6.6 Seismology of flowing and oscillating prominence threads

Mass flows in conjunction with phase speeds, oscillatory periods, and damping times might constitute an additional source of information about the physical conditions of oscillating threads. The first application of prominence seismology using Hinode observations of flowing and transversely oscillating threads was presented by Terradas et al. (2008Jump To The Next Citation Point), using observations obtained in an active region filament by Okamoto et al. (2007Jump To The Next Citation Point) discussed in Section 3.6.4.

The observations show a number of threads that flow following a path parallel to the photosphere while they oscillate in the vertical direction. The relevance of this particular event is that the coexistence of waves and flows can be firmly established, so that there is no ambiguity about the wave or flow character of a given dynamic feature: both seem to be present in this particular event. However, other interpretations for the apparent motion in the plane of the sky could be also possible, for instance, an ionization wave or a thermal front. Okamoto et al. (2007Jump To The Next Citation Point) analyzed 6 threads whose relevant measured properties are displayed in Table 1.

Table 1: Summary of geometric and wave properties of horizontally flowing and vertically oscillating threads analyzed by Okamoto et al. (2007Jump To The Next Citation Point). L thread is the thread length, v 0 its horizontal flow velocity, P the oscillatory period, V the oscillatory velocity amplitude, and H the height above the photosphere.
Thread Lthread (km) v0 (km s–1) P (s) V (km s–1) H (km)
1 3600 39 174 ± 25 16 18,300
2 16,000 15 240 ± 30 15 12,400
3 6700 39 230 ± 87 12 14,700
4 2200 46 180 ± 137 8 19,000
5 3500 45 135 ± 21 9 14,300
6 1700 25 250 ± 17 22 17,200

In their seismological analysis of these oscillations Terradas et al. (2008Jump To The Next Citation Point) started by neglecting the mass flows. Then, they interpreted these events in terms of the standing kink mode of a finite-length thread in a magnetic flux tube (see Figure 41View Image and Section 4.5). By using theoretical results by Díaz et al. (2002) and Dymova and Ruderman (2005Jump To The Next Citation Point) (see Section 4.5), Terradas et al. (2008Jump To The Next Citation Point) found that, although it is not possible to univocally determine all the physical parameters of interest, a one-to-one relation between the thread Alfvén speed and the coronal Alfvén speed could be established. This relation comes in the form of a number of curves relating the two Alfvén speeds for different values of the length of the magnetic flux tube and the density contrast between the filament and coronal plasma. Figure 68View Image shows these curves for the selection of six threads made by Okamoto et al. (2007Jump To The Next Citation Point). An interesting property of the obtained solution curves is that they display an asymptotic behaviour for large values of the density contrast, which is typical of filament to coronal plasmas and, hence, a lower limit for the thread Alfvén speed can be obtained. Take for instance thread #6. Considering a magnetic flux tube length of 100 Mm, a value of 120 km s–1 for the thread Alfvén speed is obtained.

View Image

Figure 67: Sketch of the magnetic and plasma configuration used to represent a flowing thread (shaded volume) in a thin magnetic tube. The two parallel planes at both ends of the cylinder represent the photosphere (from Terradas et al., 2008Jump To The Next Citation Point).
View Image

Figure 68: Dependence of the Alfvén velocity in the thread as a function of the coronal Alfvén velocity for the six threads observed by Okamoto et al. (2007Jump To The Next Citation Point). In each panel, from bottom to top, the curves correspond to a length of magnetic field lines of 100,000 km, 150,000 km, 200,000 km, and 250,000 km, respectively. Asterisks, diamonds, triangles, and squares correspond to density ratios of the thread to the coronal gas ζ ≃ 5, 50, 100, and 200 (from Terradas et al., 2008Jump To The Next Citation Point).

Terradas et al. (2008Jump To The Next Citation Point) next incorporated mass flows into their analysis (see Figure 67View Image). First a simple approximation was made by taking into account that the flow velocity along the cylinder, v0, enters the linear MHD wave equations through the differential operator

-∂-+ v ∂-. ∂t 0∂z

The terms coming from the equilibrium flow can, in a first approximation, be ignored because, as noted by Dymova and Ruderman (2005), inside the cylinder the terms with derivatives along the tube are much smaller than those with radial or azimuthal derivatives. By following this approach the problem reduces to solving a time-dependent problem with a varying density profile, ρ(z,t), representing a dense part moving along the tube with the flow speed. By using the flow velocities in Table 1 and after solving the two-dimensional wave equations, Terradas et al. (2008Jump To The Next Citation Point) found that the flow velocities measured by Okamoto et al. (2007Jump To The Next Citation Point) result in slightly shorter kink mode periods than the ones derived in the absence of flow. Differences are small, however, and produce period shifts between 3 and 5%. As a consequence, the curves in Figure 68View Image can be considered a good approximation to the solution of the inverse problem.

Finally, a more complete approach to the problem was followed by Terradas et al. (2008), who considered the numerical solution of the non-linear, ideal, low-β MHD equations with no further approximations, that is, the thin tube approximation was not used and the flow was maintained in the equations. The numerical results confirm the previous approximate results regarding the effect of the flow on the obtained periods and, therefore, on the derived Alfvén speed values. We must note that in this case, and because of the small value of the flow speeds measured by Okamoto et al. (2007) in this particular event, there are no significant variations of the wave properties and, hence, of the inferred Alfvén speeds, although larger flow velocities may have more relevant consequences on the determination of physical parameters in prominence threads.

In most of the examples shown here, the number of unknowns is larger that that of observed parameters. This makes difficult to obtain a unique solution that reproduces the observations. Furthermore, the inversions are performed with information that is incomplete and uncertain. The use of statistical techniques, based on bayesian inference, can help to overcome these limitations, as shown by Arregui and Asensio Ramos (2011).

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