Previous works indicate that the frequency ratio of second and first harmonics of kink oscillations has tendency towards 3 in the case of prominence threads. We aim to study the magnetohydrodynamic oscillations of longitudinally inhomogeneous promine
nce threads and to shed light on the problem of frequency ratio. Classical Sturm--Liouville problem is used for the threads with longitudinally inhomogeneous plasma density. We show that the spatial variation of total pressure perturbations along the thread is governed by the stationary Schr{o}dinger equation, where the longitudinal inhomogeneity of plasma density stands for the potential energy. Consequently, the equation has bounded solutions in terms of Hermite polynomials. Boundary conditions at the thread surface lead to transcendental dispersion equation with Bessel functions. Thin flux tube approximation of the dispersion equation shows that the frequency of kink waves is proportional to the expression alpha(2n+1), where alpha is the density inhomogeneity parameter and n is the longitudinal mode number. Consequently, the ratio of the frequencies of second and first harmonics tends to 3 in prominence threads. Numerical solution of the dispersion equation shows that the ratio only slightly decreases for thicker tubes in the case of smaller longitudinal inhomogeneity of external density, therefore the thin flux tube limit is a good approximation for prominence oscillations. However, stronger longitudinal inhomogeneity of external density may lead to the significant shift of frequency ratio for wider tubes and therefore the thin tube approximation may fail. The tendency of frequency ratio of second and first harmonics towards 3 in prominence threads is explained by the analogy of the oscillations with quantum harmonic oscillator, where the density inhomogeneity of the threads plays a role of potential energy.
Solar wind plasma is supposed to be structured in magnetic flux tubes carried from the solar surface. Tangential velocity discontinuity near the boundaries of individual tubes may result in Kelvin-Helmholtz instability, which may contribute into the
solar wind turbulence. While the axial magnetic field may stabilize the instability, a small twist in the magnetic field may allow to sub-Alfvenic motions to be unstable. We aim to study the Kelvin-Helmholtz instability of twisted magnetic flux tube in the solar wind with different configurations of external magnetic field. We use magnetohydrodynamic equations in the cylindrical geometry and derive the dispersion equations governing the dynamics of twisted magnetic flux tube moving along its axis in the cases of untwisted and twisted external fields. Then we solve the dispersion equations analytically and numerically and found thresholds for Kelvin-Helmholtz instability in both cases of external field. Both analytical and numerical solutions show that the Kelvin-Helmholtz instability is suppressed in the twisted tube by external axial magnetic field for sub-Alfvenic motions. However, even small twist in the external magnetic field allows the Kelvin-Helmholtz instability to be developed for any sub-Alfvenic motions. The unstable harmonics correspond to vortices with high azimuthal mode numbers, which are carried by the flow. Twisted magnetic flux tubes can be unstable to Kelvin-Helmholtz instability when they move with small speed relative to main solar wind stream, then the Kelvin-Helmholtz vortices may significantly contribute into the solar wind turbulence.
