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Home All issues Volume 689 (September 2024) A&A, 689 (2024) A16 Full HTML
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A&A
Volume 689, September 2024
Solar Orbiter First Results (Nominal Mission Phase)
Article Number A16
Number of page(s) 12
Section The Sun and the Heliosphere
DOI https://doi.org/10.1051/0004-6361/202450433
Published online 28 August 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1. Introduction

Space-based imaging observations with high spatial resolution, Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA; Lemen et al. 2012) and Solar Orbiter/Extreme Ultraviolet Imager (EUI; Rochus et al. 2020), have significantly shaped queries on decayless transverse (or kink) oscillations in recent years (see Nakariakov et al. 2021, for a recent review). This type of oscillation shows transverse motions of the coronal loop axis with a small amplitude (Tian et al. 2012; Zhong et al. 2022a, 2023a,b). This mode can be observed without significant damping for several oscillation cycles (Zhong et al. 2022b). Thus, this oscillation has been termed decayless to distinguish it from the standard transverse oscillations excited by solar energetic events, which show rapid damping (Nisticò et al. 2013).

Existing research confirms that decayless oscillations are a common feature in the solar corona and can exist without a noticeable energetic event (Anfinogentov et al. 2013). Anfinogentov et al. (2015) found that 90% of 21 active regions (ARs) detected with SDO/AIA showed decayless oscillations. It was observed that among 23 coronal bright points in the quiet Sun (QS) observed by SDO/AIA, decayless oscillations occurred in 70% of them (Gao et al. 2022). Recently, Shrivastav et al. (2024) confirmed that using Solar Orbiter/EUI, closed loops in coronal holes (CHs) can be seen to exhibit decayless signatures. However, Mandal et al. (2022) found that an identical loop system in an AR does not always display decayless oscillations, although the physical conditions remain approximately the same.

Decayless oscillations have been observed in loops with lengths of between about 3 and 740 Mm, and the observed periods were between about 11 s and 30 min (Li & Long 2023; Zhong et al. 2023a; Shrivastav et al. 2024). Two trends have been discovered regarding the relationship between loop length and period. One relationship is linear, implying that decayless oscillations could be interpreted as standing modes (Anfinogentov et al. 2015; Li & Long 2023). Furthermore, Duckenfield et al. (2018) found an observation of the co-existence of the fundamental and second harmonics of decayless oscillations, which can support the interpretation as standing modes. The other relationship is no correlation (Gao et al. 2022; Shrivastav et al. 2024). To identify the mode of these oscillations, there were additional efforts to investigate phase relations along the loop axis, but no significant propagating features were found (Petrova et al. 2023; Shrivastav et al. 2024). Hence, the interpretation of these decayless oscillations remains an open question.

Although a considerable number of decayless oscillations have been observed, it remains unclear as to how decayless oscillations are sustained against damping. Considering that the rapid damping mechanism of large-amplitude transverse oscillations is an inherent feature (Goossens et al. 2002; Terradas et al. 2008), one would expect to find a continuous energy input from the convection zone in order to maintain decayless oscillations for several cycles. Numerical studies have shown that decayless oscillations could be excited by a p-mode driver (Gao et al. 2023), a harmonic driver (Karampelas et al. 2017; Guo et al. 2019), supergranulation flows (Nakariakov et al. 2016; Karampelas & Van Doorsselaere 2020), and broadband drivers (Afanasyev et al. 2020; Karampelas & Van Doorsselaere 2024). However, there have also been claims that decayless oscillations are not actual kink oscillations of the full loop body. A numerical simulation found that decayless features can result from the combination of periodic brightenings and the motions produced by Kelvin–Helmholtz instabilities (KHi; Antolin et al. 2016). Recently, Lopin & Nagorny (2024) reported theoretical evidence that radial motion caused by slow magnetoacoustic oscillations in a short loop can appear as decayless transverse oscillations.

Although uncertainties remain as to the excitation and dissipation of decayless oscillations, their total energy content would be of great interest given their ubiquity. Recently, Lim et al. (2023) introduced the wave-based heating theory similar to the nanoflare heating theory (Hudson 1991). If transverse oscillations of varying frequencies generate energy flux, then the total energy flux, F, is equal to the integral of spectral energy flux, s(ω), which is the energy flux per frequency,

F = ∫ ω min ω max s ( ω ) d ω , $$ \begin{aligned} F=\int _{\omega _{\mathrm{min} }}^{\omega _{\mathrm{max} }}s(\omega )d\omega , \end{aligned} $$(1)

where s(ω) has dimensions of W m−2 Hz−1 and the limits ωmin and ωmax are the lowest and highest frequencies of transverse oscillations, respectively. If the spectral energy flux, s(ω), follows a power law of the form

s ( ω ) = s 0 ω − δ , $$ \begin{aligned} s(\omega )=s_{0}\omega ^{-\delta }, \end{aligned} $$(2)

where s0 is a scaling constant and δ is a power-law slope, then, assuming that δ <  1,

F ω ≈ s 0 − δ + 1 ω max − δ + 1 , $$ \begin{aligned} F_{\omega }\approx \frac{s_{0}}{-\delta +1}\omega _{\mathrm{max} }^{-\delta +1}, \end{aligned} $$(3)

implying that high-frequency transverse oscillations provide the dominant contribution to the heating generated by transverse oscillations. If, however, δ >  1, then

F ω ≈ s 0 δ − 1 ω min − δ + 1 , $$ \begin{aligned} F_{\omega }\approx \frac{s_{0}}{\delta -1}\omega _{\mathrm{min} }^{-\delta +1}, \end{aligned} $$(4)

and low-frequency transverse oscillations dominate the heating. Through a meta-analysis using the literature reporting decayless transverse oscillations, Lim et al. (2023) found δ of around −1.4 for decayless oscillations in the frequency range 0.002 − 0.07 Hz (corresponding to 14−500 s). This result indicated that high-frequency decayless oscillations could give the dominant heating in the corona, compared to low-frequency oscillations.

However, considering that different coronal regions have different magnetic topologies and energy losses, this analysis should be repeated for each region. Morton et al. (2016) found that the spectral slope of propagating transverse waves observed by the Coronal Multi-channel Polarimeter varied between ARs, QS, and open field regions. In this paper, we study the statistical characteristics of decayless transverse oscillations in different regions using Solar Orbiter/EUI observations in order to investigate the energy flux distribution and the power-law slope in each region. Based on the theory proposed in Lim et al. (2023), the highest frequency (ωmax) of decayless transverse oscillations is a key parameter with which to estimate the total energy flux generated by all decayless oscillations. Thus, it would be useful to identify the highest observable frequency of transverse oscillations using the high spatial and temporal resolutions of Solar Orbiter/EUI. In Section 2 we describe our data sets. Our results and a discussion are featured in Section 3. Section 4 provides our conclusions.

2. Data and analysis

2.1. Data

We consider the calibrated level-2 Solar Orbiter/EUI High Resolution Imager 174 Å (HRIEUV) data (Kraaikamp et al. 2023). Among the released data, only data sequences with a cadence of equal to or less than 3 seconds are used to expand the limit of observable high frequencies of oscillations. We note that we do not limit the upper range of periods with this criteria. We exclude observations that were already used in the study of decayless transverse oscillations (Li & Long 2023; Petrova et al. 2023; Shrivastav et al. 2024) or observations that were saturated and/or belonged to technical commissioning tests. As a consequence, we use 16 data sequences with varying spatial scales observing both QS and ARs. The details of the datasets are listed in Table 1. We note that there were three data sequences observing CHs with high cadence; one was used in Shrivastav et al. (2024) and the others were saturated. Thus, it was not possible to include CH observations in this study.

Table 1.

Details of the datasets used in this study.

We removed the jitter of the spacecraft in the level-2 data using a cross-correlation technique (Chitta et al. 2022). Although this technique has a subpixel accuracy, given that the decayless oscillations have very low amplitudes, we additionally examined the periodicity of the remaining jitter after applying the technique. We performed a wavelet analysis on the remaining jitter and found that none of the 16 data sequences had any significant periodicity.

2.2. Analysis

We analysed transverse oscillations of 41 loops in QS and 22 loops in ARs, ranging from small to large scales. Using extreme ultraviolet (EUV) images only, we considered closed loops that can be identified by the eye contrasted to the background and whose footpoints are relatively stationary throughout the observation. However, since the loop itself and its surroundings are very dynamic, some loops that do not maintain their brightness throughout their duration are included. The loop length (L) is approximated by measuring the distance (Df) between footpoints and using the relation, L = πDf/2, assuming that the loop is a semi-circle. In the case of the loops on the limb, Df is measured by considering the distance between the loop apex and the centre between two footpoints. The estimated loop lengths range from about 7 to 174 Mm (Figure 1). The average loop length in the AR is around 70 Mm, which is longer than that of around 30 Mm in the QS. This is shorter than the length of AR loops mainly observed in AIA for decayless oscillations (an average of about 220 Mm in Anfinogentov et al. 2015) and includes much longer loops than the range (about 3−30 Mm) of loop lengths considered for decayless oscillations in EUI (Li & Long 2023; Petrova et al. 2023; Shrivastav et al. 2024). We considered time-series data from an artificial slit on the loop apex (see Figure A.1 for slit positions in each data sequence). To improve the signal-to-noise ratio, we considered a slit of 5 pixels in thickness and averaged the intensity over the slit thickness.

thumbnail Fig. 1.

Histogram of the lengths of the loops in QS (blue hatched bar) and ARs (red hatched bar). The average and standard deviation of loop lengths for QS (blue), ARs (red), and all regions (black) are indicated in the panel.

Considering the low-amplitude feature of decayless oscillations and the large data samples considered in this study, tracking the transverse motions of each loop manually is arduous. To facilitate the analysis, we employed the Automatic Northumbria University Wave Tracking1 (Auto-NUWT; Morton et al. 2013; Weberg et al. 2018) code to detect the position of the loops, track their transverse motion, and extract the properties of their oscillations. To track the loop position, this code finds the position where local intensity gradients in both transversal directions are higher than a user-adjustable threshold. A constant threshold of 0.5 for unmasked time-distance maps was used in studies mostly targeting off-limb regions (Thurgood et al. 2014; Weberg et al. 2020). In contrast, the regions considered in this study are mostly observed on disc. High-resolution observations sharpen not only the coronal loop itself but also features in the background. For this reason, the intensity threshold for defining a loop is expected to be highly dependent on the region in which it is imaged. We consider the average gradient of each time-distance map as the threshold for each map.

Based on the local intensity maxima filtered out by the threshold, the fitting of the nearby intensity values with a Gaussian function is considered. From this, the centre and minor radius of the loop are detected as the Gaussian centre and the half width at half maximum, respectively (see Figure 2). The fitting is weighted by HRIEUV intensity errors (σI), which are taken as

σ I ≈ r 2 + I t exp α N , $$ \begin{aligned} \sigma _{\rm I}\approx \sqrt{r^2+\frac{It_{\mathrm{exp} }\alpha }{N}}, \end{aligned} $$(5)

thumbnail Fig. 2.

An example of detected oscillations. (a) Magnified view of the loop with slit S1 (white solid line) and its footpoints (white plus) on 2021 September 13. (b) Time–distance map made from the slit presented in panel (a). The black diamond and error bars indicate the detected loop centres and minor radius. The red dashed line outlines their sine fits that meet all criteria. The close-up view of fitted oscillations is shown in (c) and (d). In this case, error bars represent position errors.

where r = 2 DN is the detector readout noise, I is the intensity, texp is the exposure time, α = 6.85 DN photon−1 is the photons to data number (DN) conversion factor, and N is the number of pixels over which the intensity is averaged (i.e. 5 in this study; Petrova et al. 2023; Chitta et al. 2023; Gissot et al. 2023). The number of transverse pixels used in the fit could affect the convergence. We considered six different pixel sizes of 11, 13, 15, 17, 19, and 21, corresponding to the transverse scales between around 0.5 and 2 Mm. The pixel size of 13 was chosen as it best matches the loops seen in the time–distance maps.

Transverse oscillatory features of the loop centre are identified using a Fourier analysis with a 95% significance level. In order to obtain only robust transverse oscillations against potential false detected ones, we accept oscillations where they meet the following criteria.

  • 1.

    The fit should be made using more than five points (Thurgood et al. 2014). This directly gives the minimum period of oscillations, 5 s from the 1 s cadence data sequences, 10 s from the 2 s cadence, and 15 s from the 3 s cadence.

  • 2.

    The oscillation duration should be longer than one period.

  • 3.

    The percentage of time frames in which the loop centre is not detected during the total duration of the oscillation should be less than 35% (Weberg et al. 2018).

Our procedure identifies 412 single-period oscillations and 178 multi-period oscillations. The multi-period oscillations may represent a possibility of multiple harmonics as they were already discovered observationally (Duckenfield et al. 2018) and numerically (Ruderman et al. 2021; Karampelas & Van Doorsselaere 2024). To identify them, however, the phase relation along the loop axis of each oscillation should be carefully analysed, which is beyond the scope of this study. In the present study, we only consider oscillations that have a single dominant period and can be well described by a sinusoidal function, which implies the absence of damping and is mainly used in analysing decayless oscillations in previous studies (Li & Long 2023; Petrova et al. 2023; Shrivastav et al. 2024). Considering the oscillation parameters as initial guesses, we fit the data for 412 cases with a sine and a linear trend:

X ( t ) = A sin ( 2 π t P + ϕ ) + X 0 + X 1 t , $$ \begin{aligned} X(t)=A\sin {\left(\frac{2\pi t}{P}+\phi \right)}+X_0+X_1t, \end{aligned} $$(6)

where X is the transverse displacement at t, A is the displacement amplitude, P is the period, ϕ is the phase, and X0 and X1 are the constant parameters of the linear trend. In order to examine the goodness of fit and filter out the fits strictly, we consider two criteria as follows.

  • 1.

    We accept a fit if the errors (δA and δP) of the fitted parameters (A and P) are not comparable to them: ( δ A / A ) 2 + ( δ P / P ) 2 ≤ 0.7 $ \sqrt{(\delta A/A)^2+(\delta P/P)^2}\leq0.7 $ (Thurgood et al. 2014).

  • 2.

    We accept a fit if the chi-squared is less than three times the number of data points (Markwardt 2009).

After applying these criteria, 254 oscillations were retained from the original 412, and their parameters and loop information are summarised in Table 2. An example of the fitted oscillations is shown in Figure 2.

Table 2.

Parameters of 254 decayless transverse oscillations of coronal loops detected with Solar Orbiter/EUI HRIEUV.

3. Results and discussion

We find 147 oscillation events in about 71% of the QS loops (29 out of 41) and 107 oscillation events in about 86% of the AR loops (19 out of 22). These occurrence rates are consistent with the rates of about 70% from coronal bright points in QS (Gao et al. 2022) and that of about 90% in ARs (Anfinogentov et al. 2015). In this study, we consider a duty cycle defined as the duration divided by the period to indicate how long oscillations last compared to the period. The duty cycle of all oscillations ranges from 1 to 6, with an average value of about 1.5. Among 254 oscillations, 78 oscillations have a higher quality than 2. Given that the feature of decayless oscillations is no significant damping, one can expect that the quality of decayless oscillations in general would be high. However, our result shows that most decayless oscillations have a low duty cycle. This may be an effect of the fact that the oscillating loop is dynamic and its brightness is not maintained throughout the observation, as already mentioned above. No published quantitative studies have presented the duty cycle of decayless oscillations. However, through the figures showing oscillation examples, it can be seen that decayless oscillations with duty cycles ranging from approximately 1 to 7 have been observed (Anfinogentov et al. 2015; Gao et al. 2022; Li & Long 2023; Petrova et al. 2023; Shrivastav et al. 2024) and up to about 40 (Zhong et al. 2022b).

We investigate the intensity ratio (RI) of oscillating loops to backgrounds. Using the minor radius of the loop of each oscillation, we define the boundary of the loop and calculate the average intensity (Ii) inside the loop. For the background, we consider the minimum intensity outside the loop (Ie). The intensity ratio ranges from 1.1 to 2.5. The average value of QS loops is about 1.6 ± 0.02, which is slightly higher than the average ratio (1.4 ± 0.02) of AR loops. These values are overall lower than previously reported intensity ratios. Gao et al. (2022) reported an average intensity ratio of 4.1 for coronal bright points in the QS observed by SDO/AIA. For AR loops, using AIA observations, an average value of 2.5 was estimated (Anfinogentov & Nakariakov 2019; Zhong et al. 2023a). Aschwanden et al. (2003) found an average density contrast of about 3.3 (roughly 11 in an intensity contrast) for the AR loops observed by Transition Region and Coronal Explorer. We cannot come to any robust conclusions as to a specific trend in intensity ratios in the different regions because the method used to measure the loop intensity ratio is different for each study and the estimated values may depend on the imaging instruments used.

The assumption that all decayless oscillations detected in this study are standing transverse mode enables us to estimate the energy flux (F [W m−2]) and energy (E [J]) using the following formulae (Van Doorsselaere et al. 2014):

F = 1 2 f ρ ( 2 π A P ) 2 ( 2 L P ) , $$ \begin{aligned} F&=\frac{1}{2}f\rho \left(\frac{2\pi A}{P}\right)^{2}\left(\frac{2L}{P}\right), \end{aligned} $$(7)

E = ( π R 2 L ) 1 2 ρ ( 2 π A P ) 2 , $$ \begin{aligned} E&=(\pi R^{2}L)\frac{1}{2}\rho \left(\frac{2\pi A}{P}\right)^{2}, \end{aligned} $$(8)

where R, L, and ρ are the minor radius, length, and plasma density of the oscillating loop respectively, f is the filling factor, and A and P are the displacement amplitude and period of the oscillation. We use the parameters deduced from the observations for R, L, A, and P. Equations (7) and (8) are only valid for low filling factors of up to 10%, and thus we use a constant f of 10% for all oscillating loops. By assuming that all oscillating loops have the same uniform temperature (1 MK) and gravitational acceleration, we consider stratified ρ. We used the formula for ρ described in Equation (3) in Lim et al. (2023).

3.1. Statistical properties of oscillation parameters

The histograms of periods, displacement amplitudes, duration, velocity amplitudes, energy fluxes, and energies in each region are shown in Figure 3. The periods of all oscillations range from 12 to 490 s, with an average of 84 ± 4.8 s. This range falls within the period range found for all detected decayless oscillations in the literature, which is of between 11 s and 30 min (see Figure 5 in Zhong et al. 2023a and Figure 5 in Shrivastav et al. 2024). For the QS, we find periods of between 12 and 323 s with an average of 77 ± 5.3 s. The periods of AR oscillations range from 15 to 490 s, with an average value of 94 ± 8.8 s. The period ranges of QS and AR largely overlap and AR loops tend to have longer periods on average. In our study, the average loop length (72 ± 11 Mm) of AR loops is much longer than that (27 ± 3 Mm) of QS loops. In the case presented by Anfinogentov et al. (2015, hereafter T1) of decayless oscillations following a trend of oscillations, the average periods for AR and QS are around 273 and 386 s, respectively, and both of their average loop lengths are around 232 Mm (Anfinogentov et al. 2013; Duckenfield et al. 2018; Zhong et al. 2022b). In a similar case presented by Gao et al. (2022, hereafter T2), the average periods for AR and QS are around 49 and 202 s, respectively, and both of their average loop lengths are about 15 Mm (Li & Long 2023; Petrova et al. 2023; Shrivastav et al. 2024). If we assume that all decayless oscillations are the fundamental harmonic and estimate their average phase speed (2L/P), the velocities of the oscillations in this study are about 701 km s−1 for QS and 1532 km s−1 for AR. The speeds of T1 oscillations are about 1202 km s−1 for QS and 1700 km s−1 for AR. For T2 oscillations, the estimated phase speeds are 149 km s−1 for QS and 612 km s−1 for AR. From this perspective, the oscillations detected in this study are likely to be located between the two trends T1 and T2.

thumbnail Fig. 3.

Histogram of (a) the period, (b) the displacement amplitudes, (c) the duration, (d) the velocity amplitude, (e) the energy flux in log scale, and (f) the energy in log scale in QS (blue hatched bar) and ARs (red hatched bar). The average (μ) and standard deviation (σ) of each parameter for QS (blue), ARs (red), and all regions (black) are indicated in each panel. Each average value is presented with its uncertainty as σ / n $ \sigma/\sqrt{n} $, where n is the number of detected oscillations. The physical unit of each parameter is shown on the x-axis in each panel.

The displacement amplitudes of the QS oscillations are detected over a range from 12 to 287 km, which is similar to the range of decayless oscillations previously observed in the QS, which have an amplitude range of 27−365 km (Gao et al. 2022; Shrivastav et al. 2024). The amplitudes of AR oscillations range from 11 to 252 km, which is within the amplitude range of previously reported AR decayless oscillations (24−500 km; Anfinogentov et al. 2015; Mandal et al. 2022; Zhong et al. 2022a; Li & Long 2023). No flares occurred between observation durations of the AR sequences. Except for the case on 2023 March 29, where 5 C-class flares and 1 X-class flare were detected from about 12 hours to 5 hours before the observation start time, no flare activity was seen during the 24 hours before and after observations. Mandal et al. (2021) reported that solar flares enhanced displacement amplitudes compared to the preflare oscillations. In our case, although the time of the flares is not directly connected to the time when the oscillation is discovered, we compare the detected amplitudes in each AR region. We find that the amplitudes (11−162 km) in the flaring AR are smaller than those (15−252 km) in relatively stable ARs (observations on 2022 March 30 and 2023 April 7). This result may be an effect of different magnetic tension. If we assume that the coronal magnetic field strengths in ARs are similar to each other, the difference in magnetic curvature can determine the difference in magnetic tension. The curvature (the reciprocal of the minor radius of the loop) in the stable ARs is roughly 2.5 times larger than the curvature in the flaring AR. This may result in higher loop amplitudes in the stable ARs.

We also find that there is no significant difference in amplitude between QS and AR when compared to their average values (67 ± 4.1 and 66 ± 4.5 km). Given the pixel plate scale (about 100−220 km) of the observations considered in this study, we detect subresolution displacement amplitudes. Gao et al. (2022) confirmed that SDO/AIA with a pixel plate scale of 435 km can observe a decayless oscillation with an amplitude of around 27 km using a 1D model. HRIEUV, which has up to about four times higher spatial resolution than AIA, could be expected to detect oscillations with smaller amplitudes than those observed by AIA. However, Zhong et al. (2022a) reported decayless oscillations observed simultaneously when AIA and HRIEUV were aligned and showed that HRIEUV tended to underestimate the amplitude. Thus, we cannot exclude the influence of the instrument on the estimated oscillation parameters (Meadowcroft et al. 2024).

Figure 3c shows that the average duration of AR oscillations (154 ± 11.4 s) is longer than that of QS oscillations (124 ± 7.9 s). The oscillation duration could be limited to the observational duration, and four of the data sequences observing QS have a short observational duration (less than 1 min) compared to the other sequences (up to one hour). If we exclude these four sequences, the average duration of QS oscillations is 127 ± 8.1 s, implying that the average value of all QS oscillations is not affected by the observational duration. From these data, it therefore appears that decayless transverse oscillations in ARs last longer than those in QS.

We investigated the velocity amplitude (v = 2πA/P) in each region, finding a range of between about 0.7 and 138 km s−1, which is similar to the range in AR decayless oscillations (Li & Long 2023). However, most oscillations have a velocity amplitude of less than around 20 km s−1, which closely matches the findings of Nakariakov et al. (2016), Gao et al. (2022), and Shrivastav et al. (2024). We find that the average velocity amplitude of QS oscillations is somewhat higher than that of AR oscillations. This is because the displacement amplitude has a similar average value regardless of the region, whereas the period is longer in ARs.

The energy flux generated by QS oscillations ranges from about 2.0 × 10−2 to 1.3 × 105 W m−2, with an average value of 152 ± 95 W m−2. This is broader than the energy flux range reported in Shrivastav et al. (2024) (0.6 − 314 W m−2 in QS and CH) and Petrova et al. (2023) (1900 and 6500 W m−2 in QS). Among 147 oscillations, we find that only 4 have an energy flux of greater than 300 W m−2, which is the energy loss corresponding to the QS (Withbroe & Noyes 1977). The AR oscillations generate energy fluxes with a narrower range from about 0.06 to 6000 W m−2, with an average value of 147 ± 62 W m−2. This average energy flux is much smaller than previously reported (815 W m−2) in AR loops (Li & Long 2023). We do not find any single oscillations that can sufficiently heat the AR to counteract the energy loss (≈104 W m−2; Withbroe & Noyes 1977). All of the regions in this study are comparable to one another in terms of average energy flux. We would like to note that the energy fluxes estimated in Petrova et al. (2023), Li & Long (2023), and Shrivastav et al. (2024) were calculated without considering the filling factor. Below, we revisit the energy flux estimated in this study from a statistical viewpoint.

The range of estimated energies and the average value for each region are shown in Figure 3f. The energy of decayless oscillations ranges from about 1018 to 1024 erg, corresponding to femto-, pico-, and nanoflare energies, meaning that we show that there are decayless oscillations with lower energies than the decayless oscillations in the literature (1020 − 1024 erg; Lim et al. 2023). Similar to the energy flux, there is no significant difference in the average energy between QS and AR. The omnipresence in both QS and ARs of decayless oscillations and their small energy characteristics are reminiscent of small-scale brightenings believed to be miniature solar flares, such as nanoflares (Parker 1988; Winebarger et al. 2013; Chitta et al. 2021; Purkhart & Veronig 2022), EUV transient brightenings (Berghmans et al. 1998; Berghmans & Clette 1999), campfires (Berghmans et al. 2021; Chen et al. 2021), jets (Chitta et al. 2023), and explosive events (Tiwari et al. 2019). As suggested by Li & Long (2023), the decayless oscillations could be excited by small-scale flares as a consequence of the continuous motion of the footpoints in the photospheric convection. However, further study is needed to confirm this conjecture.

3.2. Correlation between oscillation parameters

We investigate the relationship between oscillation parameters in different coronal regions. Figure A.2 shows the scatter plots and Table 3 lists the correlation coefficients (CCs) between them. The CC between periods and loop lengths is 0.26 ± 0.08 for QS and −0.14 ± 0.10 for AR, which shows a considerable difference. The CCs in each region indicate no correlation, which is consistent with the results for QS (Gao et al. 2022; Shrivastav et al. 2024); however, the opposite is seen for AR (Anfinogentov et al. 2015; Li & Long 2023). We find that the CC between periods and amplitudes is 0.20 ± 0.08 for QS and 0.38 ± 0.10 for AR. Gao et al. (2022) and Shrivastav et al. (2024) presented CCs of 0.4 and 0.52 between periods and amplitudes of QS and CH oscillations, respectively. Nakariakov et al. (2016) showed that the amplitude gradually increases with the period for the decayless oscillations analysed in Anfinogentov et al. (2015), giving a CC of 0.67. We note that we calculate this CC value ourselves because it was not presented in Anfinogentov et al. (2015) or Nakariakov et al. (2016). The loop length and amplitude have no significant correlation (CC of about 0.1 for both QS and AR), which is consistent with decayless oscillations in QS (Gao et al. 2022; Shrivastav et al. 2024) and even with decaying oscillations (Nechaeva et al. 2019). Our result shows a negative correlation between period and velocity amplitude for both QS (−0.33) and AR (−0.39). Nakariakov et al. (2016) showed that self-oscillations, which is one of the possible excitation mechanisms of decayless oscillations, can give similar relationships between period and loop length (CC of 0.71) and between period and velocity amplitude (CC of −0.08) based on the oscillations presented in Anfinogentov et al. (2015). These two relationships in our study show the opposite tendency (our findings are similar to those of Gao et al. 2022; Shrivastav et al. 2024 in this respect), implying that this mechanism may be the least favourable to the oscillations detected in this study.

Table 3.

Linear Pearson correlation coefficients (CCs) between oscillation parameters as a function of the coronal region.

We find that oscillation duration is highly correlated with period but not with loop length or amplitude. This is perhaps partially caused by the fact that, necessarily, periods are shorter than durations. Considering that no significant correlation is found between period, displacement amplitude, and loop length, we can expect a linear slope with the energy flux in log scales using Equation (7) by assuming that they are independent of each other. However, a discrepancy between the expected values (−3 between period and energy flux and 2 between amplitude and energy flux) and the empirical results (−2.6 between period and energy flux and 1.2 between amplitude and energy flux) is seen. We also consider the relationship between duty cycle, intensity ratio, and oscillation parameters. These do not show significant correlations with oscillation parameters (absolute values of less than 0.4), as shown in Figure A.2. In the case of the duty cycle, period and displacement amplitude show a negative correlation of around −0.3 and the CCs of the other parameters are close to zero. Among oscillation parameters, period and duration show a positive relationship, and loop length and velocity amplitude exhibit a negative relationship, showing a higher correlation (around ±0.4) than the other parameters.

Figure 4 presents the variation in the period as a function of loop length in the present study, combined with previous reports that provide the periods and loop lengths of decayless oscillations (Wang et al. 2012; Nisticò et al. 2013; Anfinogentov et al. 2013, 2015; Duckenfield et al. 2018; Anfinogentov & Nakariakov 2019; Gao et al. 2022; Zhong et al. 2022a, 2022b, 2023a; Mandal et al. 2022; Li & Long 2023; Petrova et al. 2023; Shrivastav et al. 2024) in different regions. Most oscillations are seen to overlap with other decayless oscillations previously reported. In our study, however, a new type of oscillation with a shorter period in longer loops (about 100−200 Mm) is also found. Consequently, the phase speed of these oscillations can reach 104 km s−1, a few times higher than the typical Alfvén speed in the corona. Most of the loops where these oscillations appear are long loops that are faintly visible above the noisy background. Therefore, we cannot rule out the possibility that these short-period oscillations occur not in the long loops considered, but in a shorter loop in the background or a structure in the lower atmospheric layer. If the loop lengths for these oscillations are not overestimated, another possibility can be considered. Howson & Breu (2023) studied a 1D numerical model describing a loop with a length of about 120 Mm and found that the contribution of the lower atmosphere to the harmonics is significant when the oscillations are driven by the photospheric motions, which corresponds to decayless oscillations. In this case, the transverse motions at the loop apex are not the fundamental harmonic but are likely third harmonics. Then, by taking this into account, the phase speed is one-third of the original value and this is comparable to the coronal Alfvén speed. We note that we do not exclude other possibilities and interpretations, and to understand this more clearly, more observational and theoretical information would be needed.

thumbnail Fig. 4.

Scatter plots between period and loop length of decayless oscillations analysed in the current work (CW) and previous studies (from top to bottom in the legend; Wang et al. 2012; Nisticò et al. 2013; Anfinogentov et al. 2013, 2015; Duckenfield et al. 2018; Anfinogentov & Nakariakov 2019; Gao et al. 2022; Zhong et al. 2022a, 2022b, 2023a; Mandal et al. 2022; Li & Long 2023; Petrova et al. 2023; Shrivastav et al. 2024). The scatter plots for ARs (middle panel) and QS (right panel) are presented in log scales.

3.3. Distribution of the energy flux and oscillation frequency

Using the estimated energy flux, the logarithm of the energy flux of each oscillation with a constant frequency bin (size of 0.1 in log scale) is considered. We note that Lim et al. (2023) estimated the total energy flux per frequency bin. In tracking the centre position of a loop, even though it is a single loop, if it is interrupted by another loop or background during observation, the centre of the loop will not be continuously detected. As a result, one oscillation may be returned as multiple oscillations with similar periods. In this case, the sum of the energy flux or energy from each oscillation could be overestimated. However, it is expected that even if multiple oscillations are detected for what, in reality, is one oscillation, it will not affect the average value. Thus, in this study, we consider the average energy flux for each frequency bin.

The estimated average spectral energy flux is shown in Figure 5. The distribution has an uncertainty corresponding to the standard deviation of energy fluxes for each bin divided by the square root of the number of oscillations per bin. In order to estimate the best power-law fit of the distribution and its credible interval, we use the Solar Bayesian Analysis Toolkit (SoBAT; Anfinogentov et al. 2021). The logarithmic uncertainties of each bin are taken into account in the fit. The fitting was only considered for bins with a number of cases of greater than 1. It is shown that there is a difference between the distributions in QS and AR. The power-law slope (δ) of spectral energy flux from decayless oscillations observed in QS is around δQS = −1.59 ± 0.24 between frequency bins of about 0.003 and 0.09 Hz. In the case of AR oscillations with frequencies ranging from about 0.002 to 0.07 Hz, the slope of spectral energy flux is around δAR = −1.79 ± 0.16. This result that the slope is steeper in AR than in QS is consistent with the result in Morton et al. (2016). When we consider all observed decayless oscillations, a general power-law form has a slope of δALL = −1.59 ± 0.17 between the frequencies of about 0.002 and 0.09 Hz. Similar to the spectral slope of −1.4 obtained from previously reported decayless oscillations (Lim et al. 2023), it is clear that regardless of coronal regions, the slopes are all much less steep than the critical slope of δ = 1. This implies that, in general, high-frequency oscillations contribute more to heating than low-frequency oscillations in QS and ARs, respectively.

thumbnail Fig. 5.

Distribution, s(ω), of spectral energy fluxes as a function of oscillation frequency (top panels) and the number of oscillations for each frequency bin (bottom panels). The vertical bars show an uncertainty of ( σ ω = σ Fi / n ω i $ \sigma_{\omega}=\sigma_{Fi}/\sqrt{n_{\omega i}} $). A bin size of 0.1 has been considered. The best fits of distributions are shown as dashed lines. The fitting was only considered for bins containing more than one case. Blue, red, and black represent oscillations in the QS, ARs, and all regions, respectively. The power-law slopes are δQS = −1.59 ± 0.24, δAR = −1.79 ± 0.16, and δALL = −1.59 ± 0.17, respectively.

Figure 6 shows the average spectral energy flux of decayless oscillations in our work compared to that found in various previous studies. The list of studies considered for this combined distribution is presented in Table 1 in Lim et al. (2023). Contrary to the result of the oscillations analysed in this study, we can see that the spectral slope in QS is steeper than that in ARs. However, their slopes are still less steep than the critical slope, emphasising the significant role of high-frequency oscillations (near 0.09 Hz for QS and 0.07 Hz for ARs). We provide an assessment of the potential amount of average energy flux that can be dissipated in the different coronal regions using Equation 3. From the empirical fitting parameters of the power law in QS (s0 ≈ 9.5 × 105 and ωmax ≈ 0.09 Hz), the estimated total average energy flux carried by QS decayless oscillations is about 785 W m−2. Given the energy loss of roughly 300 W m−2 in the QS (Withbroe & Noyes 1977; Klimchuk 2006), this value indicates that the QS could be sufficiently heated by high-frequency decayless oscillation. In the case of ARs (s0 ≈ 1.1 × 106 and ωmax ≈ 0.07 Hz), the value is about 1400 W m−2, which is less than the heating requirement for ARs (about 104 W m−2). If we assume that s0 and δ are constant, we can speculate on the required minimum period for balancing the energy loss in ARs; we find this to be 6 s (0.17 Hz), which is ideally observable from HRIEUV when targeting a cadence of 1 s. In this study, one data set with a 1 s cadence was included but no oscillations were detected in that set. No conclusions about whether or not there are higher frequency oscillations can be drawn based on only one case because the observational duration of this data set was only 18 s. Thus, further high-cadence HRIEUV observations will be needed to clarify this.

thumbnail Fig. 6.

Same as Figure 5 but for the oscillations including the decayless oscillations in this study and previous studies (Anfinogentov et al. 2013, 2015; Duckenfield et al. 2018; Gao et al. 2022; Zhong et al. 2022a,b; Petrova et al. 2023; Mandal et al. 2022; Li & Long 2023; Shrivastav et al. 2024). The oscillations from previous studies are the same as those listed in Table 1 in Lim et al. (2023). The power-law slopes are δQS = −1.56 ± 0.18, δAR = −1.20 ± 0.12, and δALL = −1.31 ± 0.13, respectively.

We would like to note that the energy fluxes considered in this study are calculated assuming that the oscillating loops are very dense; that is, external densities are ignored. However, based on the result of this study and comparisons to other observational results, it seems that the plasma density between the inside and outside of the loop is not significantly different in the solar corona. By taking this into account, the estimated energy fluxes increase by a factor of 1.8. Moreover, the estimated energy flux does not correspond to the amount it dissipates. Hillier et al. (2020) presented an analytical model of the dissipation rate by assuming the KHi-induced steady-state turbulence. Using Equation (6) in Hillier et al. (2020), we find the dissipation rate to range from about 1 to 43%. If we consider the factor of 1.8 and these minimum and maximum dissipation rates for each oscillation, we approximate the dissipated energy fluxes of around 14 and 608 W m−2 in the QS and 25 and 1080 W m−2 in the ARs. In the case of ARs, our conclusion that decayless oscillations cannot sufficiently heat the AR remains unchanged, but in the QS, this result can be seen to depend on the dissipation rate. These results will be updated when more precise theoretical and numerical studies on the dissipation rate of decayless oscillations are performed.

4. Conclusions

Here, we analysed decayless oscillations in the QS and ARs observed by the Solar Orbiter/EUI HRIEUV. We identified 147 oscillations in 41 QS loops and 107 oscillations in 22 AR loops. The ranges of detected periods, displacement amplitudes, and loop lengths are comparable to those of previously reported decayless oscillations. Our statistical study of the different coronal regions reveals that periods and loop lengths are significantly different between the different regions but amplitude does not differ. We find no linear correlation between period and loop length in the QS or even in ARs. A few decayless oscillations in the QS were found to have an energy flux of greater than about 300 W m−2, which is equivalent to the energy loss in the QS. However, in ARs, there is no oscillation with an energy flux that can compensate for the energy loss. The estimated energy of decayless oscillations corresponds to the range found for femtoflares and picoflares. The distribution of spectral energy fluxes is described as a power law, showing a steeper slope in ARs than in the QS. We find that both slopes are less steep than the critical slope of 1, implying that high-frequency decayless oscillations could play a key role in coronal heating. The total average energy flux estimated from the empirical fitting shows that the QS could be sufficiently heated by decayless oscillations; however, they cannot support the heating in ARs, unless yet-unobserved oscillations are present with frequencies up to 0.17 Hz.

Our results indicate the statistical importance of high-frequency oscillations in both QS and ARs. The discovery of high-frequency oscillations depends on the temporal cadence of the instrument. Future EUI campaigns with high-cadence observations (≤1 s) will allow us to make further analyses of the energy flux distribution and its spectral slope in the future.

Acknowledgments

Solar Orbiter is a space mission of international collaboration between ESA and NASA, operated by ESA. The EUI instrument was built by CSL, IAS, MPS, MSSL/UCL, PMOD/WRC, ROB, LCF/IO with funding from the Belgian Federal Science Policy Office (BELSPO/PRODEX PEA C4000134088); the Centre National d’Etudes Spatiales (CNES); the UK Space Agency (UKSA); the Bundesministerium für Wirtschaft und Energie (BMWi) through the Deutsches Zentrum für Luft- und Raumfahrt (DLR); and the Swiss Space Office (SSO). DL was supported by a Senior Research Project (G088021N) of the FWO Vlaanderen. TVD was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 724326), the C1 grant TRACEspace of Internal Funds KU Leuven, and a Senior Research Project (G088021N) of the FWO Vlaanderen. The research benefitted greatly from discussions at ISSI. Furthermore, TVD received financial support from the Flemish Government under the long-term structural Methusalem funding program, project SOUL: Stellar evolution in full glory, grant METH/24/012 at KU Leuven. The research that led to these results was subsidised by the Belgian Federal Science Policy Office through the contract B2/223/P1/CLOSE-UP. This project DynaSun has received funding under the Horizon Europe programme of the European Union under grant agreement (no. 101131534). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union and therefore the European Union cannot be held responsible for them.

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Appendix A: Additional figures

thumbnail Fig. A.1.

Solar Orbiter/EUI HRIEUV 174 Å full FOV images for each of the 16 data sets. The image has been enhanced using the Multiscale Gaussian Normalization (Morgan & Druckmüller 2014). The observation date and start time are displayed above each image. In each panel, the white solid lines mark the positions of the artificial slits that are used for generating the time–distance maps. The white plus symbols in each panel show the approximate position of footpoints for on-disc loops and the position of the apex and the centre between two footpoints for limb loops. Some loops share the same footpoint. The details of the data set are provided in Table 1. The spatial scale is indicated as a thick white bar in each panel.

thumbnail Fig. A.2.

Scatter plots between periods, loop lengths, displacement amplitudes, duration, velocity amplitudes, energy fluxes, and energies depending on coronal regions. The ARs and QS are indicated in red and blue, respectively. A circle (triangle) represents a duty cycle of less (larger) than 2. The size of the data points is proportional to the intensity ratio.

All Tables

Table 1.

Details of the datasets used in this study.

In the text
Table 2.

Parameters of 254 decayless transverse oscillations of coronal loops detected with Solar Orbiter/EUI HRIEUV.

In the text
Table 3.

Linear Pearson correlation coefficients (CCs) between oscillation parameters as a function of the coronal region.

In the text

All Figures

thumbnail Fig. 1.

Histogram of the lengths of the loops in QS (blue hatched bar) and ARs (red hatched bar). The average and standard deviation of loop lengths for QS (blue), ARs (red), and all regions (black) are indicated in the panel.
In the text
thumbnail Fig. 2.

An example of detected oscillations. (a) Magnified view of the loop with slit S1 (white solid line) and its footpoints (white plus) on 2021 September 13. (b) Time–distance map made from the slit presented in panel (a). The black diamond and error bars indicate the detected loop centres and minor radius. The red dashed line outlines their sine fits that meet all criteria. The close-up view of fitted oscillations is shown in (c) and (d). In this case, error bars represent position errors.
In the text
thumbnail Fig. 3.

Histogram of (a) the period, (b) the displacement amplitudes, (c) the duration, (d) the velocity amplitude, (e) the energy flux in log scale, and (f) the energy in log scale in QS (blue hatched bar) and ARs (red hatched bar). The average (μ) and standard deviation (σ) of each parameter for QS (blue), ARs (red), and all regions (black) are indicated in each panel. Each average value is presented with its uncertainty as σ / n $ \sigma/\sqrt{n} $, where n is the number of detected oscillations. The physical unit of each parameter is shown on the x-axis in each panel.
In the text
thumbnail Fig. 4.

Scatter plots between period and loop length of decayless oscillations analysed in the current work (CW) and previous studies (from top to bottom in the legend; Wang et al. 2012; Nisticò et al. 2013; Anfinogentov et al. 2013, 2015; Duckenfield et al. 2018; Anfinogentov & Nakariakov 2019; Gao et al. 2022; Zhong et al. 2022a, 2022b, 2023a; Mandal et al. 2022; Li & Long 2023; Petrova et al. 2023; Shrivastav et al. 2024). The scatter plots for ARs (middle panel) and QS (right panel) are presented in log scales.
In the text
thumbnail Fig. 5.

Distribution, s(ω), of spectral energy fluxes as a function of oscillation frequency (top panels) and the number of oscillations for each frequency bin (bottom panels). The vertical bars show an uncertainty of ( σ ω = σ Fi / n ω i $ \sigma_{\omega}=\sigma_{Fi}/\sqrt{n_{\omega i}} $). A bin size of 0.1 has been considered. The best fits of distributions are shown as dashed lines. The fitting was only considered for bins containing more than one case. Blue, red, and black represent oscillations in the QS, ARs, and all regions, respectively. The power-law slopes are δQS = −1.59 ± 0.24, δAR = −1.79 ± 0.16, and δALL = −1.59 ± 0.17, respectively.
In the text
thumbnail Fig. 6.

Same as Figure 5 but for the oscillations including the decayless oscillations in this study and previous studies (Anfinogentov et al. 2013, 2015; Duckenfield et al. 2018; Gao et al. 2022; Zhong et al. 2022a,b; Petrova et al. 2023; Mandal et al. 2022; Li & Long 2023; Shrivastav et al. 2024). The oscillations from previous studies are the same as those listed in Table 1 in Lim et al. (2023). The power-law slopes are δQS = −1.56 ± 0.18, δAR = −1.20 ± 0.12, and δALL = −1.31 ± 0.13, respectively.
In the text
thumbnail Fig. A.1.

Solar Orbiter/EUI HRIEUV 174 Å full FOV images for each of the 16 data sets. The image has been enhanced using the Multiscale Gaussian Normalization (Morgan & Druckmüller 2014). The observation date and start time are displayed above each image. In each panel, the white solid lines mark the positions of the artificial slits that are used for generating the time–distance maps. The white plus symbols in each panel show the approximate position of footpoints for on-disc loops and the position of the apex and the centre between two footpoints for limb loops. Some loops share the same footpoint. The details of the data set are provided in Table 1. The spatial scale is indicated as a thick white bar in each panel.
In the text
thumbnail Fig. A.2.

Scatter plots between periods, loop lengths, displacement amplitudes, duration, velocity amplitudes, energy fluxes, and energies depending on coronal regions. The ARs and QS are indicated in red and blue, respectively. A circle (triangle) represents a duty cycle of less (larger) than 2. The size of the data points is proportional to the intensity ratio.
In the text

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