The vacuum poloidal flux functions satisfying the Grad-Shafranov equation in the flat-ring cyclide coordinate system

The Wangerin functions, which are solutions of Laplace's equation in the flat-ring cyclide coordinate system, are derived as series solutions about regular singular points μ = 0 and μ = K1 which correspond respectively to the external and internal points of plasmas, where the curve μ = const denotes the flattened cross section of plasmas and K' is the complete elliptic integral of the first kind with the modulus k'. Since the argument of the Wangerin functions is expressed as the aspect and elongation ratios of plasmas, so we can accurately treat the equilibrium problem of axisymmetric toroidal plasmas without an expansion in the aspect ratio. The Grad-Shafranov equation, which governs magnetohydrodynamic plasma equilibria, is analytically solved in the vacuum region by using the flat-ring cyclide coordinates so that the poloidal flux function is expressed as the Wangerin functions with the toroidal mode number being equal to one. (author)