5. Kinetic energy in the loop
This 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 contributions
Since 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
Each term in expression (11) represents the kinetic energy contribution E of each eigenmode .
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 frequency
In 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 and are taken to be 0.6 and 0.4 respectively. In a first approach we want to study the response to footpoint motions with the following time and x dependencies:
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.
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 () is dominated by the frequency a=2, which is even smaller than the smallest eigenfrequency .
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.
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 time
To analyse the time evolution of the kinetic energy, we look at two contributions E corresponding to the fundamental body mode () and the leaky mode () with k.
5.3.1. Pulsewise driving
In 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 a and a.
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 with the corresponding eigenfrequency as will be shown later in Sect. 6.
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 . 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 driving
In 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 a and a. In a second experiment we used a completely different series of time intervals which resulted in the Fig. 8a to 8d which radically differ from the previous figures. Nevertheless, after the driving has stopped, the kinetic energy again oscillates like with a constant amplitude.
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