Analytical stability condition for the ideal m=n=1 kink mode in a toroidal plasma with elliptic cross section

Abstract

An analytical stability condition for the ideal kink mode with toroidal mode number n=1 and (dominating) poloidal mode number m=1 in a toroidal plasma with elliptic cross section in derived from a large aspect ratio expansion of the ideal magnetohydrodynamic equations. The ellipticity (e) is treated as a small parameter independent of the inverse aspect ratio (ε), and the expansion is performed up to order ε2e in the potential energy δW. It is found that the ellipticity has a strong, destabilizing effect on the kink mode in vertically elongated tokamaks, particularly when Δq=1−q0 is small (q0 is the safety factor at the magnetic axis), whereas an ellipticity of opposite sign (horizontal elongation) is stabilizing. By means of an additional expansion, in Δq, this effect is shown to be due to a Δq-independent term in δW proportional to the ellipticity and to the poloidal beta value (βp) at the q=1 surface. Since the leading order term in δW in the absence of ellipticity is of order Δq [Bussac et al., Phys. Rev. Lett. 35, 1638 (1975)], the ellipticity term dominates δW for sufficiently small values of Δq. The analytical result is in agreement with previous, numerical results [Lütjens et al., Nucl. Fusion 32, 1625 (1992)], and leads, already for rather moderately vertically elongated plasmas, to stability limits in βp much lower than the value 0.3 valid in the limit Δq→0 for a plasma with circular cross section.