Astron. Astrophys. 325, 318-328
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Wave heating of coronal loops
driven by azimuthally
polarised footpoint motions
I. Stationary behaviour in dissipative MHD
D. Berghmans* and
Centre for Plasma Astrophysics, K.U. Leuven,
Celestijnenlaan 200 B, B-3001 Heverlee, Belgium
Received 17 January 1997 / Accepted 10 March 1997
We study the heating of coronal loops by linear resonant Alfvén waves
that are excited by photospheric footpoint motions of the
magnetic field lines. The analysis is restricted to azimuthally polarised
footpoint motions so that Alfvén waves are excited.
At the radii where
Alfvén waves, travelling back and forth along the loop,
are in phase with the footpoint motions the oscillations grow
unbounded in ideal MHD.
Inclusion of dissipation prevents singular growth and we can look at the
state in which the energy input at the photosphere
is balanced by the energy dissipated at the resonance.
The crux of our study is that the azimuthal wave number
is taken non-zero
which means that also fast waves, including quasi-modes,
can be excited by the purely azimuthally polarised footpoint motions.
In this case resonant Alfvén waves are not only excited
directly by the footpoint motions but also indirectly through coupling to
For some footpoint motions these contributions
counteract each other leading to virtually no heating (anti-resonance) while
for values corresponding to a quasi-mode the two contributions
act in concert leading to enhanced heating.
This dramatic influence of the quasi-mode is unexpected since,
in contrast to a sideways driven
loop, a loop driven at the footpoints by azimuthally polarised
footpoint motions does not need quasi-modes as energy carrier waves.
In this paper a stationary state is assumed
which allows us to determine the optimal footpoint
characteristics for heating, but does not give any
information on the time scales involved. This item is adressed
in a companion paper where an identical system is studied
time dependently using ideal MHD. This twin study
investigates the attainability of the heating scenarios and provides
an enhanced insight in the results of the present paper.
Key words: MHD - Sun: corona; oscillations - waves - methods:
*Research Assistant of the
Send offprint requests to: D. Berghmans
© European Southern Observatory (ESO) 1997