High-beta equilibria in circular, elliptical and D-shape large aspect ratio axisymmetric configurations with poloidal and toroidal flows

The Grad-Shafranov-Bernoulli system of equations is a single fluid magnetohydrodynamical description of axisymmetric equilibria with mass flows. Using a variational perturbative approach [1], analytic approximations for high-beta equilibria in circular, elliptical and D-shape cross sections in the high aspect ratio limit are found, which include finite toroidal and poloidal flows. Assuming a polynomial dependence of the free functions on the poloidal flux, the equilibrium problem is reduced to a modified Helmholtz PDE subject to homogeneous Dirichlet conditions. An application of the Green's function method leads to a closed form for the circular solution and to a series solution in terms of Mathieu functions for the elliptical case, which is valid for arbitrary elongations. To extend the elliptical solution to a D-shape domain, a boundary perturbation in terms of the triangularity is used. A comparison with the code FLOW [2] is presented for relevant scenarios.\\\\ \noindent [1] Eliezer Hameiri, \textit{Phys. Plasmas}, \textbf{20}, 024504 (2013).\\ \noindent [2] L. Guazzotto \textit{et al.}, \textit{Phys. Plasmas}, \textbf{11}, 604 (2004).