Propagating kink waves in thin twisted magnetic tubes with continuous equilibrium magnetic field

In this paper, we study kink waves in twisted magnetic tubes. In the equilibrium state there is the electrical current with constant density inside the tube directed along the tube axis. This current creates the azimuthal magnetic field with the magnitude proportional to the distance from the tube axis inside the tube and inversely proportional to this distance outside the tube. We derive the dispersion equations for propagating waves and for unstable perturbations in the long wavelength approximation. We show that there are no solutions to the dispersion equation determining the frequencies of unstable perturbations, which implies that there are no unstable long kink modes. We study the dispersion equation for propagating waves both in the case when the plasma density is larger than that in the surrounding plasma as well as when it is smaller. In the first case we obtain that, depending on the wave number, the dispersion equation for propagating waves has either no solutions, or one solution, or two solutions. In the case when there is one solution, in the approximation of very weak twist, the wave mode propagates with the phase speed slightly larger than the kink speed. This wave mode is called the accelerated kink wave. In the case when there are two solutions to the dispersion equation, one of the two solutions gives the frequency of a quasi-mode that is subjected to the Alfvén resonance outside the tube. The other solution gives the frequency of a true eigenmode of linear ideal MHD. In the approximation of very weak twist its phase speed is smaller than the kink speed. This mode is called the decelerated kink wave. In the case of rarefied tube, depending on the wave number, the dispersion equation has either one or three solutions. When there is only one solution, the mode frequency is very close to the Alfvén frequency far from the tube, so the wave mode practically coincides with the Alfvén wave. When there are three solutions, the largest frequency practically coincides with the Alfvén frequency far from the tube. Two other solutions almost coincide. In all cases the wave modes existing in the case of rarefied tube are quasi-modes that are subjected to the Alfvén resonance. A possible application of the obtained results to the solar atmospheric seismology is discussed