Electric field and flows at the edge of toroidal fusion plasmas

A series of issues in plasma physics, such as the presence of the Greenwald limit in Tokamaks and reversed-field pinches (RFP) [ 1 ], the restriction of the collisionality window for the application of the resonant magnetic perturbations (RMPs) in Tokamaks (in particular linked to the role of X-points of edge islands, e.g. in TEXTOR [ 2 ]), and again the presence of edge islands interacting with the bootstrap current in stellarators [ 3 ], seem to share the same physical mechanism, namely, 3D flows generated by edge islands interacting with the wall and embedded in a chaotic magnetic field. Detailed measurements in the RFX-mod device ( a = 0 . 46 m, R = 2 m) show that the 2D map of the edge electric field response to the magnetic perturbation, E r ( r = a,?,? ), exhibits a ripple consistent with the helicity of the perturbation. In fact, using an appropriate helical angle u = m? - n? + ? m,n , maps of E r show a sinusoidal dependence, E r = ? E r sin u , or equivalently, E r is constant on helical flux surfaces [ 4 ]. The ripple of E r is associated with a huge convective cell with v ( a ) 6 = 0 which, in RFX and near the Greenwald limit, determines a stagnation point for density and a reversal of the sign of E r [ 1 ]. Similar reversal of E r at high collisionality is found also in Alcator C-mod [ 5 ]. From a theoretical point of view, the problem is how a perturbed toroidal flux of symmetry m/n the form ? = ? 0 + ? ? sin u gives rise to an ambipolar potential ? = ? ?sin u . On the basis of a model developed with the guiding center code Orbit [ 6 ] and applied to RFX and TEXTOR, we will show that, without perturbations, the canonical toroidal momentum given by the guiding-center Lagrangian (Boozer coordinates) is conserved, ? P ? = 0. The presence of a m/n perturbation in any kind of device breakes the simmetry, ? P ? ? ?B 2 cos u , and the resulting drift is proportional to the gyroradius ? , so it is larger for ions ( ? i ? e ). Immediately an ambipolar potential arises to balance the drifts, and will possess the same symmetry as the original perturbation . The residual dependence on ? i/e makes the potential extremely sensitive to particle energy, ratio T e /T i , and collisions, which seems to fit well- established results in RFPs, Tokamaks and Stellarators.