Wave heating of coronal loops driven by azimuthally polarised footpoint motions II. The time-dependent behaviour in ideal 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 directly. 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. In the companion paper (Paper I) dissipation is included and we looked at the steady state in which the energy injected at the photospheric part of the loop is balanced by the energy dissipated in the dissipative layer around the resonance. In this paper we make an analysis in time-dependent ideal linear MHD in order to get more physical insight in the results of Paper I and to get information about the time scales involved. In the present study the azimuthal wave number is taken to be non-zero so that Alfvén and fast wave do not exist independently. In this case the heating in the resonance layer becomes a complicated interplay of the influence of the Alfvén waves excited directly at the photospheric base of the resonant layer and the influence of the quasi-modes excited indirectly. We find that the presence of these quasi-modes influence the resonance development dramatically. This is unexpected, since in contrast to a sideways driven loop, a loop driven at the footpoints by azimuthally footpoint motions does not need quasi-modes as energy carrier waves.