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

An analytical stability condition for the ideal kink mode with toroidal mode number $\text{n=1}$ and (dominating) poloidal mode number $\text{m=1}$ in a toroidal plasma with elliptic cross section in derived from a large aspect ratio expansion of the ideal magnetohydrodynamic equations. The ellipticity $\mathrm{(e)}$ is treated as a small parameter independent of the inverse aspect ratio (ε), and the expansion is performed up to order ${\epsilon }^{2}e$ in the potential energy $\mathrm{\delta W.}$ It is found that the ellipticity has a strong, destabilizing effect on the kink mode in vertically elongated tokamaks, particularly when ${\text{Δq=1−q}}_0$ is small ($q_0$ 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 $\mathrm{\Delta q,}$ this effect is shown to be due to a $\mathrm{\Delta q}$ -independent term in $\mathrm{\delta W}$ proportional to the ellipticity and to the poloidal beta value ${\mathrm{(\beta }}_p)$ at the $\text{q=1}$ surface. Since the leading order term in $\mathrm{\delta W}$ in the absence of ellipticity is of order $\mathrm{\Delta q}$ [Bussac et al., Phys. Rev. Lett. 35, 1638 (1975)], the ellipticity term dominates $\mathrm{\delta W}$ for sufficiently small values of $\mathrm{\Delta q.}$ 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 ${\beta }_p$ much lower than the value 0.3 valid in the limit $\text{Δq→0}$ for a plasma with circular cross section.