Contribution of phase-mixing of Alfv\'en waves to coronal heating in multi-harmonic loop oscillations

Kink oscillations of a coronal loop are observed and studied in detail because they provide a unique probe into the structure of coronal loops through MHD seismology and a potential test of coronal heating through the phase-mixing of Alfv\'en waves. In particular, recent observations show that standing oscillations of loops often involve also higher harmonics, beside the fundamental mode. The damping of these kink oscillations is explained by mode coupling with Alfv\'en waves. We investigate the consequences for wave-based coronal heating of higher harmonics and what coronal heating observational signatures we may use to infer the presence of higher harmonic kink oscillations. We perform a set of non-ideal MHD simulations where the damping of the kink oscillation of a flux tube via mode coupling is modelled. Our MHD simulation parameters are based on the seismological inversion of an observation for which the first three harmonics are detected. We study the phase-mixing of Alfv\'en waves that leads to the deposition of heat in the system, and we apply the seismological inversion techniques to the MHD simulation output. We find that the heating due to phase-mixing of the Alfv\'en waves triggered by the damping of the kink oscillation is relatively small, however we can illustrate i) how the heating location drifts due to the subsequent damping of lower order harmonics. We also address the role of the higher order harmonics and the width of the boundary shell in the energy deposition. We conclude that the coronal heating due to phase-mixing seems not to provide enough energy to maintain the thermal structure of the solar corona even when multi-harmonics oscillations are included, and these oscillations play an inhibiting role in the development of smaller scale structures.