Thermal equilibrium and stability of an Ohmically heated plasma

Steady-state temperature profiles in a magnetically confined tokamak type plasma are derived by balancing Ohmic power input, thermal diffusion, and radiation. A comprehensive study is presented of the impact that different electron conductivities, K₁ boundary conditions, and radiation may have on thermal stability. In particular, their influence on the bifurcation in the position of the minor radius as a function of the temperature at the center T_m is considered. Points of bifurcation correspond to marginally stable equilibria. However, it is also possible to have both branches without marginally stable equilibrium. Critical values of parameters that cause the onset of a new branch or instability are determined. Growth rates are not calculated, but domains of stabilty of all possible equilibria are determined. It is shown that Ohmically heated plasma in the classical scaling is thermally unstable to convective boundary conditions or if the plasma is cold at the edge. It is shown that a finite temperature at the edge will generate a stable, lower branch, but the upper branch will be unstable as long as p <0.5, where K_⊥(T)∼T^p.