The model considered here is an infinitely long thread of radius surrounded by a thin transition sheath of thickness in which a smooth transition from the thread to the coronal density takes place (see Figure 50). For standing kink waves, and without using the thin tube and thin boundary approximation, the normal mode period and damping ratio are functions of the relevant equilibrium parameters,et al., 2008b). The period is a function of the longitudinal wavenumber, , the transverse inhomogeneity length-scale, , and the internal Alfvén speed. Similarly for the damping ratio, except for the fact that it cannot depend on any time-scale. The long wavelength approximation further eliminates the dependence of the damping ratio. In the case of coronal loop oscillations, an estimate for can be obtained directly from the length of the loop and the fact that the fundamental kink mode wavelength is twice this quantity. For prominence threads, the wavelength of oscillations needs to be measured. Relations (40) indicate that, if no assumption is made on any of the physical parameters of interest, there are infinite different equilibrium models that can equally well explain the observations (namely the period and damping ratio). The parameter values that define these valid equilibrium models are displayed in Figure 64a, where the analytical algebraic expressions in the thin tube and thin boundary approximations by Goossens et al. (2008) have been used to invert the problem. It can be appreciated that, even if an infinite number of solutions is obtained, they define a rather constrained range of values for the thread Alfvén speed. Because of the insensitiveness of the damping rate with the density contrast for the typically large values of this parameter in prominence plasmas, the obtained solution curve displays an asymptotic behaviour for large values of . This makes possible to obtain precise estimates for the thread Alfvén speed, , and the transverse inhomogeneity length scale, . Note that these asymptotic values can directly be obtained by inverting Equations (23) and (30) for the period and the damping rate in the limit . The computation of the magnetic field strength from the obtained seismological curve requires the assumption of a particular value for either the filament or the coronal density. The resulting curve for a typical coronal density is shown in Figure 64b. Precise values of the magnetic field strength cannot be obtained, unless the density contrast is accurately known.
The transverse inhomogeneity length scale of an oscillating thread could also be estimated by using observations of spatial damping of propagating kink waves and theoretical results described in Section 5.4.2. In the context of coronal loops, Terradas et al. (2010) have shown that the ratio of the damping length to the wavelength, due to resonant damping of propagating kink waves, has the same dependence on the density contrast and transverse inhomogeneity length-scale as the ratio of the damping time to the period for standing kink waves. Similar inversion techniques to the ones explained here for the temporal damping of oscillations could be applied to the spatial damping of propagating waves.
The main downside of the technique just described is the use of thread models in which the full magnetic tube is filled with cool and dense plasma. The solution to the forward problem in the case of two-dimensional thread models is discussed in Section 5.5. The analytical and numerical results obtained by Soler et al. (2010a) using these models indicate that the length of the thread and its position along the magnetic tube influence the period and damping time of transverse thread oscillations. On the contrary, the damping ratio is rather insensitive to these model properties.
Going back to the inversion curve displayed in Figure 64a, we notice that a change in the period produces a vertical shift of the solution curve, hence the period influences the inferred values for the Alfvén speed. On the other hand, the damping ratio determines the projection of the inversion curve onto the (, )-plane. We can conclude that ignorance of the length of the thread or the length of the supporting magnetic flux tube will have a significant impact on the inferred values for the Alfvén speed (hence magnetic field strength) in the thread. On the contrary, because of the smaller sensitivity of the damping ratio to changes in the longitudinal density structuring, seismological estimates of the transverse density structuring will be less affected by our ignorance about the longitudinal density structuring of prominence threads.
An example of the inversion of physical parameters for different values of the thread length was presented by Soler et al. (2010a). When partially filled threads, i.e., with the dense part occupying a length shorter than the total length of the tube , are considered, one curve is obtained for each value of the length of the thread. The solutions to the inverse problem are shown in Figure 65a for a set of values of . Even if each curve gives an infinite number of solutions, again each of them defines a rather constrained range of values for the thread Alfvén speed. The figure shows that the ratio is a fundamental parameter in order to perform an accurate seismology of prominence threads, since different curves produce different estimates for the prominence Alfvén speed, as anticipated above. Because of the insensitiveness of the damping ratio with respect to the length of the thread, all solution curves for different lengths of the threads produce the same projection onto the (, )-plane. Hence, the same precise estimates of the transverse inhomogeneity length scale obtained from infinitely long thread models are valid, irrespective of the length of the thread. The computation of the magnetic field strength from the obtained seismological curve requires the assumption of a particular value for either the filament or the coronal density. The resulting curves for a typical coronal density and several values of are shown in Figure 65b. Here again, precise values of the magnetic field strength cannot be obtained, unless the density contrast is accurately known.
Living Rev. Solar Phys. 9, (2012), 2
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