Temporal evolution of resonant absorption in coronal loops Excitation by footpoint motions normal to the magnetic surfaces

In this paper we study the temporal evolution of linear MHD waves excited by footpoint motions using an ideal, pressureless slab model for coronal loops. We choose the footpoint motions to be polarised normal to the magnetic flux surfaces such that only fast waves are driven directly, including the so-called quasi-modes. We have derived a formal analytical solution as a superposition of eigenmodes describing the system as a function of time. The corresponding eigenvalue problem is solved numerically. This enables us to study the influence of the characteristics of the footpoint motion on the excitation of the quasi-modes. On the magnetic flux surface where the frequency of these quasi-modes equals the local Alfvén frequency, wave energy is transferred from thequasi-modes towardsAlfvén waves.We investigate the time evolution of this process inwhich small scale dissipative features are generated which can be relevant in the context of coronal heating. Special attention is given to the question whether this generation of small scale dissipative features takes place on time scales shorter than typical life times of coronal loops. Expressing the dissipation time scale as function of the length scale corresponding to the resonances, an estimate for the time when dissipation becomes important and when our ideal MHD simulation stops to be valid, can be derived. For typical dissipation coefficients and length scales, dissipation becomes important in the resonance layer in a time comparable to the life time of coronal loops.