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Prominences are intriguing, but poorly understood, magnetic structures of the solar corona. The dynamics of solar prominences has been the subject of a large number of studies, and of particular interest is the study of prominence oscillations. Ground- and space-based observations have confirmed the presence of oscillatory motions in prominences and they have been interpreted in terms of magnetohydrodynamic (MHD) waves. This interpretation opens the door to perform prominence seismology, whose main aim is to determine physical parameters in magnetic and plasma structures (prominences) that are difficult to measure by direct means. Here, we review the observational information gathered about prominence oscillations as well as the theoretical models developed to interpret small amplitude oscillations and their temporal and spatial attenuation. Finally, several prominence seismology applications are presented.
KeywordsResonant Absorption Magnetic Tube Small Amplitude Oscillation Kink Mode Large Amplitude Oscillation
Quiescent solar filaments are clouds of cool and dense plasma suspended against gravity by forces thought to be of magnetic origin. They form along the inversion polarity line in or between the weak remnants of active regions. Early observations already suggested that their fine structure is apparently composed by many horizontal and thin dark threads (de Jager, 1959; Kuperus and Tandberg-Hanssen, 1967). More recent high-resolution Hα observations obtained with the Swedish Solar Telescope (SST) in La Palma (Lin et al., 2005) and the Dutch Open Telescope (DOT) in La Palma (Heinzel and Anzer, 2006) have allowed to observe this fine structure with much greater detail (see Lin, 2010, for a review). The measured average width of resolved thin threads is about 0-3 arcsec (∼ 210 km), while their length is between 5 and 40 arcsec (∼ 3500 – 28,000 km). The fine threads of solar filaments seem to be partially filled with cold plasma (Lin et al., 2005), typically two orders of magnitude denser and cooler than the surrounding corona, and it is generally assumed that they outline the magnetic flux tubes in which they reside (Engvold, 1998; Lin, 2005; Lin et al., 2005; Okamoto et al., 2007; Engvold, 2008; Martin et al., 2008; Lin et al., 2008). This idea is strongly supported by observations which suggest that threads are skewed with respect to the filament long axis in a similar way to what has been found for the magnetic field (Leroy, 1980; Bommier et al., 1994; Bommier and Leroy, 1998). On the opposite, Heinzel and Anzer (2006) suggest that these dark horizontal filament structures are a projection effect. According to this view, many magnetic field dips of rather small vertical extension, but filled with cool plasma, are aligned in the vertical direction and the projection against the disk produces the impression of a horizontal thread.
Prominences are highly dynamic structures that display flows. These flows have been observed in Hα, UV, and EUV lines, and their study and characterization are of great interest for the understanding of prominence formation and stability, the mass supply, and the prominence magnetic field structure. In the Hα line, and in quiescent limb prominences, a complex dynamics with vertical downflows and upflows (Berger et al., 2008) as well as horizontal flows is often observed. The velocities are in the range between 2 and 35 km s-1, while in EUV lines flow velocities seem to be slightly higher. When comparing these values one should be aware that these lines correspond to different temperatures, so probably the reported flow speeds correspond to different parts of the prominence. In active region prominences, flow velocities seem to be higher than in quiescent prominences, even reaching 200 km s-1, and some of these high-speed flows are probably related with the prominence formation itself. In the case of filaments observed on the disk in the Hα line, horizontal flows in the filament spine are often observed, while in barbs flows are vertical. The range of observed velocities of filament flows is between 5 and 20 km s-1. A particular feature in these observations is the presence of counter-streaming flows, i.e., oppositely directed flows (Zirker et al., 1998; Lin et al., 2003). Because of the physical conditions of the filament plasma, all these flows seem to be field-aligned. For a thorough information about flows in prominences see Labrosse et al. (2010) and Mackay et al. (2010).
Solar prominences are subject to various types of oscillatory motions. Some of the first works on this subject were concerned with oscillations of large amplitude induced by disturbances coming from a nearby flare. Later, observations performed with ground-based telescopes pointed out that many quiescent prominences and filaments display small amplitude oscillations (Harvey, 1969). These oscillations have been commonly interpreted in terms of standing or propagating magneto-hydrodynamic (MHD) waves. Using this interpretation, a number of theoretical models have been set up in order to try to understand the prominence oscillatory behaviour. Such as we will point out in the following, the study of prominence oscillations can provide an alternative approach for probing their internal structure. The magnetic field structure and physical plasma properties are often hard to infer directly and wave properties directly depend on these physical conditions. Therefore, prominence seismology seeks to obtain information about prominence physical conditions from a comparison between observations and theoretical models of oscillations.
This review is mainly devoted to small amplitude oscillations, although a brief section deals with large amplitude oscillations. The layout of the review is the following: large amplitude oscillations are succinctly summarized in Section 2; in Section 3, the observational background about small amplitude oscillations is reviewed; in Section 4, theoretical models of small amplitude oscillations based on linear ideal MHD waves in different configurations are described; next, in Section 5, the damping of prominence oscillations produced by different mechanisms is studied from a theoretical point of view; finally, in Section 6, prominence seismology using large and small amplitude oscillations is introduced.
2 Large Amplitude Oscillations
Oscillations with velocity amplitude greater than 20 km s-1 have been observed in filaments. It was suggested that their exciter was a wave, caused by a flare, which disturbs the filament and induces damped oscillations. This hypothesis was confirmed by Moreton and Ramsey (1960), who used a refined photographic technique that permitted the observation of the propagating perturbation, with velocities in the range 500 – 1500 km s-1. In some cases, during the course of the oscillations, the filament becomes visible in the Hα image when the prominence is at rest, but when its line-of-sight velocity is sufficiently large, the emission from the material falls outside the bandpass of the filter and the prominence becomes invisible in Hα. This process is repeated periodically and for this reason this type of event is called a “winking filament”. Ramsey and Smith (1965) and Hyder (1966) studied 11 winking filaments and derived oscillatory periods between 6 and 40 min, and damping times between 7 and 120 min. They reported that there seemed to be no correlation between the period and the filament dimensions, the distance to the perturbing flare or its size. In addition, a single filament perturbed by four flares during three consecutive days oscillated with essentially the same frequency and damping time in each event. As a consequence, it was suggested that prominences possess their own frequency of oscillation.
The oscillatory velocity of the winking filaments studied by Ramsey and Smith (1965, 1966) and Hyder (1966) is quite large compared with the relevant wave speeds in prominences (namely the sound and Alfvén speeds). For this reason, one usually refers to these events as large amplitude oscillations. Recently, and thanks to space- and ground-based instruments, new observations of large amplitude oscillations have been published. The exciters seem to be Moreton or EIT waves (Eto et al., 2002; Okamoto et al., 2004; Gilbert et al., 2008) or nearby jets and subflares (Jing et al., 2003, 2006; Vršnak et al., 2007), while in other cases the oscillations are associated to the eruptive phase of a filament (Isobe and Tripathi, 2006; Isobe et al., 2007; Pouget, 2007; Chen et al., 2008) or are produced by the Alfvénic vortex shedding mechanism recently developed by Nakariakov et al. (2009). In this last case, oscillations could be a signature of the transition from a stable to an unstable situation. Although in most of the observed flare-induced filament oscillations the material undergoes vertical oscillations, Kleczek and Kuperus (1969) and Hershaw et al. (2011) have also reported horizontal oscillations. Moreover, periodic motions along the longitudinal filament axis have also been observed (Jing et al., 2003, 2006; Vršnak et al., 2007).
From the theoretical point of view, models that explain large amplitude filament oscillations are lacking. To explain the vertical motions, Anderson (1967) suggested that the disturbance coming from the flare propagates along the magnetic field and when it arrives to the filament, the material is pushed down. Hyder (1966) proposed a model which explains the vertical motions in terms of harmonically damped oscillations. The restoring force is provided by the magnetic tension, while the damping is due to coronal viscosity. Using this model, Hyder was able to calculate the strength of the vertical component of the magnetic field in the prominence. Later, Kleczek and Kuperus (1969) proposed a similar model to explain the horizontal oscillations, although in this case the damping is provided by the emission of acoustic waves. On the other hand, Sakai et al. (1987) developed a model for the formation of a prominence in a current sheet. One of the features of this model is the presence of non-linear oscillations of the current sheet. Bakhareva et al. (1992) considered a partially ionized plasma and developed a dynamical model for a solar prominence in which non-linear oscillations are present. Chin et al. (2010) have considered possible oscillatory regimes of non-linear thermal over-stability which can occur in prominences. Finally, numerical simulations (Chen et al., 2002) suggest that Moreton and EIT waves can be produced by CMEs. Then, the theoretical modelling of large amplitude oscillations excited by these events is a task that remains to be done. For a more extensive review about large amplitude prominence oscillations see Tripathi et al. (2009).
3 Small Amplitude Oscillations: Observational Aspects
Not related to flare activity.
Small velocity amplitude.
Only a restricted volume of the prominence displays periodic variations.
Regarding item 1, so far it has not been possible to identify the trigger of small amplitude oscillations. A popular conjecture about their excitation is that it lies in the periodic motions of magnetic fields caused by photospheric or chromospheric oscillations. The idea is that Alfvén waves ought to propagate upwards and that any prominence material threaded by the field should also be subject to periodic motions if there is enough energy available to overcome the inertia of the dense plasma (Harvey, 1969); this idea was later suggested by other authors too (e.g., Yi et al., 1991). Harvey made order-of-magnitude calculations to show that the ratio of prominence to photospheric oscillatory energy is around or smaller than 10-4, which indicates that this excitation mechanism is feasible. On the other hand, Harvey also noted that a few prominences in his study appear to oscillate with periods nearly twice as large as those of the photospheric oscillation. This appears to contradict the above hypothesis about the initiator of small amplitude oscillations. Much longer and shorter periods than those present in Harvey’s work have been detected afterwards (see Section 3-3), so probably this mechanism of energy transfer from the photosphere (or chromosphere), if correct, may not be the only one to cause these prominence oscillations. Mashnich et al. (2009a,b) studied the Doppler velocity field in some filaments and the underlying photosphere by means of simultaneous observations of the Hα line and the neighbouring photospheric Fe i line at 4863 .A. They detected a quasi-hourly oscillation in certain areas of the filaments and photosphere and found a good spatial correlation between them. They also reported that the parts of the photosphere displaying this oscillation are often observed below filament barbs. The spatial coincidence of this periodicity and the relation of filament barbs and the photosphere led these authors to suggest that the photosphere was the origin of these particular prominence oscillations. From an observation of a limb prominence with Hinode, Ning et al. (2009b) reported that the detected oscillatory behaviour only lasted about one period and that new oscillations appeared nearby simultaneously. These authors then concluded that the exciters or drivers of such oscillations are numerous and of small scale. The current understanding (see Section 4) is that periodic perturbations in prominences can be produced by an external impulsive agent that excites different eigenmodes of the structure or by a continuous agent, as may be the case with the 5-min photospheric and 3-min chromospheric oscillations whose influence could propagate along magnetic field lines and force motions of the prominence plasma. Recently, some evidence about the effect of the chromospheric 3-min oscillations on the corona has been found by Sych et al. (2009).
Regarding item 2, the detected Doppler velocity peak ranges from the noise level (down to 0.1 km s-1 in some cases) to 2 – 3 km s-1 (e.g., Harvey, 1969), although larger values have also been reported (e.g., Bashkirtsev and Mashnich, 1984; Molowny-Horas et al., 1999; Ning et al., 2009a). This maximum value has to be compared with the typical speeds of the prominence plasma (the sound and Alfvén speeds), which are larger than 10 km s-1 for typical prominence conditions. The main purpose of studying prominence oscillations is to obtain insight into their physics via a seismological approach (see Section 6). Therefore, the information that observations should provide are the periods, wavelength, phase, and group velocity and damping time of these phenomena. In addition, observations should also determine whether these periodic variations are standing oscillations or propagating waves, whether they affect some prominence threads or larger areas of a prominence, whether threads oscillate independently from their neighbours or which physical variables are disturbed and by which amount.
The solar origin of prominence oscillations remained controversial until the beginning of the 1990s. For example, Engvold (1981) failed to detect oscillatory motions in the velocity field of a limb prominence, although the observational setup used prevented him from reliably distinguishing velocity amplitudes below 2 km s-1, the range in which many peak values are found. In addition, Malherbe et al. (1981, 1987) recognized no oscillatory pattern in time series of line-of-sight velocities obtained with the MSDP operating on the Meudon solar tower, although positive results were later achieved using the same instrument (Thompson and Schmieder, 1991). On the other hand, the lack of small amplitude, periodic variations in signals coming from solar prominences cannot be considered a proof against the existence of these kind of processes. Harvey (1969) noted that in a sample of 68 non-active region prominences, 31% of the objects presented no significant velocity change along the line-of-sight, 28% showed apparently random line-of-sight velocity variations, and 41% presented a definite oscillatory behaviour. Analogous results were obtained for a set of 45 active region prominences. There are several reasons that may lead to the absence of periodic variations in some prominences: the velocity amplitude or its projection along the line-of-sight may be too small to stand above the instrumental noise level; or the prominence material may actually not oscillate at the time the observations are performed; or the light emitted or absorbed by various plasma elements along the line-of-sight and having different oscillatory properties may result in a noisy signal.
3.1 Detection methods
A spectroscopic observation using a slit yields restricted information on the spatial distribution of oscillations and, what is even worse, does not ensure that the same plasma elements are placed on the slit during the observing time. The first of these concerns also applies to the analysis of a twodimensional data set in which only variations in one direction are considered. Observations using a two-dimensional field of view and with high spatial resolution have diminished these worries, while allowing to study how prominence threads participate of the oscillatory motions. These observations have been conducted both with ground-based telescopes (Yi et al., 1991; Yi and Engvold, 1991; Lin et al., 2003; Lin, 2005; Lin et al., 2005, 2007, 2009) and with space-based telescopes (Okamoto et al., 2007). In addition, two-dimensional Dopplergrams have also been employed (Molowny-Horas et al., 1999; Terradas et al., 2002), although the spatial resolution of this particular observation is not good enough to appreciate the prominence thread structure.
Although most data used in the analysis of small amplitude prominence oscillations come from typical prominence lines, in some cases spectral lines or images formed at hotter temperatures have also been considered. Examples are the He i line at 584-33 Å, formed at 20,000 K (Régnier et al., 2001; Pouget et al., 2006); the Si iv and Oiv lines at 1393.76 Å and around 1401 – 1405 Å, formed at transition region temperatures (Blanco et al., 1999); and 195 Å images, with a formation temperature of 1.5 MK (Foullon et al., 2004). Cool prominences or filaments can be identified in coronal lines since the line intensity is reduced by means of two different mechanisms: absorption and volume blocking (Anzer and Heinzel, 2005). In the first case, coronal radiation coming from behind the cool structure is partially absorbed, while in the second case the volume filled with cool plasma does not contribute to coronal emission and in this region the radiative output is reduced as compared with the surrounding corona. These two mechanisms give place to a brightness reduction of coronal lines and allows us to identify the volume occupied by cool and dark structures like prominences or filaments. Arguably, oscillations in the dense prominence affect their rarer neighbourhood, so a joint investigation of the dynamics of the two media has a very promising seismological potential.
3.2 Spectral indicators
The vast majority of spectroscopic reports of prominence oscillations are based on the analysis of the Doppler velocity. Some other spectral indicators (line intensity and line width) have also been used in the search for periodic variations in prominences and, sometimes, a periodic signal has been recognized in more than one of these indicators. Landman et al. (1977) observed periodic fluctuations in the integrated line intensity and line width with period around 22 min, but not in the Doppler shift. In addition, Yi et al. (1991) detected periods of 5 and 12 min in the power spectra of the line-of-sight velocity and the line intensity. Also, Suematsu et al. (1990) found signs of a ∼ 60 min periodic variation in the Doppler velocity, line intensity, and line width. Nevertheless, the Doppler signal also displayed shorter period variations (with periods around 4 and 14 min) which were not present in the other two data sets. We here encounter a common feature of other investigations, namely that the temporal behaviour of various indicators corresponding to the same time series of spectra do not agree, either because they show different periods in their power spectra (as in Tsubaki et al., 1987) or because one indicator presents a clear periodicity while the others do not (Wiehr et al., 1984; Tsubaki and Takeuchi, 1986; Balthasar et al., 1986; Tsubaki et al., 1988; Suetterlin et al., 1997). Only rarely have the oscillations been detected in several of these spectral indicators at the same time and with the same period, which constitutes a puzzling feature of prominence oscillations. This can be caused by insufficient instrumental sensitivity or by the effect different waves have on the plasma parameters (pressure, magnetic field, ...), which in turn may give rise to perturbations of one spectral indicator alone. In addition, Harvey (1969) failed to detect periodic perturbations in the line-of-sight component of the magnetic field in a set of prominences that displayed oscillations of the Doppler velocity. He attributed this to the fact that variations in this magnetic field component were below the observational limit.
Special mention must be made of the study performed by Balthasar and Wiehr (1994), who simultaneously observed the spectral lines He at 3888 Å, H8 at 3889 Å and Ca+IR3 at 8498 Å. From this information they analyzed the temporal variations of the thermal and non-thermal line broadenings, the total H8 line intensity, the He 3888 Å to H8 emission ratio and the Doppler shift of the three spectral lines. The power spectra of all these parameters yield a large number of power maxima, but only two of them (with periods of 29 and 78 min) are present in more than one indicator.
The interpretation of the results just summarized appears difficult. First, the theoretical models (see Section 4) can give the temporal behaviour of the plasma velocity, the density, and other physical parameters, in a prominence. The observations, however, yield information on quantities such as the line intensity or the line width. Hence, a clear identification of the spectral parameters with physical variables (density, pressure, temperature, magnetic field strength) is required before further progress can be achieved. Then, the presence of a certain period in more than one signal could be used to infer the properties of the MHD wave involved. Another useful source of information could be the detection of a given period in one signal but not in the others, such as reported in some works discussed above.
3.3 Detected periods
Early observational studies of small amplitude prominence oscillations revealed a wide range of characteristic periods, ranging from a few minutes (Harvey, 1969; Wiehr et al., 1984; Tsubaki and Takeuchi, 1986; Balthasar et al., 1986), to 15 – 25 min (Harvey, 1969; Landman et al., 1977), to 40 – 90 min (Bashkirtsev et al., 1983; Bashkirtsev and Mashnich, 1984; Wiehr et al., 1984; Balthasar et al., 1986). The apparent tendency of periods to group below 10 min or in the range 40 – 90 min led to the distinction between short- and long-period oscillations to refer to these two period ranges. Later, more reports of periods in the range 10 – 40 min were published (e.g., Yi et al., 1991; Suetterlin et al., 1997; Blanco et al., 1999; Régnier et al., 2001) and the intermediate-period class emerged. However, this classification (solely based on the period value) is far from complete: Balthasar et al. (1993) observed a prominence simultaneously with the GCT and VTT telescopes in Tenerife to remove doubts about the instrumental or atmospheric origin of prominence oscillations and obtained strong power in the Doppler shift from both telescopes with period around 30 s; hence, very short-period small amplitude oscillations also exist. Furthermore, a few works in which prominences have been observed from space during extended time intervals show that very long-period oscillations also exist: Pouget et al. (2006) detected periodicities of 5 – 6 h, while Foullon et al. (2004) and Foullon et al. (2009) have observed variations in EUV filaments with periods around 12 h, and 10 – 30 h, respectively. Although the classification in terms of short-period, long-period, etc. oscillations is still in use, it does not cast any light nor gives any help with regard to the nature, origin, or exciter of the oscillations.
In some occasions, a given prominence has been observed over a few consecutive days and the outcome is that the same period seems to be recovered (Bashkirtsev and Mashnich, 1984; Mashnich and Bashkirtsev, 1990; Suetterlin et al., 1997). This seems to indicate that the overall properties of this prominence did not change much over this time interval. Similar studies have not been later carried out.
Some authors have tried to find correlations of the periods of small amplitude oscillations with other parameters. Harvey (1969) reported a correlation of the period with the unperturbed longitudinal magnetic field, such that long periods are associated with strong field strengths (Figure 3). This dependence is difficult to understand since, other parameters being equal (density, magnetic field line length, etc.), one expects just the inverse behaviour for fast MHD waves, and no dependence of the period on the magnetic field strength for slow MHD waves. Bashkirtsev and Mashnich (1993) claimed that the period of oscillation depends on solar latitude. Only periods above 40 min were included in this study and some 40 observations gathered along more than eight years were taken into account. The question then is whether this latitudinal dependence, if real, is related to the solar activity cycle or not. In a subsequent work by Mashnich and Bashkirtsev (1999) a similar latitudinal dependence was obtained for the quasi-hourly oscillations of the photosphere and chromosphere. The implications of these findings are profound and further checks are essential before their reality is firmly demonstrated.
3.4 Spatial distribution of oscillations
It now appears well established that small amplitude, periodic changes in solar prominences do not normally affect the whole object at a time, but are of local nature instead, and that this conclusion is independent of the oscillatory period. Thus, variations with a given period are seldom reported to occur over the whole prominence (see Tsubaki and Takeuchi, 1986). One case in which a periodic signal is present in all slit positions was presented by Balthasar et al. (1988), who detected long-period oscillations over the whole height of three limb prominences by placing a vertical spectrograph slit on them. In contrast, it is usually found that only a few consecutive points along the slit present time variations with a definite period, while all other points lack any kind of periodicity (e.g., Tsubaki and Takeuchi, 1986; Suematsu et al., 1990; Balthasar et al., 1993; Balthasar and Wiehr, 1994; Suetterlin et al., 1997; Molowny-Horas et al., 1997).
The works mentioned in the previous paragraph use a spectrograph slit to detect oscillations; obviously, a two-dimensional data set is much more advantageous when it comes to ascertaining which part of a prominence is affected by oscillations. Terradas et al. (2002) reported on the propagation of waves over a large region (some 54,000 km by 40,000 km in size) in a limb prominence and high spatial resolution observations with Hinode/SOT (Berger et al., 2008) also show oscillations that affect a small area of a prominence. See also the discussion in Section 3.6.4 of the work by Lin et al. (2007) that gives evidence of coherent Doppler shift oscillations over a rectangular area 3.4 arcsec × 10 arcsec in size.
Other observations with high spatial resolution have shown that individual threads or small groups of threads may oscillate independently from the rest of the prominence with their own periods (Thompson and Schmieder, 1991; Yi et al., 1991). Figure 4 displays some of the results in Yi et al. (1991). It is clear that threads 1, 4, 13, and 14 oscillate in phase with their own period, which ranges from 9 to 14 min. In addition, Tsubaki et al. (1988) obtained successively two time series of spectra by placing the spectrograph slit first at a height of 30,000 km above the solar limb and next 40,000 km above the limb. A group of vertical threads detached from the prominence main body displayed 10-7-min periodic variations at both heights, which was a first indication that threads can oscillate collectively. After these preliminary studies, much attention has been given to the detection of thread oscillations (Lin et al., 2003; Lin, 2005; Lin et al., 2005, 2007, 2009; Okamoto et al., 2007; Ning et al., 2009b,a). Apart from reporting on thread oscillations, these works have also provided detailed information about wave features such as the period, wavelength, and phase speed. Because of the importance of these quantities in the seismology of prominences, these works are discussed in more detail in Section 3.6.4.
There is also some evidence that velocity oscillations are more easily detected at the edges of prominences or where the material seems fainter, while they sometimes look harder to detect at the prominence main body (Tsubaki and Takeuchi, 1986; Tsubaki et al., 1988; Suematsu et al., 1990; Thompson and Schmieder, 1991; Terradas et al., 2002). This result has occasionally been interpreted as the consequence of integrating the velocity signals coming from various moving elements along the line-of-sight. This explanation, however, would imply the presence of broader spectral lines at the positions showing periodic variations, which is not observed, so other explanations are also possible (Suematsu et al., 1990). Mashnich et al. (2009a,b) gave evidence that different parts of filaments may support different periodicities: short-period variations (with periods shorter than 10 min) had coherence scales shorter than 10 arcsec and were detected near the edges of filaments placed close to the Sun’s central meridian. These oscillations, hence, were characterized preferentially by vertical plasma displacements. On the other hand, variations with period around 1 h occured in different positions of the filament and the size of the oscillating area was not larger than 15 – 20 arcsec. In addition, these oscillations had an amplitude that increased by an order of magnitude or more in filaments far from the solar center compared to those near the center of the Sun’s disk. Then, these oscillations showed a mainly horizontal velocity.
More information about the spatial distribution of prominence oscillations comes from Ning et al. (2009b), who detected transverse oscillations of 13 threads in a quiescent prominence observed with Hinode/SOT. These authors found that prominence threads in the upper part of the prominence oscillate independently, whereas oscillations in the lower part of the prominence do not follow this pattern. Furthermore, the oscillatory periods were short (between 210 to 525 s), with the dominant one appearing at 5 min (more information is given in Section 3.6.4). In a subsequent work, Ning et al. (2009a) used the same data set to analyze the motions of two spines in the same quiescent prominence. The spine is synonymous with the horizontal fine structure along the filament axis and is the highest part of the prominence. In the observations of Ning et al. (2009a), the spines showed drifting motions that were removed by the subtraction of a linear trend, which allowed the authors to uncover the existence of oscillations with a very similar period (around 98 min) in both structures. Further insight into the behaviour of the spines comes from a fit of a function A sin(2πt/A + A) exp(At) to the detrended data. Here A is the oscillatory amplitude, A the period, A the oscillatory phase, and -1/A the damping time. The detrended signals and the function fits are displayed in Figure 5, which includes the fitted parameters, that give the following information: from the oscillatory amplitude, the peak velocities of the spines are 1 and 5 km s-1. The periods are almost identical (96.5 and 98.5 min) and the phase difference is 149°, which means that the spines oscillated almost in anti-phase. These results about the period and phase were taken by Ning et al. (2009a) as an indication of a collective behaviour of the two structures. These authors considered an analogy with the transverse MHD oscillations of two cylinders (a problem studied by Luna et al., 2008, and discussed in Section 4.4) and concluded that a coupling of kink-like modes can give the observed behaviour. In particular, the A x mode of the system has motions resembling the anti-phase oscillatory behaviour found by Ning et al. (2009a).
3.5 Wave damping and oscillation lifetime
A visual inspection of the data sometimes reveals the existence of outstanding periodic variations and use of the FFT, or even better the periodogram (which yields an increased frequency resolution), is only necessary to derive a precise value of the period. In such cases it usually turns out that the oscillatory amplitude tends to decrease in time in such a way that the periodicity totally disappears after a few periods (e.g., Landman et al., 1977; Tsubaki and Takeuchi, 1986; Wiehr et al., 1989; Molowny-Horas et al., 1999; Terradas et al., 2002; Lin, 2005; Berger et al., 2008; Ning et al., 2009b,a), just as found in large amplitude oscillations. This is then interpreted as a sign of wave damping, although the specific mechanism has not been commonly agreed on (see Section 5 for a summary of theoretical results on this topic).
Oscillations of prominence threads also display fast attenuation. For example, Lin (2005) detected several periodicities over large areas of a filament, with maximum power at periods of 26, 42, and 78 min. Pronounced Doppler velocity oscillations with 26 min period could only be observed for 2 – 3 periods, after which they became strongly damped.
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 (1991), Molowny-Horas et al. (1997) (about which more details are given in Section 3.6.2), and Terradas et al. (2002) (see Section 3.6-3). On the other hand, Lin (2005) and Lin et al. (2007) (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. (1999) 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 Oiv 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.
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 2). 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
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 8). 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 v‖ ≃ 4.4 km s-1 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
3.6.4 Thread oscillations
Yi et al. (1991) and Yi and Engvold (1991) 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. (1991) 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 (1991) 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 (1991) 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 (2005) found a clear oscillatory pattern in time-slice diagrams along the two thin structures. She determined the following wave properties for thread T1: cp = 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).
To test the coherence of oscillations over a larger area, covering several threads, Lin et al. (2007) 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. (2002) (see Section 3.6-3).
Using data from the Swedish 1-m Solar Telescope in La Palma, Lin et al. (2009) 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 22. 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. (2009) 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. (2009) 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 centerto- 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.
4 Theoretical Aspects of Small Amplitude Oscillations: Periods and Spatial Distribution
The usual interpretation of small amplitude oscillations is that some external agent excites MHD waves in the form of periodic disturbances of the cold plasma. MHD waves can be propagating or standing. In the first case, there is a periodic disturbance of the particles of the prominence plasma that may propagate in the medium. In the second case, the wave is confined to a region with fixed boundaries, thus producing the positive interference of propagating waves. Theoretical models usually consider small amplitude perturbations superimposed on an equilibrium configuration. Then the properties of propagating/standing MHD waves are analyzed. In the case of standing waves, we usually refer to the MHD eigenmodes of the system or to the modes for short.
Following our previous discussion of observations (Section 3.4), oscillations may affect individual threads, groups of threads or even larger areas of a prominence. The wave information (period, wavelength, phase speed, damping time) obtained from the analysis of this kind of events has been presented in Sections 3.5 and 3.6. Given that the main purpose of studying prominence oscillations is to gain a more profound understanding of their nature via seismological studies, it is necessary to study these oscillations theoretically. The information one expects to derive from these works consists of the main wave properties (period, wavelength, phase speed, damping time, spatial distribution, ... ). They can then be compared with the observationally determined values. The theory also allows us to determine the temporal variation of the perturbed magnetic field strength and its orientation, the perturbed density, temperature, etc., which means that these variables constitute another source of comparison with observations that will hopefully be exploited in the near future.
Theoretical works are here divided into five groups that reflect widely different choices of prominence equilibrium models: (a) simple, “toy” prominence models (Section 4.1); (b) models in which the prominence is represented as a plasma slab of finite width surrounded by the solar corona (Section 4.2); (c) line current prominence models (Section 4-3); (d) models of infinitely long prominence threads (Section 4.4); and (e) models concerned with the oscillations of prominence threads of finite length (Section 4.5).
4.1 Oscillations of very simple prominence models
The aim of the works discussed in this section is to follow elementary arguments to derive approximations for the oscillatory period and the polarization of plasma motions of the main modes of oscillation of a prominence. Some of the obtained results correspond to MHD modes studied in more detail in other works (see Section 4.2). One of these works (Joarder and Roberts, 1992b) is concerned with a prominence treated as a plasma slab embedded in the solar corona and with a magnetic field perpendicular to the prominence main axis (Figure 27). Waves are allowed to propagate along the slab. The coordinate system introduced by Joarder and Roberts (1992b) has the x-axis pointing across the prominence (i.e., parallel to the magnetic field), the z -axis in the direction of wave propagation and the y-axis along the prominence. Three MHD modes exist in this configuration: the fast, Alfvén, and slow modes, with motions polarized in the z-, y- and x-directions, respectively. Some of the simple analogies discussed next allow us to derive approximations for the period of these modes.
The fast speed in Equation (5) and the cusp speed in Equation (8) are in general different from the phase speed and the group speed for fast and slow magnetoacoustic waves. Only for very specific directions of propagation are these quantities phase and/or group speeds. Using the same parameters as above together with vA = 28 km s-1, cs = 15 km s-1 and a prominence width equal to one tenth the length of magnetic field lines (i.e., 2xp = 2L/10 = 5000 km), Equation (4) yields the periods Pfast = 26 min, PAlfven = 30 min, and Pslow = 63 min, all of them within the range of observed intermediate- to long-period oscillations in prominences.
This string analogy points out the basic nature of a prominence’s modes of oscillation. Because there are in general three MHD modes, there is a fast hybrid mode, an infinite number of internal fast modes, and an infinite number of external fast modes (Joarder and Roberts, 1992b; Oliver et al., 1993; Roberts and Joarder, 1994). Their respective frequencies are given by Equations (11), (12), (13) and (14) with cpro and ccor, substituted by the prominence and coronal fast speeds. Something similar can be said about Alfvén and slow modes.
Anzer (2009) noted that the field line inclination is expected to be very small and, therefore, Bz1/B x ≪ 1. As a consequence, the period of x-oscillations will be much larger than that of the other two modes, polarized in the y- and z-directions.
Four oscillatory modes can be identified from these elementary considerations, but the restoring forces in the x-direction act in unison to create a single mode, so we are left with the familiar three MHD modes: fast, Alfvén, and slow.
Some values of the periods given by Anzer (2009) are similar to those in previous works: 200 min for the magnetically dominated oscillations in the x-direction, 430 min for the gas pressure driven oscillations and 20 min for the transverse, magnetically driven oscillations.
A further refinement of the string analogy (Joarder and Roberts, 1992b; Roberts and Joarder, 1994) can be introduced by noting that the magnetic field of a prominence is not at 90° with the prominence axis, contrary to the simple models of Figures 23 and 25. Instead, the prominence magnetic field makes an angle ϕ, typically around 20°, with the long axis of the slab. This is not too important for the almost isotropic fast modes, but Alfvén and slow modes propagate mainly along field lines, which in a skewed magnetic configuration are longer than 2L by a factor 1/ sin ϕ ≈ 3. Thus, the periods of these waves become larger by this same factor since the travel time needed for them to travel back and forth between the anchor points increases by 1/ sin ϕ. The result is that the hybrid Alfvén and slow modes can have periods up to 60 min and 5 h, respectively. It has been suggested that the last one may be the cause of the very long-period oscillations observed by Foullon et al. (2004); Pouget et al. (2006).
4.2 Oscillations of prominence slabs
Oliver et al. (1993) provided more insight into the nature of internal and external modes while using the non-isothermal Kippenhahn-Schlüter solution represented in Figure 25. These authors followed the evolution of fast and slow modes in the dispersion diagram when the prominence is slowly removed by taking xp → 0. They noted that the frequency of internal modes, both slow and fast, progressively grows until the modes disappear from the dispersion diagram and, therefore, only external modes remain. The presence of the prominence region thus provides physical support for the existence of internal modes. The same is true for external modes when the corona is gradually removed by making xp → L. A clear distinction then arises between the two types of modes, although it turns out that the fundamental mode is internal and external at the same time, since it survives both when the prominence and the corona are eliminated. For this reason, this mode with mixed internal and external properties was called hybrid by Oliver et al. (1993) and later string by Joarder and Roberts (1993b) because it arises in the string analogy. Nevertheless, internal and external modes are also present in the string analogy (Section 4.1), so perhaps hybrid mode is a better denomination for this solution.
From Oliver et al. (1993) it also appears that the amplitude of perturbations in the prominence is rather small for external modes, a feature that is also present in the string solutions of Figure 24. For this reason it was postulated that they would probably be difficult to detect in solar prominences and that the reported periodic variations are produced by the hybrid and internal modes. In addition, the frequency of internal modes is shown to depend on prominence properties only, while that of hybrid and external modes depends on other physical variables such as the length of field lines. This is in agreement with the approximate Equations (11) to (14).
The essential difference between the equilibrium models in Joarder and Roberts (1992b) and in Oliver et al. (1993) is that gravity is neglected in the former, which results in straight magnetic field lines, while it is a basic ingredient in the later, which results in the curved shape of field lines characteristic of the Kippenhahn-Schlüter equilibrium model. Despite the different shape of field lines, the main features of the oscillatory spectrum are similar and so the influence of gravity and field line shape on the properties of the MHD modes is not too relevant in this kind of configurations.
A study of the oscillatory modes of the Kippenhahn and Schlüter (1957) prominence model was undertaken by Oliver et al. (1992). The equilibrium model is represented in Figure 25 although the corona is omitted. This implies that this work only provides a restricted account of the MHD modes of a slab prominence since there are no hybrid and external solutions in the absence of the corona. Oliver et al. (1992) noted that the three MHD modes possess different velocity orientations. The fast mode is characterized by vertical motions. The Alfvén mode by motions along the filament long axis, and the slow mode by plasma displacements parallel to the equilibrium magnetic field, which in this configuration is practically horizontal and transverse to the prominence. The immediate consequence of this association between modes and velocity polarization is that periodic variations in the Doppler shift are more likely to be detected in filaments near the disk centre for fast modes and in limb prominences for Alfvén and slow modes, depending on the orientation of the prominence with respect to the observer. These features of the MHD modes are retained in other models in which the equilibrium magnetic field is assumed perpendicular to the filament axis (Joarder and Roberts, 1992b, 1993a; Oliver et al., 1993; Oliver and Ballester, 1995, 1996). Nevertheless, the distinction between the three MHD modes is lost when the observed longitudinal magnetic field component is taken into account (Joarder and Roberts, 1993b). Probably, there are no characteristic oscillatory directions associated to the various modes (unfortunately, the issue of velocity polarization in a skewed magnetic equilibrium model has not yet been addressed in the context of prominence oscillations). The actual velocity field in prominences can be substantially more complex than that indicated by investigations based on models with magnetic field purely transverse to the prominence slab.
The previous results rely on models in which the prominence and coronal temperatures are uniform, with a sharp jump of this physical variable from the cool to the hot region at an infinitely thin interface. A smoothed temperature transition between the two domains, representing the prominence-corona transition region (PCTR), was used by Oliver and Ballester (1996) to investigate the MHD modes of a more realistic configuration. Despite the presence of the PCTR in the equilibrium model, internal, external, and hybrid modes are still supported, just like in configurations with two uniform temperature regions. Nevertheless, the PCTR results in a slight frequency shift and in the modification of the spatial velocity distribution so as to decrease the oscillatory amplitude of internal modes inside the prominence. Hybrid modes are not so much affected by the presence of the PCTR because their characteristic wavelength is much longer than the width of the PCTR. Then, the conclusion is that the PCTR may influence the detectability of periodic prominence perturbations arising from internal modes.
Some two-dimensional equilibrium models were considered by Galindo Trejo (1987, 1989a,b, 1998, 2006). The focus of these works was in the stability properties of prominence equilibrium configurations (using the MHD energy principle of Bernstein et al., 1958) and for this reason the author concentrated in the lowest eigenvalue squared. This means that information about all other modes of the system is absent. Galindo Trejo (1987) considered four prominence models, namely those by Kippenhahn and Schlüter (1957), Dungey (1953), Menzel (1951), and Lerche and Low (1980). All these models are isothermal, i.e., they do not incorporate the corona around the prominence plasma. This implies that the important hybrid modes are absent in the analysis. In spite of this, some interesting results were obtained by Galindo Trejo (1987). Here we only mention the most relevant ones. For example, the fundamental mode of the Kippenhahn-Schlüter configuration, whose period is 16 min, has motions polarized mainly across the prominence slab, so it can be associated with the internal slow mode. On the other hand, the fundamental mode of Dungey’s model has horizontal motions mostly along the prominence axis (such as corresponds to Alfvén waves) which are more important at the top of the prominence than at the bottom. The oscillatory period ranges from 55 to 80 min. In the case of Menzel’s model, the lowest frequency eigenmode has a period of 40 min and motions whose amplitude increases with height and oriented across the prominence. The eigenmode of Lerche & Low’s solution presents a greater range of periods (17 – 50 min) and, once more, with horizontal plasma displacements transverse to the prominence axis. Two improvements of this elaborated work can be done: the inclusion of the coronal plasma and the consideration of the oscillatory properties of other modes.
In two subsequent papers the stability of the prominence model of Low (1981) was investigated. In the first one (Galindo Trejo, 1989a) a uniform magnetic field component along the prominence axis was used, whereas in the second one (Galindo Trejo, 1989b) this quantity is not uniform. The author concluded that, as long as this magnetic field component is weak, these different choices of the magnetic configuration do not influence much the period of the fundamental mode, which is in the range 3 – 7 min. The spatial distribution of motions is similar to that found by Galindo Trejo (1987) for Menzel’s and Lerche & Low’s equilibrium models.
The following paper of this series (Galindo Trejo, 1998) is concerned with the prominence model of Osherovich (1985), which is characterized by a surrounding horizontal magnetic field connected with the prominence field. Different values of the equilibrium parameters were used and as a result the fundamental mode has periods that range from 4 to 84 min. Galindo Trejo (1998) found that for small values of the longitudinal magnetic field component large velocity amplitudes predominate in the upper part of the prominence, while the opposite happens for a stronger longitudinal component. The magnetic field shear is also relevant: for a moderate (and hence non-uniform) shear, the fundamental eigenmode is in the intermediate-period range and for a uniform shear long periods are obtained.
Galindo Trejo (2006) investigated the equilibrium solution of Osherovich (1989), that is characterized by an external vertical magnetic field that allows the prominence to be placed on the boundary between two regions of opposite photospheric magnetic polarity. A wide range of periods was obtained in this work (9 – 73 min). Also, horizontal oscillatory motions either along the prominence or almost across it were found. Therefore, it seems that in most configurations studied by Galindo Trejo the fundamental oscillatory mode is a slow mode.
4.3 Oscillations of line current models
A completely different approach, based on line current models of filaments, was taken by van den Oord and Kuperus (1992), Schutgens (1997a,b), and van den Oord et al. (1998) in order to study filament vertical oscillations. They used the model introduced by Kuperus and Raadu (1974), in which the prominence is treated as an infinitely thin and long line, i.e., without internal structure. The interaction of the filament current with the surrounding magnetic arcade and photosphere was taken into account. Furthermore, both normal (NP) and inverse polarity (IP) configurations were considered. When a perturbation displaces the whole line current representing the filament, that remains parallel to the photosphere during its motion, the coronal magnetic field is also disturbed and the photospheric surface current is modified. This restructuring affects the magnetic force acting on the filament current. As a consequence, either this force enhances the initial perturbation and the original equilibrium becomes unstable, or the opposite happens and the system is stable against the initial disturbance. As a further complication, van den Oord and Kuperus (1992), Schutgens (1997a,b), and van den Oord et al. (1998) took into account the finite travel time of the perturbations between the line current and the photosphere and investigated the effect of these time delays on the filament dynamics. For both NP and IP configurations, exponentially growing or decaying solutions were found, which means that perturbations are amplified and the equilibrium becomes unstable, or that oscillations are damped in time and the equilibrium is stable.
Schutgens and Tóth (1999) considered an IP magnetic configuration in which the prominence is not infinitely thin but is represented by a current-carrying cylinder. They solved numerically the magnetohydrodynamic equations assuming that the temperature has a constant value (106 K) everywhere. The inner part of the filament is disturbed by a suitable perturbation that causes the prominence to move like a rigid body in the corona, both vertically and horizontally, undergoing exponentially damped oscillations. Horizontal and vertical motions can be studied separately since they are decoupled. It turns out that the period and damping time of horizontal oscillations are much larger than those of vertical oscillations. Some remarks about the damping mechanism at work in these models is presented in Section 5.6.
4.4 Fine structure oscillations (infinitely long thread limit)
Prominence models considered in Sections 4.1, 4.2, and 4-3 are very simplified representations of solar prominences. They provide us with information about a prominence global oscillatory behaviour, but high resolution observations (see Sections 3.4 and 3.6.4) have given us detailed information about the local oscillatory behaviour of the fine, internal structure of filaments. This has prompted the study of thread oscillations. Two situations can be considered: the simplest one is that of short waves propagating along a thread. By short we mean that the wavelength is much shorter than the thread length, so we refer to this problem as the infinitely long thread limit. On the other hand, the second situation includes propagating waves whose wavelength is comparable to or larger than the length of the thread and standing modes, whose wavelength is of the order of the length of the supporting magnetic tube and thus much larger than the thread length; the works concerned with this second kind of problem are presented in Section 4.5. Other important ingredients uncovered by observations (Sections 3.4 and 3.6.4) are the collective behaviour and the presence of flows in some oscillating threads. These features have been incorporated into some investigations and will be also discussed here.
4.4.1 Individual thread oscillations
This formula for P is based on some assumptions, namely that the thread is much longer than the wavelength, which in turn is much larger than the thread radius (this last approximation is also know as the thin tube limit). Short-wavelength propagating waves in threads have been detected by Lin et al. (2007) (see Section 3.6.4 and Figure 22). The length of the fine structure is around 20 arcsec, the reported wavelength is 3.8 arcsec, and the radius of threads is typically between 0.1 and 0.15 arcsec. We can appreciate that the assumptions made to derive Equation (23) are satisfied in this event.
4.4.2 Collective thread oscillations
Soler et al. (2009a) extracted another conclusion from Equations (26) and (27): the difference between the Alfvén (sound) speed of the threads determines the difference of the flow speeds for the existence of collective behaviour of kink (slow) modes. Therefore, when flows are present in the equilibrium, collective motions can be found even in systems of non-identical threads for very specific combinations of the two flow velocities. These velocities are within the observed values in prominences if threads with not too different temperatures and densities are considered. However, since the flow velocities required for collective oscillations must take very particular values, such a special situation may rarely occur in prominences. This conclusion has important repercussions for future prominence seismological applications, given that if collective oscillations are observed in large areas of a prominence, threads in such regions should possess very particular combinations of temperatures, densities, magnetic field strengths and flows.
4.5 Fine structure oscillations (finite length threads)
Filament threads have been modeled as magnetic flux tubes anchored in the solar photosphere (Ballester and Priest, 1989; Rempel et al., 1999) which are stacked one on top of one another in the vertical and horizontal directions, giving place to the filament body.
Many observations of oscillatory events in threads (see Section 3.6.4) cannot be accounted for by the simple models of Section 4.4 because the obtained results rely on the assumption that the thread length is much larger than the wavelength. Exceptions to this hypothesis are standing waves and propagating waves whose wavelength is of the order of or larger than the thread length. In the models presented in this section a thread is envisaged as a cold, dense condensation that fills the central part of a magnetic tube containing hot coronal plasma and anchored in the solar photosphere. Although this structure has been modeled with some complexity (Ballester and Priest, 1989; Rempel et al., 1999), only oscillations of much simpler thread configurations have been investigated so far. Because the reported thread oscillations are transverse, we here concentrate on works that investigate this kind of motions.
Using the same two-dimensional configuration, Díaz et al. (2001) performed an analytical and numerical study of the behaviour of fast modes when a proper treatment of the boundary conditions at the different interfaces of this thread configuration is included. The main conclusion is that prominence threads can only support a few non-leaky modes of oscillation, those with the lowest frequencies. Also, for reasonable values of the thread length, the spatial structure of the fast fundamental even and odd kink modes is such that the velocity amplitude outside the thread takes large values over long distances (Figure 39). Fast kink modes are associated to normal motions with respect to the thread length (i.e., in the x-direction; see Figure 38). The fundamental kink mode (simply referred to as the kink mode) has a velocity maximum at the thread centre, while its first harmonic (that is, the fundamental odd kink solution) has a node in the same position.
To study the oscillations of the above mentioned configurations, Díaz et al. (2001, 2002) developed a very general, although cumbersome procedure. However, Dymova and Ruderman (2005) considered the same problem and to simplify its study took advantage of the fact that the observed thickness of oscillating threads is orders of magnitude shorter than their length. Taking this into account, Dymova and Ruderman (2005) used the so-called thin flux tube (TT) approximation, that enables a simpler solution for the MHD oscillations of longitudinally inhomogeneous magnetic tubes. Once the partial differential equation for the total pressure perturbation is obtained, a different scaling (stretching of radial and longitudinal coordinates) of this equation inside the tube and in the corona can be performed. Following this procedure, two different equations for the total pressure perturbation inside and outside the flux tube, with well known solutions, are obtained. After imposing boundary conditions, the analytical dispersion relations for even and odd modes were derived and a parametric study was performed. A comparison between the numerical values of the periods obtained with this approach and that of Díaz et al. (2002) points out differences of the order of 1%. The only drawback of the method of Dymova and Ruderman (2005) is that it can be only applied to the fundamental mode with respect to the radial dependence.
Díaz and Roberts (2006) studied the properties of the fast MHD modes of a periodic, Cartesian thread model (see Figure 1 of Díaz and Roberts 2006). This configuration provides a bridge between a structure with a limited number of threads (studied by Díaz et al., 2005, see Figure 43) and a homogeneous prominence with a transverse magnetic field (investigated by Joarder and Roberts, 1992b, see Figure 27). Díaz and Roberts (2006) found that for thread separations of the order of their thickness the only confined modes are those in which large numbers of threads are constrained to oscillate nearly in phase. The spatial structure of these solutions is similar to that of the propagating modes of a homogeneous prominence, with small-scale deviations due to the presence of the dense threads. Their period is equal to\(\sqrt f P\), with P the period of the prominence slab and ƒ the filling factor. The system with a limited number of threads has an even shorter period and a comparison between the different configurations considered by Díaz and Roberts (2006) gives periods of 23.6 min for the homogeneous prominence, between 12.1 and 19-3 min for the system of periodic threads and 5-3 for the four-thread configuration studied by Díaz et al. (2005). Hence, the main conclusion of Díaz and Roberts (2006) is that prominence fine structure plays an important role and cannot be neglected.
Terradas et al. (2008) modeled the transverse oscillations of flowing prominence threads observed by Okamoto et al. (2007) with HINODE/SOT (Section 3.6.4). The kink oscillations of a flux tube containing a flowing dense part, which represents the prominence material, were studied from both the analytical and the numerical point of view. In the analytical case, the Dymova and Ruderman (2005) approach with the inclusion of flow was used, while in the numerical calculations the linear ideal MHD equations were solved. The results point out that for the observed flow speeds there is almost no difference between the oscillation periods when static versus flowing threads are considered, and that the oscillatory period matches that of a kink mode. In addition, to obtain information about the Alfvén speed in oscillating threads, a seismological analysis as described in Section 6.6 was also performed. Also motivated by the observations by Okamoto et al. (2007), Soler and Goossens (2011) have further studied the properties of kink MHD waves propagating in flowing threads. In good agreement with Terradas et al. (2008), the period is seen to be slightly affected by mass flows. When the thread is located near the center of the supporting magnetic tube, and for realistic flow velocities, the effect of the flow on the period is estimated to fall within the error bars from observations. On the other hand, as the thread approaches the footpoint of the magnetic structure, flows introduce differences up to 50% in comparison to the static case. The variation of the amplitude of kink waves due to the flow is additionally analysed by Soler and Goossens (2011). It is found that the flow leads to apparent damping or amplification of the oscillations. During the motion of the prominence thread along the magnetic structure, the amplitude grows as the thread gets closer to the center of the tube and decreases otherwise. This effect might be important, since it would modify the actual observed attenuation, if any physical damping mechanisms is present.
Theoretical models described in this section have considered prominence plasmas as either slabs or cylindrical magnetic flux tubes. Slab models were intended to study the global oscillation properties of prominences, while flux tube models seem to be more appropriate for their application to the fine structure of prominences. Nevertheless, the properties of modes of oscillation like the kink mode have often been first studied in Cartesian geometry and then in cylindrical configurations. A few differences that arise are relevant when comparing the theoretical results to observations.
The theoretical frequencies for the kink mode in Cartesian geometry are above the value obtained for a cylindrical equivalent with the same physical properties. This has been shown by Arregui et al. (2007b), in the context of coronal loop oscillations. By assuming that a kink mode in a cylinder can be modeled in Cartesian geometry by adding a large perpendicular wavenumber, these authors show that in that limit the cylindrical kink mode frequency is recovered. A similar analogy was used by Hollweg and Yang (1988) who derived an expression for the damping time of a surface wave in Cartesian geometry and applied their result to coronal loops in the limit of large perpendicular wave number.
The spatial distribution of the eigenfunctions also differ when one compares, e.g., the kink mode properties in Cartesian and cylindrical geometry. The drop-off rate of the transverse velocity component is faster in cylindrical flux tubes than in slabs. A cylinder is a much better wave guide. For this reason, an oscillating cylindrical thread is less likely to induce oscillations in its neigboring threads than a Cartesian thread.
5 Theoretical Aspects of Small Amplitude Oscillations: Damping Mechanisms
Temporal and spatial damping is a recurrently observed characteristic of prominence oscillations (see Section 3.5). Several theoretical mechanisms have been proposed in order to explain the observed damping. Direct dissipation mechanisms seem to be inefficient, as shown by Ballai (2003), who estimated, through order of magnitude calculations, that several isotropic and anisotropic dissipative mechanisms, such as viscosity, magnetic diffusivity, radiative losses, and thermal conduction cannot in general explain the observed wave damping. The time and spatial damping of linear non-adiabatic MHD waves has been considered by Carbonell et al. (2004, 2009), Terradas et al. (2001), Terradas et al. (2005), Carbonell et al. (2006), and Soler et al. (2007, 2008). The overall conclusion from these studies is that thermal mechanisms can only account for the damping of slow waves in an efficient manner, while fast waves remain almost undamped. Since prominences can be considered as partially ionized plasmas, a possible mechanism to damp fast and Alfvén waves could be ion-neutral collisions (Forteza et al., 2007, 2008), although the ratio of the damping time to the period does not completely match the observations. Besides non-ideal mechanisms, another possibility to attenuate fast waves in thin filament threads comes from resonant wave damping (see, e.g., Goossens et al., 2010), which needs the presence of a smooth radial profile of the Alfvén speed. This phenomenon is well studied for transverse kink waves in coronal loops (Goossens et al., 2006; Goossens, 2008) and provides a plausible explanation for quickly damped transverse loop oscillations first observed by TRACE (Aschwanden et al., 1999; Nakariakov et al., 1999).
The time scales of damping produced by these different mechanisms should be compared with those obtained from observations, that indicate that the ratio of the damping time to the period, τd/P, is of the order of 1 to 4. The theoretical approach of many works about the damping of prominence oscillations has been to first study a given damping mechanism in a uniform and unbounded medium and, thereafter, to introduce structuring and non-uniformity. This has led to an increasing complexity of theoretical models in such a way that some of them now combine different damping mechanisms. Detailed reports on theoretical studies of small amplitude oscillations in prominences and their damping can be found in Oliver (2009), Ballester (2010), and Arregui and Ballester (2011). Here, we collect the most significant aspects of the theoretical mechanisms that have been proposed to explain the observed time-scales.
5.1 Damping of oscillations by thermal mechanisms
In a seminal paper, Field (1965) studied the thermal instability of a dilute gas in mechanical and thermal equilibrium. Using this approach, the time damping of magnetohydrodynamic waves in bounded Kippenhahn-Schlüter and Menzel prominence models was studied by Terradas et al. (2001). Similar studies using prominence slabs embedded in the solar corona were undertaken by Soler et al. (2007) and Soler et al. (2009b).
5.1.1 Non-adiabatic magnetoacoustic waves in prominence slabs
Terradas et al. (2001) studied the radiative damping of quiescent prominence oscillations. They adopted a relatively simple non-adiabatic damping mechanism, by including a radiative loss term based on Newton’s law of cooling with constant relaxation time. The influence of this type of radiative dissipation on the normal modes of Kippenhahn-Schlüter and Menzel quiescent prominence models was analyzed. The normal modes of these configurations had previously been investigated by Oliver et al. (1992) and Joarder and Roberts (1993a); cf. Section 4.2. In a Kippenhahn-Schlüter prominence model, the fundamental slow mode is unaffected by radiation, but its harmonics are strongly damped. On the other hand, in a Menzel prominence configuration all slow modes are characterized by short damping times. The damping time depends on the curvature of field lines, in such a way that more curved models produce larger damping times. In both prominence models, fast modes are practically unaffected by radiative losses and have very long damping times.
A more involved analysis was performed by Soler et al. (2007) by including thermal conduction, optically thin or thick radiation, and heating in the energy equation. The prominence was modeled as a plasma slab embedded in an unbounded corona and with a magnetic field oriented along the direction parallel to the slab axis (see Figure 26); this is the equilibrium configuration of Joarder and Roberts (1992a), whose normal modes have been discussed in Section 4.2. Soler et al. (2007) found that radiation losses have a different effect on magnetoacoustic waves depending on their wavenumber. For typical values of observed wavelengths, the internal slow mode is attenuated by radiation from the prominence plasma, the fast mode by the combination of prominence radiation and coronal conduction and the external slow mode by coronal conduction. This study highlights the relevance of the coronal physical properties on the damping properties of fast and external slow modes, whereas this aspect does not affect the internal slow modes at all. For thin slabs, representing a fine thread, Soler et al. (2007) found that the fast mode is less attenuated, whereas both internal and external slow modes are not affected by non-adiabatic damping mechanisms.
Damping of magnetoacoustic waves in slab prominence models with a transverse magnetic field (see Figure 27 and Section 4.2 for a description of the normal modes) were studied by Soler et al. (2009b). The most relevant damping processes are coronal thermal conduction and radiative losses from the prominence plasma. In terms of the spatial distribution of the studied normal modes, it was found that both mechanisms govern together the attenuation of hybrid modes, whereas prominence radiation is responsible for the damping of internal modes and coronal conduction essentially dominates the attenuation of external modes. In terms of the different magnetohydrodynamic wave types, slow modes were found to be efficiently damped, with damping times compatible with observations. On the contrary, fast modes are less attenuated by non-adiabatic effects and their damping times are several orders of magnitude larger than those observed. The inclusion of the coronal medium in the analysis causes a decrease of the damping times compared to those of an isolated prominence slab, but this effect is still insufficient to obtain fast mode damping times compatible with observations.
5.1.2 Non-adiabatic magnetoacoustic waves in a single thread with mass flows
Soler et al. (2008) investigated the effects of both mass flow and non-adiabatic processes on the oscillations of an individual prominence thread modeled as an infinite homogeneous cylinder Figure 32). Thermal conduction and radiative losses were taken into account as damping mechanisms. For a discussion of the oscillatory features of this system, see Section 4.4.1.
The analysis of the damping time-scales for the different wave types shows that slow and thermal modes are efficiently attenuated by non-adiabatic mechanisms. On the contrary, fast kink modes are much less affected and their damping times are much larger than those observed. These results are compatible with the known damping properties of these waves in the absence of flows.
Although the presence of steady mass flows improves the efficiency of non-adiabatic mechanisms on the attenuation of transverse kink oscillations for propagation parallel to the flow, its effect is still not enough to obtain damping times compatible with observations.
5.1.3 Non-adiabatic magnetoacoustic waves in a two-thread system with mass flows
As concluded in Section 5.1.2, slow wave damping can be explained by thermal mechanisms. The right panel of Figure 47 shows the damping ratios of the S z and A z solutions versus the distance between the two threads. Slow modes in a threaded prominence are efficiently attenuated by non-adiabatic mechanisms. Note that τd/P is almost independent of the thread separation and the mode because the two threads oscillate independently in the S z and A z modes. Time-scales τd/P ≈ 5 are obtained, which is in agreement with previous studies (Soler et al., 2007, 2008) and consistent with observations.
Soler et al. (2009a) concluded that collective slow modes are efficiently damped by thermal mechanisms, with damping ratios similar to those reported in observations, while collective fast waves are poorly damped. This is a key point since efficiently damped transverse oscillations have been observed, which could suggest that other attenuation mechanisms could be at work.
5.2 Damping of oscillations by ion-neutral collisions
Since the temperature of prominences is typically of the order of 104 K, the prominence plasma is only partially ionized. The exact ionization degree of prominences is unknown and the reported ratio of electron density to neutral hydrogen density (see, e.g., Patsourakos and Vial, 2002) covers about two orders of magnitude (0.1 – 10). Partial ionization brings the presence of neutrals in addition to electrons and ions, thus collisions between the different species are possible. Because of the occurrence of collisions between electrons with neutral atoms and ions, and more importantly between ions and neutrals, Joule dissipation is enhanced when compared with the fully ionized case. A partially ionized plasma can be represented as a single-fluid in the strong coupling approximation, which is valid when the ion density in the plasma is low and the collision time between neutrals and ions is short compared with other time-scales of the problem. Using this approximation it is possible to describe the very low frequency and large-scale fluid-like behaviour of plasmas (Goossens, 2003).
5.2.1 Homogeneous and unbounded prominence medium
Several studies have considered the damping of MHD waves in partially ionized plasmas of the solar atmosphere (De Pontieu et al., 2001; James et al., 2003; Khodachenko et al., 2004; Leake et al., 2005). In the context of solar prominences, Forteza et al. (2007) derived the full set of MHD equations for a partially ionized, one-fluid hydrogen plasma and applied them to the study of the time damping of linear, adiabatic fast and slow magnetoacoustic waves in an unbounded prominence medium. This study was later extended to the non-adiabatic case by including thermal conduction by neutrals and electrons and radiative losses (Forteza et al., 2008). The main effects of partial ionization on the properties of MHD waves manifest through a generalized Ohm’s law, which adds some extra terms in the resistive magnetic induction equation, in comparison to the fully ionized case. Forteza et al. (2007) considered a uniform and unbounded prominence plasma and found that ion-neutral collisions are more important for fast waves, for which the ratio of the damping time to the period is in the range 1 to 105, than for slow waves, for which values between 104 and 108 are obtained. Fast waves are efficiently damped for moderate values of the ionization fraction, while in a nearly fully ionized plasma, the small amount of neutrals is insufficient to damp the perturbations.
A hydrogen plasma was considered in the above studies, but 90% of the prominence chemical composition is hydrogen and the remaining 10% is helium. The effect of including helium in the model of Forteza et al. (2008) was assessed by Soler et al. (2010b). The species present in the medium are electrons, protons, neutral hydrogen, neutral helium (He i) and singly ionized helium (He ii), while the presence of He iii is neglected (Gouttebroze and Labrosse, 2009).
The thermal mode is a purely damped, non-propagating disturbance (ωR = 0), so only the damping time, τd, is plotted (Figure 48d). We observe that the effect of helium is different in two ranges of k. For k > 10-4 m-1, thermal conduction is the dominant damping mechanism, so the larger the amount of helium, the shorter τd because of the enhanced thermal conduction by neutral helium atoms. On the other hand, radiative losses are more relevant for k < 10-4 m-1. In this region, the thermal mode damping time grows as the helium abundance increases. Since these variations in the damping time are very small, we again conclude that the damping time obtained in the absence of helium does not significantly change when helium is taken into account. Therefore, the inclusion of neutral or single ionized helium in partially ionized prominence plasmas does not modify the behaviour of linear, adiabatic or non-adiabatic MHD waves already found by Forteza et al. (2007) and Forteza et al. (2008).
5.2.2 Cylindrical filament thread model
Soler et al. (2009c) applied the equations derived by Forteza et al. (2007) to the study of MHD waves in a partially ionized filament thread modeled as an infinite cylinder with radius a embedded in the solar corona (see Figure 32). As in Forteza et al. (2007), the one-fluid approximation for a hydrogen plasma was considered. The internal and external media are characterized by their densities, temperatures, and their own relative densities of neutrals, ions and electrons. The contribution of the electrons is neglected. The coronal medium is considered as fully ionized, while the ionization fraction in the prominence plasma, \(\tilde \mu _p\), is allowed to vary.
The presence of critical wavenumbers is also found in the case of transverse kink waves (middle panels of Figure 49). Within the range of observed wavelengths, the phase speed closely corresponds to its ideal counterpart, ck = ω/k z , so non-ideal effects are irrelevant for wave propagation. The behaviour of the damping rate as a function of wavelength and ionization fraction is seen to closely resemble the result obtained for Alfvén waves, with τd/P > 10 in the range of observed wavelengths. Therefore, neither ohmic diffusion nor ion-neutral collisions seem to provide damping times as short as those observed for kink waves in filament threads. Only for an almost neutral plasma, with \(\tilde \mu _p\) > 0.95, the obtained damping rates are compatible with the observed time-scales. Just like for Alfvén waves, ohmic diffusion dominates for small wavenumbers, while ion-neutral collisions are the dominant damping mechanism for large wavenumbers.
Regarding slow waves (bottom panels of Figure 49), Soler et al. (2009c) concentrated their analysis on the radially fundamental mode with m = 1, since the behaviour of the slow mode is weakly affected by the value of the azimuthal wavenumber. Slow wave propagation is constrained by only one critical wavenumber, that strongly depends on the ionization fraction, in such a way that for k z below this critical wavenumber the wave is totally damped. More importantly, for large enough values of the ionization fraction, the corresponding critical wavelength lies in the range of observed wavelengths of filament oscillations. As a consequence, the slow wave might not propagate in filament threads under certain circumstances. As for the damping rate, it is found that ion-neutral collisions are a relevant damping mechanism for slow waves, since very short damping times are obtained, especially close to the critical wavenumber. By comparing the particular effects of ohmic diffusion and ion-neutral collisions, the slow mode damping is seen to be completely dominated by ion-neutral collisions. Ohmic diffusion is found to be irrelevant, since the presence of the critical wavenumber prevents slow wave propagation for small wavenumbers, where ohmic diffusion would start to dominate.
5.3 Resonant damping of infinitely long thread oscillations
The phenomenon of resonant wave damping in non-uniform plasmas has been extensively studied in connection to wave-based coronal heating mechanisms (Ionson, 1978) and the damping of transverse coronal loop oscillations (Hollweg and Yang, 1988; Ruderman and Roberts, 2002; Goossens et al., 2002, 2010). The mechanism relies in the energy transfer from the transverse kink mode to small scale Alfvén waves because of the plasma inhomogeneity at the transverse spatial scales of the structures. This idea was put forward by Arregui et al. (2008b), whose analysis is restricted to the damping of kink oscillations due to the resonant coupling to Alfvén waves in a pressureless (zero plasma-β) plasma. It was extended to the case in which both the Alfvén and the slow resonances are present by Soler et al. (2009e). Here we discuss the main results from these works, whose aim is to assess the damping properties of resonant absorption. For this reason, the considered configurations are based on the infinitely long thread model of Figure 31.
5.3.1 Resonant damping in the Alfvén continuum
MHD waves of axisymmetric one-dimensional cylindrical flux tubes are characterized by two wavenumbers, i.e., the azimuthal wavenumber, m, and the axial wavenumber, k z . They can have different nodes in the radial direction. Arregui et al. (2008b) concentrated their analysis on the radially and longitudinally fundamental transverse wave with azimuthal number m = 1, the kink mode. This eigenmode is consistent with the detected Doppler velocity variations (see Section 3.6.4) and their associated transverse motions, discussed in Section 4.4.1. The frequency of this mode is not influenced by the presence of a layer with small thickness, so the result of Section 4.4.1 is approximately correct; see Equation (22).
Resonant damping in the Alfvén continuum appears to be a very efficient mechanism for the attenuation of transverse thread oscillations, especially because large density contrasts and transverse plasma inhomogeneities are combined together.
5.3.2 Resonant damping in the slow continuum
Although the plasma-β in solar prominences is probably small, it is definitely non-zero. Soler et al. (2009e) showed that, in prominence plasmas, resonant damping of kink waves can additionally be produced due to the coupling to slow continuum waves. In the context of coronal loops, which are presumably hotter and denser than the surrounding corona, the ordering of sound, Alfvén and kink speeds does not allow for the simultaneous matching of the kink frequency with both Alfvén and slow continuum frequencies. Because of their relatively higher density and lower temperature conditions, this becomes possible in the case of prominence threads. Therefore, the kink mode phase speed is also within the slow (or cusp) continuum, which extends between the internal and external sound speeds, in addition to the Alfvén continuum. By considering gas pressure in the cylindrical thread model of Arregui et al. (2008b), Soler et al. (2009e) evaluated the contribution of the damping due to the slow continuum to the total resonant damping of the kink mode.
5.4 Resonant damping in partially ionized infinitely long threads
5.4.1 Temporal damping
Damping by resonant absorption in a partially ionized prominence plasma was studied by Soler et al. (2009d), who integrated both mechanisms in a non-uniform cylindrical prominence thread model in order to assess their combined effects. Partial ionization is relevant for the damping of short wavelength fast waves (Forteza et al., 2007), while resonant damping of kink waves is efficient whenever a transverse density inhomogeneity is present. The question arises on whether partial ionization affects the mechanism of resonant absorption and vice versa.
The model adopted by Soler et al. (2009d) has the magnetic and density structuring of the models used in Section 5.3 (see Figure 50), but the plasma properties are also characterized by the ionization fraction, \(\tilde \mu\). The radial behaviour of the ionization fraction in threads is unknown, so Soler et al. (2009d) assumed a one-dimensional transverse profile akin to the one employed to model the equilibrium density. The thread ionization fraction, \(\tilde \mu _p\), is considered a free parameter and the corona is assumed to be fully ionized, so \(\tilde \mu _c\) = 0.5. The non-uniform transitional layer of thickness l therefore connects two plasmas with densities ρp and ρc and ionization degrees \(\tilde \mu _p\) and \(\tilde \mu _c\). Soler et al. (2009d) used the one-fluid approximation and, for simplicity, the β = 0 limit, which excludes slow waves. The quantities η, ηC and ηH are here functions of the radial direction.
The analytical estimates described above can be verified and extended by numerically solving the full eigenvalue problem. This approach allowed Soler et al. (2009d) to additionally include Hall’s diffusion in addition to ohmic and Cowling’s dissipation. In their study, these authors first considered a configuration with an abrupt density variation across the thread boundary (that is, l = 0), which prevents resonant absorption from working. Next, they included the thin transitional layer between the thread and the corona, so that both resonant absorption and ion-neutral effects are at work.
5.4.2 Spatial damping
Motivated by the spatially damped propagating waves observed by Terradas et al. (2002) (see Section 3.6.3), the spatial damping of linear non-adiabatic magnetohydrodynamic waves in a homogenous, unbounded, magnetized, and fully ionized plasma was studied by Carbonell et al. (2006). The spatial damping in a flowing partially ionized plasma has been studied by Carbonell et al. (2010). Carbonell et al. (2006) found that the thermal (fast) wave shows the strongest (weakest) spatial damping. For periods longer than 1 s the spatial damping of magnetoacoustic waves is dominated by radiation, while at shorter periods the spatial damping is dominated by thermal conduction. Therefore, radiative effects on linear magnetoacoustic slow waves can be a viable mechanism for the spatial damping of short period prominence oscillations, while thermal conduction does not play any role. On the other hand, Carbonell et al. (2010) found that in the presence of a background flow, new strongly damped fast and Alfvén waves appear whose features depend on the joint action of flow and resistivity. The damping lengths of adiabatic fast and slow waves are strongly affected by partial ionization, which also modifies the ratio between damping lengths and wavelengths. For non-adiabatic slow waves, the unfolding in both wavelength and damping length induced by the flow allows efficient damping for periods compatible with those observed in prominence oscillations. In the case of non-adiabatic slow waves and within the range of periods of interest for prominence oscillations, the joint effect of both flow and partial ionization leads to efficient spatial damping of oscillations. For fast and Alfvén waves, the most efficient damping occurs at very short periods not compatible with those observed in prominence oscillations.
For typically reported periods of thread oscillations, resonant absorption is an efficient mechanism for the kink mode spatial damping, while ion-neutral collisions have a minor role. Cowling’s diffusion dominates both the propagation and damping for periods much shorter than those observed, while resonant absorption could explain the observed spatial damping of kink waves in prominence threads.
5.5 Resonant damping in partially ionized finite length threads
The results described in Sections 5.3, 5.4.1, and 5.4.2 indicate that, because of the coupling of kink waves to Alfvén waves, resonant absorption constitutes a very plausible mechanism for the explanation of the observed spatial and time decay of transverse oscillations. The main limitation of these studies is that they adopt a one-dimensional density model that might not be appropriate in view of the longitudinal structuring of prominence threads. This led Soler et al. (2010a) to investigate the time damping properties of two-dimensional thread models, that is, with density inhomogeneity across the thread and along the magnetic tube in which it is contained. In this study, resonant absorption and damping by partial ionization effects were considered simultaneously.
As a numerical example, in the case m = 1, Lp/L = 0.1, L = 107 m and l/a = 0.2, the damping ratio is τd/P ≈ 3.18 for a fully ionized thread (\(\tilde \mu _p\) = 0.5) and τd/P ≈ 3.16 for an almost neutral thread (\(\tilde \mu _p\) = 0.95). Note that the obtained damping times are consistent with the observations. Moreover, as seen in Section 5.4.1, the contribution of resonant absorption to the damping is much more important than that of Cowling’s diffusion, so the ratio τd/P depends only very slightly on the ionization degree and the second term on the right-hand side of Equation (35) can in principle be neglected.
Soler et al. (2010a) find that for l ≠ 0 and under the thin tube and thin boundary approximations, the period and damping time by resonant absorption have the same dependence on L e - and L e + . This means that for resonant absorption the damping ratio does not depend on these quantities. Since resonant damping dominates over Cowling’s diffusion, this leads to the conclusion that when considering both damping mechanisms, the damping ratio will be almost unaffected by the position of the prominence region within the fine structure.
The accuracy of the above analytical solutions can be assessed by numerically solving the general dispersion relation derived by Soler et al. (2010a). Here we only show the results obtained by Soler et al. (2010a) for the case in which the prominence thread is centered in the tube.
In summary, the dominant damping mechanism is resonant absorption, which provides damping ratios in agreement with the observations, whereas ion-neutral collisions are irrelevant for the damping. The values of the damping ratio are independent of both the prominence thread length and its position within the magnetic tube, and coincide with the values for a tube fully filled with the prominence plasma. A recent study that further analyses resonant damping of thread oscillations in two-dimensional equilibrium models can be found in Arregui et al. (2011). These authors additionally analyzed the influence of the density in the evacuated part of the thread. This quantity is seen to influence periods and damping times, but has little influence on the damping rate of transverse thread oscillations. The implications of some of these results for the determination of physical properties in transversely oscillating threads are discussed in Section 6.
5.6 Damping by wave leakage
However, the exact nature of the damping mechanism should be pointed out, and Schutgens and Tóoth (1999) suggest that the damping of oscillations is due to the emission of waves by the prominence, i.e., wave leakage. The damping of horizontal motions is attributed to the emission of slow waves, whereas fast waves are invoked as the cause of the damping of vertical motions. Taking into account that the main difference between this work and those of van den Oord and Kuperus (1992), Schutgens (1997a,b), and van den Oord et al. (1998) lies essentially in the cross section of the filament, it seems that the physics involved should be the same, so wave leakage should be the mechanism responsible for the accounted damping. However, in Schutgens and Tóoth (1999), the plasma-β in the prominence ranges from β > 1 in its central part to β < 0.1 at its boundary. Hence, waves emitted by the prominence into the corona propagate in a β ≪ 1 environment in which magnetic field lines are closed. Under these conditions, slow modes propagate along magnetic field lines and are unable to transfer energy from the prominence into the corona and so wave leakage in the system studied by Schutgens and Tóoth (1999) is only possible by fast waves. Then, it is hard to understand how the prominence oscillations can be damped by the emission of slow waves in this particular model, in which the dense, cool plasma is only allowed to emit fast waves. It must be mentioned, however, that the plasma-.. in the corona increases with the distance from the filament, which implies that the emitted fast waves can transform into slow waves when they traverse the β ≃ 1 region. This effect has been explored by McLaughlin and Hood (2006) and McDougall and Hood (2007); see also references therein for similar studies.
6 Prominence Seismology
Solar atmospheric seismology aims to determine physical parameters that are difficult to measure by direct means in magnetic and plasma structures. It is a remote diagnostics method that combines observations of oscillations and waves in magnetic structures, together with theoretical results from the analysis of oscillatory properties of given theoretical models. The philosophy behind this discipline is akin to that of Earth seismology, the sounding of the Earth interior using seismic waves, and helio-seismology, the acoustic diagnostic of the solar interior. It was first suggested by Uchida (1970) and Roberts et al. (1984), in the coronal context, and by Tandberg-Hanssen (1995) in the prominence context. The increase in the number and quality of high resolution observations in the 1990s has lead to the rapid development of solar atmospheric seismology. In the context of coronal loop oscillations, recent applications of this technique have allowed the estimation and/or restriction of parameters such as the magnetic field strength (Nakariakov and Ofman, 2001), the Alfvén speed in coronal loops (Zaqarashvili, 2003; Arregui et al., 2007a; Goossens et al., 2008), the transversal density structuring (Goossens et al., 2002; Verwichte et al., 2006) or the coronal density scale height (Andries et al., 2005; Verth et al., 2008).
The application of inversion techniques to prominence seismology is less developed. This is due to the complexity of these objects in comparison to, e.g., coronal loops. The recent refinement of theoretical models that incorporate the fine structuring of prominences and the high resolution observations of small amplitude oscillations have produced an increase in prominence seismology studies. Several techniques for the inversion of physical parameters have been developed that make use of observational estimates for quantities such as phase velocities, periods, damping times, and flow speeds. In general, the solution to the inverse problem cannot provide a single value for all the physical parameters of interest. However, important information about unknown physical quantities can be obtained using this method. The most relevant results of the MHD prominence seismology technique are here discussed.
The theoretical models decribed in Section 4 make use of different conceptual views of prominences, such as the string model, the slab model, and the thread model for their fine structure. Seismology efforts in the area have followed the same pattern. We describe them in increasing intricacy order, starting with a mechanical analogue (Section 6.1), followed by slab models (Section 6.2), and ending with the seismology of fine structure oscillations (Sections 6.3 to 6.6).
6.1 Seismology of large amplitude prominence oscillations
Several studies have made use of the observed characteristics of large amplitude oscillations in prominences to deduce physical parameters of these objects. The classic example is the interpretation by Hyder (1966) of the winking filament phenomenon in terms of a global mode of the prominence. This author modeled the eleven winking filament events reported by Ramsey and Smith (1965) as damped harmonic oscillators and obtained estimates of the vertical magnetic field strength in the range 2 – 30 G. More recent studies have also used large amplitude oscillations in filaments to deduce the magnetic field strength in these objects.
Vršnak et al. (2007) reported on Hα observations of periodical plasma motions along the axis of a filament. The motions were both large amplitude and large scale, with an initial displacement of 24 Mm, an initial velocity amplitude of 51 km s-1, a period of 50 min, and a damping time of 115 min. Oscillations were interpreted as a global mode of the system and the driver was thought to be the magnetic flux injection by magnetic reconnection at one of the filament legs. Although oscillatory motions along the prominence axis were also reported by Jing et al. (2003, 2006), the study by Vršnak et al. (2007) proposes an explanation for the triggering process and the restoring force, and performs diagnostics based on these interpretations.
The seismology analysis by Vršnak et al. (2007) is based on the fitting of the oscillation properties to a mechanical analogue model in terms of the classic damped harmonic oscillator equation. This analogue is first used to discard gas pressure as the restoring force, since it leads to sound speed values one order of magnitude larger than those corresponding to the typical temperature of prominence plasmas, and no signature of plasma at those temperatures was observed in TRACE EUV images. In this work a twisted flux rope model with both axial and azimuthal magnetic field components was considered and an excess azimuthal field at one of the prominence legs was assumed. This gives rise to a magnetic pressure gradient and a torque, which in turn drive a combined axial and rotational motion of the plasma. Next, an expression that relates the azimuthal Alfvén speed, v Aϕ , and the oscillatory period was obtained. From this relation, the Alfvén speed v Aϕ ∼ 100 km s-1 was inferred. By further assuming that the number density of the prominence plasma is in the range 1010 – 1011 cm-3, the azimuthal magnetic field strength results in the range 5 – 15 G. By measuring the pitch angle, Vršnak et al. (2007) additionally determined the internal structure of the flux rope helical magnetic field, from which the axial magnetic field strength was estimated to be in the range 10 – 30 G.
The twisted flux rope model was also invoked by Pintér et al. (2008) in their analysis of SoHO EUV observations of large amplitude transverse oscillations in a polar crown filament previously studied by Isobe and Tripathi (2006). Oscillations were present along a foot belonging to a larger prominence structure and occurred prior to the eruption of the full structure. Wavelet analysis tools were used to shed light into the temporal and spatial behaviour of oscillations. The filament oscillated as a rigid body with a period of 2.5 h, that was constant along the filament, but decreased in time. The line-of-sight velocity was estimated to be about a few tens of km s-1. The analysis of the spatial properties of the oscillations shows evidence of a global standing transverse oscillation, although some small scale oscillations within the structure cannot be discarded. Using the twisted flux rope model for the filament and based on the same scenario and analysis as Vršnak et al. (2007), the azimuthal Alfvén speed component was estimated to be v Aϕ = 49 km s-1 and the axial magnetic field strength in the range 2 – 10 G. In this case, the pitch angle could not be measured. By assuming a mean value of 65°, Pintér et al. (2008) estimated that the axial component of the magnetic field must be in the range 1 – 5 G.
6.2 Seismology of prominence slabs
The MHD wave properties for slab models of prominences are described in Section 4.2. Two relevant studies have made use of some of these models to infer physical properties in prominences. Their methodology is based on the identification of observed oscillations with theoretical eigenmodes.
Régnier et al. (2001) consider the possible theoretical modes that can explain their observations of oscillations in an active region filament. The slab model with a uniform and inclined magnetic field by Joarder and Roberts (1993b) is used (see Figure 29). The dispersion relations for Alfvén modes and magnetoacoustic modes are considered. They provide the frequency of six fundamental modes: the symmetric Alfvén, slow and fast kink modes and the antisymmetric Alfvén, slow and fast sausage modes, as a function of the prominence parameters. Observations provide with estimates for the width (8000 km) and length (63,000 km) of the filament. Assumptions on other parameters, such as the temperature of the filament (8000 K) and of its environment (106 K), the density of the slab (1012 cm-3), the magnetic field strength (20 G) and for the angle between the magnetic field and the long axis of the slab (25°) are made. The dispersion relations are then solved by using these parameters and the corresponding periods are obtained and classified.
Observations and Fourier analysis of Doppler velocity time series enable Régnier et al. (2001) to detect intermediate (between 5 and 20 min in this case) and long (> 40 min) period oscillations. From the comparison between the observed and calculated frequencies, an identification method of the oscillation modes in the observed filament is presented. The method makes use of the fact that the frequency ratio of the fundamental even Alfvén mode to the fundamental odd Alfvén mode only depends on the ratio of the half-with of the slab to the half-length of the filament. This quantity is measurable. The same applies to the frequency ratios involving the slow kink/sausage and fast kink/sausage modes. Parametric calculations for the frequencies as a function of the magnetic field strength and the inclination angle, while keeping the slab density constant, are next performed. A diagnostic of the observed filament is obtained by looking for the parameters values that enable the matching of theoretical and observed frequencies. By following this method, the angle between the magnetic field and the long axis of the slab is estimated to be 18°. Using this value, an algebraic relation for the magnetic field strength as a function of the slab density is derived.
A more involved and ambitious diagnostic, using the Joarder and Roberts (1993b) slab model, was performed by Pouget et al. (2006). The long duration and high temporal resolution observations with CDS/SoHO enable these authors to detect and measure the entire range of periodicities theoretically expected in a filament. In particular both the short (less that 10 min) and the long ones (more than 40 min) are detected.
The detailed analysis of three filaments is presented. The seismic inversion technique closely follows that by Régnier et al. (2001), in the sense that the first step towards the diagnostic is the use of frequency ratios between fundamental even/odd (kink/sausage) modes. These ratios only depend on the ratio of the filament half-width to its half-length. Once this ratio is measured, with a given uncertainty, Pouget et al. (2006) assume that their 16-h long observation has allowed them to observe the six modes of interest, since the slowest mode is expected at a period of 5 h, for standard prominence parameters.
The inversion method first assigns a possible triplet of measured frequencies to the 3 odd fundamental frequencies (odd Alfvén, slow sausage, and fast sausage modes). The coherence of each choice is examined against two tests. The first requires to find three corresponding even frequencies, with the condition that the even/odd frequency ratios are consistent with the measured half-width to half-length ratio. The second involves the inferred values for the density, temperature, magnetic field inclination angle, and magnetic field strength to be consistent with typical values reported in the literature. For each test, if the test was negative, the full triplet was changed and the series started again. On the contrary, if the tests succeeded, Pouget et al. (2006) considered that the six fundamental modes were identified.
The three filament observations led to coherent diagnostics and a single possible set of frequencies was found for each observation. The importance of this study is its ability to simultaneously determine the values of the inclination angle, temperature, and Alfvén speed for the same prominence. The drawback is that the modeling, as in Régnier et al. (2001), does not permit to capture the highly inhomogeneous nature of prominences.
6.3 Seismology of propagating transverse thread oscillations
Transverse thread oscillations observed by Lin et al. (2009) and discussed in Section 3.6.4 show evidence of waves propagating along individual threads. Ten of the swaying threads were chosen by Lin et al. (2009) for further investigation, and for each selected thread two or three perpendicular cuts were made in order to measure the properties of the propagating waves. Periods and amplitudes of the waves, as well as their phase velocity, were derived for each thread. Lin et al. (2009) interpreted the observed events as propagating MHD kink waves supported by the thread body. This mode is the only one producing a significant transverse displacement of the cylinder axis. In addition, it also produces short-period oscillations of the order of minutes, compatible with the observed periods (see Section 4.4.1).
The inferred values of vAp for the ten selected threads are displayed in Table 2 in Lin et al. (2009). The results show a strong dispersion, suggesting that the physical conditions in different threads were very different in spite of belonging to the same filament. This result clearly reflects the highly inhomogeneous nature of solar prominences. Once the Alfvén speed in each thread was determined, the magnetic field strength could be computed after a value for the thread density was assumed. For the analyzed events, and considering a typical value ρp = 5 × 10-11 kg m-3, magnetic field strengths in the range 0.9 – 3.5 G were obtained (see Figure 63b).
6.4 Seismology of damped transverse thread oscillations
A feature clearly observed by Lin et al. (2009) is that the amplitudes of the waves passing through two different cuts along a thread are notably different. Apparent changes can be due to damping of the waves in addition to noise in the data. The damping of prominence oscillations is a common feature in many observed events and damping time-scales provide an additional source of information that can be used when performing parameter inference using seismology inversion techniques, once a physical model that provides an explanation is available. Among the different damping mechanisms described in Section 5, resonant absorption in the Alfvén continuum seems a very plausible one and has been used to perform prominence thread seismology, using the damping as an additional source of information. In the context of coronal loop seismology, the use of damping rates in combination with oscillatory periods gives information about the transverse density structuring of coronal loops (Goossens et al., 2002; Arregui et al., 2007a; Goossens, 2008; Goossens et al., 2008).
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 (ζ, l/a)-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.
6.5 Seismology using period ratios of thread oscillations
The widespread use of the concept of period ratios as a seismological tool has been remarkable in the context of coronal loop oscillations (see Andries et al., 2009, for a review). The idea was first put forward by Andries et al. (2005) and Goossens et al. (2006) as a means to infer the coronal density scale height using multiple mode oscillations in coronal loops embedded in a vertically stratified atmosphere. In coronal loop seismology, the ratio of the fundamental mode period to twice that of its first overtone in the longitudinal direction (P1/2P2) mainly depends on the density structuring along magnetic field lines. It can therefore be used as a diagnostic tool for the coronal density scale height.
The use of the period ratio technique needs the unambiguous detection of two periodicities in the same oscillating prominence thread. Díaz et al. (2010) pointed out two main difficulties in this respect. From a theoretical point of view, the overtone with period P2 is an antisymmetric mode in the longitudinal direction, with a node in the center of the thread and two maxima located outside it. Only for sufficiently long threads, with W/L ∼ 0.1, the anti-nodes of the overtone are located inside the thread and could hence be measured in the part of the tube visible in, e.g., Hα. From an observational point of view, no conclusive measurement of the first overtone period has been reported so far in the literature, although there seem to be hints of its presence in some observations by, e.g., Lin et al. (2007), who reported on the presence of two periods, P1 = 16 min and P2 = 3.6 min in their observations of a prominence region. Díaz et al. (2010) used the period ratio from these observations to infer the value for the length of the thread ratio W/L = 0.12. Although it is difficult to estimate the length of the particular thread under consideration, assuming a value of 13,000 km, as for other threads analyzed by Lin et al. (2007), results in a magnetic tube length L ∼ 130, 000 km.
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. (2008), using observations obtained in an active region filament by Okamoto et al. (2007) discussed in Section 3.6.4.
Summary of geometric and wave properties of horizontally flowing and vertically oscillating threads analyzed by Okamoto et al. (2007). Lthread is the thread length, v0 its horizontal flow velocity, P the oscillatory period, V the oscillatory velocity amplitude, and H the height above the photosphere.
v0 (km s−1)
V (km s−1)
174 ± 25
240 ± 30
230 ± 87
180 ± 137
135 ± 21
250 ± 17
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. (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-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).
7 Open Issues
Solar prominences are among the most complicated structures in the solar corona. A full understanding of their formation, magnetic structure, and disappearance has not been reached yet, and a lot of physical effects remain to be included in prominence models. For this reason, theoretical models set up to interpret small amplitude oscillations are still poor. High-resolution observations of filaments suggest that they are made of threads whose thickness is at the the limit of the available spatial resolution. Then, one may wonder whether future improvements of the spatial resolution will provide with thinner and thinner threads or, on the contrary, there is a lower limit for thickness and we will be able to determine it in the future. The presence of these long and thin threads together with the place where they are anchored and the presence of flows along them suggest that they are thin flux tubes filled with continuous or discontinuous cool material.
This cool material is probably subject to cooling, heating, ionization, recombination, motions, etc., which, altogether, makes very difficult a proper theoretical treatment. For instance, in the case of the considered thermal mechanisms, up to now only optically thin radiation has been taken into account, while the inclusion of optically thick effects would probably be more realistic; the prominence heating mechanisms taken usually into account are tentative and “ad hoc”, while true prominence heating processes are still deeply unknown. An important step ahead would be to couple radiative transfer with magnetohydrodynamic waves as a mean to establish a relationship between velocity, density, magnetic field, and temperature perturbations, and the observed signatures of oscillations like spectral line shift, width and intensity. Partial ionization is another topic of interest for prominence oscillations since, apart from influencing the behaviour of magnetohydrodynamic waves, it poses an important problem for prominence equilibrium models since cross-field diffusion of neutral atoms can give place to flows and drain prominence material.
Another issue which still remains a mystery is the triggering mechanism of small amplitude oscillations. In the case of large amplitude oscillations, observations provide with information about the exciting mechanism, but the available observations of small amplitude oscillations show no signature of their exciting mechanism. Are these oscillations of chromospheric or photospheric origin? Are they generated inside prominence magnetic structures by small reconnection events? Are they produced by weak external disturbances coming from far away in the solar atmosphere?
The presence of flows adds another ingredient to be taken into account in the study of prominence oscillations and, up to now, we can only obtain one or two-dimensional information about the flow behaviour. It would be of great interest to collect information about the three-dimensional structure of flows and, probably, in the near future we could acquire this information by means of IRIS (http://iris.lmsal.com/).
The physical changing conditions of prominence plasmas suggest that for an in-depth theoretical study of prominence oscillations more complex models together with numerical simulations are needed. Therefore, and as a step ahead, in the next future numerical studies of the time evolution of magnetohydrodynamic waves in partially ionized flowing inhomogeneous prominence plasmas, subject to different physical processes such as ionization, recombination, etc., should be undertaken. However, a full three-dimensional dynamical prominence model involving magnetic equilibrium, radiative transfer, etc., whose oscillatory behaviour could be studied seems to be still far away in the future.
The authors acknowledge the financial support received from the Spanish MICINN/MINECO and FEDER funds under Grant No. AYA2006-07637 and AYA2011-22486. They also thank R. Soler for providing some of the figures of this review.
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