Transverse oscillations of longitudinally stratified coronal loops with variable cross-section

We consider transverse oscillations of coronal loops that have both variable circular cross-sectional area and plasma density in the longitudinal direction. The primary focus of this paper is to study the eigenmodes of these oscillations. Implementing the method of asymptotic expansions with the ratio of the loop radius to length as a small parameter, a second-order ordinary differential equation is derived describing the displacement of the loop axis. Together with the boundary conditions at the tube ends that follow from the frozen-in condition, this equation constitutes the Sturm-Liouville problem determining the eigenfrequencies and eigenmodes. Our results are relevant to the magnetoseismological method of estimating the coronal density scale height by using the observed ratio of the fundamental frequency and first overtone of loop kink oscillations. It is shown that this method is very sensitive to the tube expansion factor, which is the ratio of the tube radii at the apex and footpoints. The estimated scale height is a monotonically decreasing function of the expansion factor.