Kink waves in twisted and expanding magnetic tubes

We study kink and fluting waves in expanding and twisted magnetic flux tubes. We use the thin-tube and zero-beta plasma approximations. The equilibrium magnetic field is force free with a constant proportionality coefficient between the electrical current and the magnetic field. We derive the equation governing the kink and fluting waves in a tube. Using this equation we study the propagation of kink waves in a particular case of a magnetic tube homogeneous in the axial direction. We show that while there is only one propagating kink wave with the phase speed equal to the kink speed in an untwisted tube, in a twisted tube there are two wave modes, accelerated and decelerated. The phase speed of the accelerated wave exceeds the kink speed, while the phase speed of the decelerated wave is less than the kink speed. We also show that the standing modes are defined by the same eigenvalue problem as that in the case of an untwisted tube. Hence, the frequencies of the standing-wave modes are not affected by the twist. This implies that the seismological results based on the observation of the standing-wave mode frequencies remain valid when the twist is taken into account. The only effect of twist is the variation of the direction of polarisation of the coronal magnetic-loop displacement along the loop. As a result, an apparent node can be detected near the loop apex if only one component of the loop displacement is observed. This can lead to an incorrect conclusion that the observed coronal loop kink oscillation was the first overtone, while in fact it was the fundamental mode.