Time-dependent Tokamak equilibria are investigated for which the magnetohydrostatic equilibrium equation is satisfied at each instant and the driving terms are due to small but finite scalar resistivity. Explicit solutions are obtained under the following conditions: (a) the standard Tokamak expansion is used; (b) the temperature is assumed to be spatially constant within the plasma; (c) the toroidal field is the vacuum field; (d) the time scale is −1 ts or −2ts, where ts is the ordinary skin time and the expansion parameter. The solutions are characterized by a poloidal energy ratio of unity and four time-dependent parameters (the applied toroidal electric and magnetic fields, the ellipticity of the plasma cross-section, and the major radius) which are connected by a differential equation in time. An additional time-dependent parameter is the triangularity of the plasma cross-section. Two cases are considered with respect to the equation of state, T = const and the adiabatic equation of state. Another way of characterizing the equilibria obtained is in terms of the total energy per length W/R and the total toroidal current I. The scaling W/R ∝ I2 R−2(a(l-a))1/2, where R is the major radius and a characterizes the ellipticity, is valid throughout.
Tokamak plasma equilibrium with non-uniform current distribution