Tokamak, reversed field pinch and intermediate structures as minimum-dissipative relaxed states

The principle of minimum energy dissipation rate is utilized to develop a unified model for relaxation in toroidal discharges. The Euler–Lagrange equation for such relaxed states is solved in toroidal coordinates for an axisymmetric torus by expressing the solutions in terms of Chandrasekhar–Kendall (C–K) eigenfunctions analytically continued in the complex domain. The C–K eigenfunctions are hypergeometric functions that are solutions of the scalar Helmholtz equation in toroidal coordinates in the large-aspect-ratio approximation. Equilibria are constructed by assuming the total current J=0 at the edge. This yields the eigenvalues for a given aspect-ratio. The most novel feature of the present model is that solutions allow for tokamak, low-q as well as reversed field pinch-like behavior with a change in the eigenvalue characterizing the relaxed state.