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Review: Exponential Frequency Spectrum in Magnetized Plasmas

This is a review of a paper recently published by my group. Here, the paper is paraphrased to reach a wider audience. The plasma physics community can read the original publication, so my goal is to provide an example of current plasma physics research to the general public. The following review should be at a high school level and I appreciate any comments you may have regarding how to clarify it further. If you are interested in physics research, then I hope this helps to feed your curiosity.


Details and Download

This is a review of the paper titled, “Exponential Frequency Spectrum in Magnetized Plasmas.” I recommend that you download the published version and then follow along as you proceed below.

Basic Information:

  • Citation: D. C. Pace, M. Shi, J. E. Maggs, G. J. Morales, and T. A. Carter, “Exponential Frequency Spectrum in Magnetized Plasmas,” Phys. Rev. Lett. 101, 085001 (2008)
  • Publication: Official Journal Page (subscription required for download)
  • Download: From DavidPace.com (free, no registration required)

Color Note

The electronic copy of this paper features color figures. We were not able to secure color figures for the print version due to an administrative issue at UCLA. Most people will probably never notice this because the electronic files are becoming the dominant format for sharing and receiving papers. Still, I feel the need to point out that we are aware the figures might not make as much sense when displayed black and white, as anyone who gets an official print copy will see.



This paper concerns the measurement of fluctuating quantities in magnetized plasmas. These quantities are the density and temperature. “Magnetized plasma” means the plasma is contained within a strong magnetic field that is applied by laboratory electromagnets. We calculate the power of these fluctuations and plot them according to the corresponding frequency. These are known as power spectra and they are commonly measured or calculated in turbulence research. We observe a power spectrum, P, that exhibits an exponential dependence on frequency, P(f) ∝ exp(-f/fs), where fs is a scaling factor that depends on the plasma properties.

The exponential spectrum is found to result from the presence of particular pulses in the raw measurements. These pulses have a Lorentzian shape. Analytically, the power spectrum resulting from a Lorentzian shaped pulse is exponential. This fact, coupled with results from two different experiments, lead us to claim that Lorentzian pulses are the cause of exponential power spectra. Similar looking spectra already present in the research literature lead us to suggest that this behavior is a universal feature of plasma turbulence.


Experimental Setup

The important aspect of the two experiments presented is that each one isolated a gradient. The temperature filament experiment features a temperature gradient with a constant background density. The limiter-edge experiment features a density gradient with a minimal temperature gradient. Instead of writing a lengthy description of the LAPD machine, I'll just provide two links that will be helpful: LAPD Website, LAPD device from my thesis.


Power Spectra

exponential spectrum
Figure 1a from the paper, showing an exponential spectrum from our temperature filament experiment.
The figure to the right is Fig. 1a from the paper and it provides an example of an exponential spectrum from one of our experiments. The solid black line is the experimental result and the solid red (lower line) line is a model result. The dashed blue lines are parallel and provide perspective. In a semi-log representation, i.e., when plotted with a logarithmic y-axis and a linear x-axis, an exponential behavior will appear as a straight line. Notice that the “Measured” line features a straight line appearance under the peaks that rise out on occasion.

A simple expression to demonstrate the “linear” behavior (I'll call the spectra plots linear because that is the manifestation of the exponential) of the semi-log plotted spectra is given below. Notice that the expression has the form of a line, mx + b, where in our case the frequency is the independent variable. This shows that plotting an exponential spectrum in a semi-log manner results in a linear display. Furthermore, the slope of this linear region is related to the scaling frequency mentioned previously.

We state that the linear behavior in a semi-log plot is a simple way to detect, or at least be suspicious of, an exponential dependence in power spectra. The problem is that most papers present spectra in a log-log format (the x-axis is also logarithmic). The first paragraph of the paper says

“Significant experimental and theoretical effort has been devoted to the identification of universal trends in the spectrum of fluctuations; a strong motivation being the early work by Kolmogorov [1]. Influenced by this pioneering work that predicts algebraic spectral dependencies, the published experimental plasma studies typically display the data in log-log scale.”

Kolmogorov made many important contributions to the study of turbulence. The work referenced in our paper came out in 1941 and suggested that a particular type of turbulence should result in power spectra with an algebraic dependence on frequency. An algebraic dependence will look like a straight line if plotted in log-log format. A lot of research seeks to demonstrate that this theory and the algebraic dependence correctly describe plasma turbulence. This may be the case, and both types of spectra can exist simultaneously, but it is more difficult to catch exponential spectra in the literature when all the plots are log-log.


Lorentzian Pulses

Lorentzian shape
Generic example of a Lorentzian pulse. Important parameters for the experiment are the FWHM and the center (to).
The existence of the exponential is tied to the presence of Lorentzian shaped pulses, or bursts, in the raw measured signals. It is shown that the power spectrum of a Lorentzian pulse is exponential in frequency, see equation (2). Power spectrum is defined as the absolute value of the square of the Fourier transform, which is given in equation (2). A sample Lorentzian shape is shown to the right. The Full Width at Half Maximum (FWHM) is relevant to the paper because we define the width of a single Lorentzian pulse, τ, as τ = 1/2 FWHM. This width is related to the slope of the power spectrum according to τ = 1 / (2πfs).

The relation between the pulse widths and the slope of the power spectra allows us to verify that they are related. We can either take a spectrum and predict the Lorentzian width, or we can take a Lorentzian pulse and use its width to predict the behavior of the spectrum. Such a comparison is done for both experiments and the resulting values agree. The spectrum of one experiment leads to a pulse width that is accurate for that experiment (the two experiments have different values from each other).


Relation to Transport

Having demonstrated the relationship between Lorentzian pulses and exponential power spectra, it remains to tie this to anomalous transport. Plasma physics has a well developed classical theory of transport that describes how energy and particles move in plasma due to particle collisions. The reality of the situation is, however, that particles and energy often move through plasmas much more readily than classical theory predicts. This enhanced level of transport is called anomalous transport. I have touched on this in another post. If the turbulent spectrum caused by Lorentzian pulses can be related to anomalous transport, then that might provide new avenues of research into how to control or stop it. Controlling plasma transport is a key element to making fusion energy (as in electricity) possible.

The contour below is figure 3 from the paper. This is a contour of the power spectrum (color levels) versus time (x-axis) for the temperature filament experiment. During the early times prior to about 5.5 ms, the transport is classical (not shown) and only a coherent mode is present. The coherent mode is also seen in figure 1a as the peak near 30 kHz. In the figure below, the coherent peak is a sloping green-yellow band. At about 5.5 ms this band disappears into a very wide swath of green-yellow that represents a much broader spectrum. The individual spectra that generate that area are exponential.

When the spectrum becomes exponential, the measurements begin to show Lorentzian pulses. The white trace within the contour shows bursty features at times after 5.5 ms. This data is the fluctuating component of ion saturation current, which is mostly density. At these times the transport is anomalous. It remains to quantify how much transporting the pulses can do.

power spectrum versus time
A spectrogram (color contour), power spectrum through time, shows the exponential spectrum appearing at the same time the raw signal (white trace) begins to show pulses.


The paper concludes with a summary of what I've written above. Anomalous transport is often observed in situations that generate broadband power spectra. Our experiments show that these spectra can be exponential and not necessarily algebraic. Furthermore, the existence of an exponential power spectrum can be traced to the presence of Lorentzian pulses in the measured data. While we have some ideas about what causes these pulses, that remains an unsolved problem.

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Initial Posting: Friday, 22 August 2008
Last Updated: Wednesday, 03 September 2008