Non-linear equations for the evolution of the amplitude and rotation frequency of low poloidal mode number magnetic islands interacting with external fields are derived within a two fluid magneto-hydrodynamic description. The inhomogeneous boundary value problem for the perturbed magnetic flux function is solved numerically and analytically without 'tenuous plasma' approximations, explaining the screening effect on the penetrated field driven by external coils. The island rotation is governed by the toroidal momentum balance equation which includes the electrodynamical forces due to the resonant external fields and viscous damping. It is shown that, in addition to perpendicular viscosity, a neoclassical parallel viscous damping force, important for large islands, arises from the broken axisymmetry of the magnetic field. The effects of viscosity on the non-linear evolution of the width of a rotating island are found to be different from previous results. Implications for the problems of feedback control of rotating islands are discussed.
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