1 Introduction

Photospheric bright points (BPs) are dynamic, small-scale, bright structures in the solar photosphere, usually appearing in dark intergranular lanes. They are associated with magnetic elements in magnetograms and have a magnetic-field strength of about 1 kG. BPs are typically circular (Berger et al., 1995; Bovelet and Wiehr, 2003), while some of them have elongated shapes (Kuckein, 2019). Their equivalent diameter is about 100 – 300 km, and their lifetime is about 90 seconds – 10 minutes (Berger and Title, 1996; Utz et al., 2010; Keys et al., 2011, 2014). In addition, BPs usually have a brightness contrast of 0.8 – 1.8 relative to the mean intensity of the photosphere (Sánchez Almeida et al., 2004).

BPs were first discovered in G-band observations of the photosphere in the 1970s (Dunn and Zirker, 1973), and later Mehltretter (1974) found that they represent magnetic-flux concentrations by comparing Ca ii K images with magnetograms. BPs are therefore also called magnetic bright points, and they are considered as the photospheric footpoints of magnetic-flux tubes (e.g. Berger et al., 1998; De Pontieu, 2002). A process named convective collapse is believed to be their formation mechanism (Spruit, 1979), which has received some observational support (e.g. Solanki et al., 1996; Bellot Rubio et al., 2001; Nagata et al., 2008).

Since BPs can be regarded as tracers of photospheric magnetic-flux tubes, many studies have suggested that BPs may host a variety of MHD waves (Martínez González et al., 2011; Jess et al., 2012b; Mumford, Fedun, and Erdélyi, 2015; Jafarzadeh et al., 2017). Given that the dissipation of MHD waves is among the most promising mechanisms for heating the solar corona, it is of great significance to study MHD waves associated with BPs or photospheric magnetic elements. Thus far, possible observational manifestations of kink waves (Stangalini, Berrilli, and Consolini, 2013; Stangalini, Giannattasio, and Jafarzadeh, 2015) and Alfvén waves (Jess et al., 2009) have already been found in BPs (or magnetic elements). However, sausage waves have not been unambiguously identified in them.

Sausage waves (or sausage modes) in a flux tube are characterized by a sausage-like boundary of the tube, first defined by Defouw (1976). A detailed discussion of this wave mode is given in Section 4.1. There are many studies focusing on the theoretical understanding and seismological applications of sausage modes (e.g. Chen et al., 2015; Guo et al., 2016; see also the review by Li et al., 2020). In observations, sausage modes in the photosphere have been widely studied in magnetic pores and sunspots (e.g. Dorotovič, Erdélyi, and Karlovský, 2008; Morton et al., 2011; Grant et al., 2015; Freij et al., 2016). Very recently, possible signature of sausage-mode slow waves is also identified in a plage region (Ji et al., 2021). Also, Morton et al. (2012) found evidence for the simultaneous existence of sausage and kink modes in fine magnetic structures in the chromosphere. In the corona, short-period, quasi-periodic pulsations (QPPs) in flares are also generally interpreted as sausage waves (e.g. Nakariakov, Melnikov, and Reznikova, 2003; Van Doorsselaere et al., 2011; Tian et al., 2016; see also the review by Zimovets et al., 2021).

In this article, we report the existence of in-phase oscillations of area and average intensity in BPs using high-resolution TiO images for the first time. Our findings can be regarded as possible evidence of sausage modes in photospheric flux tubes. In fact, magnetoacoustic waves characterized by intensity or line-of-sight (LOS) velocity oscillations have been found in BPs before (Jess et al., 2012a,b; Stangalini et al., 2013; Jafarzadeh et al., 2017). However, these previous studies could not tell whether these waves were sausage waves, because either they did not consider BP areas or the area variations were too weak. Martínez González et al. (2011) studied four magnetic patches in circular-polarization images obtained by the Imaging Magnetograph eXperiment (IMaX) onboard Sunrise, and they found that the iso-flux area oscillates with time. However, the time-varying periods suggested that the oscillations may be caused by granular motions rather than a characteristic oscillation mode. Fujimura and Tsuneta (2009) also focused on oscillations of intergranular magnetic structures (IMSs) in a magnetogram and interpreted these oscillations as sausage and/or kink waves. However, the lifetimes of these IMSs are all greater than one hour, which is much longer than the typical lifetime of BPs. Kolotkov et al. (2017) also detected long-period (80 – 230 minutes) quasi-periodic oscillations of the average magnetic field and total area of a small magnetic structure. More recently, Keys et al. (2020) studied the rapid variations of magnetic fields in BPs and suggested that such changes are consistent with the behavior of MHD waves, but they did not discuss in detail the changes of intensity.

This article is organized as follows: In Section 2, we describe our observation and data-analysis methods, and the key results are presented in Section 3. We discuss the wave-mode identification in Section 4 before giving the final conclusion and some remarks in Section 5.

2 Observations and Data Analysis

The data used in this study were acquired with the 1.6-meter Goode Solar Telescope (GST: Cao et al., 2010) at the Big Bear Solar Observatory (BBSO). We used the Broadband Filter Imager (BFI) installed on the GST to observe the TiO band (7057 Å) from 16:28 to 17:59 UT on 6 August 2018, with a cadence of 15 seconds. The observed regions include the active region NOAA 12717 at \((501^{\prime \prime }, -222^{\prime \prime }) \) from 16:28 to 17:33 and the western lobe of a coronal hole at \((-260^{\prime \prime }, 2^{\prime \prime })\) from 17:33 to 17:59.

The bandwidth of the filter is 10 Å. The field of view (FOV) of the TiO data was \(77^{\prime \prime }\times 77^{\prime \prime } \) and the pixel size was \(0.034^{\prime \prime } \). During the observation, the seeing condition was better than \(3^{\prime \prime }\), which is good for the identification of BPs. Speckle reconstruction and a high-order adaptive optics (AO) system were used to reach the spatial-resolution limit of \(0.1^{\prime \prime }\). The data were also destretched to remove residual atmospheric-seeing effects.

We first selected isolated BPs in the FOV by eye. Liu et al. (2018) and Kuckein (2019) divided BPs into isolated ones, which are individual, and non-isolated ones, which display a splitting or merging behavior in observations. Obviously, it is more suitable to use isolated BPs to search for MHD waves, as for non-isolated BPs their splitting and merging behaviors may affect our identification of waves. For the selected BPs, we used contours with a fixed intensity-threshold level to identify and track them. Then, we calculated the area and average intensity of these BPs at each time and examined if there was any oscillation signal. Finally, four of the BPs are found to be well isolated, and there is also possible evidence of sausage waves inside. Their positions and shapes are shown in Figure 1. We named them BP1 to BP4. When calculating their average intensity and area, we carefully removed the unrelated contours nearby (we can see them around BP1, BP2, and BP4 in Figure 1) by manually zooming on the region of interest for every frame. We summarize the properties of the four BPs in Table 1.

Figure 1
figure 1

TiO images with BPs marked by boxes. The lower images below show an enlarged view of the red box regions, where BPs are identified and tracked with red contours.

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Table 1 Observational details of the BPs. The area is averaged over time, and C.C. represents the correlation coefficient of average intensity and area.
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3 Result

Figure 2 presents the time series of the area and average intensity of these four BPs, and we can see obvious oscillations for two to four cycles. It can also be seen that for each BP, the crests and troughs for these two features appear at nearly the same time, which suggests that the oscillations of area and average intensity are in-phase. Among these BPs, BP1 has the most oscillation cycles, with four clear crests in the time series of area, at 105, 195, 315, and 465 seconds. These instants correspond well to crests in the curve of average intensity. In addition, there is also a good correspondence between the moments of each trough. Figure 3 gives ten snapshots of BP1 in Panels A – J, each corresponding to a moment of a wave crest or a trough. We can clearly see that the BP’s area oscillates with time.

Figure 2
figure 2

Time series of average intensity and area of the four BPs. Each row corresponds to one BP, and the right two columns show the temporal variations of the average intensity and area, respectively. Here, we use the intensity-fluctuation percentage with respect to the mean value of the background. (Animation of this figure is available in the Supplementary Materials.)

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Figure 3
figure 3

(A – J): snapshots of BP1 marked by green arrows at ten different time. Approximately, every three adjacent panels correspond to an oscillation cycle. (K): wavelet-analysis results of detrended time series (shown in Panel K1) of average intensity. The results include the wavelet power spectrum (Panel K2) and the global power spectrum (Panel K3). Darker colors represent stronger powers. The dashed lines indicate a significance level of 95%. (L): the same as Panel K but for the detrended time series of area.

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Then, we detrended the time series of average intensity and area by subtracting a three-minute running average before performing wavelet analysis. The detrended series and wavelet analysis results are shown in Panels K and L of Figure 3. The peak global wavelet powers for the average intensity and area are at around 150 seconds, which could be seen as the oscillation period. For the other BPs, the periods can be estimated to be 100 – 200 seconds. The period ranges that we found here are comparable to the results in some previous studies, which interpreted the oscillations that they found in BPs as magnetoacoustic waves (Jess et al., 2012a,b). We note that oscillations with longer periods, above four minutes, were previously detected in small magnetic structures (Fujimura and Tsuneta, 2009; Martínez González et al., 2011; Kolotkov et al., 2017).

In order to further confirm the in-phase relation between oscillations of average intensity and area, we drew scatter plots of the relationship between them and calculated the correlation coefficients, as shown in Figure 4. It can be seen that the correlation coefficients are all above 0.68. For BP2 and BP4, the coefficients are around 0.9, showing a good positive correlation. Meanwhile, we also performed a lagged cross-correlation (LCC) analysis. From Figure 5, we can see that the coefficients for all BPs reach the peak when the time lag is zero. All the results confirm the in-phase relation between average intensity and area.

Figure 4
figure 4

Scatter plots of the relationship between average intensity and area for the four BPs, with solid lines representing linear fits. Note that the average intensity and area are both detrended.

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Figure 5
figure 5

Lagged cross-correlations between average intensity and area. The time lag is between the detrended time series of area and average intensity.

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A possible question is whether the correlation or the in-phase relation is caused by our analysis method, i.e. using contours with a fixed intensity threshold to identify BPs and calculate their parameters. If we consider the total intensity rather than the average intensity in the contours, we might tend to obtain an in-phase relation, since it appears that the total intensity will always increase (or decrease) along with the increase (or decrease) of the area. Hence, here we chose to use the time series of the average intensity. Additionally, we studied for more than four isolated BPs with the same method, and we found that there are cases without such an oscillation pattern. In conclusion, the in-phase relation is not a result of our data-processing method, but more likely associated with some characteristic oscillation modes.

We will show in the next section that the observed in-phase oscillations might be explained as sausage waves in BPs.

4 Discussion

4.1 Sausage-Mode Waves in a Flux Tube

In the solar atmosphere, magnetic-flux tubes are formed with magnetic-field lines clumping into tight bundles. They can be regarded as channels guiding MHD waves. A flux tube is usually assumed as a straight cylinder described by cylindrical coordinates \((r, \phi , z)\). The tube boundary can oscillate in different geometrical patterns, forming various wave modes including sausage modes, kink modes, fluting modes, and torsional (Alfvén) modes (Roberts, 2019). For sausage modes, the axis of the tube is motionless, while the cross-sectional area fluctuates with time, resulting in a sausage-like axisymmetric perturbation of the tube boundary. Hence, sausage waves are axisymmetric waves, belonging to magnetoacoustic waves. According to the relationship between phase speed \(c_{\mathrm{p}} \) and sound speed \(c_{\mathrm{s}} \) in the tube, we can divide the sausage wave into a fast mode and a slow mode. The former has a faster phase speed than the sound speed, while the latter has a slower phase speed. It is assumed here and hereafter that the characteristic speeds inside and outside the tube satisfy the ordering for “photospheric conditions” in the sense of Figure 3 of Edwin and Roberts (1983). In particular, \(\beta \) inside the tube is assumed to be less than unity. This assumption on the internal plasma \(\beta \) is generally accepted for photospheric BPs, even though \(\beta \) is difficult to measure directly (e.g. Shelyag et al., 2010; Cho et al., 2019).

Based on ideal MHD equations, setting the radial-velocity perturbation \(v_{r} = A(r) \cos (kz-\omega t) \), we can obtain (Freij et al., 2016, without considering azimuthal components of the velocity perturbance and magnetic field):

$$ b_{z}=\frac{B_{0}}{\omega }\frac{1}{r} \frac{\partial (rA(r))}{\partial r}\sin (kz-\omega t)\,, $$
(1)
$$ p_{1}=c_{\mathrm{s}}^{2}\rho _{1}=- \frac{\omega \rho _{0}c_{\mathrm{s}}^{2}}{(c_{\mathrm{s}}^{2}k^{2}-\omega ^{2})} \frac{1}{r}\frac{\partial (rA(r))}{\partial r}\sin (kz-\omega t)\,, $$
(2)

where \(b_{z} \) is the longitudinal magnetic-field perturbation, \(B_{0}\) and \(\rho _{0}\) are the background magnetic field and the density, \(p_{1}\) is the pressure perturbation, and \(\rho _{1}\) is the density perturbation.

Comparing Equations 1 and 2, we can state that if \(c_{\mathrm{s}}^{2}k^{2}-\omega ^{2}>0\) (i.e. the phase speed \(c_{\mathrm{p}}=\omega /k\) is smaller than the sound speed \(c_{\mathrm{s}}\), corresponding to a slow sausage mode), then \(b_{z}\) and \(\rho _{1}\) (or \(p_{1}\)) will be 180∘ out of phase (i.e. in anti-phase). On the contrary, if \(c_{\mathrm{s}}^{2}k^{2}-\omega ^{2}<0\) (i.e. the phase speed is larger than the sound speed, corresponding to the fast sausage mode), then the perturbations will be in-phase.

Since the magnetic flux in a flux tube is constant, we have \(B_{0}S_{0}=(B_{0}+b_{z})(S_{0}-S_{1})\). Here, \(S_{0} \) and \(S_{1} \) represent the unperturbed and perturbed cross-sectional areas, respectively. Ignoring \(b_{z}S_{1}\), we can obtain \(B_{0}S_{1}=-b_{z}S_{0}\), or

$$ \frac{S_{1}}{S_{0}}=-\frac{b_{z}}{B_{0}}\,. $$
(3)

Hence, the area perturbation \(S_{1}\) and the longitudinal field perturbation \(b_{z}\) are in anti-phase, and we can find the change of the magnetic field from the change of the area. This was also used by Grant et al. (2015).

As a result, for the slow sausage mode, fluctuations of area and density are in-phase; whereas for the fast sausage mode, the fluctuations are in anti-phase. These simple phase relations have also been confirmed by other theoretical investigations (Moreels, Goossens, and Van Doorsselaere, 2013; Moreels and Van Doorsselaere, 2013).

Next, we will discuss the impact of the opacity effect and associate density with the continuum intensity, so that we can obtain the phase relation between fluctuations of intensity and area for fast and slow sausage modes.

4.2 Phase Relation Between Fluctuations of Intensity and Area

The TiO data used in our study could be regarded as continuum images (see, e.g., LeBlanc, 2010), so the intensity that we observed could be written as

$$ I=\int _{0}^{\tau }\frac{\sigma T(\tau )^{4}}{\pi }\mathrm{e}^{-\tau } \mathrm{d}\tau \,, $$
(4)

where \(\sigma \) is the Stefan–Boltzmann constant, and \(T(\tau )\) is the local temperature at an optical depth of \(\tau \). Thus, the observed intensity fluctuations come primarily from two sources: the change of temperature and the change of optical depth. The latter depends on the density and temperature along the line-of-sight, which is known as the opacity effect (Fujimura and Tsuneta, 2009).

If the oscillations of intensity and area in BPs are caused by the opacity effect, then the oscillations will be in anti-phase. In Figure 6, a BP is shown as a laterally expanding flux tube, and the layer of \(\tau =1\) is marked with the dashed line. Suppose that from time \(t_{1}\) to \(t_{2}\), the \(\tau =1\) layer moves to a lower height, resulting in a reduced area of the BP. In the deeper layer with a higher temperature, the intensity would increase, which means that the intensity and area should be in anti-phase. Clearly, this is contrary to our observation.

Figure 6
figure 6

The change of intensity and area caused by the opacity effect in a flux tube. The dashed lines represent the \(\tau =1\) surface, corresponding to the BP that we see in the TiO image. From time \(t_{1}\) to \(t_{2}\), the surface moves to a lower height. The two arrows in the middle indicate that the height increases from the bottom up and temperature increases from the top down.

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Without considering the opacity effect, we can assume that the surfaces that we see are at the same height, as shown in Figure 7. Another assumption that we made is that the BP intensity increases from time \(t_{1} \) to \(t_{2} \), and it is caused by the increase of temperature. The higher temperature will lead to a decrease in the absorption coefficient \(\kappa _{\nu }\) at a frequency \(\nu \), which is proportional to \([1-\exp \left (-\mathrm{h}\nu /\mathrm{k_{B}}T\right )]\), where \(\mathrm{h}\) and \(\mathrm{k_{B}}\) are the Planck constant and Boltzmann constant (Rutten, 1995), respectively. In order to ensure

$$ \tau =\int _{r_{0}}^{\infty }\kappa _{\nu }\rho (r) \mathrm{d}r=1\,, $$
(5)

the density should increase. Here, \(\rho (r)\) represents the density at height \(r \), and \(r_{0} \) represents the height of the surface that we see. Our observation shows that the fluctuations of intensity and area are in-phase. If the area increases at the same time, the density will be in-phase with the area, which is a characteristic of the slow sausage wave (see Section 4.1). Hence, the in-phase oscillations of intensity and area that we have observed could be possible evidence of slow sausage waves in BPs.

Figure 7
figure 7

The change of intensity and area caused by the sausage waves. The \(\tau =1\) surface remains at the same height from time \(t_{1}\) to \(t_{2}\). The two arrows in the middle indicate that the height increases from the bottom up and temperature increases from the top down.

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However, if we consider both the opacity effect and the presence of sausage modes, the result will be different. In Figure 8, from \(t_{1}\) to \(t_{2}\), the sausage waves enlarge the cross-sectional area of the flux tube, and the magnetic field will decrease. If the opacity effect moves the \(\tau =1\) layer to a lower height at the same time, we also expect in-phase oscillations of intensity and area, but now we cannot distinguish which mode the sausage waves are. The total pressure equilibrium at the same height is given by

$$ p_{\mathrm{i}}+\frac{B_{\mathrm{i}}^{2}}{2\mu _{0}}=p_{\mathrm{e}}+ \frac{B_{\mathrm{e}}^{2}}{2\mu _{0}}\,, $$
(6)

where \(p_{\mathrm{i}}\) and \(p_{\mathrm{e}}\) are the internal and external thermal pressure, \(B_{\mathrm{i}}\) and \(B_{\mathrm{e}} \) are the internal and external magnetic field, and \(\mu _{0}\) is the permeability. Equation 6 indicates that the decreasing magnetic field inside the tube will lead to an increasing thermal pressure, which is proportional to \(\rho T\). If the density \(\rho \) increases, the waves will be slow mode; otherwise they will be fast mode. For specific determination, it may be necessary to introduce some appropriate photospheric atmosphere models (e.g. Fang, Tang, and Xu, 2006), which will be our next step. Nevertheless, no matter which mode the waves actually are, the results that we observed can be seen as evidence for sausage waves in BPs.

Figure 8
figure 8

The changes of intensity and area caused by both the opacity effect and sausage waves. The \(\tau =1 \) surface moves to a lower height from time \(t_{1} \) to \(t_{2} \). The two arrows in the middle indicate that the height increases from the bottom up and temperature increases from the top down.

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5 Conclusion and Remarks

In this study, we used the TiO images obtained from the BBSO/GST on 6 August 2018 to carry out a detailed analysis of several isolated photospheric BPs. For the four selected BPs, the changes of their area and average intensity with time show obvious oscillations and an in-phase relation.

In order to interpret these results, we presented a simple theoretical discussion, ruling out the opacity effect alone and identifying the observed oscillations in BPs as sausage modes. Assuming that the fluctuations are only caused by sausage waves, we can identify the oscillations in BPs as slow sausage modes, which have been found in some previous observations of magnetic pores (e.g. Grant et al., 2015; Freij et al., 2016). Our discovery provides strong support for the existence of sausage waves in a photospheric flux tube. Although the presence of sausage waves has also been observed in magnetic pores or sunspots, BPs are better tracers for the flux tubes, and they are more ubiquitous in the photosphere. Additionally, BPs are much smaller than pores and sunspots, which makes them more easily buffeted by surrounding granular motions. Since the dissipation of sausage modes is generally believed to have an important contribution to the heating of the chromospheric atmosphere (Grant et al., 2015; Yu, Van Doorsselaere, and Goossens, 2017; Chen et al., 2018; Keys et al., 2018; Gilchrist-Millar et al., 2021), our findings could be helpful for understanding chromosphere heating.

Our future work will first focus on the mode identification by further addressing the opacity effect. We plan to introduce an appropriate photospheric atmosphere model to determine this. Furthermore, our research is still limited to single-channel observations of the photosphere. We plan to combine more data in different wavelengths or spectral observations to study the propagation of sausage waves in BPs to the upper atmosphere. Moreover, the Daniel K. Inouye Solar Telescope (DKIST: Rast et al., 2021) can provide observations of the photosphere at an unprecedented high resolution. By using the oncoming data from DKIST, we may find more evidence of MHD waves in BPs and carry out a more detailed study.