The External Kink Mode in Diverted and Limited Tokamaks

The disruptive instability in diverted tokamaks when the safety factor q at the 95\% flux surface, $q_{95}$, is below 2.0 is shown to be a resistive kink. The mode is a counterpart to the ideal mode that explains the corresponding disruption in limited cross sections when $q_a$, the safety factor at the plasma boundary, lies just below a rational value m/n. Experimentally, the 2/1 kink instability is unstable for ${q_a<{2}$. However, for diverted plasmas, $q_a$ is formally infinite and the ideal theory would predict stability. Yet, the disruptive limit occurs in practice when $q_{95}$, reaches 2. It is shown from numerical calculations in L-mode equilibria that a resistive kink mode is destabilized by the rapidly increasing resistivity at the plasma edge when ${q_{95}<{2}$, but ${q_a>>{2}$. The resistive kink behaves much like the ideal kink but the growth rates scale with a fractional power of the resistivity near the q = 2 surface; the exponent transitions smoothly between fractional values up to the ideal scaling. The model also explains an observed discrepancy in the limiter case where the onset actually occurs when $q_a$ is slightly above 2.0.