When you turn a bottle of mineral water upside down, the water falls out even though the atmospheric pressure of 15 lbs./sq. in is certainly strong enough to sup­port the water. This happens because of an instability called the Rayleigh-Taylor instability, which is illustrated in Fig. 5.5. If the bottom surface of the water remained perfectly flat, it would be held by the atmospheric pressure. However, if there is a small ripple on the surface, there is slightly less water pressing on the top of the ripple than elsewhere, and the balance between the weight of the water above the ripple and the atmospheric pressure is upset. The larger the ripple grows, the greater is the unbalance, and the ripple grows faster. Eventually, it grows into a large bubble which rises to the top, allowing water to flow out under it. If you hold the end of a straw filled with water, the water does not fall out because surface ten­sion prevents the interface from deforming like that. A similar instability occurs in a plasma held by magnetic pressure, as we’ll soon see.

Instabilities occur because of positive feedback. There are many examples of this in real life. Microphone screech occurs because the loudspeaker feeds into the microphone the tone to which it is most sensitive. The audio system amplifies that tone, making it louder in the speaker, which then drives the microphone harder. Forest fires are instabilities. A small fire dries the wood around it so that it catches fire more readily. The larger fire then dries a larger amount of wood near it, which then starts to burn, and the instability spreads like… a wildfire. Stock market insta­bilities can go both ways, as a rise or fall in the market induces more people to buy or sell. A more subtle instability creates snow cups when a field of snow melts or sublimes, as can be seen in Fig. 5.6.

If the sun shines evenly on a perfectly flat surface of snow, it should melt evenly, retaining a smooth surface. It never does, because there are ripples in the snow. A depression in the snow will cause some sunlight to scatter onto its walls, heating them before reflecting out into space. The deeper the hole is, the more light will deposit energy into it to hasten the melting. A snow cup can be started by a twig or pebble, which, being dark, will absorb more heat. But instabilities will always start and grow because there is always some imperfection or noise in the system. It just takes longer if the system starts out being almost perfect.

The main obstacle to making a leak-proof magnetic bottle is instability. There are many instabilities, and the first step is to know your enemy. This first instability, however, was known from the beginning because it is similar to the well-known Rayleigh-Taylor instability in hydrodynamics. A plasma weighs almost nothing, so the instability is driven not by gravity but by pressure. To see how this works, we have to extend the concept of E x B drifts to drifts caused by other forces. Or, you can skip the next two diagrams and move on to see how this instability is stabilized.

Fig. 5.7a is the same as Fig. 5.4 except that the small gyrations have been sup­pressed, and only the guiding center drift due to an electric field is shown. In Parts (b) and (c), the E-field has been turned to different directions, and the drifts have rotated correspondingly. In Part (c), the E-field applies a downward force on the ions. If we apply to the ions another type of downward force, such as a pressure force, the ions will also drift to the left, as shown in Fig. 5.7d. Note that the elec­trons and ions now drift in opposite directions. The reason that the electric-field drifts are the same for both species is that both the electric force and the Lorentz force of the magnetic field depend on the sign of the charge, and these two depen­dences cancel each other. The pressure force, however, is in the same direction regardless of charge, so this cancelation does not take place, and the pressure drift depends on the sign of the charge.

Figure 5.8a shows a part of the plasma boundary when it is perfectly smooth, like the first drawing of a water bottle in Fig. 5.5. The upper part is plasma, and the lower part is vacuum, containing only the magnetic field. The plasma pressure is held back by the magnetic pressure, just as the water in Fig. 5.5 is supported by the atmospheric pressure. The force that now tries to push the denser fluid into the less dense fluid is now the plasma pressure rather than gravity. The pressure force, according to Fig. 5.7, causes ions to drift to the left and electrons to the right. As long as the plasma surface is straight and smooth, these drifts are perfectly harm­less, and the magnetic field prevents the plasma from leaking out. Now suppose there is a small ripple in the surface, like the one in the Rayleigh-Taylor instability

Fig. 5.8 Development of a Rayleigh-Taylor instability in a plasma for water. What happens is shown in Fig. 5.8b. The ions, drifting to the left, pile up on the right side of the ripple, and the electrons, drifting right, pile up on the left side. These charges create an E-field pointing to the left, as shown in Fig. 5.8b. From Fig. 5.7b, we see that this E-field causes both ions and electrons to drift upwards, thus enhancing the ripple. The ripple or bubble then grows unstably, with the magnetic field forcing its way into the plasma, ejecting the plasma outwards in a way reminiscent of Fig. 5.5. The plasma escapes from the magnetic trap by orga­nizing itself to create electric fields which can push it out! Since in the long run the magnetic field has basically changed its place with the plasma, with the field on the inside and the plasma on the outside, this instability is also called an interchange instability.