Effect of paramagnetism on the global stability of straight screw pinches

An analytical expression for the eigenvalue of the global m = 1 kink mode in a straight screw pinch with arbitrary profiles of current density and axial magnetic field is derived in a long wavelength approximation of the incompressible ideal MHD equations. While the influence of the current profile is related to the inductance l i of the plasma column, as has been shown previously (Nucl. Fusion 24 (1984) 1357), it is found here that the effect of the axial magnetic field profile can be expressed in terms of the poloidal beta value β p of the pinch. In turn, β p is related to a possible paramagnetism ( β p < 1) or diamagnetism ( β p > 1) of the plasma. Considering the range of stable eigen-values, the Kruskal-Shafranov condition q(a) > 1, independent of l i and β p , is recovered in the tokamak regime B z , ≫ B ϕ . In the regime B z ⪅ B ϕ , on the other hand, the stability condition is strongly dependent on both l i and β p . Optimal stability conditions in this regime are obtained for l i , ≪ 1 (hollow current distribution), β p ≪ 1 (strong paramagnetism) and B z (a) ≪ B ϕ (a) (low q(a)). In this case, wavelengths of up to fourteen times the plasma radius become stable, and also the growth rates of the unstable wavelengths are considerably reduced as compared to the situation with l i , β p ∼ 1 and B z ∼ B ϕ , or compared to the growth rates obtained in the tokamak regime. The results obtained are consistent with recent experimental observations in 'ultra low q' discharges.