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Astron. Astrophys. 335, 329-340 (1998)

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1. Introduction

Since Skylab it is known that the extremely hot corona (roughly 3 K) is a complex medium, structured by the magnetic field in myriads of dynamic coronal loops. Moreover the X-ray images show that these magnetic loops have the largest heating requirements.

The high conductivity and the relatively high mass density of the photospheric plasma provide an effective photospheric anchoring of the coronal magnetic field lines. The photospheric footpoints of the magnetic field lines are forced to follow the convective motions. If these footpoint motions are 'slow' (in comparison with the Alfvénic transit time along the loop), the coronal flux tubes are slowly twisted and braided. The magnetic stresses, which are built up that way, and the small length scales in between fieldlines of different polarity, lead to magnetic reconnection and hence to a conversion of magnetic energy into heat (Parker 1972).

In contrast, footpoint motions which are 'fast' in comparison with the Alfvénic transit time, generate magnetosonic waves and Alfvén waves. Due to the steep density gradients at the photospheric edges these MHD waves are reflected back and forth along the length of the loop. The loop is then expected to act as a leaking, resonant cavity for MHD waves (Hollweg 1984), in which dissipation is enhanced by means of turbulence (Gomez 1990), resonant absorption (Goossens 1991) and/or phase mixing (Heyvaerts & Priest 1983).

Plasma heating by resonant absorption of Alfvén waves hinges on the existence of a continuous part of the Alfvén spectrum in linear ideal MHD. Tataronis & Grossmann (1975) were the first to give a basic theory on plasma heating by these Alfvén continuum waves. Chen & Hasegawa (1974) and Kappraff & Tataronis (1977) elaborated on Alfvén continuum waves as effective heaters of fusion plasmas. Shortly after, Ionson (1978) suggested that resonant absorption might effectively heat the solar corona.

A lot of work, both analytically and numerically, was done on sideways excitation of these resonant Alfvén waves where a wave impinges laterally on the loop (Poedts et al. 1989; Poedts & Kerner 1992; Steinolfson & Davila 1993; Ofman & Davila 1995, 1996; Wright & Rickard 1995). When a loop is perturbed by a broad band spectrum on its side surface, it will respond at a discrete set of frequencies of fast waves which may resonantly excite Alfvén waves in turn. Hence the plasma-driver coupling was found to be very efficient due to the effect of the present quasi-modes.

However it is important to see that sideways excitation by an externally impinging fast wave can only yield a minor contribution to the heating of a coronal loop by resonant absorption. Due to the enhanced density the interior Alfvén speed must be smaller than the exterior Alfvén speed. Therefore only fast waves which are exponentially decaying on their way to the loop can resonantly excite Alfvén waves inside the loop. This suggests that fast waves originating from within the loop must be the prime contribution. Such fast wave can be excited by e.g. a reconnection event inside the loop or by the photospheric motions of the footpoints of the magnetic field lines.

Recently, more attention is paid to the fact that coronal loops are finite and bounded by the photospheric plasma. Strauss & Lawson (1989) considered MHD simulations of resonant absorption in an incompressible cylindrical plasma that is excited at his footpoints. This is one of the first papers in which the consequences of line-tying is discussed. Later on, it was shown analytically that line-tying completely changes the character of the basic MHD waves occuring in a coronal loop as compared to those in a periodic system. MHD waves of mixed nature occur: the waves consist of large amplitude Alfvén components in the corona and fast components with a small but rapidly varying amplitude in the photospheric boundary layers (Goedbloed & Halberstadt 1994). The subsequent paper by Halberstadt & Goedbloed (1995) discusses the coupling in the stationary state between fast and Alfvén waves in a cylinder with helical magnetic field for which the footpoint excitation has both an Alfvén wave polarization and a fast wave polarization.

Berghmans & De Bruyne (1995) and Ruderman et al. (1997) studied the direct excitation of resonant Alfvén waves by azimuthally polarized footpoint motions in ideal and dissipative MHD respectively. Berghmans & Tirry (1997) and Tirry & Berghmans (1997) showed how the results can be drastically changed when the coupling to fast waves is taken into account. On the other hand Berghmans, De Bruyne & Goossens (1996) investigated the temporal behaviour of the sausage fast wave excited by radially polarized footpoint motions, whereas Tirry, Berghmans & Goossens (1997) revealed the importance of the quasi-mode in this indirect driving of resonant Alfvén waves.

So far, most of the analyses focussed on a periodic driving or a driving by one pulse. In the present paper we want to figure out how the results alter for randomly driven footpoint motions. We start to consider a loop that is driven by a train of pulses with random time-intervals in between and to look at the behaviour of the fast waves within the loop without coupling to Alfvén waves.

The paper is organized as follows. In the next section, the physical model with the relevant equations and boundary conditions is discussed. In Sect. 3 we repeat the important steps in the derivation of the analytical solution, which describes the temporal behaviour of the excited MHD waves, as given by Tirry, Berghmans & Goossens (1997). The solution is written as a superposition of eigenmodes. In Sect. 4 we classify these eigenmodes as leaky or body modes, whereas in Sect. 5 we look how the input energy is spread over the different modes. Sect. 6 is devoted to obtain analytically a relationship between the mean value of the kinetic energy contribution of each eigenmode and the eigenfrequency. In Sect. 7 the results are discussed. Finally in Sect. 8 we give a summary.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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