A semi-empirical scaling law for the β-limit in tokamaks

A limited numerical optimization of the current and pressure profiles in tokamaks of various shapes and aspect ratios has led to a simple scaling law for the maximum volume averaged β which can be reached before the onset of ideal MHD instabilities: β < (2.0?2.5)μ₀I/Ba, where I is the total currrent, B the toroidal field at the centre of the plasma and a the minor plasma radius in the equatorial plane. The limit is set by the appearance of an n = 1 free boundary instability, neglecting the possible stabilizing effect of a shell. The highest β's attained in tokamaks equipped with strong auxiliary heating are shown to be very near the limit given by this scaling law in spite of their very different conditions of operation. Possible implications are discussed.