Ll. Lodestro et Ld. Pearlstein, ON THE GRAD-SHAFRANOV EQUATION AS AN EIGENVALUE PROBLEM, WITH IMPLICATIONS FOR Q-SOLVERS, Physics of plasmas, 1(1), 1994, pp. 90-95
It is shown that the Grad-Shafranov equation for toroidally symmetric
ideal-magnetohydrodynamic (MHD) equilibria is a conventional albeit no
nlinear eigenvalue problem. That this has been generally overlooked wi
th limited consequences has been made possible by the existence of a s
cale-invariant transformation of the equation. If the safety factor q
is chosen in place of the toroidal field as one of the free flux funct
ions specifying the source (numerical Grad-Shafranov solvers with this
capability are called ''q solvers''), the eigenvalue is 1 and the sca
le-transformation factor drops out of the problem. It is shown how thi
s is responsible for the. numerical problems that have plagued a class
of q solvers, and a simple remedy is suggested. This has been impleme
nted in Livermore's toroidal equilibrium code (TEQ), and as an example
, a quasistatically evolved vertical event is presented.