Analytical solution of the Grad Shafranov equation in an elliptical prolate geometry

Abstract

The analytical solution in toroidal coordinates of the Grad Shafranov equation has been at the origin of the tokamak breakthrough in the fusion development. Unfortunately, the standard toroidal coordinates have a circular poloidal section, which does not fit the elongated cross-section of the present tokamak experiments. In axisymmetry, the vacuum Grad Shafranov equation coincides with the Laplace equation for the toroidal component of the vector potential. In the present paper the solutions for the Laplace equation and that for the vacuum Grad Shafranov equation are tackled in the elliptical prolate toroidal cap-cyclide coordinates framework. The following report of the geometrical properties and of the metric of these coordinates allows us to work out the analytical solution of both equations in terms of the Wangerin functions.

Analytical Solution of the Three-Dimensional Laplace Equation in Terms of Linear Combinations of Hypergeometric Functions

“…Starting from [19], the vacuum Grad-Shafranov equation are tackled in the elliptical prolate toroidal cap-cyclide coordinates framework.…”

“…where k and k 1 are respectively the parameter and the complementary parameter of the elliptic integrals. By varying the parameter k, the coordinate transformation (11) describes a large set of quite different geometries [19]. For µ → 0, independently of k, the geometry resembles the standard toroidal geometry; for larger values of µ the shape of the constant µ surfaces depend on the value of the k parameter.…”

“…The work of Crisanti [19] opens up the possibility of creating a reconstructive code of equilibrium based on an elongated geometry. Wangerin functions were first evaluated in [4].…”

“…Wangerin functions were first evaluated in [4]. However, to use them in a budget code, an independent evaluation and a cross-check will be required [19]. Furthermore, it will be necessary to evaluate the derivatives, to derive the poloidal magnetic field and find the Green's function for this [19] geometry.…”

“…In this article, starting from the recent work of Crisanti [19], we transform the Laplace equation into the Heun equation, and based on the work of A.M. Ishkhanyan [25], we provide an analytical solution of the Laplace equation for some parameter values. Since the Laplace equation and the Grad-Shafranov equation differ by one sign, the procedure presented in this manuscript can be repeated to find the analytical solution of the Grad-Shafranov equation.…”