New magnetohydrodynamic equilibria by secondary bifurcation

For planar fields, the Grad–Shafranov theory of magnetohydrodynamic equilibria is reduced to solving Δψ+kG (ψ) =0 for some function G (ψ), where Δ is the two-dimensional Laplacian and k is an ’’amplitude’’ of G. This equation is solved on a rectangular region of aspect ratio l, with ψ=0 on the boundary, and G satisfying the conditions, G (0) =0, G′ (0) =1, G″ (0) =0, and G‴ (0) ≠0. Since G (0) =0, the basic state, ψ≡0, is a solution for all values of the parameter λ≡ka2, where a is the x dimension of the rectangle. Other equilibria, which are called the primary states, branch from the basic state at the eigenvalues λ=λmn(l) of the linearized theory. The eigenvalues are the bifurcation points of the basic state. They become multiple eigenvalues at special values of l=l̃. A previously developed perturbation method is used to show that when l varies from l̃, the multiple eigenvalues split into their constituent simple eigenvalues, and into secondary bifurcation points on some of the primary states that branch from these simple eigenvalues. The solutions that branch from the secondary bifurcation points are the new magnetohydrodynamic equilibria. They are calculated near the secondary points, and for l near l̃, by the perturbation method.