Vertical displacements and MHD marginal stability in fusion tokamak plasmas

This vertical stability analysis is carried out on a tokamak plasma for nuclear fusion, where plasma is confined through magnetic flux surfaces that are analytically described as constant u surfaces. The first magnetic surface where plasma is contained is the elliptical surface u = u_b, which is then surrounded by a vacuum region, where a magnetic separatrix, u = u_X, is present. This latter is a magnetic flux surface with two magnetic X-points produced by additional external currents (divertor tokamak configuration), that also cause a vertical elongation of the confined plasma. Everything is then confined inside a toroidal containment chamber, that in our case study will be assumed to stand sufficiently far away from the plasma’s boundary, so that it does not interfere with plasma stability. Work[Yolbarsop_2022_Plasma_Phys._Control._Fusion_64_105002] analyzed vertical stability for the elongated confined plasma with an ideal-MHD model, where perturbations are guided by a current sheet forming on the plasma’s boundary and grow with very fast time-scales. An unstable behavior for plasma localized inside the first magnetic surface, u = u_b, is found, where perturbation grows in time as e^(gamma*t), gamma as the growth rate. When plasma is instead extended up to the magnetic separatrix, the effect of the current sheet at the plasma’s boundary changes from destabilizing to stabilizing. This transition in plasma’s stability leads us to search a flux surface u = u_marg, where marginal stability occurs. The solution to our problem is carried out through the ideal-MHD energy principle. The stability problem is reduced to the solution of an eigenvalue problem, where gamma^2 is the eigenvalue to be found and its sign determines whether our system is stable or not. Since gamma^2 is a real number, consequently a positive value determines an unstable behavior of the system, whereas if negative, gamma becomes purely imaginary with null real part, therefore producing stability with an oscillatory behavior in time. The problem of finding the marginal stability surface is thus reduced to search the flux surface u = u_marg for which gamma^2 is zero. The normal mode formulation, that has been used for the problem, can be used to define the correlation between the eigenvalue gamma^2 and the perturbed potential energy δW, as gamma^2 = δW( ̃ψ∗, ̃ψ)/K( ̃ψ∗, ̃ψ), where ̃ψ is the perturbed flux solution and also the eigenfunction of our eigenvalue problem. K( ̃ψ∗, ̃ψ) is instead the perturbed kinetic energy. Deriving the evolution of the perturbed potential energy δW as function of u_c, starting from u = u_b up to u = u_X, allow us to locate the marginal stability surface, u = u_marg. Unfortunately, this formulation brings to the resolution of a ill-conditioned system, causing the obtained results to have a level of accuracy decreasing with |u| = |u_c|; making it necessary to deal with extrapolation methods in the last part of this work in order to obtain the all δW(u_c ) curve to find the marginal stability flux surface.