Direct excitation of resonant torsional Alfvén waves by footpoint motions

The present paper studies the heating of coronal loops by linear resonant Alfvén waves that are excited by the motions of the photospheric footpoints of the magnetic field lines. The analysis is restricted to torsionally polarised footpoint motions in an axially symmetric system so that only torsional Alfvénwaves are excited. For this subclass of footpointmotions, the Alfvén and cusp singularities are absent from the analysis which means that resonant coupling between global modes of the loop and localised oscillations is avoided. Instead, the focus is on the resonances due to the finite extent of the loop in the longitudinal direction: at the radii where Alfvén waves travelling back and forth along the length of the loop are in phase with the footpoint motions, the oscillations grow unbounded in ideal MHD. Inclusion of electrical resistivity and viscosity as dissipation mechanisms prevents singular growth and we can look at the steady state in which the energy injected at the photospheric part of the loop is balanced by the energy dissipated at the dissipative layer around the resonance. In this sense,we show that the direct excitation of Alfvén waves by torsionally polarised footpoint motions leads to a very efficient heating mechanism for coronal loops, even without resonant coupling to global modes.