Coronal loop oscillations: energy considerations and initial value problem

Context. Flares and eruptions in the solar corona generate oscillations of loops which have been interpreted as eigenmodes ( mainly the fundamental kink mode, although other modes can also be excited). From the theoretical point of view the excitation of the tube eigenmodes due to an initial disturbance has not been studied in much detail. Aims. The main aim of this work is to calculate for a given initial disturbance the amount of energy that is deposited in the trapped fast mode oscillation, how it depends on the initial perturbation and how it is distributed among the different eigenmodes ( kink and fluting and also the longitudinal harmonics). Methods. We calculate, using analytical expressions, the amplitude and the energy of the oscillation of the magnetic tube for different kinds of initial excitations. Results. We find that external excitations deposit a small amount of energy in the tube. We show that fluting modes have quite small energies in comparison with the energy of the kink mode ( around three orders of magnitude for the first fluting mode). On the contrary, the longitudinal fundamental mode and the longitudinal harmonics have energies of the same order of magnitude. In addition, we find that the loop length and density contrast can be important factors that determine the amount of energy that is trapped by the loop. Conclusions. The energy deposited in loops is typically six orders of magnitude smaller than the energy of the initial disturbance ( for external excitations). However, it strongly depends on the distance of the initial perturbation and also on the loop properties ( length and density). Fluting modes in coronal loops are very difficult to excite. Longitudinal harmonics are in principle more easily excited.