Standing kink oscillations of thin twisted magnetic tubes with continuous equilibrium magnetic field

In this article we study standing kink waves in twisted magnetic tubes. We use the cold plasma and thin tube approximation. We assume that the plasma density is constant inside and outside the tube. We also assume that the magnetic twist is weak and take the ratio of the azimuthal and axial component of the magnetic field to be of the order of ratio of the tube radius and tube length. The azimuthal component of the magnetic field is proportional to the distance from the tube axis inside the tube, and inversely proportional to this distance outside the tube. Using the method of asymptotic expansions we derived the governing integral equation that determines the eigenfrequencies and eigenmodes of the tube kink oscillations. In the approximation of a very weak twist, we calculated analytically the corrections to the frequencies of the fundamental mode and first overtone of a straight magnetic tube related to the presence of twist. The analytical results are compared with the numerical results obtained using the full set of linear ideal magnetohydrodynamic equations. We also calculated the ratio of frequencies of the fist overtone and fundamental mode. We found that the magnetic twist enhances this ratio for moderate values of the density ratio, and reduces this ratio for large values of the density ratio. In general, the deviation of the frequency ratio from 2 caused by the magnetic twist is comparable to that found in simultaneous observations of the fundamental mode and first overtone of the coronal loop kink oscillations. Finally, we studied the eigenmode polarization. We found that, in a particular case of linear polarization, the polarization direction rotates along the tube