tokens: ). These structures are lying mainly horizontally with respect to z = 0. This behavior is due to the fact that twist introduces local dips and mass tends to accumulate around these dips. However, at the center of the flux rope, even in the case with the strongest twist, the magnetic field lines are essentially horizontal, and therefore the dense material does not reorganize much along these lines, producing the horizontal structuring. Figure 16. Same as in Figure 14 but for N t = 4.3. Figure 17. Same as in Figure 14 but for N t = 8.5. 8. CONCLUSIONS AND DISCUSSION The time evolution of a cold prominence initially embedded in a magnetic flux rope has revealed interesting characteristics of the structure. We have concentrated on sustained prominences, i.e., detached from the photosphere. The different numerical simulations have shown how the magnetic field evolves in response to the dense prominence that is pulled downward by gravity. The various twisted flux ropes considered in this work, together with the embedded prominences, show a different evolution. We have shown that in some cases they are clearly evolving toward a stationary state. These quasi-stationary 3D solutions can be used in the future for other purposes such as the analysis of winking filaments produced by external perturbations like Moreton waves, EIT waves, or nearby flares and CMEs. During the relaxation to the quasi-stationary equilibrium, the system goes through several periodic vertical oscillations. In the linear regime, i.e., when the amplitude of oscillation is small in comparison with the local Alfvén speed, the process of continuum damping transfers the excess of energy in the system to the PCTR. It is in this highly inhomogeneous layer where the process of phase mixing takes place. This energy redistribution explains the attenuation of the position of the CM of the prominence body. Thus, at least in the linear regime the resonant absorption process is crucial in the relaxation of the structure. The energy in the PCTR is eventually dissipated by the numerical scheme used to perform the simulation, and this dissipation can be hardly avoided owing to the continuous generation of small spatial scales with time by the phase-mixing mechanism. Fortunately, although the numerical dissipation produces an energy loss, this does not necessarily mean that the attenuation of the CM is dominated by this artificial dissipation. In fact, it is known that a small dissipation does not change the damping time (Poedts Kerner 1991 ) of the quasi-mode (i.e., the resonantly damped global mode); it affects the dissipation in the layer but not the global attenuation (see Terradas et al. 2006a , for the linear time evolution with dissipation). The different convergence tests performed in the present study point in this direction. Wave leakage is not relevant in our model and does not contribute to the attenuation of the position of the CM. The problem studied here demonstrates the necessity
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