A Statistical Study of Short-period Decayless Oscillations of Coronal Loops in an Active Region

From the online animation, we can see that coronal loops exhibit highly dynamic, such as drifting motions, rotation movements, transverse and longitudinal oscillations. Here, we focus on the transverse oscillations of fine-scale coronal loops, particularly for the decayless oscillations with periods shorter than 200 s. Some brightest loops are not selected to avoid the saturated pixels.

In order to enhance the appearance of small-scale oscillations, a smooth window of 200 s is applied to the original map (cf. Mandal et al., 2022; Zhong et al., 2022). The smooth window is reasonable, since we are only interested in the short periods, i.e., $<$200 s. Figure 3 (a) plots the time-distance map after removing a smooth version along the slit S1 in Figure 2 (c), which are made from the image sequences at HRI_EUV 174 Å. Here, the starting point of the time-distance analysis is indicated by the green symbol of ‘$\ast$’. Totally, 13 coronal loop oscillations (the detail can be seen in the Appendix A) are found in this time-distance map, as indicated by cyan circles, which represent the central or boundary positions of oscillating loops. Generally, the loop centers are determined by Gaussian fitting (e.g., Wang et al., 2012; Zhong et al., 2022). However, it could be difficult to apply this method when a series of overlapping loop structures simultaneously appear in the time-distance map (Anfinogentov et al., 2015; Goddard et al., 2016; Gao et al., 2022). Thus, we manually identified those brightest/weakest pixels as loop centers or edges. It should be pointed out that although some coronal loops reveal oscillations, they also show strongly vortex motions, and we have to abandon them, i.e., those coronal loops in the region centered at x$\approx$1500 s and y$\approx$15 Mm. These thirteen coronal loops show transverse oscillations without significant decaying, which could be regarded as decayless oscillations. We also note that some coronal loop oscillations drift significantly, such as loops 6 and 13, while some others drift weakly, i.e., loops 4 and 7. Therefore, the decayless oscillation is fitted by a widely adopted function, namely, the sine function with a linear trend but without the decaying term (cf. Anfinogentov et al., 2013, 2015; Li et al., 2020b; Zhang, 2020; Gao et al., 2022), as indicated by Equation 1:

where $A_{m}$ stands for the displacement amplitude, $P$ is the oscillatory period, $\phi$ and $A_{0}$ represent the initial phase and initial position of the oscillating motion, while $k$ is constant and denotes to the drifting velocity of the oscillating loop in the plane-of-the-sky (e.g., Ning et al., 2009; Li et al., 2018b; Zhang et al., 2022). The fitting results are indicated by the magenta curves, and they appear to match well with the oscillatory positions of coronal loops. Using the derivative of Equation 1, the velocity amplitude ($v_{m}$) of the decayless oscillations could be derived (cf. Li et al., 2022; Petrova et al., 2022), that is, $v_{m}=2\pi A_{m}/P$. Some key oscillatory parameters of loop oscillations are listed in Table LABEL:tab_loop1.

In order to take a closer look at the oscillating loops in the corona, we arbitrarily extracted the HRI_EUV 174 Å images to show fine-scale coronal loops during the time interval of decayless oscillations, as indicated by two red vertical lines in Figure 3 (a), which happens to cross the oscillating loops 1, 6 and 12. In Figure 3 (b) and (c), we show EUV images at HRI_EUV 174 Å at two fixed instances of time, which contain three oscillating loops, as marked by red curves. The cyan line represents the slit center, which has a constant width of about 675 km (5 pixels), and it is almost perpendicular to the oscillating loop. The green symbol of ‘$\ast$’ indicates the starting point of the time-distance analysis in panel (a). Based on the HRI_EUV 174 Å images displayed in Figure 3 (b) and (c), the distance between double footpoints of the coronal loop can be estimated. If we assume a semi-circular profile for the coronal loop (cf. Tian et al., 2016; Gao et al., 2022; Li & Chen, 2022; Mandal et al., 2022) and consider the projection effect owing to the derived position (e.g., Aschwanden et al., 2002; Kumar et al., 2013; Li, 2017), the deprojected loop length ($L$) could be estimated, as listed in Table LABEL:tab_loop1.

Using the same method (see details in the Appendix A), we manually extracted 6 coronal loop oscillations from slit S2 in region R1 and 10 coronal loop oscillations from slit S3 in region R2, as shown in Figures 4 and 5, respectively. The oscillatory positions of these coronal loop oscillations are manually marked by cyan circles, and they are well fitted by Equation 1, as indicated by magenta curves, implying that they are all decayless oscillations. Some oscillating loops are also shown in the HRI_EUV 174 Å images (see panels (b) and (c) in Figures 4 and 5), which have been outlined by red curves. Then, the loop length can also be estimated by assuming the semi-circular shape (cf. Tian et al., 2016; Gao et al., 2022; Mandal et al., 2022). Coronal loop oscillations with periods longer than 200 s are not considered in this study, for instance, the coronal loop oscillation in the region at around x$\approx$500$-$1000 s and y$\approx$10 Mm in Figure 5 (a).

At last, a total of 111 coronal loop oscillations are extracted from thirteen slits in the two regions R1 and R2 using the HRI_EUV 174 Å observation. Table LABEL:tab_loop1 lists some key oscillatory parameters based on the fitting results, i.e., the loop length, the oscillatory period, the displacement amplitude, and the velocity amplitude.

Assuming that all those decayless oscillations of coronal loops are fundamental mode of standing kink waves, the kink speed ($c_{k}$) of transverse oscillations could be determined by the loop length ($L$) and the oscillatory period ($P$), as shown in Equation 2:

Then, the magnetic field strength ($B$) inside oscillating loops could be roughly expressed in Equation 3 (e.g., Nakariakov et al., 2003; Nisticò et al., 2013; Li et al., 2017; Anfinogentov & Nakariakov, 2019; Gao et al., 2022; Tan, 2022; Zhang et al., 2022):

Where $\mu_{0}$ represents the magnetic permittivity of free space, $\mu_{0}=4\pi~{}\times~{}10^{-7}$ N A^-2 in the international system of units (SI), and $\widetilde{\mu}~{}\approx~{}1.27$ stands for an average molecular weight in the solar corona (e.g., Nakariakov & Ofman, 2001; Zhang et al., 2020). $\rho_{e}$ and $\rho_{i}$ are the external and internal plasma densities of the coronal loop, respectively. In our study, a typical value of the internal plasma density of the coronal loop is applied, such as $\rho_{i}~{}=~{}1.67~{}\times~{}10^{-12}$ kg m^-3 ($\sim$10⁹ cm^-3), and the density contrast ($\rho_{e}/\rho_{i}$) is assumed to be equal to 1/3 (cf. Gao et al., 2022; Petrova et al., 2022).

Next, we can estimate the wave energy flux of the observed coronal loop oscillations based on the MHD wave seismology (see, Nakariakov & Verwichte, 2005; Nakariakov & Kolotkov, 2020, for the review articles). Using Equations 4 and 5, the time-averaged wave energy flux ($E$) carried by the kink-mode MHD wave could be calculated in the oscillating loop (e.g., Goossens et al., 2013; Van Doorsselaere et al., 2014; Yuan & Van Doorsselaere, 2016a; Li et al., 2022; Petrova et al., 2022):

where $b$ stands for the Lagrangian perturbation of magnetic fields, namely magnetic field perturbation, which could be estimated by the Lagrangian displacement vector (cf. Yuan & Van Doorsselaere, 2016a), $A_{m}$ and $v_{m}$ represent the displacement and velocity amplitudes of decayless oscillations, and $L$, $B$, $c_{k}$, and $\rho_{i}$ are the loop length, the magnetic field strength, the kink speed, and the plasma density of oscillating loops, respectively. The estimated parameters of decayless oscillations are listed in Table LABEL:tab_loop1. A detailed estimation for the energy flux ($E$) is given in Appendix B.

In total, 111 decayless oscillations are analyzed here, which allowed us to establish the distribution of some key parameters. Figure 6 presents histograms of the loop length (a), the oscillatory period (b), the displacement amplitude (c), and the velocity amplitude (d). Lengths of oscillating loops are measured in the range between about 10.5 Mm and 30.2 Mm, with a median length of 14.4 Mm and an averaged length of 15.1 Mm. They are much shorter than previous measurements of several hundreds Mm in coronal loop oscillations (e.g., Aschwanden et al., 2002; Anfinogentov et al., 2013, 2015; Goddard et al., 2016; Nechaeva et al., 2019), while our estimations are quite similar to those measured in coronal bright points with a length range from 14 Mm to 42 Mm (cf. Gao et al., 2022). Oscillatory periods are in the range of $\sim$11$-$185 s, with a median/averaged period of 40/49 s, which is similar to that detected in decayless oscillations of shorter coronal loops (Petrova et al., 2022), but they are smaller than those in longer coronal loops (Nakariakov et al., 1999; Anfinogentov et al., 2015; Goddard et al., 2016; Mandal et al., 2022). Displacement amplitudes are found to range from roughly 51 km to 488 km, with a median/averaged value of 156/186 km, which is roughly equal to that (170 km on average) in decayless oscillations of coronal loops, but it is larger than those of decayless oscillations in coronal bright points, i.e., ranging from about 27 km to 133 km with an average of 65 km (Gao et al., 2022). Velocity amplitudes are measured in about 5.3$-$118.8 km s^-1, with a median/averaged speed of 25.6/29.8 km s^-1. Our measurements are similar to those observed in high-frequency decayless waves in coronal loops (Petrova et al., 2022). However, they are much larger than previous findings (several km s^-1) obtained in decayless oscillations of loop-like structures (e.g., Tian et al., 2012; Anfinogentov et al., 2013; Nakariakov et al., 2016; Gao et al., 2022; Li et al., 2022).

Figure 7 shows histograms of the kink speed (a), the magnetic field strength (b), the magnetic field perturbation (c), and the energy flux (d). Kink speeds of oscillations loops are $\sim$330$-$1910 km s^-1 with an averaged speed of 780 km s^-1. Magnetic field strengths in oscillating loops are estimated to about 4.1$-$25.2 G with an averaged strength of 9.8 G, which is similar to previous findings using the MHD coronal seismology in coronal loops, i.e., a few Gs to several tens Gs (e.g., Nakariakov & Ofman, 2001; Aschwanden et al., 2002; Yang et al., 2020b; Gao et al., 2022; Zhang et al., 2022). Magnetic field perturbations in coronal loops are found to be about 0.07$-$1.5 G with an averaged perturbation of 0.37 G. Finally, energy fluxes taken by decayless oscillations are estimated to range from roughly 7 W m^-2 to 9220 W m^-2 with an averaged flux of 820 W m^-2, the median flux is only 380 W m^-2, implying the lower energy is dominant. The oscillatory and physical parameters estimated from decayless oscillations are summarized in Table 3.

Figure 7 (d) shows that the energy flux carried by decayless oscillations has a broad range, and they concentrated on the lower energy-end. Hence, we plot the frequency distribution of energy fluxes in log-log space, as shown in Figure 8. It can be seen that the frequency distribution of energy fluxes appears to follow a power-law behavior, which could be expressed by Equation 6:

where $E$ represents the energy flux carried by decayless oscillations, dN is the number of events in the energy interval [E, E+dE], C and $\alpha$ (power-law index) stands for two constants, which are determined by the maximum likelihood estimation (e.g., Clauset et al., 2009; Verbeeck et al., 2019; Lu et al., 2021). This maximum likelihood method automatically returns a break of the power-low model, as indicated by the vertical line in Figure 8. The dropping of the frequency distribution at higher frequencies is mainly due to the observational fact that a large number of decayless oscillations carried weaker energies are ignored in this study, because we only study these decayless oscillations with periods shorter than 200 s. Therefore, the power-law index is as hard as about 1.12, which is far away from the theoretical expectation, i.e., slightly less than 2.0 (e.g., Hudson, 1991; Crosby et al., 1993; Su et al., 2006; Ning et al., 2007; Li et al., 2013, 2016; Ryan et al., 2016).

In this subsection, we will establish a statistically significant relationship between several key parameters, as shown in Figure 9. The displacement and velocity amplitudes showed weakly dependence on oscillatory periods or loop lengths (e.g., Nakariakov et al., 2016; Gao et al., 2022), and thus we do not analyze their statistical scaling in this study.

Figure 9 (a) presents the scatter plot between oscillatory periods and loop lengths in norm-norm space. Similar to previous findings (Anfinogentov et al., 2015; Goddard et al., 2016), the oscillatory period linearly (indicated by a magenta line) increases with the length of oscillating loops. Their linear Pearson correlation coefficient (cc.) is as high as 0.98, confirming their strong correlation. Figure 9 (b) shows the scatter plot between oscillatory periods and energy fluxes in log-log space. We find that, despite significant scattering of the two parameters, the energy flux decreases with the growth of the oscillatory period. The negative value of the Pearson correlation coefficient (-0.77) indicates their negative correlation, and the magenta line represents a linear fit.

In Figure 9 (c) we show scatter plots between velocity (black diamonds) and displacement (cyan diamonds) amplitudes and energy fluxes in log-log space. It can be seen that the energy flux linearly (indicated by a magenta line) increases with the speed of velocity amplitudes. However, it is almost not dependent on the displacement amplitude, which shows quite large scattering. The Pearson correlation coefficients confirm that the energy flux is strongly dependent (i.e., 0.97) on the velocity amplitude, but it has weakly dependence (i.e., 0.23) on the displacement amplitude. Figure 9 (d) presents scatter plots between magnetic field perturbations (black squares) or strengths (cyan squares) and energy fluxes in log-log space. From which, we can see that the energy flux linearly increases with the magnetic field perturbation, as indicated by the magenta line, and their Pearson correlation coefficient is as high as 0.97. Although some scattering, the energy flux appears to increase with the magnetic field strength, and the Pearson correlation coefficient between them is 0.75.

Transverse oscillations of coronal loops were first discovered in 1999 by TRACE (Aschwanden et al., 1999; Nakariakov et al., 1999), and they raised a great interest in connection with MHD waves and the coronal seismology (e.g., Li et al., 2020; Nakariakov & Kolotkov, 2020; Van Doorsselaere et al., 2020; Yang et al., 2020a, b; Nakariakov et al., 2021; Pascoe et al., 2022; Zhong et al., 2022). Transverse loop oscillations have been classified into decaying and decayless oscillations (e.g., Nisticò et al., 2013). In our observations, the decayless oscillations could also be explained by MHD waves. The slow magnetoacoustic wave can be first excluded because it belongs to the longitudinal-mode wave and is often parallel to the oscillating loop (Wang, 2011; Yuan et al., 2015; Kolotkov et al., 2019; Nakariakov et al., 2019). Moreover, the local sound speed ($v_{s}\approx 152\sqrt{\frac{T}{MK}}$) in the coronal loop (cf. Nakariakov & Ofman, 2001; Kumar et al., 2013; Li et al., 2017) is estimated to be about 152 km s^-1, considering that our observations are taken from the HRI_EUV 174 Å channel that has a formation temperature of about 1 MK. The local sound speed is much slower than the estimated kink speed, as can be seen in Table 3.

Kink speeds are estimated to be 330$-$1910 km s^-1, with an averaged speed of 780 km s^-1. The median speed is about 710 km s^-1, which is roughly equal to the averaged speed. The estimated kink speeds are much smaller than that requires for the global sausage-mode wave. For this kind of wave, the speed is expected to be in the range of $\sim$3000$-$5000 km s^-1 (e.g. Nakariakov et al., 2003; Inglis et al., 2008). Moreover, the global sausage wave is generally seen in the thicker and denser plasma loop, namely, the density contrast inside and outside the plasma loop should be large enough, i.e., a density contrast of 42 in the flaring loop (cf. Tian et al., 2016). Obviously, the oscillating loops in our study are not satisfied with such high density contrast, although we cannot determine them due to the limited observation. But previous observations have suggested that the typical density contrast of coronal loops are about 2$-$10 (Aschwanden et al., 2002; Gao et al., 2022; Petrova et al., 2022).

Herein, the measured speeds are very close to the typical kink speed of global kink-mode waves (Nakariakov et al., 2021). The estimated magnetic field strengths are in the range of 4.1$-$25.2 G, with an average of 9.8 G. Our estimations based on the MHD coronal seismology are quite similar to previous findings derived from the kink oscillation of coronal loops, i.e., several Gs to a few tens Gs (Nakariakov & Ofman, 2001; Aschwanden et al., 2002; Long et al., 2017; Zhang et al., 2022). Therefore, the small-scale transverse oscillations observed in coronal loops could be regarded as decayless kink-mode oscillations. Moreover, a strong correlation is found between oscillatory periods and loops lengths (Figure 9 (a)), confirming the explanation of decayless oscillations as the standing kink-mode wave in coronal loops.

Benefitting from the high-temporal resolution (i.e., $\sim$3 s) imaging observations observed by SolO/EUI at the HRI_EUV 174 Å passband, the averaged period of decayless kink oscillations is estimated to 49 s, while the median period is only 40 s. It is shorter than previous detections in kink oscillations of coronal loops observed by SDO in AIA 171 Å, namely, hundreds of seconds (see, Anfinogentov et al., 2015; Goddard et al., 2016; Nechaeva et al., 2019, for statistical reults). Thus, the decayless kink oscillations are regarded as short-period oscillations (cf. Petrova et al., 2022). Based on the observational fact that oscillatory periods of kink oscillations are always dependent on their loop lengths (Anfinogentov et al., 2015; Goddard et al., 2016; Nechaeva et al., 2019), the short-period kink oscillations should correspond to shorter loops, which is consistent with our measurements (i.e., the averaged loop length is estimated to 15.1 Mm and the median loop length is 14.4 Mm). The measured loop lengths are much shorter than previous findings in coronal loop oscillations with longer periods, which could be as large as several hundreds Mm (e.g., Aschwanden et al., 2002; Anfinogentov et al., 2013).

The averaged displacement amplitude of decayless oscillations is 186 km, with a median value of 156 km. This is quite close to previous findings (170 km on average) in decayless kink oscillations (Anfinogentov et al., 2013, 2015). Obviously, the apparent displacement amplitudes do not gradually decrease with the shorten of oscillatory periods, which agrees with previous findings, for instance, the oscillating amplitude does not reveal any dependence on only one parameter such as oscillatory periods or loop lengths, but it could be determined by both of them (Nakariakov et al., 2016; Gao et al., 2022; Petrova et al., 2022). The displacement amplitudes measured here are mostly equal to the pixel size of HRI_EUV 174 Å (135 km) images, which are reasonable. This is because sub-pixel displacement amplitudes have been reported in decayless oscillations in AIA 171 images, and they are demonstrated to be reliable (see, Anfinogentov & Nakariakov, 2016; Anfinogentov et al., 2022; Zhong et al., 2021; Gao et al., 2022). The short oscillatory period would lead to a fast velocity amplitude. As being expected, velocity amplitudes of decayless oscillations are found to 5.3$-$118.8 km s^-1 with an averaged speed of 29.8 km s^-1, which are similar to that of high-frequency decayless waves (Petrova et al., 2022). On the other hand, the velocity amplitude in our study is much larger than previous observations of decayless oscillations observed by SDO/AIA, and it seems to exceed the velocity amplitude of decaying oscillations. Therefore, the previous classification (Nisticò et al., 2013) such as low-amplitude decayless and high-amplitude decaying based on velocity amplitudes is not applicable in this work, and we regarded them as small-scale decayless oscillations.

Kink-mode MHD waves, one of the possible candidates for coronal heating, have been intensively studied since their first detection (see, Van Doorsselaere et al., 2020; Nakariakov et al., 2021, for reviews). Decayless kink oscillations are found to be persistent and omnipresent in solar atmospheres, i.e., coronal loops (Anfinogentov et al., 2013, 2015), prominence threads (Okamoto et al., 2007; Li et al., 2022), coronal bright points with loop-like profiles (Gao et al., 2022). Thus, they could provide continuous energy input that is required to heat the solar corona. However, previous observations found that the dissipating energy carried by decayless kink oscillations were not enough to heat the ambient plasmas in the solar corona (Klimchuk, 2015; Li et al., 2022; Gao et al., 2022). On the other hand, a recent high-temporal observation from HRI_EUV 174 Å images indicates that the short-period (or high-frequency) decayless kink oscillations take significant energy to supply for heating the quiet corona, even if they only consider the kinetic energy (cf. Petrova et al., 2022).

Given simultaneously consideration of the kinetic and magnetic energies (see, Equation 5), we estimate the energy flux carried by decayless kink oscillations at shorter periods in an active region using the HRI_EUV 174 Å images with a time cadence of 3 s. It should be pointed out that we have assumed that the decayless kink oscillations had the same damping mechanism with decaying kink oscillations. And thus, the estimated energy fluxes could also regard as energy losses from decayless kink oscillations, which might be contribution to the coronal heating, as seen in Appendix B. The energy flow in the case of decayless oscillations looks like the following: unknown driver $\rightarrow$ standing kink oscillations $\rightarrow$ quick damping $\rightarrow$ plasma heating. Such process is continuing when the unknown driver always operates, manifested as ‘decayless oscillations’. However, the unknown driver is still an open issue, we will discuss this in Section 5.3. The energy flux is estimated to $\sim$7$-$9220 W m^-2 with an average of 820 W m^-2, the median value is as low as 380 W m^-2, suggesting that the lower energy is dominant. The minimum energy flux (7 W m^-2) taken by the decayless kink oscillation corresponds to a longer oscillatory period (169 s), but the amplitude and the magnetic field are rather low, particularly the velocity amplitude is only 5.3 km s^-1, and the magnetic field perturbation is as weak as 0.07 G, as shown in Table LABEL:tab_loop1 (slit S1, loop 12) and Figure 3. The same case of a lower energy flux (40 W m^-2) has the longest oscillatory period (185 s) and the weakest magnetic field strength (4.1 G) in this statistic, as seen in Table LABEL:tab_loop1 (slit S2, loop 6) and Figure 4. In contrast, the maximum energy flux (9220 W m^-2) carried by the decayless kink oscillation corresponds to a shorter oscillatory period (26 s), while the amplitude and the magnetic field are stronger, particularly the velocity amplitude can reach 118.8 km s^-1, and the magnetic field perturbation is as strong as 1.5 G, as seen in Table LABEL:tab_loop1 (slit S1, loop 6) and Figure 3. The same case of a higher energy flux (3970 W m^-2) also has the shortest oscillatory period (11 s) and the strongest magnetic field strength (25.2 G) in this statistical study, as seen in Table LABEL:tab_loop1 (slit S3, loop 7) and Figure 5. This is consistent with the scatter plots in Figure 9, which shows that the energy flux is strongly dependent on both the oscillatory period and the magnetic field. However, the maximum energy flux of 9220 W m^-2 is still not enough to heat the AR corona, such as 10⁴ W m^-2 (cf. Withbroe & Noyes, 1977). Moreover, the higher energy flux (i.e., $>$1000 W m^-2) is not dominant (Figure 7), and the energy flux estimated by the expression for the fast kink MHD wave (i.e., Equation 5) is always overestimated (Goossens et al., 2013). In summary, the practical energy carried by the decayless kink MHD wave is not sufficient for heating the AR corona.

Previous observations suggested that the decaying oscillation could be impulsively excited by an external solar eruption that can be clearly observed, such as solar flares and jets, EUV waves, or flux ropes (e.g., Antolin & Verwichte, 2011; Kumar et al., 2013; Reeves et al., 2020; Zhang et al., 2020; Mandal et al., 2022; Zhang et al., 2022). Similarly, the decayless oscillation may be triggered by an impulsive energy release (Nakariakov et al., 2021). On the other hand, the decayless oscillation often lasts for several wave periods or even longer (Tian et al., 2012; Anfinogentov et al., 2015). Hence, the external driver should be continuous to keep the decayless oscillation for a long time, as seen in Appendix B. Now, several theoretical models have been proposed to illustrate the excitation/driver of decayless oscillations, for instance, it might be triggered by the fast magnetoacoustic wave train (Liu et al., 2011; Wang et al., 2012), or it could be excited by the coronal rain caused by a catastrophic cooling process (Verwichte et al., 2017), or it may be a self oscillation that is driven by the slipping interaction between oscillating loops and steady external medium flows (Nakariakov et al., 2016, 2022). However, the triggered eruptions are difficult to detect, largely due to their fine-scale structures, as shown in Figure 10, which presents an overview of our observation.

Figure 10 (a) shows full-disk light curves in the X-rays from 03:05 UT to 04:10 UT on 17 March 2022, which are recorded by SolO/STIX and GOES, respectively. Here, the observed time of GOES has been corrected to the SolO time stamps. There were not solar flares appearing in the HRI_EUV observation during 03:18:00$-$04:02:51 UT, as indicated by the yellow shadow. We note that a small flare erupted at about 03:11 UT, which occurred before the HRI_EUV observation. The solar flare is estimated to be a B7 class based on the STIX counts²²2https://datacenter.stix.i4ds.net/view/plot/lightcurves. It was observed by the STIX channels at 4$-$10 keV and 15$-$25 keV, but it was impossible to determine the flare location due to the weak counts. Herein, we plot the normalized EUV light curve (normalized to their maximal intensity) integrated over the active region with a FOV of $\sim$278$\times$278 Mm² observed by SDO/AIA at 131 Å, as shown by the cyan line in panel (a). To save the computing time, we use AIA binned (4$\times$4) images with a time cadence of 120 s. Similar to STIX and GOES fluxes, the flare peak also appears in the local EUV flux, indicating that the small flare occurred inside the AR. To identify the flare source, we draw the AIA 131 Å image within a small FOV containing the AR, as shown in Figure 10 (b). The small flare exhibits a bright loop, as marked by the pink arrow. However, we should state that the flare has disappeared before the beginning of decayless oscillation observations. Moreover, most of oscillating loops are far away from the flare source, especially for these oscillating loops in the R1 region. Therefore, the small flare has little impact on the decayless oscillations in our study. Our results are similar to the previous finding, for instance, the solar flare could increase the oscillatory amplitude, but it had little effect on the nature and periods of decayless oscillations (cf. Mandal et al., 2021).

Figure 10 (c) presents the FSI 304 Å image at 03:50:00 UT during our observation. From which, we do not find any coronal mass ejection (CME) or prominence eruptions, implying that the decayless oscillation is impossible to be excited by the precursor of a large-scale solar eruption, i.e., the solar flare, the CME or the erupted prominence. Figure 10 (d) and (e) shows the EUV images with sub-fields (marked by the cyan box in panel (c)) that including the AR measured in wavelengths of HRI_Lyα 1216 Å and FSI 304 Å. Similarly, we do not see any small-scale eruptions in these two EUV channels. However, the two EUV images have lower spatial or temporal resolutions, and the observational data is always discontinuous, making it hard to synthesize the animation. Therefore, it is impossible to exclude the possibility that the decayless oscillations are excited by small-scale eruptions, particularly for those small-scale eruptions with short durations, i.e., campfires (Berghmans et al., 2021; Chen et al., 2021), ultraviolet bright points (Li, 2022), or fast repeating jets (Chitta et al., 2021; Hou et al., 2021), since they are ubiquitous in the solar corona. On the other hand, the animation of eui_174.mp4 shows that a large number of thread-like structures reveal continuous movements, which might be regarded as steady flows. These steady flows could continuously interact with the oscillating loops, causing decayless oscillations (Nakariakov et al., 2016). In a word, we can conclude that the decayless kink oscillations at short periods could not be excited by large-scale solar eruptions, but they might be triggered by fine-scale solar eruptions, or they could be self oscillations caused by slipping interactions between external flows and oscillating loops.