DOI: 10.1051/0004-6361:200810318

## Magnetohydrodynamic waves in a compressible magnetic flux tube with elliptical cross-section

**R. Erdélyi and R. J. Morton**

Solar Physics and Space Plasma Research Centre (SP

Received 3 June 2008 / Accepted 6 November 2008

** Abstract ***Aims. *The propagation of magnetohydrodynamic (MHD) waves in a
finite, compressible magnetic flux tube with an elliptical
cross-section embedded in a magnetic environment is
investigated.*Methods. *We present the derivation of the general dispersion
relation of linear magneto-acoustic wave propagation for a
compressible magnetic flux tube with elliptical cross-section in a
plasma with finite beta. The wave modes of propagation for the *n*=0
(symmetric) sausage and *n*=1 (anti-symmetric) kink oscillations are
then examined within the limit of the thin flux tube
approximation.*Results. *It is shown that a compressible magnetic tube with
elliptical cross-section supports slow and fast magneto-acoustic
waves. In the thin tube approximation, the slow sausage mode and the
slow and fast kink modes are found in analogue to a circular
cross-section. However, the kink modes propagate with different
phase speeds depending on whether the axial displacement takes place
along the major or minor axis of the ellipse. This feature is
present in both the slow and the fast bands, providing two infinite
sets of slow kink modes and two infinite sets of fast kink modes,
i.e. each corresponding cylindrical mode splits into two sets of
modes due to the ellipticity. The difference between the phase
speeds along the different axis is dependent on the ratio of the
lengths of the two axes. Analytical expressions for the phase speeds
are found. We show that the sausage modes do not split due to the
introduced ellipticity and only the phase speed is modified when
compared to the appropriate cylindrical counterpart. The percentage
difference between the periods of the circular and elliptical
cross-sections is also calculated, which reaches up to for
oscillations along the major axis. The level of difference in period
could be very important in magneto-seismological applications, when
observed periods are inverted into diagnostic properties (e.g.
magnetic field strength, gravitational scale height, tube expansion
parameter). Also shown is the perturbation of focal points of the
elliptical cross-section for different modes. It is found that the
focal points are unperturbed for the sausage mode, but are perturbed
for all higher modes.

**Key words:**magnetohydrodynamics (MHD)

**--**plasmas

**--**Sun: corona

**--**Sun: photosphere

**--**waves

**©**

*ESO 2009*