Self-consistent equilibrium calculation through a direct variational technique in tokamak plasmas

A self-consistent equilibrium calculation, valid for arbitrary aspect ratio tokamaks, is obtained through a direct variational technique that reduces the equilibrium solution, in general obtained from the 2D Grad–Shafranov equation, to a 1D problem in the radial flux coordinateρ. The plasma current profile is supposed to have contributions of the diamagnetic, Pfirsch–Schlüter and the neoclassical ohmic and bootstrap currents. An iterative procedure is introduced into our code until the flux surface averaged toroidal current density 〈JT〉, converges to within a specified tolerance for a given pressure profile and prescribed boundary conditions. The convergence criterion is applied between the 〈JT〉 profile used to calculate the equilibrium through the variational procedure and the one that results from the equilibrium and given by the sum of all current components. The ohmic contribution is calculated from the neoclassical conductivity and from the self-consistently determined loop voltage in order to give the prescribed value of the total plasma current. The bootstrap current is estimated through the full matrix Hirshman–Sigmar model with the viscosity coefficients as proposed by Shaing, which are valid in all plasma collisionality regimes and arbitrary aspect ratios. The results of the self-consistent calculation are presented for the low aspect ratio tokamak Experimento Tokamak Esférico. A comparison among different models for the bootstrap current estimate is also performed and their possible limitations to the self-consistent calculation is analysed.