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Review: Spontaneous Thermal Waves in a Magnetized Plasma

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 what current plasma physics research is working on 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.

The paper does not have separate sections because the style is set by the journal. This review does feature sections, and the relevant place within the paper will be cited to make it easier to follow along. Links to additional material are provided, but these are always items I found through a quick web search and it is definitely possible to find additional information on your own. This particular journal enforces a four page limit on papers because its goal is to disseminate the newest results quickly. I am happy to add to this webpage when contacted since we could not add anything else to the paper.


Details and Download

This is a review of the paper titled, "Spontaneous Thermal Waves in a Magnetized Plasma." 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, “Spontaneous Thermal Waves in a Magnetized Plasma,” Phys. Rev. Lett. 101, 035003 (2008)
  • Publication: Official Journal Page (subscription required for download)
  • Download: From DavidPace.com (free, no registration required)


Paper Begins: This section is marked in the paper.

The interesting result presented in this paper concerns the appearance of oscillations in the electron temperature of the plasma. The unique aspect of these oscillations is that they arise without us forcibly driving them, hence the “spontaneous” part of the title. Our experiment consists of a narrow filament of hot plasma surrounded by a much larger and much colder plasma. This filament serves as a resonance cavity for the temperature oscillations. Resonance cavity, in this case, means that one-fourth of the wavelength of the thermal wave (i.e., the wave behind the temperature oscillations) fits inside the heated plasma filament. The entire situation is analogous to thermal waves that have been previously studied in solids, liquids, and gases.



Paper Begins: “In recent years the venerable topic of thermal (diffusion) ...”

Thermal waves are also known as diffusion waves. An example of diffusion can be observed by placing a droplet of colored dye into a glass of clear water. The spreading of the color throughout the liquid is by the process of diffusion. As another example, smells diffuse through the air. That is why there is a delay between opening a can of fresh coffee and smelling it from afar (unfortunately, bad smells also diffuse, though they seem to do it more quickly than pleasant smells).

The rate at which something diffuses depends on the material through which it is diffusing. Our droplet of dye diffuses through water much more quickly than it does through molasses. Thermal waves propagate (diffuse) through materials based on the thermal conductivity of the material. This relationship allows people to measure the thermal conductivity of a material by driving thermal waves within it. A thermal wave may be generated by applying a heat pulse to a material. Reference [5] from the paper leads to an experiment in which a laser is fired at an absorbing wall. The wall warms while the laser is incident upon it, and then cools when the laser is off. Gas inside a chamber next to the wall is heated and cooled similarly, leading to thermal waves. The resonant cavity in that case is set by the physical container of the gas. This is a forced thermal wave. Its properties lead to the calculation of various conduction and diffusion properties of the enclosed gas.

Thermal waves have been observed in plasmas previously, this paper is not claiming to be the first observation. Previous observations in magnetized plasmas, however, were instances of forced thermal waves. Tokamaks have created thermal waves by cycling their electron cyclotron heating (ECH). The behavior of the thermal wave can be used to study how energy moves through the plasma, which further helps scientists who are trying to learn how to better contain energy in a tokamak so that nuclear fusion and energy production may be obtained.

Studying thermal waves in magnetized plasmas is very different from studying them in other states of matter. A magnetized plasma is not symmetric. The background magnetic field causes the plasma to behave differently along it than it does in the directions perpendicular to it. One consequence of this behavior is that a thermal wave will have a parallel wavelength (i.e., wavelength in the direction of the applied, background magnetic field) that is much longer than the perpendicular wavelength. In a tokamak this does not pose a problem because the parallel direction is circular since the device is toroidal. For a basic plasma experiment in a linear device it is difficult to generate such a wave because the boundary conditions significantly increase the complexity. We show later in the paper that the ¼-wavelength of the waves in our experiment is roughly 8 meters. In this case, a basic plasma machine would have to be at least 8 meters long in order to provide an environment in which such a wave could exist. There are few such machines in the world.


Dispersion Relation

Paper Begins: “The parallel and transverse complex wave numbers of a ...”

The fundamental equation for describing a wave is known as its dispersion relation. This is an expression in which the relationship between the wavelength, λ, and the frequency, f, is given. Notice that in the paper we use wave number, k, and frequency ω. These are related to each other by the following expressions: k = 2π/λ and ω = 2πf. The reason for using k and ω are not important in this treatment.

parallel wave number
Part of equation (1) from the paper: the parallel wave number.

Equation (1) writes out the dispersion relation separately for directions parallel to the magnetic field and perpendicular to it. The parallel component is shown to the right. Notice that this is a complex number. The most important part of this equation is that it contains the parallel thermal conductivity of the plasma, κ||. By measuring the thermal wave's frequency and wavelength, and then separately measuring the plasma density (n), we can solve for the thermal conductivity. The value of such a diagnostic is that the thermal conductivity of the plasma is traditionally a difficult measurement.


Experimental Setup

Paper Begins: “The experiment is performed in the upgraded large ...”

electron beam crystal
The lanthanum hexaboride crystal is shown alongside a common ballpoint pen.

I have provided details on the LAPD device elsewhere (see this). The hot plasma filament is created by firing an electron beam into a cold LAPD plasma. The standard LAPD plasma is five times too hot for this experiment, so the beam is fired into the plasma that remains after the machine turns off. This plasma exists for a few hundred milliseconds. Our entire experiment takes place within 15 milliseconds. The source of the electron beam is a single crystal of lanthanum hexaboride (LaB6). The LaB6 crystal and mount is shown in the image at right. The diameter of the crystal surface is 3 millimeters and this entire surface emits electrons when heated to 1800 Kelvin.

Under most conditions the firing of an electron beam into a plasma results in a complicated system of waves and behaviors due to both the fast electrons of the beam and the extra density created when these electrons ionize any neutrals that may be present. We run the electron beam in such a way that these processes are minimized and may be ignored. First, the density of the background plasma is kept high enough that the fast electrons from the beam collide with the plasma and slow down. When colliding they serve to heat the plasma, which is our desired result. Since they collide so much, they do not propagate down into the region in which our measurements are made1. Second, the energy of the electron beam is set below the ionization energy of the background neutral gas. When the electrons emitted by the beam strike a neutral atom they do not have enough energy to knock out an electron and cause ionization. The primary benefit of this setup is that we are able to use an electron beam to generate a temperature change without affecting the density.



Paper Begins: “The electron temperature is obtained as a continuous ...”

electron temperature
Figure 1a from the paper: Temporal evolution of the electron temperature (solid black, top) along with a corresponding theoretical result (dashed red) and the injected beam current.

Figure 1 of the paper provides evidence for the thermal wave in the form of coherent oscillations in electron temperature. Figure 1a is reproduced to the right, and figure 1b (see the paper) is a zoomed in view of the oscillating part. The thing to take away from figure 1a is that the beam current (solid black trace at the bottom) does not have any oscillations in it. The beam current is proportional to the injected power or direct heating. One way to forcibly drive a thermal wave is to oscillate this input heat source. The fact that the beam current does not oscillate, yet the temperature does, is support for the spontaneous nature of this wave.

The code comparison (dashed red curve) is important for its own reason. Our group uses a computer model to predict how the temperature of the filament evolves in time and space. Recall, the LAPD is a 20 meter long machine along the direction of the magnetic field so there is a lot of room for the plasma to change. The code, or model, accurately predicts how the plasma evolves when it exhibits classical transport. It is possible to make this statement because the expressions for describing classical transport (i.e., the way in which particles and energy move in a plasma due to collisions between the particles) have been known for many years. Previous work done by the group focused on comparing a classical plasma with the results of the model. These experiments confirmed the accuracy of the model and can be found in references [12] and [13]. Reference [13] provides more detail for the model.

The fact that the model result in figure 1a agrees with the measurement during the times of temperature oscillations (∼ 2 ms), shows that the plasma is behaving classically in the presence of the thermal wave. If the plasma were not classical while the thermal wave is present, then the relationship between the thermal conductivity and the wave would not hold. In a non-classical plasma, the thermal wave does not necessarily allow for the determination of the thermal conductivity. This is why we needed to show that the plasma is classical in our experiment.

When a plasma does not display classical transport it is said to be displaying anomalous transport. Unfortunately for fusion researchers, the anomalous transport of a plasma tends toward the too much transport side. Anomalous transport in a tokamak is the situation encountered when energy and particles leave the confining magnetic field too quickly, thereby preventing the density and temperature needed to attain fusion conditions. Transport is major research thrust in modern fusion research. To oversimplify ITER, the idea behind it is that anomalously high transport levels can be partially overcome by making the tokamak so large that even escaping particles and energy will be confined long enough to achieve sustained fusion.

Paper Begins: “As expected from (1), the relatively large value of the ...”

Figure 2 is another display to help verify the existence of the thermal wave. If we look at the perpendicular component of the dispersion relation, we will find that the imaginary part of the wave number is going to be very large. This will be evident to people familiar with the numbers, so don't worry if the size of this imaginary wave number is not obvious to you. It is large, and this means that the radial extent of the thermal wave should be very small. The radial dimension of the wave is strongly damped. The damping is so strong that the wave should only be observed within a range of roughly 2 mm. For technical, and frustrating, reasons, we have to use Langmuir probes that are 1 mm in diameter (such a probe tip is shown in the first image here). As such, our resolution in measuring the radial wavelength is not good enough. Demonstrating the radial extent of the thermal wave requires a discussion of its amplitude.

Figure 2 plots the amplitude of the thermal wave versus radial position. The damping of the wave predicts that the damping coefficient will be 0.09 cm. From figure 2 we measure the damping coefficient to be 0.10 cm, which is an excellent agreement. Additionally, the electron temperature is plotted over the wave amplitude to show that the filament itself is much wider than the region of wave oscillation. If these regions were similar in extent, then it would suggest that perhaps the temperature oscillations are not a thermal wave.

Paper Begins: “The transport code can be used further to understand the ...”

Figure 3 is a pure theory result and it is used to add discussion of the parallel wavelength of the thermal wave. In keeping with my initial statement that little math will be included in this popular review, the details of this figure will be omitted.

Since this is a ¼-wavelength resonator, the length of the heated plasma filament must be able to fit ¼ of the parallel wavelength of the thermal wave. The parallel wavelength is related to the frequency of the wave, which allows a model to be constructed in which the wavelength is calculated based on the measured frequency of the electron temperature oscillations. To further add to the evidence in support of this being an actual thermal wave, we need to use the measured frequency to predict the quarter-wavelength value. If this value is longer than the entire machine, then it is unlikely we have observed a true thermal wave. Figure 3 shows that the calculated quarter-wavelength is about 8 meters and that the length of the heated filament is roughly the same. Since the length of the filament comes entirely from the code and the quarter-wavelength is calculated using measured parameters, this agreement is strongly in support of a true thermal wave observation.



Paper Begins: “Although the detailed mechanism that triggers the ...”

If other experiments in plasmas and other states of matter can generate a thermal wave, then what is causing it in this experiment? We do not know. Future work will focus on the cause and on ways in which new types of plasma diagnostics can be designed around this principle. It is also suggested that thermal waves may exist in the corona of the Sun. If that turns out to be the case, then it might be possible to study the behavior of the solar corona with diagnostics that detect oscillations in electron temperature.

That's it for the physics. The final paragraph acknowledges financial support we have received. Part of my pay came from the Center for Multiscale Plasma Dynamics, which is listed as the DOE Fusion Science Center Cooperative Agreement No. DE-FC02-04ER54785.

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Bad smells only seem to diffuse more quickly, I assure you all smells travel through air at the same speed.
1 Technically, they do still propagate throughout the plasma. The collisions cause them to thermalize before they make it very far. This means that at the location of our measurements it is impossible to distinguish between the electrons from the beam and those from the background plasma. The various instabilities and waves that typically result from fast electrons are due to the non-thermal contribution of the beam electrons, which we do not have.

Initial Posting: Wednesday, 23 July 2008
Last Updated: Friday, 22 August 2008