Wave heating of coronal loops driven by azimuthally polarised footpoint motions I. Stationary behaviour in dissipative MHD

We study the heating of coronal loops by linear resonant Alfvén waves that are excited by photospheric footpoint motions of the magnetic field lines. The analysis is restricted to azimuthally polarised footpoint motions so that Alfvén waves are excited.At the radii whereAlfvénwaves, travelling back and forth along the loop, are in phase with the footpoint motions the oscillations grow unbounded in ideal MHD. Inclusion of dissipation prevents singular growth and we can look at the steady state in which the energy input at the photosphere is balanced by the energy dissipated at the resonance. The crux of our study is that the azimuthal wave number is taken non-zero which means that also fast waves, including quasi-modes, can be excited by the purely azimuthally polarised footpoint motions. In this case resonant Alfvén waves are not only excited directly by the footpointmotions but also indirectly through coupling to fast waves. For some footpoint motions these contributions counteract each other leading to virtually no heating (anti-resonance) while for values corresponding to a quasi-mode the two contributions act in concert leading to enhanced heating. This dramatic influence of the quasi-mode is unexpected since, in contrast to a sideways driven loop, a loop driven at the footpoints by azimuthally polarised footpoint motions does not need quasi-modes as energy carrier waves. In this paper a stationary state is assumed which allows us to determine the optimal footpoint characteristics for heating, but does not give any information on the time scales involved. This item is adressed in a companion paper where an identical system is studied time dependently using ideal MHD. This twin study investigates the attainability of the heating scenarios and provides an enhanced insight in the results of the present paper.