Finite pressure ballooning mode stability in toroidal equilibriums

In this paper the effect of finite pressure on the ballooning instability in toroidal magnetohydrodynamics (MHD) equilibria of steep boundary stellarators and tokamaks is examined. Ballooning modes tend to arise near the place where the local shear vanishes and the normal curvature (the curvature component perpendicular to the flux surface, pointing away from the magnetic axis) is negative. It is shown how the pressure gradient determines the position of the shearless points, and demonstrated in detail how this effect explains the existence of second stability in tokamaks. For large aspect ratio circular cross-section tokamaks the second stability condition is found to scale as α = const S/sup 1.25/. Stellarators are inherently more stable because of the negative vacuum shear, which at moderate pressure gradients allows the zero shear point to localize on the inner side of the flux surface. However, at high pressure gradients the Pfirsch--Schlueter current produces a positive mean shear when the total toroidal current on a flux surface is zero. This causes the zero shear point to localize on the outer edge near the vertical extremes of the flux surface. This effect, together with helical contributions to the helical curvature, allows for ballooning instability to arise. At higher pressure gradients, with zero net toroidal current, an unstable ballooning mode which localizes to within a helical period always arises where the normal curvature is unfavorable