This is a limit on the plasma pressure that a tokamak’s magnetic field can hold. Unlike the Greenwald limit, this criterion is rigorously calculated from ideal MHD (MagnetoHydroDynamics) theory. The quantity that measures the balance between the pressure and magnetic forces is called В (beta). Since В is used in many scien­tific disciplines, especially in medicine, I had refrained from defining it until it was necessary. It is now necessary. Beta is the ratio between plasma pressure and magnetic pressure:

Plasma pressure Magnetic pressure

The plasma’s pressure is the product of its density and its temperature, and the magnetic pressure is proportional to the square of the field strength B. These quanti­ties are not constant over a cross section of the plasma, so a reasonable definition would be to take the average pressure and divide it by the average magnetic field before the plasma is created. The last proviso is needed because the plasma is diamagnetic, so its very presence decreases the B-field inside it. Since the B-field is the most expensive component, В is a measure of the cost effectiveness of a tokamak. It has a value below 10%, typically 4-5%.

The value of В has been shown to depend on the toroidal current I divided by the plasma radius a and the magnetic field strength B. Figure 8.19 shows how data from different tokamaks all fall on the same line if plotted against I / aB. It is con­venient, then, to introduce a normalized В, called BN, which would apply to all tokamaks, regardless of their values of I, a, and B:

В x a x B
I

The Troyon limit (Troyon et al. [30])9 is when BN is about 3.5. A numerical formula is given in footnote 10. Figure 8.17 shows how well the experiments in different tokamaks obey the Troyon limit, above which disruptions are likely to occur.