Grad-Shafranov Equation in Fractal Dimensions

Abstract

The main aim of this paper is to discuss the influence of fractal dimensions on the behavior of the solutions of the Grad-Shafranov equation. Our study is based on the product-like fractal measure approach constructed by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal continuum media. The fractal Grad-Shafranov equation gives the possibility to analyze, in a toroidal fusion reactor, the plasma equilibrium in fractal dimensions. Examples of the exact equilibrium solution are given for both the vacuum case outside the plasma and the toroidally shaped spheromak. Note: PACS numbers 05.45.Df: Fractals; 28.52.−s: Fusion reactors; 52.30.Cv: Magnetohydrodynamics; and 52.55.Ip: Spheromaks.