The Grad-Shafranov Equation in Cap-Cyclide Coordinates: The Heun Function Solution

The Grad-Shafranov plasma equilibrium equation was originally solved analytically in toroidal geometry, which fitted the geometric shape of the first Tokamaks. The poloidal surface of the Tokamak has evolved over the years from a circular to a D-shaped ellipse. The natural geometry that describes such a shape is the prolate elliptical one, i.e., the cap-cyclide coordinate system. When written in this geometry, the Grad-Shafranov equation can be solved in terms of the general Heun function. In this paper, we obtain the complete analytical solution of the Grad-Shafranov equation in terms of the general Heun functions and compare the result with the limiting case of the standard toroidal geometry written in terms of the Fock functions.