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. (2007) analyzed 6 threads whose relevant measured properties are displayed in Table 1.
|Thread||(km)||(km s–1)||(s)||(km s–1)||(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. (2008) 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 41 and Section 4.5). By using theoretical results by Díaz et al. (2002) and Dymova and Ruderman (2005) (see Section 4.5), Terradas et al. (2008) 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 68 shows these curves for the selection of six threads made by Okamoto et al. (2007). 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.
Terradas et al. (2008) next incorporated mass flows into their analysis (see Figure 67). First a simple approximation was made by taking into account that the flow velocity along the cylinder, , enters the linear MHD wave equations through the differential operator
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, , 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. (2008) found that the flow velocities measured by Okamoto et al. (2007) 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 68 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- 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).
Living Rev. Solar Phys. 9, (2012), 2
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