Negative-energy Waves in the Vertical Threads of a Solar Prominence

Figure 1 shows the observations of the prominence at around 02:52 UT. The prominence is noticeable in absorbing structures observed at EUV wavelengths of EUV 193 and 171 Å (see panels (a) and (c)), while appearing bright in Hα observations (see panels (b) and (d)). Notably, the Hα observations are reversed, indicated by a negative sign. Numerous vertical threads within the prominence are discernible. These vertical threads were rooted on the solar disk nearby the solar limb. It should be noted that a bright feature appeared near the base of these vertical threads, marked by the blue arrow in panel (c) (also see in the animation of Figure 1), suggesting a heating event in that region. An eruption took place on the right-hand side close to the prominence around 03:00 UT, which may be related to the subsequent eruption of the prominence. Following this event, the prominence became active and unstable and then began to erupt, as evident in the animation of Figure 1. During the onset of prominence eruption, some vertical threads exhibited a transverse swaying motion, accompanied by the lifting of material along the vertical threads from their base. Figures 2(b) and (c) exhibit the swaying motion of the vertical thread at two different moments, with the black dotted lines outlining the thread's boundary. The conspicuous transverse wavelike structures are clearly identifiable. Half of the projected wavelength of the wavelike structure in panel (b) is determined to be 11.6 Mm, while the projected full wavelength of the wavelike structure in panel (c) is derived to be 22.5 Mm. Therefore, the mean projected full wavelength of these two wavelike structures is approximately 22.9 Mm. These transverse motions can be considered signatures of transverse MHD kink waves traveling along the prominence threads (Lin et al. 2009; Li et al. 2022b). Simultaneously, we also observe some materials flowing along these swaying threads from their base, which is associated with the heating event. Panel (d) displays the time–distance diagram reconstructed by a series of NVST Hα images along the blue line AB in panel (a). In this diagram, inclined stripes representing the flowing material are marked by the inclined dotted lines, and their velocities are determined to be approximately 26.7, 26.9, 28.9, and 26.2 km s⁻¹, with a mean velocity of about 27 km s⁻¹. This also implies that the vertical threads of the prominence were experiencing significant velocity shear. Additionally, it is found that the speed of flowing material in certain locations exhibits an increasing trend, as indicated by the blue dotted lines in panel (d).

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To conduct a detailed study of these swaying motions representing MHD kink waves, we make time–distance diagrams along the path perpendicular to their propagation direction. Figures 3(a) and (b) display the time–distance diagrams along the line CD in Figure 2(a), reconstructed by NVST Hα images and SDO/AIA 193 Å images, respectively. From panels (a) and (b) and the animation in Figure 3, numerous periodic oscillations in bright or dark structures can be identified, representing the signals of MHD kink waves. Many oscillations can be identified over several periods, and they have a comparable oscillation period (see the animation in Figure 3). It should be noted that the bright signals in the Hα and 193 Å channels are considered as the gap between two vertical threads. Panels (c1)–(c3) and (d1)–(d3) display several time–distance diagrams along the three yellow solid lines in Figure 1(a), representing different heights of the vertical threads. Panels (c1)–(c3) are reconstructed from NVST Hα observations, while panels (d1)–(d3) correspond to SDO/AIA 193 Å. Based on panels (c1) and (c2), the width of the dark stripes can be identified as approximately 0.5 Mm. There are many remarkable and similar oscillating signals in both time–distance diagrams. The dark and bright structures are coupled in these oscillating signals, and the bright signals are more distinct in Hα images. Given this fact, we consider these bright signals as representing the swaying motions or MHD kink waves of the vertical threads. The yellow plus signs in panel (c2) outline one signal of the transverse swaying motion with an entire period, which has been plotted in the corresponding SDO/AIA 193 Å image in panel (d2). The signal of the swaying motions in 193 Å observations is thicker than that in Hα observations, possibly due to the lower spatial resolution in 193 Å observations. By identifying the peaks and troughs of these signals (marked by different white vertical solid lines), we could estimate their periods to be 16.2, 13.8, and 14.4 minutes, respectively. These periods are comparable to the ones (10–16 minutes) of oscillations found in a quiescent prominence by Li et al. (2022b).

On the other hand, we provide an in-depth analysis of the swaying motions associated with panels (c1) and (c2). Figure 4 illustrates the evolution of the swaying motion associated with panel (c1), named as SMa, while Figure 5 is for the evolution of the swaying motion associated with panel (c2), named as SMb. Panels (a1)–(a6) of Figures 4 and 5 display time–distance diagrams along six parallel cuts at a distance of 0.55 Mm from each other, as indicated by six light-blue and red lines in Figure 2(a), representing heights from higher to lower positions. This setup could capture the propagation of a single transverse swaying motion passing through different heights from bottom to top along the vertical threads. It is found that there was a time delay of the peaks among these wave structures marked by the yellow and blue plus signs in Figures 4 and 5, suggesting that these swaying motions correspond to traveling waves, a phenomenon also observed in coronal loops (e.g., Li et al. 2023). According to the peaks of these wave structures, we plot the height of the peaks for each swaying motion, represented by the height of each cut with respect to time, as shown in the white box in panels (a6) in Figures 4 and 5. Subsequently, we fit these data points with a linear line outlined by the yellow lines. The inclination of the fitted line represents the phase speed of this traveling wave, and we obtain that the phase speeds of SMa and SMb are 23.0 ± 2.4 km s⁻¹ and 29.0 ± 3.9 km s⁻¹, respectively. These phase velocities are comparable to the projected velocities of the flowing materials. Panels (b1)–(b6) of Figures 4 and 5 outline the displacements of corresponding wavelike structures. We fit them by using the following equation:

$$\begin{eqnarray}&&D(t)={A}_{m}\sin \Space{0ex}{0.25ex}{0ex}(\displaystyle \frac{2\pi }{P}t+\varphi \Space{0ex}{0.25ex}{0ex})+{kt}+{D}_{0},\end{eqnarray} \tag{ 1 }$$

in which A_m , P, and φ represent the displacement amplitude, the oscillatory period, and the initial phase of the oscillatory motion, respectively. Parameters k and D₀ are fitting parameters associated with a linear background. Table 1 exhibits the key parameters of the transverse swaying motion. The uncertainties arise from the estimated displacements, which are regarded as 2 pixels in the images.

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According to Table 1, the projected displacement amplitudes (A_m ) and oscillatory periods (P) of SMa range from 0.39 ± 0.04 Mm to 0.58 ± 0.04 Mm and from 14.8 ± 0.5 minutes to 17.5 ± 0.3 minutes, respectively. For SMb, these values range from 0.42 ± 0.05 Mm to 0.59 ± 0.05 Mm and from 13.8 ± 0.5 minutes to 14.5 ± 0.4 minutes, respectively. Therefore, the projected velocity amplitudes (v_am = 2π ∗ A_m /P) can be derived (see the seventh column of Table 1), ranging from 2.76 ± 0.38 km s⁻¹ to 3.47 ± 0.30 km s⁻¹ with a mean of 3.29 ± 0.38 km s⁻¹ for SMa and from 3.19 ± 0.49 km s⁻¹ to 4.26 ± 0.38 km s⁻¹ with a mean of 3.80 ± 0.49 km s⁻¹ for SMb. These velocity amplitudes are consistent with the findings of previous studies (Lin et al. 2009; Li et al. 2022b), confirming that these transverse swaying motions are classified as small-amplitude oscillations. On the other hand, the ratio of the displacement amplitude to the thread's cross-section radius (approximately 0.25 Mm, as seen in Figures 1(b) and 3(c1) and (c3)) is larger than unity. Therefore, these transverse waves exhibit nonlinear oscillations (Ruderman et al. 2010; Ruderman & Goossens 2014). An intriguing finding is that A_m , P, and v_am exhibit an increasing trend from D_a6/D_b6 to D_a1/D_b1. It is worth noting that the positions of D_a6/D_b6 to D_a1/D_b1 represent a progression from lower to higher cuts as the swaying motions propagate upward. This implies that the projected displacement amplitudes, oscillatory periods, and projected velocity amplitudes of SMa and SMb increase over time. Combining the previous finding that there appears to be a significant increase in speed in some locations (marked by blue dotted lines in panel (d) of Figure 2), we speculate that the increase in flow speed could lead to an increase in the amplitude of the swaying motion. This phenomenon would be linked to shear flow and negative-energy wave instabilities (Ryutova 1988; Ruderman & Goossens 1995).

Based on the oscillatory period (P) and projected phase velocity(V_pph), it also could obtain the projected wavelengths (see the eighth column of Table 1), which range from 20.4 ± 2.8 Mm to 24.2 ± 2.9 Mm for SMa and from 24.0 ± 4.1 Mm to 25.2 ± 3.9 Mm for SMb. It also exhibits an increasing pattern over time. The mean projected wavelength is 23.3 ± 3.5 Mm, which is consistent with that (22.9 Mm) derived by the above direct measure. It also confirms that the bright signals could be used to represent the swaying motions or MHD kink waves of the vertical threads. It is also found that SMb exhibits a larger projected velocity amplitude and phase velocity in comparison to SMa. This suggests that higher transverse swaying motion in the vertical threads corresponds to a larger projected velocity amplitude and phase velocity. Such variations indicate the presence of distinct local magnetic or plasma properties at different heights within the vertical threads and also imply that the prominence has been subject to significant velocity shear.

Figure 6 exhibits the observations from CHASE/HIS at around 04:07 UT. Panel (a) displays the composite map constructed using a set of scanning spectrum data at Hα center from 04:07:29 to 04:07:33 UT. This moment is indicated by the white vertical line in Figure 2(d), coinciding with the upflowing material. Panel (b) shows the Hα spectrum along the slit marked by the black line in panel (a), which is located in the prominence. To better display the prominence spectrum, we also reverse the intensity in the spectrum map. Panel (c) displays the corresponding map of FWHM of the profiles, taken at the same observational time as panel (a). We utilize the mean intensity within the wavelength range from 6560.4 to 6561.4 Å, and from 6564.2 to 6565.0 Å as the baseline for the line profiles. The FWHM is calculated as the width of the profile where the intensity exceeds half of the maximum amplitude. The FWHM in the prominence region is almost less than 1 Å. The blue solid line corresponds to the line AB in Figure 2(a), which is for the time–distance diagram of Figure 2(d). Two yellow asterisks on the line correspond to two yellow asterisks in Figure 2(d). Therefore, we consider that the plasma in the white box could be related to flowing material. The FWHM in the white box is determined to be 0.72 ± 0.03 Å, while in the higher region marked by the red box, where lower optical thickness is expected, it is 0.61 ± 0.03 Å. The errors are represented by standard deviations. The similar values of the FWHMs in these two regions suggest that the profile of the white box could be regarded as under the optically thin approximation hypothesis. Panel (d) shows the average intensity of the spectrum over the white box, while the error bars represent the corresponding standard deviations. The shape of the profile (diamond) is mostly single peaked, which supports the optically thin assumption. The asymmetry of the profile also reveals a background containing some additional redshifted components. Based on the optically thin approximation hypothesis, we use a single- and double-Gaussian function to fit the spectrum curve, respectively. The red dashed line represents the result derived by a single-Gaussian function, with a Doppler velocity of 4.07 ± 0.07 km s⁻¹. The blue lines represent the result derived by a double-Gaussian function, where the two dotted blue lines correspond to the slow and fast components and the solid blue line corresponds to the combination of the two components. It can be derived that the Doppler velocities of the slow and fast components are −0.21 ± 1.83 km s⁻¹ and 16.37 ± 2.44 km s⁻¹, respectively. We believe that these fast components are closely associated with the upflowing plasma. Therefore, the line-of-sight velocities of the fast components are considered to represent the line-of-sight velocities of the upflowing plasma.

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Next, by combining the projected velocity on the sky plane of approximately 27 km s⁻¹, we can deduce that the full velocity of the upflow plasma is 31.5 km s⁻¹, with an inclination angle of 31° relative to the plane of the sky. Considering that these lifting materials propagated along the vertical threads, it can be inferred that the vertical threads have an inclination angle of 31° relative to the plane of the sky. On the other hand, the transverse swaying motions also propagated along these vertical threads. Then, the velocity amplitude and phase velocity can be corrected for the projected effect. For SMa, the corrected mean velocity amplitude and phase velocity are 3.84 ± 0.44 km s⁻¹ and 26.9 ± 2.8 km s⁻¹, respectively. For SMb, the corrected velocity amplitude and phase velocity are 4.44 ± 0.57 km s⁻¹ and 33.9 ± 4.6 km s⁻¹, respectively. The mean wavelength of these oscillatory motions can be corrected to be about 28.4 ± 4.3 Mm.

As mentioned earlier, these transverse swaying motions can be considered as signatures of transverse MHD kink-mode waves (Okamoto et al. 2007; Lin et al. 2009; Li et al. 2022b). Assuming that these prominence threads are composed of numerous magnetic flux tubes configured as straight and homogeneous cylinders filled with filament-like plasma, and given that the density within the prominence is significantly higher than the external coronal density, the Alfvén speed (v_A) in the thread can be estimated using the following equation (Lin et al. 2009):

$$\begin{eqnarray}&&{v}_{{\rm{A}}}={V}_{\mathrm{ph}}/\sqrt{2},\end{eqnarray} \tag{ 2 }$$

where V_ph is the phase velocity of the MHD kink-mode wave represented by transverse swaying motions. Using the corrected phase velocities, we can estimate the Alfvén speeds to be around 19.0 ± 2.0 km s⁻¹ and 24.0 ± 3.3 km s⁻¹ for SMa and SMb, respectively. It should be noted that SMa and SMb are located in the same series of vertical threads at different heights, with SMa below SMb. The difference in Alfvén speed between the two MHD kink-mode waves may be related to the varying plasma density at different heights within the same vertical threads. Assuming that the same vertical threads (the same magnetic flux tubes) have similar magnetic field strength and that the magnetic field strength does not vary much between the two heights, we can utilize the Alfvén speed equation (${v}_{{\rm{A}}}=B/\sqrt{{\mu }_{0}\rho }$) to deduce that the density in SMa is approximately 1.6 times denser than that in SMb.

On the other hand, assuming a density similar to that of prominences or coronal rains for the thread, within the range of ρ = (1.67−16.7) × 10⁻¹¹ kg m⁻³ (Stellmacher & Wiehr 1997; Labrosse et al. 2010; Wang et al. 2018; Antolin et al. 2021), and considering a mean Alfvén velocity of 21.5 km s⁻¹ within the threads, the magnetic field strength of these threads is estimated to be approximately 1–3 G. The time-averaged wave energy flux (〈F〉) carried by these kink MHD waves in multiple magnetic flux tube structures can be estimated by the following equation (Morton et al. 2012; Goossens et al. 2013; Van Doorsselaere et al. 2014):

$$\begin{eqnarray}&&\langle F\rangle =\displaystyle \frac{1}{4}\ast f\ast {V}_{\mathrm{ph}}\Space{0ex}{0.25ex}{0ex}(\rho {v}_{\mathrm{am}}^{2}+\displaystyle \frac{{b}^{2}}{{\mu }_{0}}\Space{0ex}{0.25ex}{0ex})\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&b\approx B\displaystyle \frac{2\pi }{\lambda }{A}_{m},\end{eqnarray} \tag{ 4 }$$

where f, V_ph, v_am, b, μ₀, λ, and A_m represent the filling factor, phase velocity, velocity amplitude, Lagrangian perturbation of the magnetic field, magnetic permeability of vacuum, wavelength of oscillation, and displacement amplitude, respectively. Van Doorsselaere et al. (2014) proposed that the energy flux in multiple kink waves should be corrected with a filling factor. Taking the parameter values derived from the above analysis (V_ph = 30.4 km s⁻¹, ρ = (1.67–16.7) × 10⁻¹¹ kg m⁻³, v_am = 4.14 km s⁻¹, B = 1–3 G, λ = 28.4 Mm, A_m = 0.59 Mm, μ₀ = 4π × 10⁻⁷ N/A²) and considering a unit filling factor for these prominence threads, the energy flux carried by these kink MHD waves could be estimated to be approximately (0.32–3.1) × 10⁴ erg s⁻¹ cm⁻². On the other hand, when also considering the correction factor for filling, the kinetic energy flux of the upward flow could be calculated by the equation of F_fl = $\tfrac{1}{2}\rho {v}_{\mathrm{fl}}^{2}\ast {v}_{\mathrm{fl}}\ast f$, where v_fl is the corrected velocity (31.5 km s⁻¹) of upflow material. Then, the energy flux of the upward flow could be estimated to be approximately (5.2–52.2) × 10⁴ erg s⁻¹ cm⁻². We can see that the energy carried by these swaying motions is only a few percent of that of the upward flow.