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Astron. Astrophys. 321, 935-944 (1997)

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1. Introduction

In spite of numerous theoretical investigations and observational campaigns no conclusive explanation has been found for the long-standing problem of the heating of the solar corona. There is a general consensus that the magnetic field plays a dominant role in this problem and heating by dissipation of magnetic waves, Alfvén waves in particular, is one of the strong proponents. The basic idea of the wave heating mechanism is still the same as in the late forties (Biermann 1946 , Schwarzschild 1948) enriched with a leading role of the magnetic field: magnetic waves which are generated by the (overshooting) turbulent convective motions in the photosphere transport the required energy through the chromosphere to the corona, guided by the magnetic field ; in the corona these waves then dissipate their energy in the magnetic coronal loops. In this paper, we concentrate on the most difficult part of the whole scenario: the dissipation of the wave energy in the magnetic loops.

The popularity of Alfvén wave heating is based on observational evidence, e.g. from the non-thermal broadening of transition and coronal spectrum lines (Zirker 1993) which is ascribed to Alfvén waves (Doschek et al. 1976 , Cheng et al. 1979 , O'Shea et al. 1996), and on theoretical investigations which showed that Alfvén waves can be dissipated efficiently under coronal conditions (Hollweg 1991 , Poedts et al. 1989a, 1989b, 1990a, 1990b). However, most theoretical studies of Alfvén wave heating are done in the framework of linearized MHD. In other words, usually it is assumed that the waves are nothing but small-amplitudes perturbations of a fixed background. This assumption may yield a good approximation in the bulk part of a coronal loop but it is challenged in the resonance layers themselves where the fields develop a nearly-singular behaviour, due to the very high magnetic Reynolds numbers in the hot coronal plasma. The high wave amplitudes in the resonant layers may induce nonlinear effects which may yield a different picture of the wave heating mechanism. The objectives of the present paper are

1. to show that the dynamics of the resonant layer is indeed very nonlinear, even for very small amplitudes of the incident waves;

2. to show that the resonant dissipation mechanism may be less efficient than concluded from linear theory due to nonlinear mode coupling and the variation of the background fields in time.

The resonant dissipation of the wave energy is simulated numerically. We here consider side-ways excitation as opposed to footpoint excitation, which has been studied in a previous paper (Poedts & Boynton 1996). In the next section, the physical model, i.e. the cylindrical coronal loop model and the nonlinear MHD equations, used for the present investigations is given and motivated. In order to appreciate the results of numerical simulations, some information on the applied discretisation techniques is required. This information is given in Sect.  3. The results of the numerical simulations of the time-dependent nonlinear MHD behaviour of an externally driven cylindrical coronal loop model are given in Sect.  4. In Sect.  5, the consequences of these nonlinear simulations for solar coronal loop heating are discussed.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998