Numerical simulations of sawteeth in tokamaks have been carried out using reduced magnetohydrodynamics and a simple transport model. The electron temperature is evolved self-consistently, including Ohmic heating and a highly anisotropic thermal diffusivity. The sawtooth period and collapse time found in the simulations for Lundquist numbers S below 107 compare favourably with experimental results from small and medium size ohmically heated tokamaks. The sawtoothing is found to be sensitive to the values of the transport coefficients. In particular, the perpendicular viscosity must be comparable to, or larger than, the perpendicular thermal diffusivity for distinct relaxation oscillations to occur. To study the scaling with S, the viscosity and perpendicular thermal conductivity have been scaled as 1/S, and S has been varied. Modifications of the equilibrium, caused by the sawteeth, play an important role in the scaling of the collapse time and period with the Lundquist numnber. The self-consistently computed q-profiles are very flat in the central region where q ≈ 1. Outside the low-shear region, the shear rises sharply. This q-profile allows a resistive mode to be turned on quickly with a high growth rate. The deviation of the central q from unity over the sawtooth cycle decreases with increasing S, and the collapse time shows a weaker dependence than the S½ scaling of Sweet and Parker.
Simulations of tokamak disruptions including self-consistent temperature evolution