Context. Previous works indicate that the frequency ratio of second and first harmonics of kink oscillations tends towards 3 in the case of prominence threads. This is not a straightforward result, so it requires adequate explanation. Aims. We aim to study the magnetohydrodynamic oscillations of longitudinally inhomogeneous prominence threads and to shed light on the problem of the frequency ratio. Methods. The 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 Schrödinger equation, where the longitudinal inhomogeneity of plasma density stands for the potential energy. The Schrödinger equation appears as the equation of quantum harmonic oscillator for a parabolic profile of plasma density. Consequently, the equation has bounded solutions in terms of Hermite polynomials. Results. Boundary conditions at the thread surface lead to a transcendental dispersion equation with Bessel functions. The thin flux tube approximation of the dispersion equation shows that the frequency of kink waves is proportional to the expression α(2n + 1), where α is the density inhomogeneity parameter, and n the longitudinal mode number. Consequently, the ratio of the frequencies of second and first harmonics tends to 3 in prominence threads. The numerical solution of the dispersion equation shows that the ratio decreases only slightly for thicker tubes in the case of less longitudinal inhomogeneity of the 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 in the frequency ratio for wider tubes, and therefore the thin tube approximation may fail. Conclusions. 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 as the potential energy.

Context. It is known that hydrodynamic triangular jets (i.e. the jet with maximal velocity at its axis, which linearly decreases at both sides) are unstable to anti-symmetric kink perturbations. The inclusion of the magnetic field may lead to the stabilisation of the jets. Jets and complex magnetic fields are ubiquitous in the solar atmosphere, which suggests the possibility of the kink instability in certain cases. Aims. The aim of the paper is to study the kink instability of triangular jets sandwiched between magnetic tubes (or slabs) and its possible connection to observed properties of the jets in the solar atmosphere. Methods. A dispersion equation governing the kink perturbations is obtained through matching of analytical solutions at the jet boundaries. The equation is solved analytically and numerically for different parameters of jets and surrounding plasma. The analytical solution is accompanied by a numerical simulation of fully non-linear magnetohydrodynamic (MHD) equations for a particular situation of solar type II spicules. Results. Magnetohydrodynamic triangular jets are unstable to the dynamic kink instability depending on the Alfvén Mach number (the ratio of flow to Alfvén speeds) and the ratio of internal and external densities. When the jet has the same density as the surrounding plasma, only super-Alfvénic flows are unstable. However, denser jets are also unstable in a sub-Alfvénic regime. Jets with an angle to the ambient magnetic field have much lower thresholds of instability than field-aligned flows. Growth times of the kink instability are estimated to be 6−15 min for type I spicules and 5−60 s for type II spicules matching with their observed lifetimes. The numerical simulation of full non-linear equations shows that the transverse kink pulse locally destroys the jet in less than a minute in type II spicule conditions. Conclusions. Dynamic kink instability may lead to the full breakdown of MHD flows and consequently to an observed disappearance of spicules.