FORWARD MODELING OF STANDING KINK MODES IN CORONAL LOOPS. II. APPLICATIONS

Event V (Verwichte et al. 2013) was observed at AR 11283 in the AIA 171 Å channel on 2011 September 06. AR 11283 was associated with a Hale-class βγ or βδ sunspot; the general β (bipolar) magnetic configuration formed a bundle of distinct coronal loops connecting the opposite polarities. It crossed the central meridian on the previous day and was well-exposed for AIA observation on 2011 September 06 (Figure 1). Two or more loops oscillated transversely in response to a GOES-class X2.1 flare, which started at 22:12 UT and peaked at 22:20 UT. A fainter loop (indicated by the green dashed line in Figure 1, corresponding to loop # 2 in Verwichte et al. 2013) oscillated for about four wave cycles. It did not fade out, or become significantly brighter during the kink oscillation, and therefore, it is chosen for further investigation. In our study, the latter three wave cycles were selected for modeling.

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Verwichte et al. (2013) performed 3D stereoscopy and gave a loop geometry with a length of L₀ = 160 Mm and a radius of a = 0.85 Mm. The plasma temperature was assumed to be the nominal response temperature (0.8 MK) of the AIA 171 Å bandpass, since this loop was only visible in this channel (Verwichte et al. 2013). The electron density was estimated at a lower limit at ${n}_{\mathrm{ei}}=0.7\;\cdot \;{10}^{9}\;{\mathrm{cm}}^{-3}$. The loop oscillated with a period of about 2.0 minutes and an amplitude at 1.9 ± 1.0 Mm (about 1.0a–3.4a). The relevant measurements are summarized in Table 1 (Event V).

We model this loop with a length of L₀ = 160 Mm and a radius of a = 0.85 Mm. The internal electron density is set to ${n}_{\mathrm{ei}}=1.0\;\cdot \;{10}^{9}\;{\mathrm{cm}}^{-3}$, while the density ratio is defined as ${n}_{\mathrm{ei}}/{n}_{\mathrm{ee}}=3.0$. The loop is filled with plasma at ${T}_{{\rm{i}}}=0.8\;\mathrm{MK}$, 1.5 times hotter than the ambient plasma. We used ${B}_{{\rm{i}}}={B}_{{\rm{e}}}=30\;{\rm{G}}$ for both internal and external magnetic field strength.⁵ This equilibrium state has internal acoustic and Alfvén speeds ${C}_{\mathrm{si}}=150\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and ${V}_{\mathrm{Ai}}=2100\;\mathrm{km}\;{{\rm{s}}}^{-1}$, and ${C}_{\mathrm{se}}=120\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and ${V}_{\mathrm{Ae}}=3600\;\mathrm{km}\;{{\rm{s}}}^{-1}$ as the corresponding external speeds. The oscillation period is about 126 s; and the amplitude ξ₀ is about 2.0 Mm (2.4a).

The horizontal mode is modeled with parameters listed in Table 2 (Model V). The viewing angle [30°, 130°] (see Paper I, for definition) matches the loop orientation very well (Figure 1). The synthetic image is interpolated into AIA resolution and aligned by matching the center of the baseline at [2263, 2153]. Then the aligned synthetic image is then rotated by an angle of 3° clockwise.

Figure 2 displays the time-distance plots along the slits labeled in Figure 1 (in counterclockwise order). The oscillations at various parts of the loop are in phase with each other and exhibit amplitude variation along the loop. The loop motions are traced manually (red crosses in Figure 3), and then the time series of loop displacement was fitted with a sinusoidal function, as presented in Aschwanden & Schrijver (2011), but without the damping term. Figure 3 plots the fitted amplitude and phase for Event V. The same procedure is applied to both Event A and Event W. We also measure the amplitude and phase from the synthetic time-distance plots as displayed in Figures 2, 5, and 8, and plot them in Figures 3, 6, and 9, respectively. In the synthetic time-distance plots, we simply track loop positions by finding the maximum intensity within each time step.

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The selected loop in Event V is clear from background contamination, therefore we measure the oscillation along the slit at s = 0.5L₀ in detail. We fit Gaussian functions to the intensity profiles across the loop at s = 0.5L₀ and extract the loop displacement, flux, and width (full width at half maximum) variations. And then we compare them with synthetic kink oscillations; see the results in Section 3.3.

Event A was studied in detail by Aschwanden & Schrijver (2011). On 2010 October 16, a GOES-class M2.9 flare occurred at active region AR 11112. The excited coronal wave was observed to propagate to the northwest of AR 11112 and swept across the extended flare ribbons (Kumar et al. 2013). A bundle of coronal loops was located at a distance of about 0.32 R_☉ to the disk center (about 230 Mm northwest to AR 11112, Figure 4). Sequential brightening of the flare ribbons provided a good estimate of the Alfvén transit time, and therefore the external Alfvén speed of the loop system was roughly measured (Aschwanden & Schrijver 2011). Two or more adjacent loops oscillated for about three to four cycles, no significant damping was observed. Moreover, the loop displacement appeared to exhibit a saw-tooth pattern (Figure 5), rather than sinusoidal curves, as usually observed (Aschwanden et al. 2002).

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Event A was claimed to be a vertical transverse loop oscillation observed by the AIA 171 Å channel (Aschwanden & Schrijver 2011). The loop length measured 163 Mm; and the radius was about 2.5 ± 0.3 Mm. A bundle of loops connected two opposite polarities that were not associated with any sunspots. A potential field extrapolation gave a field strength of about 6 G at the loop apex, while the field value was measured at 4.0 ± 0.7 G using MHD seismology (Aschwanden & Schrijver 2011). The loop was filled with a plasma of density of about 2 · 10⁸ cm⁻³ and a temperature of about 0.6 MK. The oscillation period was about 6.3 minutes and the amplitude was about 1.7 ± 0.4 Mm. The measured parameters are listed in Table 1 (Event A).

We model the loop with a semi-toroidal geometry of a length at L₀ = 160 Mm and a radius at a = 2.5 Mm. The internal and external magnetic fields are ${B}_{{\rm{i}}}=4.0\;{\rm{G}}$ and ${B}_{{\rm{i}}}=4.1\;{\rm{G}}$, respectively. The loop is filled with a plasma of density at 2.2 · 10⁸ cm⁻³, 12 times denser than the ambient plasma. The loop temperature is set at 0.57 MK, 1.5 times hotter than the background. The plasma β is about 0.054 and 0.003 for the internal and external plasma, respectively. This configuration gives a kink mode solution with P₀ = 6.7 minutes, obtained by solving the dispersion relationship (Equation (13) in Paper I). The oscillation amplitude is set at ξ₀ = 4.5 Mm (1.8a).

We construct both horizontal and vertical kink wave models for this loop (Table 2, Model A). The best matching viewing angle is [32°, 135°]. The center of the baseline is placed at [6462, −2748]; the synthetic image is interpolated in to AIA resolution and rotated by an angle of 5° clockwise.

Figure 5 illustrates the time-distance plots along selected slits normal to the loop spine (Figure 4). The loop oscillations are coherently in phase along the loop, which confirms that the kink oscillation is an established eigenmode of the loop. The synthetic kink wave exhibits similar motions (Figure 5, middle and right columns). The horizontal mode finds intensity maxima when the loop oscillates to extreme positions, while the vertical mode reaches maxima when the loop crosses the equilibrium position. The phase and amplitude of the oscillation as functions of loop coordinates are measured and plotted in Figure 6.

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AR 11121 was associated with a Hale-class α sunspot group with a unipolar magnetic configuration, observed on the east limb of the solar disk on 2010 November 03. A GOES-class C3.4 flare started at about 12:12 UT and excited two EUV waves (see, e.g., Liu & Ofman 2014) or wave trains (e.g., Yuan et al. 2013). A magnetic flux tube that connected the main spot and another polarity was quickly filled up with hot and dense plasma (Figure 7). A kink loop oscillation was excited by the mass ejecta and exhibited non-coherent motions. Event W was a sporadic transverse oscillation of a flaring loop observed in the SDO/AIA channels that are sensitive to hot plasma emissions (131 Å, ∼10 MK). The loop supported possible higher longitudinal overtones, and perhaps vertical polarization of a kink wave (White et al. 2012).

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White et al. (2012) performed 3D stereoscopy loop reconstruction combining the STEREO-B EUVI 195 Å bandpass and the SDO/AIA 131 Å channel. This procedure, using a low (∼1.6 MK) and a high (∼10 MK) temperature channel, may have overestimated the loop length by a factor of two, therefore we measure the loop length by fitting a projected semi-torus to the loop (Figure 7(a)), and obtain a loop length of L₀ = 240 Mm. Differential emission measure (DEM) analysis using the forward-fitting technique (Aschwanden et al. 2013) gives the loop radius a = 3.8 Mm, electron density ${n}_{\mathrm{ei}}=3.2\cdot {10}^{9}\;{\mathrm{cm}}^{-3}$, and loop temperature ${T}_{{\rm{i}}}=10\;\mathrm{MK}$. The loop oscillated back and forth about every 5 minutes with an amplitude of about 4.7 Mm (1.2a).

To identify the overtone number, we constructed models of n = 2 and 3 overtones with options of either vertical and horizontal polarization (Figure 7). For the n = 2 overtone, we use ${B}_{{\rm{i}}}=11\;{\rm{G}}$ and ${B}_{{\rm{e}}}=17\;{\rm{G}}$ as internal and external magnetic field strength. The flux tube is filled with plasma of ${n}_{\mathrm{ei}}=2.5\cdot {10}^{9}\;{\mathrm{cm}}^{-3}$ and ${T}_{{\rm{i}}}=10\;\mathrm{MK}$, 4 times denser and 2 times hotter than the background. Therefore, the internal and external plasma β are about 1.7 and 0.08, respectively, which are reasonable values for flaring loops (see, e.g., Van Doorsselaere et al. 2011a). In this configuration, the internal acoustic and Alfvén speeds are ${C}_{\mathrm{si}}=520\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and ${V}_{\mathrm{Ai}}=480\;\mathrm{km}\;{{\rm{s}}}^{-1}$, while the external speeds are ${C}_{\mathrm{se}}=370\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and ${V}_{\mathrm{Ae}}=1600\;\mathrm{km}\;{{\rm{s}}}^{-1}$, respectively. The kink mode solution gives the period P₀ = 301 s; the amplitude is set at ξ₀ = 1.5 Mm (0.5a).

For the n = 3 overtone, the internal and external magnetic field values are ${B}_{{\rm{i}}}=9\;{\rm{G}}$ and ${B}_{{\rm{e}}}=15\;{\rm{G}}$, respectively. The loop is filled with plasma at ${n}_{\mathrm{ei}}=3.0\;\cdot \;{10}^{9}\;{\mathrm{cm}}^{-3}$ and ${T}_{{\rm{i}}}=10\;\mathrm{MK}$, 2 times denser and 2 times hotter than the ambient plasma. Then the typical speeds are ${C}_{\mathrm{si}}=520\;\mathrm{km}\;{{\rm{s}}}^{-1}$, ${V}_{\mathrm{Ai}}=360\;\mathrm{km}\;{{\rm{s}}}^{-1}$, ${C}_{\mathrm{se}}=370\;\mathrm{km}\;{{\rm{s}}}^{-1}$, and ${V}_{\mathrm{Ae}}=870\;\mathrm{km}\;{{\rm{s}}}^{-1};$ and the internal and external plasma beta are 2.5 and 0.22, respectively. P₀ = 277 s is the period of the kink mode solution. We used an oscillation amplitude at ξ₀ = 1.5 Mm (0.5a).

The synthetic images of both n = 2 and n = 3 modes are interpolated into AIA resolution and aligned with the AIA FOV by matching the center of the loop baseline at [−8847, −3929]; then they are rotated by an angle of −120° clockwise.

Figure 8 illustrates the time-distance plots extracted from the AIA 131 Å observations and synthetic views of the n = 3 horizontal and vertical overtones. We compare the oscillation amplitude and phase distribution along the loop coordinate and attempt to identify the loop node (Figure 9).

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Event V is a horizontally polarized fundamental kink mode, given the simple geometry and projection (Figure 1). The time-distance plots of the synthetic data are consistent with the observation (Figure 2): the oscillations at different positions of the loop are coherently in phase, and the emission intensity maxima are found when the loop oscillates to the extreme positions.

Figure 4 compares the difference images of Event A and synthetic data of the n = 1 horizontal and vertical polarizations. The vertical mode agrees better with the observation: it stretches the loop geometry and the oscillation phase remains unchanged in this viewing angle (Figure 4(d)). In the horizontal mode, the oscillation phase jumps by 180° at the right leg due to the line of sight (LOS) effect. The maxima of emission intensity could not be effectively measured in observation, therefore, no further information could be extracted from the intensity variation owing to the contamination of other loops (Figure 5).

In Event W, one leg of the loop blended into the background and could not be effectively identified; however, the rest of the loop gave a possible geometry for the missing leg (Figure 7(a)). By comparing the difference images of Event W and Model W, we could tell that both the n = 3 horizontal and vertical modes agree with the observation (Figure 7), while the n = 2 modes do not give the right position of the node. The left panel of Figure 8 illustrates that the emission intensity reached maxima when the loop oscillated to extreme positions; it implies that the horizontal polarization is more likely to be the right mode. In the follow-up analysis of this event, we henceforth only consider the n = 3 modes.

Figure 3 compares the amplitude and phase distribution of Event V and Model V. Model V reproduces the general trend of amplitude distribution along the loop and successfully matches the location of maximum amplitude. The phase extracted in Model V is constant at the selected loop coordinate, while those measured in the observation scatters around the synthetic values.

Figure 6 presents the case of Event A and Model A. The n = 1 vertical mode reproduces Event A better: both the general profile and the position of maximum displacement. Again, the match between the phases of the observation and models is excellent. However, close to the footpoint, the horizontal mode exhibits a 180° phase jump, as also illustrated in Figure 4. Near the footpoint, the observational errors are large in the phase, and thus this can not be used to distinguish the mode.

Figure 9 studies the case of Event W and Model W. Since we already determined the longitudinal overtone (see Section 3.1), only two polarizations of the n = 3 modes are plotted. The amplitude finds a minimum at a node around s/L₀ = 0.54−0.59; the horizontal mode reproduces this minimum at a close position. The phase jump is also well-modeled by both modes. By considering the difference images (Figure 7) and the profile of the oscillation amplitude (Figure 9(a)), we conclude that the n = 3 horizontal mode agrees better with the Event W than the other modes.

In Paper I, we show that a quadrupole term in the kink mode could deform the loop cross-section, and thus the loop width is liable to a periodic modulation at half of the kink mode period. In spectral observations, the non-thermal spectral line broadening, caused by the quadrupole term, leads to line intensity suppression at loop edges, so it further enhances the effective loop width modulation. In imaging observations, the line intensity suppression at loop edges does not exist; however, if the loop displacement is large enough, this effect could also be observed. In Event V, the loop of interest was clear from background contamination, and it had an displacement at the order of two loop radii. Therefore, Event V is selected to demonstrate the loop width modulation effect. We extracted the time series of the loop displacement, normalized flux and width variation at s = 0.5L₀, and measured the oscillation period with a Lamb-Scargle periodogram (see, e.g., Scargle 1982; Horne & Baliunas 1986; Yuan et al. 2011).

The loop oscillated back and forth about every 2 minutes, with an amplitude of about 2 Mm (Figures 10(a) and (b)). The modeled loop oscillation reproduces a similar amplitude and periodicity (Figures 11(a) and (b)).

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The normalized flux also has periodic variations, and the power spectrum exhibits a prominent peak at 1.8 ± 0.2 minutes (Figures 10(c) and (d)), which has a false alarm probability (FAP; see Horne & Baliunas 1986; Yuan et al. 2011, for definition) or a significance level less than 0.05. The peak value is consistent with the oscillation period of the kink oscillation, if we consider the 1σ error bar (1.6–2.0 minutes versus 1.9–2.4 minutes) and the coarse resolution of the spectra. The modulation depth is about 20% of the average loop intensity. In the synthetic loop oscillation, the flux also shows the period of the kink oscillation, but also its harmonics at 0.52 minutes and 1.0 minute. The strongest periodicity is at 0.52 ± 0.02 minutes, this may be due to the complex motions of the quasi-rigid kink oscillations, the quadrupole terms, and the breaking of symmetry due to the LOS effect. The lack of this period in the observation may be caused by lower time resolution, therefore we do a four-point moving average on the times series and recalculate the power spectrum (blue lines in Figures 11(c) and (d)). Now the periodicity at 2.1 ± 0.3 minutes becomes more prominent and is more consistent with observations.

The loop width was also measured and appears to vary with a clear periodicity. The amplitude is about 0.15 Mm (0.17a), about 15% of the loop displacement. The order of magnitude is consistent with the measurement in Paper I. Two peaks in the spectrum are measured at 2.3 ± 0.5 minutes and 1.0 ± 0.1 minutes, respectively, although they are below the value of the 95% confidence level, but the periodicities are clearly seem in the time series, albeit for only 2–3 cycles. In the synthetic data, these two peaks are significantly measured. Other higher harmonics are also seen. As we have much better time resolution, we are able to measure more details of kink oscillations, which is beyond the detectability of modern instrument.