On the Grad–Shafranov equation as an eigenvalue problem, with implications for q solvers

It is shown that the Grad–Shafranov equation for toroidally symmetric ideal-magnetohydrodynamic (MHD) equilibria is a conventional albeit nonlinear eigenvalue problem. That this has been generally overlooked with limited consequences has been made possible by the existence of a scale-invariant transformation of the equation. If the safety factor q is chosen in place of the toroidal field as one of the free flux functions specifying the source (numerical Grad–Shafranov solvers with this capability are called ‘‘q solvers’’), the eigenvalue is 1 and the scale-transformation factor drops out of the problem. It is shown how this is responsible for the numerical problems that have plagued a class of q solvers, and a simple remedy is suggested. This has been implemented in Livermore’s toroidal equilibrium code (TEQ), and as an example, a quasistatically evolved vertical event is presented.