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 and Lorentzian Pulses in Magnetized Plasmas”. The published version of the paper can be downloaded from the following links.

Basic Information:

• Citation: D. C. Pace, M. Shi, J. E. Maggs, G. J. Morales, and T. A. Carter, “Exponential Frequency Spectrum and Lorentzian Pulses in Magnetized Plasmas,” Phys. Plasmas 15, 122304 (2008)
• Download: From DavidPace.com (free, no registration required)
• Statements Required by the Publisher: Copyright (2008) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.
  The following article appeared in Phys. Plasmas 15, 122304 (2008) and may be found at http://link.aip.org/link/?PHP/15/122304.

Previous Work

Figure 14a from the paper, example of a pulse in the time series data.

The basic idea behind all of this work is that we observe pulses, or spikes, in the plasma. These pulses are not always present, but whenever they are, the power spectrum of the data features an exponential dependence in frequency. The figure to the right provides an example of a pulse that was measured in one of the experiments. There are some background fluctuations, but the labeled pulse is clearly a unique feature of this signal. This is figure 14a from the paper.

Figure 6 from the paper, example of an exponential spectrum (such a spectrum appears as a straight line when plotted semi-log as in this figure).

The first paper (PDF) focused on demonstrating the connection between the pulses and the exponential spectra. The paper discussed in this review provides additional evidence for this connection by way of new modeling work. Some natural questions arise from the statements made thus far and are addresses by the more recent paper, including:

• Can other pulse shapes generate exponential spectra (how do you know these pulses are Lorentzian)?
• Is it possible to observe an exponential spectra with just about any combination of pulse-like features?
• The data appear to show some non-symmetric pulses, can these be accounted for in the resulting spectra?

Modeling Lorentzian Pulses

The three questions above can all be addressed through analytic models of the pulses and spectra. Beginning with the question of whether non-Lorentzian shapes can generate exponential, the answer is yes. The argument in favor of the observed pulses being Lorentzian is that other pulse shapes, while potentially generating exponential spectra, do not produce the agreement between the pulse width and the exponential decay constant. As the PRL showed, there is a relationship between the temporal width of a Lorentzian pulse and the exponential decay constant of the power spectrum it produces. Presently, only the Lorentzian shape satisfies this relationship.

Figure 16 from the paper, showing the different spectral shapes resulting from different pulses.

This initially seems to indicate that the hyperbolic-secant squared is a viable candidate for the pulse shape. It turns out that this is not the case, however, because the relationship between the pulse time-width and the exponential decay of the spectrum is not consistent. The spectra plotted in the figure all come from pulses with the same temporal width. The exponential parts of the Lorentzian and hyperbolic-secant squared spectra exhibit different slopes (the exponential part of the hyperbolic-secant squared spectrum is plotted as the dashed black line). This means that the spectrum produced by the hyperbolic-secant squared pulse would not allow for the calculation and prediction of the pulse width. In the experiments, however, we have shown that calculating the exponential behavior of the spectra allows for the determination of the widths of the pulses that generated it. Therefore, while this is not an exhaustive study of every single mathematical pulse shape in existence, it does show that there are no obvious examples of pulses other than Lorentzians that generate the same agreement between pulse and spectral characteristics.

Figure 17 from the paper, showing the different spectral shapes resulting from pulse sets with different distributions of time widths.

The figure shows that the power spectrum from the Narrow set exhibits exponential behavior over a greater frequency range than that of the Broad set. The Broad set loses its exponential behavior at lower frequencies. The relevance to the experiments is that since we measure exponential spectra over wide frequency ranges (see figure 6 as shown above), that implies the pulses in the experiment feature a narrow distribution of temporal widths. A consequence of this result is that if the pulses feature such consistent widths, then there is likely a generation mechanism that can explain the production of the pulses themselves. There are measurements of hundreds of pulses for a given experimental setup, and thousands of pulses overall, so the observation that exponential spectra are only observed when these pulses are reproducible and consistent in width is a powerful conclusion. In fact, it appears as though a completely random collection of pulses does not spontaneously result in an exponential power spectrum. The pulses need to feature the same shape and similar time widths.

Figure 14b from the paper, example of a Lorentzian fit to the pulse from figure 14a.

Figure 18a from the paper, fitting a skewed Lorentzian pulse to a measured pulse.

What is the Relevance?

In a physics paper the relevance of the work is established in the beginning. In a review like this, however, it works better to explain the physics first and then put the results into context. This work experimentally observes pulse structures of a particular shape that only arise when the system has transitioned to a turbulent state (that is all shown in the PRL). The structures exhibit consistent time widths, implying that the generation mechanism is constant during their production. Turbulence is generally associated with the loss of coherent waves and the production of a broad spectrum of fluctuations. Here, however, it is shown that a very narrow-band behavior (the narrow distribution of pulse widths) is required to observe the broadband feature (the wide ranging exponential frequency behavior). This is a unique result that may be the beginning of new developments in plasma turbulence and transport. The bigger picture still needs to be determined. For example, what causes the pulses? If they are produced by a non-linear interaction of electromagnetic plasma waves, then it is likely that this same behavior occurs in other plasmas, both in space and in the laboratory. As such, we suggest that the pulses and exponential power spectra are a universal feature of plasma turbulence. If a Lorentzian shape is the result, then perhaps we can now work backwards to describe plasma turbulence with a set of solvable equations. What differential equations feature Lorentzian pulses as their solution? Perhaps these same equations can describe plasma transport in a new way. Plasma transport is important for understanding phenomena such as the solar wind and aurora in addition to the present challenge of developing nuclear fusion as an energy source.