Anisotropic E × B shearing rate in a magnetic island

We derive the E × B shearing rate associated with vortex flow inside a macroscopic magnetic island (MI) in axisymmetric toroidal geometry. Due to the elongation of the MI and incompressibility of the E × B flow, the shearing rate near X-points is much lower than that near the mid-plane (x-axis of the local Cartesian coordinate) of the MI on the same flux surface. Furthermore, the rate formally vanishes at the X-points where the local poloidal magnetic field associated with the MI stagnates. This calculation of E × B shearing profile and, in particular, minimal E × B shear near the X-points is consistent with the recent experimental finding that turbulence tends to spread into an MI through regions around the X-points [K. Ida et al., Phys. Rev. Lett. 120, 245001 (2018)] and can contribute to more thorough quantitative interpretation of the results from experiments and simulations.