The period ratio P_1/P_2 of torsional Alfv\'en 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 P_1/P_2 of these waves is obtained for finite plasma-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. 
H. Ebadi, S. Shahmorad, S. Vasheghani Farahani
astro-ph.SR