## 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 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 In addition, it is also relevant to look at the contribution of the kinetic energy averaged in time: where ## 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 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
## 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
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 () is dominated by the frequency a=2, which is even smaller than the smallest eigenfrequency . ## 5.2.2.The results corresponding to the driving pulse of duration are pictured in Fig. 3. If we compare these figures with Fig. 2, we notice a few differences.
First of all, the total energy in the loop is much larger than in
the first case ( Secondly, for the same reason, modes with eigenfrequency around
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
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 ## 5.3.1. Pulsewise drivingIn a first step we again drive the loop with the pulse of the form
(13). For the body mode the time evolution of
the energy excited by the driving pulse is plotted on Figs. 5a and 5b
respectively for the cases
If When the driving pulse lasts only for approximately 0.26 time units
(
Figs. 5c and 5d show the time evolution of the kinetic energy contribution of the leaky mode with . Just like in the case of the body mode, the kinetic energy is influenced by the footpoint motion in the beginning and afterwards the energy oscillates like . Note once more that the amount of energy stored in this leaky mode is much smaller than the energy of the body mode. 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 and
in the cases
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 |