Behavior of perturbed plasma displacement near regular and singular X -points for compressible ideal magnetohydrodynamic stability analysis

Abstract

The ideal magnetohydrodynamic (MHD) stability analysis of axisymmetric plasma equilibria is simplified if magnetic coordinates, such as Boozer coordinates (ψT radial, i.e., toroidal flux divided by 2π, θ poloidal angle, ϕ toroidal angle, with Jacobian g∝1∕B2), are used. The perturbed plasma displacement ξ⃗ is Fourier expanded in the poloidal angle, and the normal-mode equation δWp(ξ⃗*,ξ⃗)=ω2δWk(ξ⃗*,ξ⃗) (where δWp and δWk are the perturbed potential and kinetic plasma energies and ω2 is the eigenvalue) is solved through a 1D radial finite-element method. All magnetic coordinates are however plagued by divergent metric coefficients, if magnetic separatrices exist within (or at the boundary of) the plasma. The ideal MHD stability of plasma equilibria in the presence of magnetic separatrices is therefore a disputed problem. We consider the most general case of a simply connected axisymmetric plasma, which embeds an internal magnetic separatrix—ψT=ψTX, with rotational transform ι̷(ψTX)=0 and regular X-points (B⃗≠0)—and is bounded by a second magnetic separatrix at the edge—ψT=ψTmax, with ι̷(ψTmax)≠0—that includes a part of the symmetry axis (R=0) and is limited by two singular X-points (B⃗=0). At the embedded separatrix, the ideal MHD stability analysis requires the continuity of the normal plasma perturbed displacement variable, ξψ=ξ⃗∙∇⃗ψT; the other displacement variables, the binormal ηψ=ξ⃗∙(∇⃗θ−ι̷∇⃗ϕ) and the parallel μ=−gξ⃗∙∇⃗ϕ, can instead be discontinuous everywhere. The permissible asymptotic limits of (ξψ,ηψ,μ) are calculated for the unstable (ω2<0) eigenvectors, imposing the regularity of δWp, δWk, and ξ⃗ at the embedded separatrix and at the edge separatrix. An intensified numerical radial mesh following Boozer magnetic coordinates is set up; it requires a logarithmic fit to the rotational transform near the embedded magnetic separatrix, a minimum distance between the radial mesh and both separatrices, and finally an extended spectrum of poloidal mode numbers in the Boozer angle. The numerical results are compared “a posteriori” with the permissible asymptotic limits for the perturbed displacement: the radial displacement variable ξψ is found to be always near its most unstable asymptotic limit, while the full range of permissible asymptotic behaviors can be obtained for the binormal and the parallel displacement variables.