3.6 Wavelength, phase speed, and group velocity

To derive the wavelength (λ) and phase speed (cp) of oscillations, time signals at different locations on the prominence must be acquired. The signature of a propagating wave is a linear variation of the oscillatory phase with distance. Hence, when several neighbouring points are found to oscillate with the same frequency, one can compute the Fourier phase of the signal at each of the points and check whether it varies linearly with distance. If it does, this gives place to a wave propagation interpretation and the wavelength can be calculated. This approach has been followed by Thompson and Schmieder (1991Jump To The Next Citation Point), Molowny-Horas et al. (1997Jump To The Next Citation Point) (about which more details are given in Section 3.6.2), and Terradas et al. (2002Jump To The Next Citation Point) (see Section 3.6.3). On the other hand, Lin (2005Jump To The Next Citation Point) and Lin et al. (2007Jump To The Next Citation Point) (see Section 3.6.4) detected wave propagation along threads by studying Doppler velocity variations at fixed times. They observed a sinusoidal variation of the Doppler shift with distance along the thread, which allowed them to compute the wavelength. Moreover, the phase velocity of the oscillations can be derived from the inclination of the coherent features in the Doppler velocity time-slice diagrams. Other authors have followed less strict methods to calculate these wave parameters.

It must be mentioned that observations of wave propagation in slender waveguides or plane wave propagation in a uniform medium do not provide the actual value of the wavelength (λ), but its projection on the plane of the sky, which is shorter than λ. And if a slit or some points along a straight line are used, then the computed wavelength is the projection of λ on the slit or the line. The observationally measured period and wavelength can in turn be used to calculate the phase speed, but since the observational wavelength is a lower limit to λ, this observational phase speed is also a lower limit to cp (Oliver and Ballester, 2002). Hence, even if it is not explicitly mentioned, the values of λ and cp quoted here are observationally derived lower bounds to the actual values.

The results presented in this section are grouped in four parts, the first three of them in increasing order of complexity of the data analysis; the fourth one is devoted to thread oscillations. The reported wavelength values cover a range from less than 3000 km (for waves propagating along some threads) to 75,000 km (for waves propagating in a large area of a quiescent prominence). These numbers must be taken into account in the theoretical study of these events.

3.6.1 Simple analyses

Malville and Schindler (1981) observed a loop prominence some 90 min before the onset of a nearby flare and detected periodic changes with a wavelength along the loop of 37,000 km. This value, together with the period of 75 min, results in a phase speed of about 8 km s–1.

Subsequent reports, which we now describe, are based on sheet-like prominences. Thompson and Schmieder (1991) detected periodic variations with periods between 3.5 and 4.5 min in a filament thread. They then computed the Fourier phase of the points along the thread and, after confirming its linearity from a phase versus distance plot, the value λ ≃ 50,000 km was derived, from which the phase speed is cp ≃ 150 –200 km s–1. In other works (e.g., Tsubaki and Takeuchi, 1986; Tsubaki et al., 1987, 1988; Suematsu et al., 1990) the signal in some consecutive locations along the slit has been found to be in phase. Although this seems to indicate that the wavelength of oscillations is much larger than the distance between the first and last of those points, this may not be necessarily true and a proper determination of the wavelength requires computing the Fourier phase corresponding to the oscillatory period.

Blanco et al. (1999Jump To The Next Citation Point) detected 15 – 20 min periodic variations corresponding to a pulse travelling with a speed of 170 km s–1. Such a large phase velocity is hard to reconcile with the typical speeds in a prominence, but it must be taken into account that this result has been obtained using Si iv and O iv lines, which are formed at transition region temperatures. Still, Blanco et al. (1999) mention that the sound speed in the prominence-corona transition region must be considerably faster than 170 km s–1, which leads them to conclude that the detected wave is of fast or Alfvénic character. Assuming a density of 1010 cm–3, a magnetic field of 8 G is derived.

Foullon et al. (2004Jump To The Next Citation Point) analyzed the intensity on a set of points along the main axis of a filament in 195 Å images. A time-space plot shows a clear oscillatory pattern at one end of the filament (around position 25 in Figure 9View Imagea). The oscillatory phase, displayed in Figure 9View Imageb, presents oscillatory fronts that are well correlated along the filament, meaning that the oscillations of neighbouring points along the filament are almost in phase. This is true in particular for positions around 25, although in positions around 5 and 10 the phase presents a linear trend in neighbouring points, which can be interpreted as wave propagation along the filament axis. Lower bounds to the wavelength and phase speed in this area could be determined as explained above. It is remarkable that the most pronounced periodic intensity variations, those around position 25, were detected during 6 days, which suggests that they suffered very little damping or were excited continuously during this time span.

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Figure 9: (a) Time-space diagram of the 195 Å image intensity along the main axis of a filament. The vertical black stripes are caused by the lack of observational data. (b) Spatial distribution of the Fourier phase (gray coloured contours) (from Foullon et al., 2004Jump To The Next Citation Point).

Berger et al. (2008) used high-resolution observations of limb prominences made by SOT on Hinode and detected oscillations that do not affect the whole prominence body. They considered three horizontal time slices at heights separated by 4.7 Mm and detected the presence of coherent oscillations in the three slices (Figure 2View Image). A phase matching of the sinusoidal profiles of these oscillations results in a vertical propagation speed (i.e., phase speed) around 10 km s–1. Again this value comes from a projection on the plane of the sky and is therefore a lower bound of the real value.

3.6.2 An elaborate one-dimensional analysis

Molowny-Horas et al. (1997Jump To The Next Citation Point) took into account the projection effects in their analysis of the Doppler velocity along the spectrograph slit. They detected periodic velocity variations with period of 7.5 min some 7300 km along the slit and found that the Fourier phase of the velocity at this period changes linearly with distance (Figure 10View Image). The value λ ≥ 20,000 km was derived. The corresponding phase speed is cp ≥ 44 km s−1.

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Figure 10: Relative Fourier phase as a function of position along the slit for several sets of consecutive points with similar oscillatory period: (a) 10.0 min, (b) 7.5 min, (c) 12.0 min, and (d) 4.0 min. The variation of the phase in (a) is not linear and so this oscillatory feature is not interpreted as a true signal. Regarding (c) and (d), the phase varies linearly with position, but the number of points involved is too small to make a firm conclusion. Finally, the phase in (b) displays a very robust linear dependence with distance, so this is interpreted as a signature of wave propagation (from Molowny-Horas et al., 1997Jump To The Next Citation Point).

To obtain the group velocity of this event, Molowny-Horas et al. (1997) performed a wavelet analysis of the same set of data, which revealed the presence of a train of 7.5-min waves in the slit locations (Figure 8View Image). Moreover, the time of occurrence of the train of waves increases linearly along the slit, which agrees with the assumption of a propagating disturbance. The velocity of propagation along the slit can then be computed and the value −1 v∥ ≃ 4.4 km s is obtained. Taking into account that v∥ is the projection of the group velocity, vg, on the slit, one concludes that the above value provides a lower limit for the group velocity, so vg ≥ 4.4 km s−1.

3.6.3 A two-dimensional analysis

This section is devoted to review the work by Terradas et al. (2002Jump To The Next Citation Point), that stands out among all other works in which wave properties have been determined since in this one a fully two-dimensional analysis is carried out. Figure 11Watch/download Movie shows a time series of Hβ filtergrams of the prominence studied by Terradas et al. (2002Jump To The Next Citation Point). The corresponding time series of the Doppler signal is presented in Figure 12Watch/download Movie.

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Figure 11: avi-Movie (3010 KB) Hβ line center images of a quiescent prominence observed with the VTT of Sacramento Peak Observatory. Images have been coaligned and a persistent drift towards the left has been suppressed. The thick white line displays the prominence edge and the solar photosphere is at the top (from Terradas et al., 2002Jump To The Next Citation Point).

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Figure 12: avi-Movie (11684 KB) Temporal evolution of the Doppler velocity in all points of the field of view of Figure 11Watch/download Movie (from Terradas et al., 2002Jump To The Next Citation Point).

The data used by Molowny-Horas et al. (1999) were re-analysed by Terradas et al. (2002Jump To The Next Citation Point) and clear evidence for propagating and standing waves was uncovered. These authors started from the Doppler velocity, which in many areas of the two-dimensional field of view can be very well fitted by a damped sinusoid (Figures 6View Image and 7View Image). The subsequent analysis was performed in a rectangle (black box in Figure 13View Image) that includes an area in which the correlation coefficient of the fit is large. The period of the oscillations in this rectangle is quite uniform and with a value around 75 min. First, Terradas et al. (2002Jump To The Next Citation Point) conducted an analysis of the phase along two straight lines inside the rectangle. Along the continuous line in Figure 13View Image, it is found that waves emanate from a point and propagate away from it (Figure 14View Image). It is clear both from the raw and the fitted signals in Figure 14View Image that the slope of wave propagation to the left is larger than that to the right. To derive the wavelength, Terradas et al. (2002Jump To The Next Citation Point) plotted the Fourier phase associated to the most relevant period in the Fourier spectrum (i.e., the one with 75 min period) along the selected path (right panel of Figure 14View Image). There is an almost linear decrease of the phase between positions 5 and 30, a linear increase between positions 50 and 62 and a region of roughly constant phase in between. The first two patterns correspond to propagation to the left and right along the path, such as was pointed out from the first two panels of Figure 14View Image, while the third pattern is caused by standing wave motions. The slope of a straight line fitted to the Fourier phase in each of the regions with wave propagation gives the wavelength of oscillation (projected on the selected path) which is around 75,000 km and 70,000 km for propagation to the left and right, respectively. The corresponding phase velocities are around 17 km s–1 and 15 km s–1.

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Figure 13: Two-dimensional Doppler velocity distribution at a given time in a quiescent prominence. The signal in the black rectangle can be fitted by a damped sinusoid with a high correlation coefficient (see Figure 7View Image). Two paths (straight continuous and dashed lines) were selected. The continuous white line represents the approximate position of the prominence edge. The photosphere is slightly outside the image top (from Terradas et al., 2002Jump To The Next Citation Point).
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Figure 14: Doppler velocity versus position and time along the solid path in Figure 13View Image. Left: raw Doppler signal; middle: fitted exponentially damped sinusoid; right: Fourier phase associated to the 75 min periodicity (from Terradas et al., 2002Jump To The Next Citation Point).

Another interesting feature of this data set can be discerned by considering the dashed path in Figure 13View Image. A representation of the Doppler velocity versus position and time (Figure 15View Image) shows that, at least for the first half of the observational time, positive and negative velocities seem to alternate in phase separated by a region, around position 25, with nearly zero amplitude. This pattern suggests that rather than a propagating feature, the signal in this area behaves like a standing wave with two regions completely out of phase. The Fourier phase (right panel of Figure 15View Image) is practically constant in a small region around position 10 and in a larger region for positions greater than 30, which indicates that there is no signal propagation in these locations. The phase difference between positions 10 and 50 is close to π, which, together with the fact that between these points the amplitude takes low values, is in close agreement with the standing wave picture and so a tentative identification of nodes and antinodes is possible. The estimated distance between the two antinodes visible in the left panel of Figure 15View Image is around 22,000 km. This implies that the (projected) wavelength of the standing wave is about 44,000 km and the corresponding phase speed is 10 km s–1. These values are about half those obtained for propagation along the other selected path and are a consequence of the anisotropic propagation of the perturbation.

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Figure 15: Doppler velocity versus position and time along the dashed path in Figure 13View Image. Left: raw Doppler signal; right: Fourier phase associated to the 75 min periodicity (from Terradas et al., 2002Jump To The Next Citation Point).

In addition to the identification of standing and propagating wave features in the prominence, Terradas et al. (2002Jump To The Next Citation Point) went on to perform an investigation of the two-dimensional distribution of the wavelength and phase speed. They started by plotting the Fourier phase for the most relevant period in the Fourier spectrum at each point (Figure 16View Image), which shows that a deep global minimum is found around the central position of the plot. This particular phase structure is an indication that motions have their origin at the position of the minimum and propagate, although in an anisotropic way, from this point. Terradas et al. (2002Jump To The Next Citation Point) gave a much more clear interpretation of the two-dimensional phase by plotting the wavevector field (Figure 17View Image), computed as the gradient of the Fourier phase. The arrows in this figure indicate the direction of wave propagation, their length being proportional to the modulus of the wavenumber, k. The projection of the phase velocity on the plane of the sky (computed from cp = ω∕k) is also displayed in Figure 17View Image. The analysis of the wavevector field shown in this figure clearly indicates that motions seem to be generated in a narrow strip close to positions x = 35 – 50 and y = 20 – 30 and spread out from this region. It is remarkable that the direction of the propagating waves from the source region is essentially parallel to or towards the prominence edge, revealing the anisotropic character of the observed wave propagation. The values of the phase velocity in Figure 17View Image are also quite different for both directions, being greater for the direction parallel to the edge, with cp ∼ 20 km s− 1, than for the direction perpendicular to the edge, with cp ∼ 10 km s−1. This is an indication of the possible existence of some wave guiding phenomenon, which shows a preferential direction of propagation. Note the good agreement between the values of the phase velocity in the directions parallel and perpendicular to the edge and those derived from the analysis of the two selected paths based on Figures 14View Image and 15View Image.

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Figure 16: Fourier phase associated to a period around 75 min (that is, the one corresponding to the largest peak in the Fourier spectrum) for the rectangular region selected in Figure 13View Image, both as a contour and as a surface plot. The selected paths are also displayed with continuous and dashed straight lines. Note that cuts of the Fourier phase along these two paths give rise to the Fourier phase displayed in Figures 14View Image and 15View Image (from Terradas et al., 2002Jump To The Next Citation Point).
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Figure 17: Arrows represent the wavevector field computed from the gradient of the Fourier phase displayed in Figure 16View Image, where the length of the arrows is proportional to the modulus of the wavevector. The phase velocity is shown with the help of different levels of grey and black and white colors (from Terradas et al., 2002Jump To The Next Citation Point).

3.6.4 Thread oscillations

Yi et al. (1991Jump To The Next Citation Point) and Yi and Engvold (1991Jump To The Next Citation Point) used two-dimensional spectral scans and investigated the presence of periodic variations of the Doppler shift and central intensity of the He i 10,830 Å line in two filaments. Yi et al. (1991Jump To The Next Citation Point) performed a first examination of the data and found oscillations with well-defined periods along particular threads in each prominence. For this reason, Yi and Engvold (1991Jump To The Next Citation Point) plotted the Doppler velocity versus position for different times in a given thread, so that a periodic spatial structure would directly yield a measure of the wavelength. Instead of this pattern, an almost linear variation of the velocity along the thread was found and consequently a value of λ much larger than the length of the threads, some 20,000 km in the two cases considered, was reported. Given that the periods are between 9 and 22 min, the corresponding phase speed is cp ≫ 15 km s−1. This result suggests that the thread is oscillating in the fundamental kink mode (whose wavelength is of the order of the length of the supporting magnetic tube, that is, around 100,000 – 200,000 km; see Section 4.4.1), rather than being disturbed by a travelling wave. Let us mention that, in general, this analysis may be misleading since the velocity signal does not generally consist of the detected periodic component only, but it is made of this component mixed with other velocity variations. If the periodic component is weak, then the method used by Yi and Engvold (1991Jump To The Next Citation Point) may fail because the signature of the propagating wave is masked by the rest of the signal.

In the analysis of the Doppler velocity in two threads (denoted as T1 and T2) belonging to the same filament, Lin (2005Jump To The Next Citation Point) found a clear oscillatory pattern in time-slice diagrams along the two thin structures. She determined the following wave properties for thread T1: c p = 60 km s–1, λ = 22, 12, 15 arcsec, and P in the range 2.5 – 5 min (the 4.4 min period being particularly pronounced). For thread T2, the wave properties are: cp = 91 km s–1, λ = 38, 23, 18 arcsec, and P in the range 2.5 – 5 min (the 5-min period being particularly pronounced).

The previous study by Lin (2005Jump To The Next Citation Point) is followed by a much more profound one in which the two-dimensional motions and Doppler shifts of 328 features (or absorbing “blobs”) of different threads are examined (see also Lin et al., 2003Jump To The Next Citation Point). Forty nine of these features are observed to flow along the filament axis with speeds of 5 – 20 km s–1 while oscillating in the line-of-sight at the same time with periods of 4 – 20 min (see Figure 18Watch/download Movie). To simplify the examination of oscillations, Lin (2005Jump To The Next Citation Point) computed average Doppler signals along each thread and found that groups of adjacent threads oscillate in phase with the same period. This has two consequences: first, since the periodicity is outstanding in the averaged signal for each thread, the wavelength of oscillations is larger than the length of the thread. Again the interpretation of this result is that the threads oscillate in their fundamental kink mode. Second, in this data set threads have a tendency to vibrate collectively, in groups, rather than independently.

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Figure 18: mpg-Movie (7124 KB) Hα line center images of a quiescent filament observed with the Swedish Solar Telescope in La Palma. The small-scale structures display the characteristic filament counter-streaming motions and undergo simultaneous transverse oscillations, detected as periodic Doppler variations (from Lin et al., 2003Jump To The Next Citation Point).

Horizontally flowing threads that undergo simultaneous transverse oscillations have not only been detected by Lin et al. (2003) and Lin (2005Jump To The Next Citation Point), but also by Okamoto et al. (2007Jump To The Next Citation Point) using SOT on Hinode. A Ca ii H line movie shows continuous horizontal thread motions along an active region prominence (cf. Figure 19Watch/download Movie). This movie also shows that the threads suffer apparently synchronous vertical oscillatory motions. An example of this phenomenon is shown in Figure 20View Image. Six threads displaying the same behaviour were studied and periods in the range 135 – 250 s were measured. The thread flow velocities range from 15 to 46 km s–1 and the vertical oscillation amplitudes range from 408 to 1771 km. These values are, of course, minimum estimates. A particularly interesting feature of these oscillations is that points along each thread oscillate transversally with the same phase. To reach this conclusion, a given thread is selected and several cuts along its length are considered. A representation of the signal as a function of time reveals that oscillations are synchronous along the entire length of the thread (Figure 21View Image). Once more this points to the kink mode as the responsible for the oscillations, as first pointed out by Van Doorsselaere et al. (2008a).

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Figure 19: mpg-Movie (5825 KB) Ca ii H line images taken with Hinode/SOT that shows ubiquitous continuous horizontal motions along the prominence threads at the top right of the image. These threads also oscillate up and down as they flow (from Okamoto et al., 2007Jump To The Next Citation Point).
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Figure 20: Close-up view of a flowing thread displaying transverse oscillations. The measured flow speed is 39 km s–1, the amplitude of vertical oscillations is 900 km and the period is 174 s (from Okamoto et al., 2007Jump To The Next Citation Point).
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Figure 21: Example of a prominence thread undergoing synchronous oscillations along its entire length (all images are shown in negative contrast). (a) The ends of the considered thread are marked by the two arrows. S1 to S5 indicate the locations used to make the height versus time plots shown in panels (b) to (f). (b) – (f) Height-time plots for the locations indicated in (a). Maximum and minimum amplitudes occur at nearly the same time for all locations (from Okamoto et al., 2007Jump To The Next Citation Point).

Hα observations conducted with the Swedish 1-m Solar Telescope by Lin et al. (2007Jump To The Next Citation Point) allowed to detect waves propagating in some selected threads. Figure 22View Image serves to illustrate the data analysis procedure for one thread. Here the line intensity shows no coherent behaviour (Figure 22View Imagea), while the line-of-sight velocity presents some inclined features caused by waves propagating along the thread; two such features are labelled 1 and 2 in Figure 22View Imageb. Figure 22View Imagec is another way of presenting Figure 22View Imageb and is useful to illustrate more clearly the wavy character of the line-of-sight velocities along an individual thread. Two shorter time sequencies of Doppler velocity are extracted from Figure 22View Imagec and shown in Figures 22View Imaged and e. It is clear that oscillations are of small amplitude since the Doppler shift has an amplitude of 1 – 2 km s–1. The power spectra of two of the curves in Figure 22View Imagec (shown in Figures 22View Imagef and g) yield wavelengths of the oscillatory pattern of, respectively, 3.8 arcsec and 4.7 arcsec. The phase velocity of the oscillations can be derived from the inclination of the features appearing in the Doppler time-slice diagrams of Figure 22View Imageb. The phase velocities thus obtained correspond to, respectively, 8.8 and 10.2 km s–1. Lin et al. (2007Jump To The Next Citation Point) found similar evidence of travelling waves in eight different threads. The mean phase velocity and period (obviously affected by the projection effect) are 12 km s–1 and 4.3 min. Periods between 3 and 9 min were found; longer period oscillations could not be detected in the data set used in this work because of its limited duration (18 min).

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Figure 22: (a) and (b) Time-slice diagrams of the Hα line intensity and Doppler shift along a filament thread. (c) Data of panel (b) shown as a set of curves instead of as a contour plot. Each curve represents the Doppler velocity along the thread for a fixed time (frame). (d) and (e) Signals from panel (c) for some selected times (frames). (f) and (g) Power spectra of the Doppler shift along the thread for two times. Large peaks help identify the wavelength of propagating oscillations (from Lin et al., 2007Jump To The Next Citation Point).

To test the coherence of oscillations over a larger area, covering several threads, Lin et al. (2007Jump To The Next Citation Point) averaged the line-of-sight velocity in a 3.4 arcsec × 10 arcsec rectangle containing closely packed threads. The averaged Doppler signal (left panel of their Figure 4) displays a very clear oscillation. In addition, the power spectrum of this signal has a significant power peak at 3.6 min. Thus, the conclusion is that neighboring threads tended to oscillate coherently in this rectangular area, possibly because they were separated by very short distances. This signal averaging could be analogous to acquiring data with poor seeing, such as in Terradas et al. (2002Jump To The Next Citation Point) (see Section 3.6.3).

Using data from the Swedish 1-m Solar Telescope in La Palma, Lin et al. (2009Jump To The Next Citation Point) performed a novel analysis of thread oscillations by combining simultaneous recordings of motions along the line-of-sight and in the plane of the sky, which provides information about the orientation of the oscillatory velocity vector. From the measurements of swaying motions in the plane of the sky, several threads in this work presented travelling disturbances whose main features were characterized (period, phase velocity, and oscillatory amplitude). These parameters were obtained following the procedure of Figure 22View Image. Moreover, two of the previous threads also showed Doppler velocity oscillations with a period similar to that of the swaying motions, so that the threads had a displacement that was neither in the plane of the sky nor along the line-of-sight. By combining the observed oscillations in the two orthogonal directions, Lin et al. (2009Jump To The Next Citation Point) derived the full velocity vectors. They suggested that thread oscillations were probably polarized in a fixed plane that may attain various orientations relative to the local reference system (for example, horizontal, vertical, or inclined). Swaying motions are most clearly observed when a thread sways in the plane of the sky, while Doppler signals are strongest for oscillations along the line of sight. In the case of the two analyzed threads, a combination of the observed velocity components yielded an orientation of the velocity vectors of 42° and 59° with respect to the plane of the sky. Once the heliocentric position of the filament was taken into account, these angles transformed into oscillatory motions which were reasonably close to the vertical direction. Lin et al. (2009Jump To The Next Citation Point) alerted that this conclusion is only based on two cases and that it is not possible to draw any general conclusion about the orientation of the planes of oscillation of filament threads. In fact, Yi and Engvold (1991) found no center-to-limb variations of the velocity amplitude of threads displaying Doppler velocity oscillations, so they concluded that there did not seem to be a preferred direction of oscillatory motions in their data set.

Ning et al. (2009b) analyzed the oscillatory behaviour of 13 threads in a quiescent prominence observed with Hinode/SOT. They found that many prominence threads exhibited vertical and horizontal oscillatory motions and that the corresponding periods did not substantially differ for a given thread. In some parts of the prominence, the threads seemed to oscillate independently from one another, and the oscillations were strongly damped. Some of the oscillating threads presented a simultaneous drift in the plane of the sky with velocities from 1.0 to 9.2 km s–1. The reported periods were short (between 210 to 525 s), with the dominant one appearing at 5 min. Peak to peak amplitudes were in the range 720 – 1440 km and the phase velocity varied between 5.0 and 9.1 km s–1.


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