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Astron. Astrophys. 335, 329-340 (1998) 5. Kinetic energy in the loopThis section is divided in three subsections. In the first one we derive an analytical expression for the kinetic energy in the coronal loop. Then, in the second subsection, we study the kinetic energy as function of the frequency to analyse the amount of energy in both body and leaky modes. In the third subsection we focus on the kinetic energy of the modes as function of time. 5.1. Kinetic energy contributionsSince the footpoint motions are assumed to be purely radially polarized with k, only the x-component is excited, so that the total kinetic energy inside the loop is given by Since and we can rewrite the expression for the total kinetic energy simply as where Each term in expression (11) represents the kinetic energy
contribution E of each eigenmode In addition, it is also relevant to look at the contribution of the kinetic energy averaged in time: where P is a period sufficiently longer than any period characteristic to the system or to the footpoint motion. 5.2. Kinetic energy as function of frequencyIn this subsection we study how the energy is spread over body and leaky modes. As an instructive example, rather than a realistic model, we look
at a loop with dimensions L and b. In this case the
characteristics manifest themselves the clearest. The density
parameters which is chosen in order to simulate an instant 'kick' at the loop's feet. As the duration of the driving pulse is determined by the value of a, the dominant frequency in the power spectrum of the pulse is given by a. In what follows we look at the kinetic energy distribution for three different values of a. 5.2.1.
In Fig. 2 we depict the time-averaged kinetic energy contributions (up to a constant factor) of the body modes and leaky modes as a function of the corresponding eigenfrequency. Each spike in these figures corresponds to a dot in Fig. 1 representing a body or leaky mode. As clearly seen, a lot more energy is stored in the excited body modes than in the leaky modes. In consequence, there is more energy stored in the coronal loop than there is radiated away in the coronal environment. In the case where k such that the body modes couple to the Alfvén waves, this means that a good base is formed for resonant absorption as dissipation mechanism.
As a second point we can remark that predominantly small
frequencies are excited. We could have expected this result since the
power spectrum of the driving pulse ( 5.2.2.
The results corresponding to the driving pulse of duration
First of all, the total energy in the loop is much larger than in the first case (a). The reason is that the dominant frequency in the power spectrum of the pulse is higher and hence more kinetic energy is put into the loop. Secondly, for the same reason, modes with eigenfrequency around a are most efficiently excited. Beside these differences, there is also an important similarity. Again most of the energy is stored in the body modes and as a consequence there is not that much energy lost in the coronal environment of the loop. 5.2.3.
Driving the system with a pulse with a results in Fig. 4. The total kinetic energy again increases and the peak of kinetic energy moves to higher frequencies. Again the by far largest contribution of energy comes from the body modes whereas the leaky modes are almost not excited.
Thus again a good base for resonant absorption as dissipation mechanism is formed. 5.3. Kinetic energy as function of timeTo analyse the time evolution of the kinetic energy, we look at two
contributions E corresponding to the fundamental body mode
( 5.3.1. Pulsewise drivingIn a first step we again drive the loop with the pulse of the form
(13). For the body mode
If a the driving pulse lasts for approximately 0.5
dimensionless time units and after that time there is no more external
driving. We notice that after that point the kinetic energy on Fig. 5a
shows a regular oscillation. After the driving has stopped the energy
of the eigenmode oscillates like When the driving pulse lasts only for approximately 0.26 time units (a), the excited energy is a lot smaller (see Fig. 5b). This is a consequence of the fact that the previous driving pulse, with a, approximates the considered eigenoscillation better than the driving pulse with a, as one can see in Fig. 6.
Figs. 5c and 5d show the time evolution of the kinetic energy
contribution of the leaky mode with Finally, as a check, we can compare the time evolution of the kinetic energy (Fig. 5) with the time-averaged kinetic energy as function of the eigenfrequencies (Figs. 3 and 4): the amplitudes of the final oscillations in Fig. 5. are in agreement with the amplitudes of the spikes in Figs. 3 and 4 corresponding to the considered eigenfrequencies. 5.3.2. Stochastic drivingIn a next step we drive the system by a succession of identical pulses with random time intervals in between. We take the single pulses equal to the model pulse used in Sect. 5.2. In what follows we consider a driving with 10 pulses. One can compare this situation with a succession of 10 situations equal to the one in Sect. 5.2. The first pulse generates an oscillation of the eigenmodes. But after the random time interval again the same pulse is given. Dependent on the time in between these two pulses, the second pulse will be in phase or rather out of phase with the generated oscillation. So the amplitude will be increased or decreased respectively under influence of the footpoint motion. In this way each of the 9 subsequent pulses in turn will increase or decrease the kinetic energy. The final result can take various forms. In Fig. 7a to 7d we
plotted the time evolution of the kinetic energy contribution
corresponding with the frequencies
We are interested in the mean value of this final amplitude since it determines the amount of energy which can be stored in the coronal loop (in case of a body mode). By imposing a well defined probability distribution function for the time intervals in between the pulses, we analytically derive an expression for the mean value of the final amplitude of the kinetic energy contribution as function of the corresponding frequency and the distribution parameters. ![]() ![]() ![]() ![]() © European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 ![]() |