Fast magnetohydrodynamic oscillation of longitudinally inhomogeneous prominence threads: an analogue with quantum harmonic oscillator
arXiv:1403.6027 · doi:10.1051/0004-6361/201323225
Abstract
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 prominence 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 Schrödinger 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 α(2n+1), where αis 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.
arXiv comment8 pages, 7 figures (accepted in A&A)
References in corpus (9)
- Physics of Solar Prominences: II - Magnetic Structure and Dynamics
- Physics of Solar Prominences: I - Spectral Diagnostics and Non-LTE Modelling
- Swaying threads of a solar filament
- Transverse oscillations of flowing prominence threads observed with Hinode
- Observation of multiple sausage oscillations in cool postflare loop
- Prominence seismology using small amplitude oscillations
- Magnetohydrodynamic kink waves in two-dimensional non-uniform prominence threads
- Torsional oscillations of longitudinally inhomogeneous coronal loops
- Radio seismology of the outer solar corona