A theory of the relaxation of tokamak discharges

Tokamak equilibria are considered as extremum magnetic-energy states. Two quadratic invariants ∫Ψd³x, and ∫A.Bd³x are found to yield finite-beta equilibria in toroidal geometry. The theory predicts ‘bell’-shaped current profiles for relaxed discharges at finite beta. A ‘main-sequence’ curve of q_l/q₀ versus β_p is presented for circular discharges with low q_l and is found to form a limiting curve for experimental data.