Jet Snake Magnetohydrodynamic Equilibria

Magnetohydrodynamic (MHD) equilibrium states with a three-dimensional helical core that display the characteristics of a saturated ideal internal kink mode are computed to model snake structures that have been observed in the JET tokamak (Weller et al 1987 Phys. Rev. Lett. 59 2303). The equilibrium states are calculated with a peaked pressure profile and a weak to moderate reversed core magnetic shear with a minimum safety factor qmin near unity in the neighbourhood of the mid-radius of the plasma. Snake equilibrium states are computed in the range 0.94 < qmin < 1.03. This range aligns with linearly unstable ideal MHD internal kink solutions of the purely axisymmetric branch of the equilibrium states. The energy difference between the bifurcated axisymmetric and helical snake equilibrium solutions is minimal. One very important novelty is that the helical structures are computed with an equilibrium code developed for three-dimensional (3D) stellarator applications in a tokamak context and cannot be obtained with standard Grad–Shafranov equation solvers. (Some figures in this article are in colour only in the electronic version) Snake structures have been reported in the JET tokamak with pellet injection [1] and also spontaneously due to central impurity accumulation [2]. The standard theoretical picture proposed for the formation of snakes is described by assuming that a pellet is ablated inside the q = 1 rational magnetic island (or that impurities accumulate at this same place in the spontaneously generated snakes) [3]. We provide an alternative perspective. The pellet ablates on-axis, causing the plasma to cool locally. This displaces the plasma current channel radially outwards producing a hollow current profile with the minimum value of q (the safety factor which corresponds to the inverse rotational transform) approaching unity off-axis. Under these conditions, the plasma becomes unstable to an ideal internal magnetohydrodynamic (MHD) kink mode which quickly saturates. The large density due to the pellet provides the allure of the snake. The spontaneous snake occurs because the impurities accumulate on axis that radiate cooling the plasma centre leading to similar conditions as with pellet injection. In this work, we demonstrate that three-dimensional (3D) snake equilibria exist in theory even with an imposed axisymmetric plasma boundary. It should be noted that it has been difficult to distinguish snake structures from multiple harmonic modes in JET experiments [4]. The application of stellarator equilibrium codes to tokamaks was pioneered by Garabedian where the NSTAB code obtained bifurcated solutions with local 3D structures near low-order rational surfaces that were interpreted as indicators of magnetic island formation [5]. The calculations we undertake are more global in character. We have previously applied the procedure to compute 3D helical equilibria to model TCV [6] and MAST [7] in which the sawteeth disappear, but continuous mode structures [8, 9] and long-lived modes [10] remain in which we prescribe the plasma mass or pressure profles that are very flat in the plasma core as reported in these experiments. Saturated internal kinks have been reported in NSTX [11] and sawteeth change from kink-like to quasiinterchange modes in DIII-D as the plasma boundary is varied from oval to bean-shaped [12]. These phenomena could also be potentially described with the model we propose. ITER hybrid scenario equilibria with similar profiles are also predicted to develop 3D internal structures [13]. The snake equilibria, on the other hand, have more peaked pressure profiles. The formulation of 3D MHD states is based on the minimization of the plasma energy W in which the magnetic flux surfaces are constrained to be nested with a single 0029-5515/11/072002+06$33.00 1 © 2011 IAEA, Vienna Printed in the UK & the USA Nucl. Fusion 51 (2011) 072002 Letter magnetic axis. The static conditions can be described by W = ∫ ∫ ∫ dx ( B2 2μ0 + p‖(s, B) − 1 )