A statistical study of propagating MHD kink waves in the quiescent corona

The Coronal Multi-channel Polarimeter (CoMP) has opened up exciting opportunities to probe transverse MHD waves in the Sun's corona. The archive of CoMP data is utilised to generate a catalogue of quiescent coronal loops that can be used for studying propagating kink waves. The catalogue contains 120 loops observed between 2012-2014. This catalogue is further used to undertake a statistical study of propagating kink waves in the quiet regions of the solar corona, investigating phase speeds, loop lengths, footpoint power ratio and equilibrium parameter values. The statistical study enables us to establish the presence of a relationship between the rate of damping and the length of the coronal loop, with longer coronal loops displaying weaker wave damping. We suggest the reason for this behaviour is related to a decreasing average density contrast between the loop and ambient plasma as loop length increases. The catalogue presented here will provide the community with the foundation for the further study of propagating kink waves in the quiet solar corona.


INTRODUCTION
The presence of MHD waves in the solar atmosphere is now well established (e.g., Ofman et al. 1997;Erdélyi et al. 1998;DeForest & Gurman 1998;Aschwanden et al. 1999;Nakariakov et al. 1999;Schrijver et al. 1999;De Moortel et al. 2000;Williams et al. 2001Williams et al. , 2002Goossens et al. 2002;Marsh et al. 2002;Ofman & Aschwanden 2002;Katsiyannis et al. 2003;Wang et al. 2003;Verwichte et al. 2005;Tomczyk et al. 2007;Tomczyk & McIntosh 2009;Morton et al. 2012Morton et al. , 2016Morton et al. 2019). Of all the MHD wave modes observed in the solar atmosphere, some of the most interesting are the transverse waves. They are thought to be critical in transferring energy from the turbulent convection in the photosphere to the solar corona.
The most common transverse wave in the corona appears to be the kink mode (the presence of torsional modes are more difficult to determine; however, there Corresponding author: Ajay K. Tiwari ajaynld13@gmail.com is some evidence for such motions, e.g. Kohutova et al. 2020). The kink mode has, to date, been observed in the corona in three variants: (decaying) standing kink waves, 'decayless' standing kink waves, and propagating kink waves. The standing kink waves were the first transverse wave modes to be observed in active region coronal loops (Nakariakov et al. 1999;Aschwanden et al. 1999), found with the Transition Region and Coronal Explorer (TRACE -Handy et al. 1999). These observations, and the launch of the Solar Dynamics Observatory (SDO -Pesnell et al. 2012), heralded a new era in the exploration and understanding of the physical properties of the solar corona through standing kink modes (Nakariakov & Kolotkov 2020). The standing kink modes are typically observed in active region coronal loops following an eruptive process (Stepanov et al. 2012;Zimovets & Nakariakov 2015;Goddard et al. 2016). The excitation mechanism of these standing kink waves is believed to be nearby eruptions or plasma ejections (rather than a blast shock wave ignited by a flare, as previously thought, e.g., Zimovets & Nakariakov 2015), which leads to a displacement of the coronal loops from their equi-arXiv:2105.12451v1 [astro-ph.SR] 26 May 2021 librium position. These waves are found to be rapidly damped, with the damping being attributed to the phenomenon of resonant absorption or mode coupling (e.g. Ionson 1978;Hollweg 1984;Ruderman & Roberts 2002;Goossens et al. 2002;Aschwanden et al. 2003). More recently, Goddard et al. (2016) produced a catalogue of standing kink modes observed with the SDO Atmospheric Imaging Assembly (AIA - Lemen et al. 2012), which was later extended by Nechaeva et al. (2019). A study of the relationship between damping time and amplitude indicated that a non-linear damping mechanism might also contribute to the observed damping. Van Doorsselaere et al. (2021) suggested that the observed relationship could be explained by uni-turbulence, a form of generalised phase-mixing Van Doorsselaere et al. 2020).
Secondly, there has been the discovery of 'decayless' standing kink wave modes (Tian et al. 2012;Wang et al. 2012;Anfinogentov et al. 2013Anfinogentov et al. , 2015Nisticò et al. 2013) in active region loops. These low-amplitude (< 1 Mm) oscillations do not appear to damp in time and are seen for several cycles. In some cases, the wave amplitudes are shown to gradually grow (e.g., Wang et al. 2012).
Finally, it was demonstrated that there are persistent and ubiquitous fluctuations in the Doppler velocities of coronal emission lines, which propagate at Alfvénic speeds and follow magnetic field lines (Tomczyk et al. 2007; Morton et al. 2019;Yang et al. 2020a,b). These motions have been interpreted as propagating kink waves and have also been observed with SDO/AIA (e.g. McIntosh et al. 2011;Thurgood et al. 2014;Weberg et al. 2018Weberg et al. , 2020. There have been several studies to reveal the properties of the propagating kink waves, finding that the power spectra of the velocity fluctuations can be described with a power law, and also show an enhancement of power at 4 mHz (e.g. Tomczyk & McIntosh 2009;Morton et al. 2015Morton et al. , 2016Morton et al. 2019). The excitation mechanism for the propagating waves is believed to be the random shuffling of magnetic elements in the photosphere due to convection, although this mechanism appears only to be able to explain the high-frequency part of the observed power spectrum (Cranmer & Van Ballegooijen 2005). Observational and theoretical studies provide evidence which indicates that mode conversion of p-modes may play a role in exciting some fraction of the observed waves (Cally 2017;Morton et al. 2019). Moreover, the origin of the low-frequency velocity fluctuations is still unclear, although Cranmer (2018) suggest that reconnection resulting from the evolution of the magnetic carpet may be the source.
The damping and dissipation of the propagating kink waves have not yet received as much attention as the standing modes. To date, there has only been a single observational case study that has been analysed. Verth et al. (2010) were the first to highlight the wave damping of the event presented in Tomczyk & McIntosh (2009), suggesting resonant absorption could provide a reasonable description of the observed behaviour. A number of other studies have also used this event as a case study (e.g. Verwichte et al. 2013b;Pascoe et al. 2015;Tiwari et al. 2019;Montes-Solís & Arregui 2020).
The focus of many previous studies has been on the (decaying and 'decayless') standing kink waves observed in active regions, with many statistical studies revealing the typical properties of these modes (e.g., Anfinogentov et al. 2015;Nechaeva et al. 2019). Given that the quiescent corona occupies a larger volume of the Sun's atmosphere than active regions and is omnipresent over the solar cycle, it is vital to understand the nature of the propagating kink waves that exist there and the waves' role in heating the quiescent coronal plasma. However, to date, there has been little focus on the propagating kink waves observed in quiescent corona. This paper attempts to fill some of that gap in our knowledge and provides a catalogue of suitable quiescent coronal loops that can be used for studying the propagating kink waves. In generating this catalogue, an overview of some of the typical propagating kink wave properties in the quiescent Sun is also provided. This paper also serves as a natural extension to the study by Tiwari et al. (2019).
The paper is structured as follows: In Section 2 the details of data and the analysis methods used are provided. Section 3 presents the main results and discusses the findings. The paper is concluded in Section 4.

Observations
The data are obtained from the Coronal Multi-channel Polarimeter (CoMP -Tomczyk et al. 2008), a coronagraph which observes the off-limb corona between 1.05 R and 1.35 R . CoMP is an imaging spectropolarimeter, and provides images of the corona taken at three different wavelength positions centred on the 10747Å Fe XIII coronal emission line (referred to as three-point measurements). The data were selected from CoMP observations taken between 2012-2016. The data sets in which there were more than 135 (near)contiguous data frames is identified by a manual inspection on the CoMP 1 data web-page. The dates are given in Table 2. The data sets from each selected date have a temporal cadence of 30 seconds (some with a small number of missing frames, < 5%) and spatial sampling of ∼ 4. 5. The Doppler velocity data products derived from fitting a Gaussian model to the line profile from the three-point measurements are the focus of this study. Details of the procedure used to estimate the Doppler velocities are given in Tian et al. (2013), and an assessment of their uncertainties is performed in Morton et al. (2016). A time-series of Doppler velocity images of the corona is used for this study. In cases where frames are missing, linear interpolation is performed to fill the gaps. Further registration of the Doppler images within each time-sequence is undertaken via cross-correlation, with further details given in Morton et al. (2016).
The analysis of the propagating kink waves requires the measurement of the wave propagation direction. Hence, a data product called a wave angle map is also derived, which gives the relative direction of propagation for the velocity signal within each pixel. The basis of the wave angle calculation requires a coherencebased approach for the analysis of the velocity signals, with general details discussed in McIntosh et al. (2008) and its use on CoMP data is discussed in Tomczyk & McIntosh (2009), Morton et al. (2015) and Tiwari et al. (2019). The strategy is to use the coherence between the Doppler velocity time-series of each pixel and its neighbouring pixels to obtain islands of coherence above a threshold value. The direction of wave propagation is calculated by a straight line fit through the islands, which minimises the sum of perpendicular distances from the points to the line. Performing this operation for each pixel of the Doppler velocity images gives the wave angle map. A sample wave angle map is shown in the centre panel of Figure 1.

Selection of loops for study
The selection of the loops from the CoMP data is a critical step in the analysis of the waves. For each of the data sets, suitable systems of coronal loops are identified. A lower limit of 50 Mm is placed on the lengths of the loop systems selected, whereby length refers to their visible length in the CoMP field of view 2 . This limit is required to preserve a high signal-to-noise level in the Doppler velocity time-series. Smaller loops are closer to the occulting disk, where the signal suffers from a high scattering of photons that leads to poor estimates for the Doppler velocity and hence increased noise in timeseries. The imposition of a minimum loop length also ensures an appropriate sampling in the k-direction in Fourier space, which is required for further analysis. A few such selected loops are shown in the right panel of Figure 1. The loops are chosen by manually identifying closed-loop structures first in the Doppler and intensity image sequences and later also in the wave angle map. The closed-loop structures as arcades of loops that appear semi-circular are identified, with each leg starting near the occulting disk.
The second criteria for loop selection require that the loops should be orientated such that the longitudinal direction of the magnetic field is close to being positioned in the plane-of-sky. The geometry and orientation of the loops are identified by performing magnetic field extrapolations using the Potential Field Source Surface (PFSS -Schrijver & DeRosa 2003) software. PFSS extrapolations performed using line-of-sight magnetogram data obtained from SDO's Helioseismic and Magnetic Imager (HMI -Scherrer et al. 2012). The extrapolations provide us with a schematic geometry and orientation of the loops in the plane-of-sky. The extrapolated field lines visibly agree with the loop structures observed in the coronal EUV images obtained by SDO/AIA and with the intensity images obtained by CoMP. Furthermore, the loops are selected to avoid loops within the cores of active regions (i.e. rooted in or near sunspots). Some of the trans-equatorial loops identified were located in the extended plage region of active regions on the visible solar disk. The observed loops are assumed to be rooted in network regions and are not part of active region loop systems. The left panel of Figure 1 displays the location of the apex positions, with respect to the solar limb, of all the loops used within this study.
Due to the low spatial resolution of CoMP, the effect of line-of-sight integration, and projection effects, there is an issue of disambiguation of individual loops in CoMP observations. Hence, the focus is on the wave signals in the system of coronal loops, as opposed to an individual structure.

Wave parameter estimation
For each system of coronal loops, a number of wave paths are extracted. A wave path is defined as being a contiguous set of pixels through the loop system, starting and finishing at the occulting disk. A pixel is selected within the loop system, and the wave angle map is utilised to map out a path, selecting the subsequent pixel based on the angle of propagation. The path is followed until the occulting disk is reached. A square box-car median filter of width two pixels was applied to the wave angle map to try and suppress some of the noise and led to the improved tracing of wave-paths. The pixel locations for each wave path are then used to extract the relevant velocities from the Doppler velocity images for each frame in the image sequence. A cubic interpolation maps the velocities from the selected wave paths onto (x, t) space. For each wave path, the neighbouring five wave-paths on either side of the original wave path are also extracted. The result of this is timedistance diagrams along the coronal loops systems. The longer loops lead to additional issues when tracing them because the wave angle suffers from more significant uncertainties closer to the apex of these loops. This arises because the wave angle is being poorly estimated near the upper boundaries of the wave angle map, primarily due to lower signal-to-noise in these regions arising from fainter coronal emission. In such cases, only the wave path for half a loop is obtained. The half-loop length is defined for each loop, obtained by finding the point of inflexion for the traced trajectory of the wave-path (except for the longer loops where only a half-loop is already traced).
The Doppler velocity time-distance diagram for each half-loop is subject to a two-dimensional Fourier Transform. The Fourier components are used to separate the inward and outward components of the wave propagation and provide velocity power spectra as a function of wavenumber-frequency (k − ω).
The propagation speed for the waves is calculated in a manner similar to Tomczyk & McIntosh (2009), Morton et al. (2015), Morton et al. (2019) and Tiwari et al. (2019). It is straightforward to filter ei-ther the inward or outward waves from the Fouriertransformed Doppler velocity time-distance diagrams by setting Fourier components equal to zero. The inverse Fourier transform of the filtered Fourier components then provides a Doppler velocity time-distance diagram containing only the inward or outward propagating waves. The cross-correlation between the time-series at the centre of the wave path and the neighbouring time-series along the path is calculated from these filtered Doppler velocity time-distance diagrams. The location of the peak of the cross-correlation function gives the time-lag between the signals and is determined by fitting a parabola to the peak. The observed lags as a function of the position along the wave path are fit with a linear function, and the gradient gives the propagation speed of the wave.
A feature of the propagating kink waves that is of particular interest is to estimate is the observed damping of these waves. The damping can be measured through analysis of the power ratio of the outward to inward propagating waves, P (f ) ratio . As mentioned, this has been performed previously by a number of authors for a single case-study (Verth et al. 2010;Verwichte et al. 2013b;Pascoe et al. 2015;Tiwari et al. 2019; Montes-Solís & Arregui 2020).
The velocity power as a function of frequency for the inward, P in (f ), and outward P out (f ), component of the waves are obtained by summing the velocity power spectra in the k-direction. For each loop, the inward and outward power spectra are averaged over the neighbouring wave paths to suppress the variability in the power spectra. From this one-dimensional averaged wave power, the ratio of the outward to inward power, P (f ) ratio , is determined by taking the ratio of the power at corresponding positive and negative frequencies.
Following Verth et al. (2010), the function to model the ratio of the power spectra is defined as follows: where L is the half-loop length, v ph is the propagation speed and ξ is the equilibrium parameter (or quality factor) that provides a measure of the strength of the wave damping. The factor P out /P in can be interpreted as the power ratio at the loop footpoint.
Estimates for ξ are obtained by fitting the model power ratio given by Equation 1 to the data, using a maximum likelihood approach. The associated confidence intervals on the model parameters were estimated by utilising the Fisher Information. For a detailed discussion on the statistics of the power ratio and the maximum likelihood approach, see Tiwari et al. (2019, their Section 3.4).

RESULTS AND DISCUSSION
In total propagating kink waves in 120 individual quiescent loops observed with CoMP are analysed. For each loop, estimates for the loop length, the propagation speed, the power ratio at the loop footpoint (P out /P in ) and the equilibrium parameter (ξ) are obtained. The various parameters that were obtained are listed in Table 2. In the following subsections, a summary of the main properties of the propagating kink waves is provided.

Loop lengths
First, a summary of the typical wave-path lengths is provided, which is used as a proxy for the loop length. In the left panel of Figure 2, the distribution of the halfloop lengths is shown. The half-loop lengths for the traced coronal loops are in the range of 50-600 Mm. The distribution peaks at around 150-200 Mm, and most of the loops observed are between 50-250 Mm. The number of longer loops is low, as it becomes increasingly difficult to trace the longer loops due to the limited field-of-view of the CoMP instrument. As well as the mentioned issue with visibility of the lower portions of the loops, the observed loop lengths suffer from projection effects, although it is hoped that the selection criteria for the loops minimise this (see Section 2.2). Due to the reasons mentioned above, in the case of propagating waves, a looplength always means half the looplength.

Propagation speeds
The distribution of propagation speeds is shown in the right panel of Figure 2, with the measured speeds distributed between 200-800 km s −1 and peaking around 400-600 km s −1 . This is consistent with the various propagation speeds reported in the literature (Tomczyk et al. 2007;Tomczyk & McIntosh 2009;Liu et al. 2014;Morton et al. 2015;Tiwari et al. 2019;Yang et al. 2020a,b). The propagation speed values obtained are averaged over the outward and inward wave propagation speeds. There is some evidence that the outward and inward velocities are different, which can be explained by the presence of flows along the coronal loops. However, the methodology for the measurement of the wave propagation speed is currently not sensitive enough to quantify this, apart from in extreme cases (e.g., in coronal holes, see Morton et al. 2015). The presence of flows leads to modification of the resonant damping of the kink waves, as described by Soler et al. (2011); consequently, this would require a change in the model for the power ratio that has been used. However, the influence of flows is neglected until they can be inferred more readily.

Power ratio
The power ratio factor, P out /P in , defined in Equation 1 is essentially a measure of the power of the waves entering the corona at each footpoint of the loops. The distribution of the estimated values of power ratio is shown in the left panel of Figure 3, and has a mean value of 1.29 ± 0.04. While the footpoint power ratio does not provide any information about the driving mechanism, it can be used as a proxy for measuring the energy input at each footpoint of the loop. The driving mechanism of propagating kink waves are thought to be one that acts globally due to the ubiquitous nature of these waves ). Hence, one would expect that the energy entering the corona through each footpoint will be approximately equal, unless each set of footpoints is located in regions with dissimilar magnetic field strengths. The mean value of the footpoint power ratio supports this hypothesis and is in agreement with the results from previous studies (Verth et al. 2010;Tiwari et al. 2019). The scatter around the value of 1 could indicate that in some regions of the atmosphere, the driver is weaker/stronger than in others.
However, examination of the behaviour of the power ratio as a function of the length of the coronal loop reveals the footpoint power ratio exhibits a decreasing trend with loop length (right panel of Figure 3). The power ratio starts at values close to 1.5 for the shorter loops and tends towards one as the loop length increases. A potential explanation for this trend could be that, for  shorter loops, the wave power injected at each footpoint is different, possibly due to different excitation mechanism or due to differences in the frequency/strength of the driver. Why this should be the case for shorter loops only is not evident.
Instead, it is suggested that the enhanced power ratio for shorter loops is an artefact of the analysis method. If the spatial wavelength of the oscillation is on the order of, or greater than, the length of the wave path used in the analysis, then there can be a leakage of power into negative/positive values of k. As an example, the wavelength of the kink modes is v ph P , where v ph is the phase speed, and P is the period. Using the values associated with the observed waves, a period of 300 s and phase speed in the range 200-600 km s −1 then gives wavelengths in the range 60 -180 Mm. Hence, the wavelengths are comparable to the lengths of the shorter loops selected here. The leakage of power from one quadrant of the power spectrum to another might be able to explain the observed deviation from 1. In future work, analysis can be modified to negate the impact of this power leakage, which can be avoided by fitting the power spectrum in ω-k space instead of just frequency.

Equilibrium parameter
Perhaps the most interesting parameter estimated here is the equilibrium parameter or quality factor (ξ), which quantifies the damping rate of the waves. The dis-tribution of ξ presented in the left panel of Figure 4 for the positive values. The values of ξ are between 0.89 and ∼ 298; hence they occupy a wide range of values (relative to the standing modes, see Morton et al. 2021). The distribution illustrates that ∼ 80 % of the positive ξ observations fall in the range of (0.89,30). Hence, the propagating kink waves can be strongly damped or very weakly damped. The median value is ∼ 11, and a mean value of ∼ 18, which is greater than that found for the standing kink modes (see Morton et al. 2021, for a full discussion of the importance of the observed values of the equilibrium parameter).
It should also be noted that of the 108 loops identified and studied, 31 of them show signs of power amplification, with a negative value of ξ. These are observed only for short loops (less than the half-loop length of 350 Mm). At present, it is unclear whether these results are physical. As already mentioned, short loops typically contain lower signal-to-noise velocity time-series due to their proximity to the occulting disk. However, it has been shown by Soler et al. (2011) that flows can play an important role and can lead to amplification of waves. The amplification of waves is always in competition with wave damping mechanisms. However, as discussed, the values of ξ are typically large, which implies a weak damping. Hence, in loops with weak damping, there is an improved chance to observe any amplification that may arise from flows or other mechanisms. These shorter loops with negative ξ need further investigation.

Equilibrium parameters relationship with half-loop length
Tiwari et al. (2019) gave evidence in favour of dependence between the equilibrium parameter (ξ) and the loop length, although there were only seven loops analysed in that study. The additional measurements made here enables us to examine this dependence further. As mentioned, the loops selected for this study correspond to loops being orientated such that the longitudinal axes are predominantly in the plane-of-sky. It implies that the longer loops reach higher altitudes in the corona. The right panel of Figure 4 displays a scatter plot, revealing a range of equilibrium parameters are possible for all loop lengths. The equilibrium parameter also shows a distinct behaviour with increasing loop length. As indicated by the results in Tiwari et al. (2019), as loop length increases, there is an increase in the value of the damping equilibrium parameter.
In order to show this relationship, whether several simple models of the formξ = f (L) could describe the data is examined. The models examined were: constant, linear, quadratic, square root and log. Each model also contained a constant term. The models were fit to the data assuming that a Normal distribution describes the likelihood of the data; hence the negative log-likelihood of the form is minimised: where ξ i is the observed value,ξ(L) is the model prediction, and σ i is the uncertainty on ξ i . The out-of-sample prediction error is estimated to test the ability of each model to describe the data. First,leave-one-out crossvalidation is utilised, using the mean value of the negative log-likelihood as a measure of the test error. It turns out all models (except the constant) show a similar ability to describe the data, with the quadratic and linear models performing the best, although the difference between the non-constant models is small. Moreover, the Akaike Information Criteria calculation supports the results from cross-validation and confirms that the constant model performs the worst. In the right panel of Figure 4, the results from the quadratic model is displayed. The uncertainty on the model curve is calculated by performing 10,000 bootstrap simulations of the fitting procedure, then using the percentile method to estimate the point-wise confidence intervals. The increase in ξ with loop length is evident. It is noted that the data also appear to show there is a lower bound to the possible values of ξ for the loops that increases with loop length.
The implication of the increase in ξ with loop length is that the propagating kink waves are subject to a reduced rate of damping for longer loops. As discussed in Tiwari et al. (2019), a physical explanation for the apparent decrease in damping rates can be made. Assuming that resonant absorption is the mechanism acting to provide the observed frequency-dependent damping, the quality factor for kink modes is given by where R is loop radius, l is the thickness of the density inhomogeneity layer, ζ = ρ i /ρ e is the ratio of the internal and external densities of the magnetic flux tube, respectively, and α is a constant whose value describes the gradient in density across the resonant layer. It is suggested that the key factor in understanding the observed behaviour would be the density ratio between the internal and external plasma. Quiescent coronal loops should be subject to similar heating rates; hence the rate of associated chromospheric evaporation is similar. If this is true, then the average density of the longer loops is likely to be less than those of shorter loops. This will lead to the density ratio (ρ i /ρ e ) for longer loops being, on average, smaller than for the shorter loops compared to the ambient plasma. As a basic examination of this premise, the scaling laws for dynamic loops derived in Bradshaw & Emslie (2020) is utilised. It is noted that the scaling laws are derived under the assumption of constant pressure along the loop, which may limit the applicability of the results to short, hot loops. From their Eq. (45), the loop apex density, n m , is related to the heating rate, E H , by The loop number density can be thought of as a combination of an initial number density (n 0 ) (due to some basal heating, E H0 ), which is assumed to be equal to ambient plasma (n e ), plus an additional density, n 1 , from the evaporation of the chromospheric/transition region due to some heating event (E H1 ) associated with the loop, i.e. n m = n 0 (E H0 ) + n 1 (E H1 ).
The density ratio ζ can be defined as: (2.4 × 10 −15 ) 4/7 M n e L −3/7 This expression suggests that the density ratio may depend upon the length of the coronal loop, with the over-density of the loop decreasing as the loop length increases. Substituting the expression for ζ into Eq. 2 to estimate the quality factor as a function of loop length. The values of ξ by providing some reasonable values for the quiet Sun parameters are calculated, assuming: the energy flux is F H ∼200 W m −2 = 2×10 5 ergs s −1 cm −2 ; the electron density is n e = 10 8.5 cm −3 ; the Mach number is 0.1 (corresponding to flows of ∼ 10 km/s); α = 2/π; and l/R = 1. The dependence of the quality factor on loop length from the scaling law theory is shown in Figure 4. While it does not match the curve from the model fitting, the results support our physical explanation for an increase in the equilibrium parameter as a function of loop length.
It is always worthwhile drawing comparisons between related results. To our knowledge, there is only one previous estimate for the damping rate of Alfvénic modes in the quiescent corona (excluding Verth et al. 2010;Tiwari et al. 2019), which was estimated in Hahn & Savin (2014) 3 . They provide estimates for the damping lengths of the waves, finding a broad distribution (up to 500 Mm) with a median value between 100-200 Mm. In order to provide a comparison to their results, the estimated quality factors should be converted to damp-ing lengths. The damping length (L d ) can be calculated using where λ prop = v ph P . Substituting typical values of the period (P ) of the waves observed by CoMP (100 -1000 s) and the measured phase speeds (see Table 2) in this expression given the damping lengths. Figure 5 shows the estimated damping lengths as a function of the period for three different values of v ph , using the median value of ξ from measurements in this study. The damping lengths reported in Hahn & Savin (2014) are comparable to the estimated damping lengths for the shorter period waves from the CoMP observations ( Figure 5). The damping lengths given by Hahn & Savin (2014) will themselves, of course, be related to waves with a particular range of periods. However, it is not straightforward to ascertain the periods of the waves encapsulated in the non-thermal line-widths.
A value of ξ from the information in Hahn & Savin (2014) can also be estimated. Although the assumed values of propagation speed, v ph , are not given in Hahn & Savin (2014), by inverting the given equation for the energy flux, F , namely, where δv is the velocity amplitude from the nonthermal widths, gives the propagation speed. The given values of energy flux (5.5 × 10 5 ergs s −1 cm −2 ), non-thermal widths (30 km s −1 ) and electron density (5 × 10 8 cm −3 ) is also utilised to find v ph = 550 km s −1 . Hence, using the damping lengths of 100-200 Mm, the quantity ξP = 180 -360 s. The Hinode/EIS data used in their study is integrated over 60 s. Assuming that only waves with periods less than 60 s contribute to the line broadening (which is a very conservative assumption), then ξ = 3-6. These values are likely overestimates for their study but are broadly in agreement with the range of values found in this study (e.g. Figure 4).

Comparison with observations of different modes of kink waves
Before concluding, a light-touch comparison between the properties of the propagating kink waves observed here and the previous studies of the two standing kink modes, i.e. damped and decay-less, is also provided. The comparison is worthwhile to highlight that the standing and propagating modes are found in coronal loops with significantly different plasma conditions. For the damped standing kink waves, the catalogue of events compiled by Goddard et al. (2016) and Nechaeva et al. (2019), using data from SDO/AIA is used. All the observations from these data sets which did not have any associated period or damping time information are removed. The total number of cases after this selection is 103 events over the course of solar cycle 23.
In the catalogue of standing kink waves, the loop length is estimated under the assumption that the loops are close to the semi-circular shape by either measuring the projected distance between the footpoints or by the apparent height. For each oscillation, the amplitude of the initial displacement and initial oscillation amplitude was given, along with the period. The mode of the standing kink waves here is assumed to be the fundamental.
The kink speed, which is not provided in the catalogue of Nechaeva et al. (2019), is calculated as follows: where c kink is the kink speed, L is the half loop length, and P is the period of the waves observed. Furthermore, the equilibrium parameter is also calculated by using the measured damping time of the oscillations and the periods, i.e. ξ = τ /P .
For the decay-less kink waves, the catalogue put together by Anfinogentov et al. (2015, see their Table 1) is shown 5 . For the damped waves, the median value of the equilibrium parameter is also presented.
In terms of the length of coronal loops which support the oscillations, the loops studied in this paper here have a similar distribution to those from the damped and decay-less standing wave studies. The propagating kink wave catalogue also contains several longer loops. The reader is reminded that the loop lengths measured by CoMP and SDO/AIA are not directly comparable. The CoMP instrument has an occulting disk that obscures the corona below 1.05 R Sun . Hence the measurements for the coronal loops' length do not begin near the footpoints, and it is likely an underestimation of the length of the CoMP loops (a rough correction being the addition of 70 Mm to the given values). On the other hand, SDO/AIA images show the solar-disk, often making the coronal footpoints of the loop visible in the images.
One of the main difference between the two loop populations is the measured kink speed. In the case of standing waves, the estimated kink speeds have medians of ∼1300 km s −1 and ∼1700 km s −1 for damped and decay-less, respectively. While for the propagating waves, the median value of propagation speed is substantially smaller at ∼480 km s −1 . The contrast of these values reflects the sizeable differences in the magnetic field strengths between the regions where these waves are observed and somewhat the different densities. The electron density in the quiet Sun (10 8−9 cm −3 ) is less than that of active regions (10 9−10 cm −3 ), which indicates the magnetic field strength must be substantially weaker in the quiescent Sun. This is borne out by estimates of the magnetic field strength, which in the active region coronal loops lies in the range of 4-30 Gauss (e.g. Nakariakov & Ofman 2001;White & Verwichte 2012) 6 , 5 The loop length in case of the standing and the decayless kink waves are the full semi-circular loop lengths as seen in AIA/SDO, while in the case of the propagating kink waves the loop lengths refer to the half-loop length as seen in the CoMP FOV 6 It should be noted that the magnetic field measurements obtained by coronal magneto-siesmology provide an underestimate of the magnetic field values (Verwichte et al. 2013a).
while the magnetic field of the quiet Sun loops are estimated to be between 1-9 Gauss (Morton et al. 2015;Long et al. 2017;Yang et al. 2020b,a). The difference in plasma parameters will ultimately bring about differences in how the waves/oscillations evolve as a function of time and/or distance. In fact, in the companion paper, Morton et al. (2021), a comprehensive discussion on the implications of differences in the equilibrium parameter found for the standing waves (ξ median = 1.8) and the propagating waves (ξ median = 11.4) is provided.

SUMMARY AND CONCLUSION
The details of a catalogue of quiescent coronal loops observed with the CoMP instrument are provided, all of which show evidence for the presence of propagating kink waves. The catalogue is used to undertake a statistical study of the propagating kink waves providing estimates for the damping rate and propagation speeds of the waves, presenting some details of how the propagating kink waves evolve. It is found that the equilibrium parameter, which quantifies the degree of wave damping, has a broad range of values, which indicates that in some of the coronal loops, the propagating kink waves are only weakly damped (this is aspect is discussed further in Morton et al. 2021). The damping length of the propagating kink waves is also estimated, which is found to be broadly comparable to the previous estimates of (Hahn & Savin 2014). Moreover, the study also finds that there is a relationship between the degree of damping and loop length, with waves propagating along longer loops typically experiencing reduced damping, verifying claims of Tiwari et al. (2019). The suggested reason for this behaviour is related to longer loops having a lower average density contrast, potentially due to limits on the amount of mass that can be evaporated during heating events associated with a given heating rate.
The study also reports the amplification of waves, the source of which is unclear at this time.
A brief comparison of the observed properties of the propagating waves to the standing modes is also presented. Notable differences between propagation speed and damping rates are found, with the contrast being due to the dissimilar plasma and magnetic environments of the two populations of loops that support the waves. The standing kink waves have been reported predominantly in loops with at least one footpoint in an active region; however, the propagating kink waves have been reported to be ubiquitous in the solar corona.
It is envisaged that the catalogue of propagating kink waves will provide the community with the foundation for further study of propagating kink waves in the quiet solar corona 7 . Many potential studies can exploit the propagating kink waves to further probe the plasma conditions in the quiescent loops, with the potential to incorporate density measurements from the Fe XIII line pair that CoMP also observes and provide estimates of magnetic field and flows through magneto-seismology (Morton et al. 2015;Yang et al. 2020b). This will ultimately enable us to develop a clear picture of how the propagating kink waves evolve in the quiescent corona and determine their role in plasma heating. Moreover, it is emphasised that there is a need for 3D MHD sim-ulations of kink wave propagation in quiescent coronal loops to aid our understanding of the role of resonant absorption in the damping of propagating kink waves.