It is also easy to imagine that this is a process that occurs abundantly in the universe. There are many objets that are composed of charged particles with permeating magnetic fields. Some examples include accretion disks, the ISM, and star forming regions.
Image source here
Above is an artists perception of an accretion disk.
Thus one can see the extreme value in studying magnetohydrodynamics and turbulence associated with it.

To talk a little more quantitatively about MHD, let me introduce some important terms. One very important quality of turbulence is its Mach number. There are two types of Mach numbers: sonic and Alfvenic. The Sonic Mach number relates the relative importance of gas pressure to magnetic pressure. Supersonic conditions mean that magnetic pressure dominates and Subsonic conditions mean that gas pressure dominates. The actual number scheme is quite simple. If the Sonic Mach number is greater than 1, the medium is Supersonic and conversely if the Sonic Mach number is less than 1, the medium is Subsonic. Sonic turbulence creates waves that propagate throughout a medium. A great analogy for these waves is a stormy sea and a calm sea. The story sea equates to the Supersonic case and the calm sea relates to the Subsonic case; even though it is "calm", there are still waves and perturbations.

The Alfven Mach number, on the other hand, is a measure of the strength of the magnetic field. The same basic notation applies where an Alven Mach number less than 1 is Sub-Alfvenic, and if it is greater than 1, it is Super-Alfvenic. For the Sub-Alfvenic case, the magnetic field lines shape the plasma and the opposite is true for the Super-Alfvenic case. Hence a Sub-Alfvenic Mach number corresponds to a strong magnetic field and Super-Alfvenic corresponds to a weak magnetic field. The way these numbers manifest themselves in the research is by the symbols B and C_s, where B represents the magnetic field and C_s represents the speed of sound. The Mach numbers are defined to be v/B and v/C_s where v is a constant. Thus, for B<1, the Alfvenic Mach number >1 and hence Super-Alfvenic.

The goal of this summer's research was to take simulations of MHD Turbulence and study certain statistics of it and determine whether these statistics correspond to anything significant or not. The simulations are presented in data cubes that are 512x512x512 pixels. In each pixel, there is information about the velocity in the x,y,and z direction, the magnetic field in the x,y, and z direction, and the density. Hence we have data that is a 512x512x512x7 array. Personally, I analyzed three different data cubes. One that took self gravity into account and two that did not. For the cubes that did not have self gravity, I dealt with one cube that had a strong magnetic field and one that had a weak magnetic field. However, one problem with data cubes is that although Astrophysical phenomenon are 3 dimensional, we can only look at their 2 dimensional projections. Thus, we must only look at 2D projections of these data cubes. In order to visualize what this whole process looks like, below is a picture of a data cube. This cube shows velocity, where orange represents a greater velocity.

In each of the data cubes, there is a magnetic field parallel to the x direction, as can be seen in the above picture. We look at these 2D projections in two lines of sight (LOS): one parallel and one perpendicular to the magnetic field. Once these projections are attained, we look at statistics of these projections and look for interesting features. The particular statistics that I considered were line profiles, centroids, skewness and kurtosis, and the structure function which will be talked about below.

The first step to analyze the data cubes is to compute the line profiles. These are created using the local velocity as the center of a gaussian, the speed of sound as the standard deviation, and the local density as height. Thus, for one individual pixel, we arrive at something that looks like:

What the line profile represents in an actual system is the intensity as a function of frequency. We use the density to represent the intensity because we assume that the line intensity is proportional to the density (Esquivel A. et. al. 2007, MNRAS, 381, 1733-1744). This is something that we can actually observe, therefore it is an useful plot to make. In addition to using the density, the plot makes use of velocity, becuase the frequency and velocity can be related via the doppler shift. Here the velcity is normalized with the speed of sound. So in essence, we have a plot of Intensity as a function of frequency.

Before we go any further, we must take into account the effects of resolution on our analysis. In a real system, our telescopes have a finite resolution. So while our cube is 512³ pixels, to a given instrument we might be observing with, the resolution might be much lower. Thus, we must test our analysis for many different resolutions. A beautiful example of how resolution comes into play can be seen in the following images.

The image on the left is a slice from a data cube looking at the local density at a resolution of 512x512 pixels. The image on the right is the same slice but with a resolution of 64x64 pixels. The features in this image can still be discerned, yet with far less detail.

So now that we have these line profiles at different resolutions, we can start to analyze them. The first thing that I looked at was the velocity centroid. The velocity centroid is essentially the weighted center of the gaussian profile. The equation of the velocity centroid is:
One can see that we weight the velocity with the density to obtain our weighted center. What we do now is calculate the velocity centroid for each pixel in different resolutions. And in addition to that, we do this in different lines of sight, denoted by the subscript z in the above equation; one parallel and one anti-parallel to the magnetic field. With this data we construct histograms of these centroids. Here are 4 examples:


Immediately we can recognize the effects of resolution. The red line represents a resolution of 64x64, and the black line represents a resolution of 16x16. We can see that the lower resolution histogram is more broad due to the smoothing out of information. One other interesting thing to note is that for the self gravity case, average centroid is near 0, whereas without self gravity, the center is usually not centered around 0.

Further analysis of the data cubes was done using the structure function statistic. The structure function (SF) is what is known as a two point statistic. What it does is measure the difference between values at a set radius on a 2D map. This is useful for our research becuase when we look at 2D projections of our data cubes, we have a map that is 512x512 pixels. The structure function is given by

where &phi is some parameter, i.e. velocity centroid. For our studies, we used a second order structure function, hence n=2. Conceptually, the structure function measures how much values change as points get further from a given point. Therefore, we get a measure of the amount of correlation for a given parameter. For each map, we compute the SF in a direction parallel and perpendicular to the magnetic field. Then, we plot contours of the SF at some given value in each of the directions called isocontours. Once we do this, we look for anisotropies in the isocontours. Specifically, we are looking for elongation in the direction parallel to the magnetic field becuase this means that there is a stronger correlation in that direction. Theoretically, this is the physics occuring, and we hope that our statistics will reflect this. Below are the plots showing the isocontours in the x line of sight (LOS).

Self Gravity                                   No Self Gravity B=0.1 c_s=0.1             No Self Gravity B=1.0 c_s=0.1





And in the z LOS:

Self Gravity                                   No Self Gravity B=0.1 c_s=0.1             No Self Gravity B=1.0 c_s=0.1






There are many things that we can glean from these plots. One of the most obvious and important is that for all of the plots in the Xlos, there is a very noticeable elongation in the parallel direction. This is fantastic because this is precisely what we expect and hoped for. In addition to this, we can note that the anisotropies in the non self gravity case are greater when we have a stronger magnetic field. This is also a great result as we expect this to be true. With the stronger magnetic field, there is a greater confinement of the flow of the plasma and hence, should be more correlated.

Also interesting to note is that for the z LOS, almost all of the isocontours have anisotropies in the perpenidcular direction. This is also an interesting result and the most likely reason for this are the shocks that are created perpendicularly to the field lines. These shocks are created becuase the flow is faster along the field lines and therefore condensations will mostly occur perpendicularly. In addition to shocks, there could also be effects from gravity, but not necessarily.

As we have seen, there are many statistics that can be applied to a data cube. We have tested velocity centroid, dispersion, skewness, and kurtosis histograms, as well as the same parameters used in the structure function to see if they will tell us anything of use and value when we actually observe. Althoguh there are many more statistics to test, and there are people testing them, each of these statistics we tested returned information about the underlying physics which is exactly what we wanted to find. With these statistics, and the help off many others, one can start to look at astrophysical processes and understand something about the physics, specifically the turbulence, of the medium.