Interchange instability

A basic magnetic mirror. The magnetic lines of force (green) confine plasma particles by causing them to rotate around the lines (black). As the particles approach the ends of the mirror, they see an increasing force back into the center of the chamber. Ideally, all particles would continue to be reflected and stay within the machine.

Magnetic confinement systems attempt to hold the plasma within a vacuum chamber using magnetic fields. The plasma particles are electrically charged, and thus see a transverse force from the field due to the Lorentz force. When the particle's original linear motion is superimposed on this transverse force, its resulting path through space is a helix, or corkscrew shape. Such a field will thus trap the plasma by forcing it to flow along the lines.[4]

One can produce a linear field using an electromagnet in the form of a solenoid wrapped around a tubular vacuum chamber. In this case, the plasma will orbit the lines running down the center of the chamber and be prevented from moving outward towards the walls. This does not confine the plasma along the length of the tube, and it will rapidly flow out the ends. Designs that prevented this from occurring appeared in the early 1950s and experiments began in earnest in 1953. However, all of these devices proved to leak plasma at rates far higher than expected.[5]

In May 1954, Martin David Kruskal and Martin Schwarzschild published a paper demonstrating two effects that meant plasmas in magnetic fields were inherently unstable.[6] One of the two effects, which became known as the kink instability, was already being seen in early z-pinch experiments and occurred slowly enough to be captured on movie film. The topic of stability immediately gained significance in the field.

The other instability noted in the paper considered an infinite sheet of plasma held up against gravity by a magnetic field. It suggested there would be behaviour similar to that in classical physics when one heavy fluid is supported by a lighter one, which leads to the Rayleigh–Taylor instability. Any small vertical disturbance in an initially uniform field would result in the field pulling on the charges laterally and causing the initial disturbance to be further magnified. As large sheets of plasma were not common in existing devices, the outcome of this effect was not immediately obvious. It was not long before a corollary became obvious; the initial disturbance resulted in a curved interface between the plasma and the external field, and this was inherent to any design that had a convex area in the field.

In October 1954 a meeting of the still-secret Project Sherwood researchers was held at Princeton University's Gun Club building. Edward Teller brought up the topic of this instability and noted that two of the major designs being considered, the stellarator and the magnetic mirror, both had large areas of such curvature and thus should be expected to be inherently unstable. He further illustrated it by comparing the situation to jello being held together with rubber bands; while such a setup might be created, any slight disturbance would cause the rubber bands to contract and eject the jello. This exchange of position appeared to be identical to the mirror case in particular, where the plasma naturally wanted to expand while the magnetic fields had an internal tension.

No such behaviour had been seen in experimental devices, but as the situation was considered further, it became clear it would be more obvious in areas of greater curvature, and existing devices used relatively weak magnetic fields with relatively flat fields. This nevertheless presented a significant problem; a key measure of the attractiveness of a reactor design was its beta, the ratio of magnetic field strength to confined plasma - higher beta meant more plasma for the same magnet, which was a significant factor in cost. However, higher beta also implied more curvature in these devices, which would make them increasingly unstable. This might force reactors to operate at low beta and be doomed to be economically unattractive.

As the magnitude of the problem became clear, the meeting turned to the question of whether or not there was any arrangement that was naturally stable. Jim Tuck was able to provide a solution; the picket fence reactor concept had been developed as a solution to another problem, bremsstrahlung losses, but he pointed out that its field arrangement would be naturally stable under the conditions shown in the Kruskal/Schwarzschild paper. Nevertheless, as Amasa Bishop noted;

The suggestion of the picket-fence concept did little, however, to dispel the atmosphere of gloom which settled over the conference, toward its end. The stability criterion which had been suggested by Teller was admittedly tentative, but it seemed dismayingly reasonable; perhaps more important, it brought the whole of the Sherwood program face to face with a problem which threatened its very existence.

The correctness of the simplified model was then called into question and led to further study. The answer appeared at a follow-up meeting at Berkeley in February 1955, where Harold Grad of New York University, Conrad Longmire of Los Alamos and Edward A. Frieman of Princeton presented independent developments that all proved the effect to be real, and worse, should be expected at any beta, not just high beta. Further work at Los Alamos demonstrated that the effect should be seen in both the mirror and stellarator.

The effect is most obvious in the magnetic mirror device. The mirror has a field that runs along the open center of the cylinder and bundles together at the ends. In the center of the chamber the particles follow the lines and flow towards either end of the device. There, the increasing magnetic density causes them to "reflect", reversing direction and flowing back into the center again. Ideally, this will keep the plasma confined indefinitely, but even in theory there a critical angle between the particle trajectory and the axis of the mirror where particles can escape. Initial calculations showed that the loss rate through this process would be small enough to not be a concern.[4]

In practice, all mirror machines demonstrated a loss rate far higher than these calculations suggested.[5] The interchange instability was one of the major reasons for these losses. The mirror field has a cigar shape to it, with increasing curvature at the ends. 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]

In the 1950s, the field of theoretical plasma physics emerged. The confidential research of the war 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.[7] 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 derived an energy principle that rigorously proved that the change in potential must be greater than zero for a system to be stable.[8] 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.[9] 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 Kruskal and Schwarzschild's papers a tremendous amount of theoretical work has been accomplished that handle multi-dimensional configurations, varying boundary conditions and complicated geometries.

Studies of planetary magnetospheres with space probes has helped the development of interchange instability theories, especially the comprehensive understanding of interchange motions in Jupiter and Saturn’s magnetospheres.

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.

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.

• Plasma stability
• Magnetic mirror
• Fusion power