The local ideal three dimensional (3-D) magnetohydrodynamic (MHD) stability for the Wendelstein VII-X (W VIl-X) configuration is studied. A volume averaged beta limit of 5% is confirmed with a nearly optimal pressure profile using two methods to calculate the parallel current density: the magnetic method that uses magnetic information about the configuration (in particular, the condition of charge conservation, ∇.j = 0, is explicitly used in the resolution) and the geometric method that uses the geometry of the configuration itself. It is shown that the ballooning stability does not depend on the method of calculating the parallel current. In contrast, the value of the Mercier criterion depends sensitively on which method is used. Not only is the geometric method not sensitive to resonant surfaces (in particular, the surface ιp = 1/6), but there is a systematic error in the Mercier criterion for nonresonant surfaces when an insufficient number of modes is used to calculate the equilibria numerically with a spectral method. However, this systematic error does not change the averaged beta limit of W VIII-X because the ballooning stability is more stringent than the Mercier stability for this configuration
Methods for measuring 1/1 error field in Wendelstein 7-X stellarator