The theory of the production of a steady state current in the Alfvén resonance region is developed for an incompressible resistive plasma in axisymmetric toroidal geometry. A steady state current associated with externally excited Alfvén waves is produced by a nonlinear wave mixing process due to the nonlinearities of the medium. The steady state current is enhanced in the Alfvén resonance region, where the amplitudes of the waves have steep gradients. A net total current emerges at second order in an expansion that assumes a toroidal configuration with a large aspect ratio. Because the frequencies of the excited waves belong to the ideal MHD continuous spectrum, we have analysed the properties of the spatial singularities of the continuum eigenmodes. We have found that, as has been previously discovered for a compressible plasma, the singularities are of the essential type and become logarithmic only for special cases of the equilibrium, for instance, for an up/down symmetric configuration. We have shown that these types of singularities are recovered in the asymptotic limit of vanishing resistivity in the solutions of the resistive MHD equations.; As a prelude to the toroidal problem, we have investigated the production of a steady state current in the Alfvén resonance region in a Cartesian slab geometry. It is found that an inhomogeneous equilibrium magnetic field is required for a net current to arise. In the course of the analysis, we have demonstrated the existence of the shear Alfvén and cusp continua in a cylindrical screw pinch geometry. We have demonstrated the existence of the shear Alfvén continuum for a two dimensional equilibrium in Cartesian geometry in the case of a spatially symmetric density. We have also shown that the singular eigenfunctions belonging to the ideal MHD spectrum are recovered by taking the zero resistivity limit of the resistive eigenfunctions if no boundary conditions are imposed in a planar sheet pinch.
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