1/10/2024 0 Comments Calculate magnetic field of coilsIn order to extend the elliptical solution to a D-shaped domain, a boundary perturbation in terms of the triangularity is used. An application of the Green's function method leads to a closed form for the more ยป circular solution and to a series solution in terms of Mathieu functions for the elliptical case, which is valid for arbitrary elongations. Assuming a polynomial dependence of the free functions on the poloidal flux, the equilibrium problem is reduced to an inhomogeneous Helmholtz partial differential equation (PDE) subject to homogeneous Dirichlet conditions. Using a variational perturbative approach, analytic approximations for high-beta equilibria in circular, elliptical, and D-shaped cross sections in the high aspect ratio approximation are found, which include finite toroidal and poloidal flows. The Grad-Shafranov-Bernoulli system of equations is a single fluid magnetohydrodynamical description of axisymmetric equilibria with mass flows.
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