tokens: could be directly understood from the second term on the RHS of Eq. ( 34 ). Also the damping due to the plasma resistivity is clearly shown by the presence of the third term on the RHS of Eq. ( 34 ). Since the viscosity is small for coronal conditions, the deceleration is not very strong in the long wavelength limit, but in the long run would gradually become significant. 3. Discussions and conclusions In the solar atmosphere MHD waves interact with the plasma. When a gradient in the density is present, a wave guide is created that guides waves leading to the appearance of collective modes. In this work a magnetized cylindrical structure whose density is higher than its surroundings has been taken into consideration. The interplay of the equilibrium twist and rotation exhibits interesting features of the nature of the torsional wave that are of interest for coronal seismology. We considered a twisted and rotating magnetic flux tube, which resembles a solar spiral or tornado, and studied the influence of its helical properties on other dissipating parameters linked to the oscillations of the torsional waves. The analytic expressions obtained for the dependence of the phase speed of the torsional wave on the equilibrium twist and rotation enabled a quantitative study of the interplay of various actors on its propagation. The comparison of the explicit expressions obtained for the phase speed of the torsional wave in the present study with the implicit version, in all cases proved adequate for the robustness of the explicit expressions. This is reflected in the figures where the numerical results are compared with analytic solutions. The full dispersion relation for axisymmetric oscillations in the long wavelength limit has been obtained by showing the joint effects of the equilibrium twist and rotation with the viscosity and gravity (Eq. ( 19 )). An explicit expression for the dependency of the torsional wave frequency in the presence of viscosity, equilibrium, and environmental conditions on the wave number has been obtained (Eq. ( 20 )). This explicit dispersion relation also gives the damping and deceleration of the torsional wave guided by solar plasma structures. The fine consistency between the full dispersion relation and the explicit expressions for the frequency, damping, and speed of the torsional wave justifies the model and results. The magnetic twist owes its efficiency on the torsional wave frequency to not only the flux-tube rotation and coronal conditions, but also to the viscosity and gravity. In a zero plasma- β structure that is non-rotating, the magnetic twist acts very weakly on the torsional wave frequency (Fig. 3 ). If in such conditions the viscosity and solar gravity become negligible, the magnetic twist, no matter how large, would be completely neutral without any effect on the frequency. The damping of the torsional wave due to viscosity in solar structures is proportional to the strength of the magnetic twist and
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