The period ratio P1/P2$P_{1}/P_{2}$ of torsional Alfvén waves with steady flows in spicules

The aim here is to model the standing torsional oscillations in solar spicules in the presence of density stratification, magnetic field expansion, and steady flows. By implementing cylindrical geometry, the eigenfrequencies, eigenfunctions, and the period ratio P1/P2$P_{1}/P_{2}$ of these waves is obtained for finite plasma-β$\beta$. The shifts created by the steady flow justifies the divergence of the observed period ratio for the first and second periods from the number 2.