In this paper we studied the excitation of fast MHD waves in a coronal loop driven by radially polarized footpoint motions. The main topic was to extend previous analyses, in which the driving was assumed to be harmonic, to a stochastic driving. As a first step we assumed the loop to be driven by a train of identical pulses with random time intervals in between. The solution is written as a superposition of eigenmodes, which can be classified to be either 'leaky' or 'body' modes. The leaky modes radiate their energy away from the loop, whereas the body modes, which are trapped in the loop cavity, may resonantly couple to Alfvén waves and in this way dissipate their energy.
Firstly, by analysing the response to a kick at the loop's feet, we found that most of the input energy is stored in the body modes. Hence driving at the loop's feet forms a good base for resonant absorption as heating mechanism.
Secondly, by imposing the time intervals between the pulses to satisfy a gamma distribution, we analytically derived a relation between the mean value of the kinetic energy contribution of each eigenmode and the corresponding eigenfrequency. The remarkable peaked structure, obtained in the case of a very peaked gamma distribution, reflects very well the power spectrum of the footpoint driving.
Also the output of the several numerical simulations were in agreement with the predicted analytical result for the final amplitude after the driving with a train of pulses.
By considering gamma distributions with greater variance, we showed how the peak structure disappears in the variation of the mean value as function of the frequency. In order to determine the efficiency of resonant absorption in case of this stochastic driving at the loop's feet, it is very important to know where the eigenfrequencies of the body modes are located in comparison with the peaks in a figure like Fig. 9. Hence parametric studies for loops with different 'realistic' dimensions with 'realistic' parameters for the gamma distribution (and others) should be done. On the other hand, it is also important to find out how efficient the energy is transferred from the body modes to the resonant Alfvén waves under the stochastic footpoint driving. This implies to study the case with k which is the main topic of De Groof et al. (1998).
© European Southern Observatory (ESO) 1998
Online publication: June 12, 1998