study these statistcs(Ms and Ma respectively).
The simulations were generated by the ideal MHD equations in a periodic box
given by:
A few Important Definitions
It is important to keep in mind that turbulence is governed by nonlinear
dynamics. A nonlinear system is one in which the behavior can not be
expressed as a simple sum of its parts. In most nonlinear systems, many
assumptions can not be made and thus it can be difficult to predict behavior.
This is the reason why statistical numerical tools are so useful for studies
of MHD turbulence. These tools can help us gain
important information about a system that is very difficult or impossible to obtain via
direct solutions.
Through out my paper and this website the terms supersonic, subsonic,
super-Alfvenic, and sub-Alfvenic will be used often. It is important to
understand what these terms mean in order to gain a physical understanding of
what is occuring in the turbulence of the ISM.
One of the most important distinctions in turbulence is that of supersonic and
subsonic. Supersonic means that the magnetic pressure dominates while
subsonic is where the gas pressure dominates. Supersonic turbulence creates a
"shock wave" that travels throughout the system while in subsonic turbulence,
no such wave is created, and a perturbation must travel between individual
waves. One can think of this as a "calm sea" and a "stormy sea". The calm
sea would ne the subsonic turbulence and the supersonic turbulence would be
the stormy sea.
Other important terms are sub-Alfvenic and super-Alfvenic turbulence and are
related to the magnetic field of the system.
Sub-Alfvenic implies that the magnetic
field lines shape the surrounding plasma and thus are for models with strong
magnetic fields. In super-Alfvenic models, the flow of plasma
shapes the magnetic field lines which suggest a weak magnetic field.
The Earth's magnetic field lines shaped by the solar wind is
an excellent example of super-Alfvenic MHD
Image source
Sonic and Alfvenic mach numbers are also important ways of discussing
turbulence and are directly related to the above definitions (hence the names
sonic and Alfvenic)! Mach number is a dimensionless measure of relative speed. Sonic Mach number is defined as the
ratio of the speed of an object relative to a fluid to the speed of "sound" in that
medium. For our purposes," sound" here is any perturbation in the system. Thus
for Alfvenic mach number, our "sound" will be governed by the magnetic field.
Values of sonic mach number and Alfvenic mach number (Ma and Ms) will the
tell us how subsonic, supersonic or sub-Alfvenic and super-Alfvenic a model
is. For example a model with Ma=0.7 (corresponding to b=1) and Ms=0.7
(corresponding to p=1) is sub-Alfvenic and subsonic
while a model with Ma=7.0 (corresponding to b=0.1) and Ms=7.0 (corresponding
to p=0.01) is one that is super-Alfvenic and supersonic.
Correlations
In their simplest implications, correlation plots can be used to determine
if there is any dependence between two quantities. If there is, the dependence
can sometimes be shown as a functional dependence and an equation can be
generated to govern this behavior. I made several correlation plots for the
final snapshots of my data cubes. The following is only for the case of
density vs. velocity.
From plots such as these we can tell many things about the behavior of the
turbulence in the ISM. The plots shown here are for density vs. velocity
squared (specific kinetic energy). Figures are organized supersonic to
subsonic going down and super-Alfvenic to sub-Alfvenic going across. For
supersonic sub-Alfvenic turbulence density clumps are much more structured
then they would be if the turbulence was
subsonic in a weak magnetic field. The shocks of supersonic turbulence
compress the density clumps while the magnetic field acts as a "net" which can
speed up perterbations These correlations are just as one might
suspect in that the general trend for all models is a increasing velocity with
decreasing density.
Bispectrum
Although this summer I did spend a lot of time looking at statistics and
correlations of MHD turbulence, the most interesting part of my
research was to examine the bispectrum. The bispectrum is the three-point
correlations function in Fourier space. Put simply, the bispectum is a type of
statistic used in cases of nonlinear interactions in order to search for like
frequencies. The bispectrum improves the power
spectrum by not throwing away phase information that could be used to characterize the
signal. The bispectrum allows us to search for multiple complex frequencies
that may not be apparently related. It has been used many times to analyze
gravitational waves and study large-scale structure of the universe but has
never been applied for MHD turbulence. Thus, I got to examine the bispectrum
for turbulence for the first time and analyze the results.
What is the Bispectrum?
It is often convient to
think of the bispectrum as it is related to the power spectrum. The power
spectrum can be defined as:
Similarly the bispectrum can be defined as:
Where k1 and k2 are two different wave numbers in Fourier space, A(k) is the
original time series data split into L segments where L is a power of two.
Of course, there are many other ways to define the bispectrum and power
spectrum but these are the definitions that we used in our code to calculate
bispectrum for MHD turbulence. In our simulations we take the Fourier
transform of each A(k) in the above equation, however since we have continuous
data, we need not divide it into subsegments. The results of this give us a signal containing multiple
wave numbers that have related frequencies. The below figure illustrates
this.
The Bispectrum of Compressible MHD Turbulence
We explored the bispectrum of density and column density for the last snapshot
of the data cubes for density and column density. The below figures show
color and colorless contours of the bispectrum for k1 vs. k2. The first
column represents density, the second represents column density parallel to
the magnetic field (the x direction) and the third column represents column
density perpendicular to the magnetic field (y direction).
Conclusions
Over the summer I investigated several different statistics of density
structures of compressible MHD turbulence. I looked at the skewness and
kurtosis as well as several correlations as they related to density and column
density. I also examined the bispectrum of turbulence, a technique that has
been used extensivly in the study of large scale structure, yet has never been
employed for turublence. I found:
1) Subsonic models are more Gaussian then supersonic for density and column
density. The opposite is true for log density and column density.
2) Correlations of density and column density verse quantities such as
velocity, magnetic energy and mach number can give several insights
into the structure of clumps in turublent clouds.
a) Correlations of density vs. velocity show interesting clumps for
high densities, yet is still an exponential decay as would be expected.
b) For both sub-Alfvenic and super-Alfvenic models, the magnetic
energy generally seems to increase with decreaseing density. Another trend is that the magnetic energy seems to
increase much more rapidly for subsonic turbulence.
3) The bispectrum is a useful technique when applied to turbulence.
a) There are strong correlations for cases where k1=k2
b) There are virtually no correlations for k1 not equal to k2 for
subsonic cases.
c) There are correlations with compressible turbulence for all k1
and k2 however, k1=k2 remains the strongest.
d) It is apparent that the introduction of a magnetic field enhances
correlations.
