Interchange instability

The interchange instability is a type of plasma instability similar to the Rayleigh–Taylor instability in fluid dynamics, but is driven by the gradients in the magnetic pressure due to the curvature of the external magnetic field applied on the plasma.[1] The name of the instability refers to the action of the plasma changing position with the magnetic field lines (i.e. an interchange of the lines of force in space[2]) without significant disturbance to the geometry of the external field.[3] The instability causes flute-like structures to appear on the surface of the plasma, and thus the instability is also known as the flute instability.[1][2] The interchange instability is a key issue in the field of fusion energy, where magnetic fields are used to confine a plasma in a volume surrounded by the field.

The basic concept was first noted in a famous 1954 paper by Martin David Kruskal and Karl Schwarzschild, which demonstrated that a situation similar to the Rayleigh–Taylor instability in classic fluids existed in magnetically confined plasmas. The problem can occur anywhere where the magnetic field is concave with the plasma on the inside of the curve. Edward Teller gave a talk on the issue at a meeting later that year, pointing out that it appeared to be an issue in most of the fusion devices being studied at that time. He used the analogy of rubber bands on the outside of a blob of jelly; there is a natural tendency for the bands to snap together and eject the jelly from the center.

Most machines of that era were suffering from other instabilities that were far more powerful, and whether or not the interchange instability was taking place could not be confirmed. This was finally demonstrated beyond doubt by a Soviet magnetic mirror machine during an international meeting in 1961. When the US delegation stated they were not seeing this problem in their mirrors, it was pointed out they were making an error in the use of their instrumentation and when that was considered it was clear their experiments were also being affected by the same problem. This led to a series of new mirror designs, as well as modifications to other designs like the stellarator to add negative curvature. These had cusp-shaped fields so that the plasma was contained within convex fields, the so-called "magnetic well" configuration.

In modern designs, the interchange instability is suppressed by the complex shaping of the fields. In the tokamak design there are areas of "bad curvature" but particles within the plasma spend only a short time in those areas before being circulated to an area of "good curvature". Modern stellarators use similar configurations, differing largely in how that shaping is created.

Basic concept

Most magnetic confinement systems attempt to contain the plasma within a magnetic field inside a certain volume of space within a vacuum chamber. The particles in the plasma are electrically charged, and orbit the lines of force. The goal is to shape the fields so that the particles are not able to touch the walls of the container and escape confinement. As the electrons are much lighter than the ions, they orbit the lines more tightly. Additionally, the fields must be arranged to keep the ions and electrons mixed, so charge separation does not occur.[4]

One of the simplest such confinement systems is the magnetic mirror. This consists of a cylinder with conductors wrapped around the outside. When current flows in the conductors, a field is set up with the magnetic lines running along the open center of the cylinder. The particles orbit these lines, which keeps them away from the walls of the cylinder. Unfortunately, this arrangement, a basic solenoid, does not confine their movement along the lines towards the ends. To address this, the mirror adds more windings to either end of the cylinder, squeezing the field together in those locations. That causes the particles to reflect off the area of increasing field, bouncing back and forth between the two ends of the cylinder.[4]

The resulting field has a distinct curvature to it, with the overall shape of a cigar. The interchange instability applies to the areas in the center of the long axis, where the field is convex (with the plasma on the concave side). When the plasma is located in its design location, the electrons and ions are roughly mixed. However, if the plasma is displaced, the non-uniform nature of the field means the ion's larger orbital radius takes them outside the confinement area while the electrons remain inside. It is possible the ion will hit the wall of the container, removing it from the plasma. If this occurs, the outer edge of the plasma is now net negatively charged, attracting more of the positively charged ions, which then escape as well.[4]

This effect allows even a tiny displacement to drive the entire plasma mass to the walls of the container. The same effect occurs in any reactor design where the plasma is within a field of sufficient curvature, which includes the outside curve of toroidal machines like the tokamak and stellarator. As this process is highly non-linear, it tends to occur in isolated areas, giving rise to the flute-like expansions as opposed to mass movement of the plasma as a whole.[4]

History

In the 1950s, the field of theoretical plasma physics emerged. The confidential research of the war time years became declassified and allowed the publication and spread of very influential papers. The world rushed to take advantage of the recent revelations on nuclear energy. Although never fully realized, the idea of controlled thermonuclear fusion motivated many to explore and research novel configurations in plasma physics. Instabilities plagued early designs of artificial plasma confinement devices and were quickly studied partly as a means to inhibit the effects. The analytical equations for interchange instabilities were first studied by Kruskal and Schwarzschild in 1954.[5] They investigated several simple systems including the system in which an ideal fluid is supported against gravity by a magnetic field (the initial model described in the last section). In 1958, Bernstein derives an energy principle that rigorously proves that the change in potential must be greater than zero for a system to be stable.[6] This energy principle has been essential in establishing a stability condition for the possible instabilities of a specific configuration. In 1959, Thomas Gold attempted to use the concept of interchange motion to explain the circulation of plasma around the Earth, using data from Pioneer III published by James Van Allen.[7] Gold also coined the term “magnetosphere” to describe “the region above the ionosphere in which the magnetic field of the Earth has a dominant control over the motions of gas and fast charged particles.” Marshall Rosenthal and Conrad Longmire described in their 1957 paper how a flux tube in a planetary magnetic field accumulates charge because of opposing movement of the ions and electrons in the background plasma. Gradient, curvature and centrifugal drifts all send ions in the same direction along the planetary rotation meaning that there is a positive build-up on one side of the flux tube and a negative build-up on the other. The separation of charges established an electric field across the flux tube and therefore adds an E x B motion, sending the flux tube toward the planet. This mechanism supports our interchange instability framework, resulting in the injection of less dense gas radially inward. Since the Kruskal and Schwarzschild’s paper a tremendous amount of theoretical work has been accomplished that handle multi-dimensional configurations, varying boundary conditions and complicated geometries. The motivation for new technologies in World War II and the Cold War also directly led to the advanced rocketry and satellite technology developed in the space race. This boded well for experimentalists who now had the means and support to build satellites that could orbit the Earth. Pioneers like James Van Allen, Louis Frank and others in space physics were finally able to acquire in situ data from Earth’s magnetosphere. Ever since the establishment of NASA in 1958, the program has overseen the bulk of American space exploration. Dozens of unmanned missions have been completed and many are still ongoing. From inner earth orbit to the interstellar medium, these missions have become essential in our understanding of the planets and the properties of their magnetospheres. Without them, the study of interchange instability would be much more brief and much less developed. These unmanned missions, that have now reached every planet in our solar system, have enabled a more comprehensive understanding of interchange motions in Jupiter and Saturn’s magnetospheres. Jupiter has had two major orbital missions: Galileo (launched in 1989) and Juno (launched in 2011 and currently operating in orbit). Saturn currently has the Cassini-Huygens probe in orbit. Cassini was launched in 1997 and has been at Saturn since 2004.

Instability in a plasma system

The single most important property of a plasma is its stability. MHD and its derived equilibrium equations offer a wide variety of plasmas configurations but the stability of those configurations have not been challenged. More specifically, the system must satisfy the simple condition

where ? is the change in potential energy for degrees of freedom. Failure to meet this condition indicates that there is a more energetically preferable state. The system will evolve and either shift into a different state or never reach a steady state. These instabilities pose great challenges to those aiming to make stable plasma configurations in the lab. However, they have also granted us an informative tool on the behavior of plasma, especially in the examination of planetary magnetospheres.

This process injects hotter, lower density plasma into a colder, higher density region. It is the MHD analog of the well-known Rayleigh-Taylor instability. The Rayleigh-Taylor instability occurs at an interface in which a lower density liquid pushes against a higher density liquid in a gravitational field. In a similar model with a gravitational field, the interchange instability acts in the same way. However, in planetary magnetospheres co-rotational forces are dominant and change the picture slightly.

Simple models

Let’s first consider the simple model of a plasma supported by a magnetic field B in a uniform gravitational field g. To simplify matters, assume that the internal energy of the system is zero such that static equilibrium may be obtained from the balance of the gravitational force and the magnetic field pressure on the boundary of the plasma. The change in the potential is then given by the equation: ? If two adjacent flux tubes lying opposite along the boundary (one fluid tube and one magnetic flux tube) are interchanged the volume element doesn’t change and the field lines are straight. Therefore, the magnetic potential doesn’t change, but the gravitational potential changes since it was moved along the z axis. Since the change in is negative the potential is decreasing. A decreasing potential indicates a more energetically favorable system and consequently an instability. The origin of this instability is in the J × B forces that occur at the boundary between the plasma and magnetic field. At this boundary there are slight ripple-like perturbations in which the low points must have a larger current than the high points since at the low point more gravity is being supported against the gravity. The difference in current allows negative and positive charge to build up along the opposite sides of the valley. The charge build-up produces an E field between the hill and the valley. The accompanying E × B drifts are in the same direction as the ripple, amplifying the effect. This is what is physically meant by the “interchange” motion. These interchange motions also occur in plasmas that are in a system with a large centrifugal force. In a cylindrically symmetric plasma device, radial electric fields cause the plasma to rotate rapidly in a column around the axis. Acting opposite to the gravity in the simple model, the centrifugal force moves the plasma outward where the ripple-like perturbations (sometimes called “flute” instabilities) occur on the boundary. This is important for the study of the magnetosphere in which the co-rotational forces are stronger than the opposing gravity of the planet. Effectively, the less dense “bubbles” inject radially inward in this configuration. Without gravity or an inertial force, interchange instabilities can still occur if the plasma is in a curved magnetic field. If we assume the potential energy to be purely magnetic then the change in potential energy is: . If the fluid is incompressible then the equation can be simplified into . Since (to maintain pressure balance), the above equation shows that if the system is unstable. Physically, this means that if the field lines are toward the region of higher plasma density then the system is susceptible to interchange motions. To derive a more rigorous stability condition, the perturbations that cause an instability must be generalized. The momentum equation for a resistive MHD is linearized and then manipulated into a linear force operator. Due to purely mathematical reasons, it is then possible to split the analysis into two approaches: the normal mode method and the energy method. The normal mode method essentially looks for the eigenmodes and eigenfrequencies and summing the solutions to form the general solution. The energy method is similar to the simpler approach outlined above where is found for any arbitrary perturbation in order to maintain the condition . These two methods are not exclusive and can be used together to establish a reliable diagnosis of the stability.

Observations in space

The strongest evidence for interchange transport of plasma in any magnetosphere is the observation of injection events. The recording of these events in the magnetospheres of Earth, Jupiter and Saturn are the main tool for the interpretation and analysis of interchange motion.

Earth

Although spacecraft have travelled many times in the inner and outer orbit of Earth since the 1960s, the spacecraft ATS 5 was the first major plasma experiment performed that could reliably determine the existence of radial injections driven by interchange motions. The analysis revealed the frequent injection of a hot plasma cloud is injected inward during a substorm in the outer layers of the magnetosphere.[8] The injections occur predominantly in the night-time hemisphere, being associated with the depolarization of the neutral sheet configuration in the tail regions of the magnetosphere. This paper then implies that Earth’s magnetotail region is a major mechanism in which the magnetosphere stores and releases energy through the interchange mechanism. The interchange instability also has been found to have a limiting factor on the night side plasmapause thickness [Wolf et al. 1990]. In this paper, the plasmapause is found to be near the geosynchronous orbit in which the centrifugal and gravitational potential cancel exactly. This sharp change in plasma pressure associated with the plasma pause enables this instability. A mathematical treatment comparing the growth rate of the instability with the thickness of the plasmapause boundary revealed that the interchange instability limits the thickness of the boundary.

Jupiter

Interchange instability plays a major role in the radial transport of plasma in the Io plasma torus at Jupiter. The first evidence of this behavior was published by Thorne et al. in which they discovered “anomalous plasma signatures” in the Io torus of Jupiter’s magnetosphere.[9] Using the data from Galileo’s energetic particle detector (EPD), the study looked at one specific event. In Thorne et al. they concluded that these events had a density differential of at least a factor of 2, a spatial scale of km and an inward velocity of about km/s. These results support the theoretical arguments for interchange transport. Later, more injections events were discovered and analyzed from Galileo. Mauk et al. used over 100 Jovian injections to study how these events were dispersed in energy and time.[10] Similar to injections of Earth, the events were often clustered in time. The authors concluded that this indicated the injection events were triggered by solar wind activity against the Jovian magnetosphere. This is very similar to the magnetic storm relationship injection events have on Earth. However, it was found that Jovian injections can occur at all local time positions and therefore can’t be directly related to the situation in Earth’s magnetosphere. Although the Jovian injections aren’t a direct analog of Earth’s injections, the similarities indicate that this process plays a vital role in the storage and release of energy. The difference may lie in the presence of Io in the Jovian system. Io is a large producer of plasma mass because of its volcanic activity. This explains why the bulk of interchange motions are seen in a small radial range near Io.

Saturn

Recent evidence from the spacecraft Cassini has confirmed that the same interchange process is prominent on Saturn. Unlike Jupiter, the events happen much more frequently and more clearly. The difference lies in the configuration of the magnetosphere. Since Saturn’s gravity is much weaker, the gradient/curvature drift for a given particle energy and L value is about 25 times faster. Saturn’s magnetosphere provides a much better environment for the study of interchange instability under these conditions even though the process is essential in both Jupiter and Saturn. In a case study of one injection event, the Cassini Plasma Spectrometer (CAPS) produced characteristic radial profiles of plasma densities and temperatures of the plasma particles that also allowed the calculation of the origin of the injection and the radial propagation velocity. The electron density inside the event was lowered by a factor of about 3, the electron temperature was higher by an order of magnitude than the background, and there was a slight increase in the magnetic field.[11] The study also used a model of pitch angle distributions to estimate the event originated between and had a radial speed of about 260+60/-70 km/s. These results are similar to the Galileo results discussed earlier.[9] The similarities imply that the Saturn and Jupiter processes are the same.

See also

References

  1. 1 2 J., Goldston, R. (1995). "19 - The Rayleigh-Taylor and flute instabilities". Introduction to plasma physics. Rutherford, P. H. (Paul Harding), 1938-. Bristol, UK: Institute of Physics Pub. ISBN 0750303255. OCLC 33079555.
  2. 1 2 Frank-Kamenetskii, D. A. (1972), "Interchange or Flute Instabilities", Plasma, Macmillan Education UK, pp. 98–100, doi:10.1007/978-1-349-01552-8_32, ISBN 9781349015542, retrieved 2018-09-08
  3. ↑ Southwood, David J.; Kivelson, Margaret G. (1987). "Magnetospheric interchange instability". Journal of Geophysical Research. 92 (A1): 109. doi:10.1029/ja092ia01p00109. ISSN 0148-0227.
  4. 1 2 3 4 Fowler, T.K.; Post, Richard (December 1966). Scientific American. Vol. 215 no. 6. pp. 21–31. Missing or empty |title= (help)
  5. ↑ Kruskal, Martin David; Schwarzschild, Martin (1954-05-06). "Some instabilities of a completely ionized plasma". Proc. R. Soc. Lond. A. 223 (1154): 348–360. doi:10.1098/rspa.1954.0120. ISSN 0080-4630.
  6. ↑ Bernstein, I. B.; Frieman, E. A.; Kruskal, Martin David; Kulsrud, R. M. (1958-02-25). "An energy principle for hydromagnetic stability problems". Proc. R. Soc. Lond. A. 244 (1236): 17–40. doi:10.1098/rspa.1958.0023. ISSN 0080-4630.
  7. ↑ Gold, T. (1959). "Motions in the magnetosphere of the Earth". Journal of Geophysical Research. 64 (9): 1219–1224. doi:10.1029/jz064i009p01219. ISSN 0148-0227.
  8. ↑ DeForest, S. E.; McIlwain, C. E. (1971-06-01). "Plasma clouds in the magnetosphere". Journal of Geophysical Research. 76 (16): 3587–3611. doi:10.1029/ja076i016p03587. ISSN 0148-0227.
  9. 1 2 Thorne, R. M.; Armstrong, T. P.; Stone, S.; Williams, D. J.; McEntire, R. W.; Bolton, S. J.; Gurnett, D. A.; Kivelson, M. G. (1997-09-01). "Galileo evidence for rapid interchange transport in the Io torus". Geophysical Research Letters. 24 (17). doi:10.1029/97gl01788@10.1002/(issn)1944-8007.galijov1. ISSN 1944-8007.
  10. ↑ Mauk, B. H.; Williams, D. J.; McEntire, R. W.; Khurana, K. K.; Roederer, J. G. (1999-10-01). "Storm-like dynamics of Jupiter's inner and middle magnetosphere". Journal of Geophysical Research: Space Physics. 104 (A10): 22759–22778. doi:10.1029/1999ja900097. ISSN 0148-0227.
  11. ↑ Rymer, A. M.; Smith, H. T.; Wellbrock, A.; Coates, A. J.; Young, D. T. (2009-08-13). "Discrete classification and electron energy spectra of Titan's varied magnetospheric environment". Geophysical Research Letters. 36 (15): n/a–n/a. doi:10.1029/2009gl039427. ISSN 0094-8276.
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