In the vicinity of a magnetic island in tokamaks, the toroidal symmetry in the magnitude of the magnetic field |B| is broken. This leads to enhanced radial transport fluxes and momentum dissipation. The radial electric field can be determined from the quasineutrality condition, or equivalently the momentum equation, on the island magnetic surface. The equation that governs the radial electric field is nonlinear and can have bifurcated solutions. This may suppress turbulence fluctuations and improve plasma confinement. The theory remains valid for a rotating island with an appropriate re-interpretation of the radial electric field.