*Astron. Astrophys. 321, 935-944 (1997)*
## 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.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998
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