Periodic Mesoscale Density Structures Comprise a Significant Fraction of the Solar Wind and Are Formed at the Sun

Mesoscale density structures in the solar wind are often periodic, with f ∼ 0.1–5 mHz. They are trains of advected density structures with radial length scales of ∼100–10,000 Mm. While studies have shown that these periodic density structures (PDSs) are often formed at the Sun and released into the solar wind, it is unknown what percent of the solar wind at 1 AU is comprised of PDSs from the Sun, as opposed to periodicities formed through dynamics en route. We expand on Kepko et al. (2020, https://doi.org/10.1029/2020ja028037) which analyzed 25 years of in situ solar wind proton data, and include here alphas to examine the compositional characteristics of PDSs. Compositional changes, such as the alpha‐to‐proton ratio (α/p), are frozen into the solar wind plasma low in the corona, and so do not evolve as the solar wind advects and fills the Heliosphere. We find a broad occurrence enhancement in both the proton and α/p distributions between 1 and 3 mHz, centered near ∼2.1 mHz, and demonstrate that this distribution can be modeled assuming ∼30% of the solar wind segments contain a PDS. We find a distinct distribution below 1 mHz, with markedly different α/p characteristics. The α/p indicates that both populations are from the Sun, with likely different generation mechanisms. We conclude by summarizing mechanisms at the Sun that could produce periodic mass release, namely, periodic magnetic reconnection. The lower frequency PDSs likely involve reconnection at S‐web arcs including the heliospheric current sheet, while the higher frequency PDSs may be driven by p‐mode‐related transverse coronal oscillations.

PDSs have been studied in both the time (Viall, Kepko, & Spence, 2009) and spatial domains (Kepko et al., 2020;Viall et al., 2008).Here, the spatial domain is defined simply as the radial length scale, L r (i) = 'v r (i) × Δt, for each solar wind measurement, where v r (i) is the solar wind radial velocity of the time i with sampling time Δt = t(i) − t(i − 1).Working in the spatial domain accounts for any compression or relaxation of the density structures that may have occurred in transit.This was confirmed by Viall et al. (2008), who found a higher occurrence rate of periodicities in length scale than the comparable Viall, Kepko, and Spence (2009) results in time.All of these studies have found narrow bands of recurrent time and length scales that were determined to occur more often than others.Kepko et al. (2020) analyzed 25 years of Wind spacecraft solar wind proton density data to calculate the occurrence distributions (ODs) of statistically significant PDSs in length using 9,072 Mm windows (roughly equivalent to 6 hr at slow solar wind velocity).They found recurrent bands of L ∼ 130-140  Mm in the slow wind, equivalent to frequencies of f ∼ 3.0 and 2.3 mHz in the rest frame of Earth's magnetosphere or a spacecraft, and additional bands near 230 and 300 Mm in both fast and slow wind.Notably, Kepko et al. (2020) showed over 25 years (i.e., approximately two solar cycles) that the evolution of these bands changed abruptly near solar "terminator" events that mark the end of a Hale magnetic cycle (McIntosh et al., 2015(McIntosh et al., , 2019)).
These studies of occurrence rates relied on creating ODs of statistically significant frequencies or, equivalently, length scales.In all of the above studies, a frequency was defined as statistically significant if it had power above a background spectra and simultaneously passed an independent test for phase coherence.In general, a detection of a statistically significant frequency can be due to a true periodicity in the solar wind, either generated en route as the solar wind advects or created at the Sun (Viall et al., 2021), or from "false positives" that are inherent when applying a statistical confidence test and defined by the confidence threshold applied.When studying many different spectra, the overall shape of the OD of statistically significant frequencies depends on the ability to accurately model the background spectra.For example, if a method consistently overestimated the background spectral fit at high frequencies, this would effectively create an artificially high confidence threshold at those frequencies, and the occurrence rates would be reduced over that frequency range.Importantly, identifying localized occurrence enhancements on top of this OD is insensitive to both the overall shape of the ODs, and also to the assumed background spectral model (Kepko et al., 2020).Hence, previous studies have focused on the localized enhancements in occurrence rates (bands) that are in addition to this overall trend (e.g., Kepko & Viall, 2019;Viall et al., 2008).Yet, even under different assumptions about the spectral background fit, the overall OD of significant frequencies consistently exhibited a broad enhancement between 100 and 400 Mm (1-3 mHz) (Kepko et al., 2020).This suggests that the overall shape of the statistically significant frequency distribution is physically meaningful and could be used to gain more information about the characteristics of PDSs.
Recently, Di Matteo et al. (2021) studied the false positive distribution of significant frequencies under a variety of background spectral fit assumptions.Using these techniques, we are now able for the first time to quantify the 10.1029/2023JA031403 3 of 20 overall distribution of PDSs in Wind proton number density measurements, not just the narrow enhancements as was done in previous studies.We also significantly extend previous work to include longer window lengths in both time and space, providing further confidence that both the localized bands of enhancements and an overall broad enhancement exists.Finally, we also include α/p, the first time the alphas have been statistically studied in the context of PDSs.
With these results, we finally answer long-standing questions related to PDSs, including: (a) How much of the solar wind at 1 AU is comprised of PDSs of any origin (i.e., above the false positive rate)?(b) How much of the solar wind at 1 AU is comprised of PDSs of solar origin?and (c) Of the full 0.1-5 mHz distribution of frequencies, how many separate populations exist, possibly indicating different generation mechanisms?The answer to (a) requires the thorough background fitting routines of Di Matteo et al. (2021), while answers to (b) and (c) additionally require α/p statistics, which is the crucial piece for examining how often PDSs come from the Sun, as opposed to generated en route.We conclude by using these results to speculate on the solar processes that could be responsible for the generation of PDSs.

Methods
We follow the data preparation and analysis that is thoroughly described in Kepko et al. (2020), and briefly summarized here.The primary two differences from that previous study are the addition of longer data segments and the addition of alpha measurements.Our study uses 25 years of Wind SWE solar wind proton and alpha measurements (Ogilvie et al., 1995), with density and velocity moments computed with the bi-Maxwellian technique of Kasper et al. (2006) that separates the proton and alpha distributions.We calculate a length series in the radial direction, L r (i), straightforwardly as L r (i) = v r (i) × (ut(i) − ut(i − 1)).This length series has an uneven sampling rate that is directly related to the solar wind speed at time ut(i), with higher speeds having larger ΔL, and slower speeds having smaller ΔL.We therefore interpolate L(i) to produce a "fast" length series, with fixed ΔL f = 56.7 Mm, and a "slow" length series, with ΔL s = 35.4Mm, using the average observed fast and slow solar wind speeds.We segment the data into window lengths of 6, 12, 18, 24, and 36 hr in time, and 9,072, 18,144, 27,216, 36,288, and 54,432 Mm in length, shifting each subsequent window by 1/36th of the window length (e.g., 10 min for 6 hr windows, and 252 Mm for 9,072 Mm windows).We keep from the "fast" length series those segments with   ≥ 550 km/s and keep from the "slow" length series segments those with   < 550 km/s.This separation of "fast" and "slow" segments is not a physics-based classification; rather, it derives from the inherent need for an evenly sampled series for spectral analysis, and designed to avoid having undersampled "fast" segments and oversampled "slow" segments.
We perform spectral analysis on these segments using the multitaper method (MTM) (Thomson, 1982) to calculate spectral power.We make no a priori assumptions about the shape of the background spectra, only that it can be fit by a bending power law (BPL) (Di Matteo et al., 2021;Vaughan et al., 2011) where β and γ represent the spectral slope at low and high frequencies, respectively, f b is frequency at which the bend occurs, and N is a normalization factor.The BPL collapses to a power law for very low and very high (near the Nyquist) values of f b .We then perform an amplitude test, the so-called narrowband test, for spectral enhancements at specified confidence levels above this background fit.

Results
We applied our spectral analysis technique to every segment of solar wind density data that passed the quality checks, identical to those defined in Kepko et al. (2020).Once we have determined the statistically significant spectral peaks in each data segment, we calculate ODs of these significant frequencies and length scales for each window length.While the previous statistical studies of in situ Wind data examined only 9,072 Mm and 6 hr windows, we include here multiple window lengths in both time (6,12,18,24,and 36 hr) and length (9,072,18,144,27,216,36,288,and 54,432 Mm), and further expand by including ODs for n α /n p , which have never been previously shown.There are currently no statistics on the duration of a PDS event, other than that empirically we find 6-hr segments are better for identifying high frequency PDS, while the lower frequency PDS (≲1 mHz) are best observed at longer window lengths.
Di Matteo et al. (2021) had shown that misfitting of the spectra happens primarily within NWf ray of the zero and Nyquist frequencies, where N is the number of points of the analyzed interval, W is the half-width of the main lobe of the spectral window, and f ray = 1/(NΔt) is the Rayleigh frequency.Note that the quantity NW is often referred to as the time-half-bandwidth product.Increasing the window length reduces f ray = 1/(NΔt), and hence reduces the area of misfitting, thereby allowing us to determine how the ODs could be affected by misfitting of the spectral background near these boundaries.We used NW = 3 for the L = 6, 12 and L = 9,072, 18,144 windows, and NW = 4 for the longer windows.For reference, NWf ray = 0.14 mHz for 6-hr segments, and NW = 0.03 mHz for 36 hr segments.
Note that for the length segments, the spectral analysis is performed in inverse length scale space, equivalent to inverse time (frequency).For convenience in interpretation, we plot for the length scale ODs the inverse, which is length in Mm.The inverse Nyquist for length is therefore simply 2 × ΔL, which is 70.8 Mm for "slow" and 113.4 Mm for "fast."ODs are binned in integers of the 9,076 Mm and 6 hr Rayleigh frequencies, and longer window lengths are binned in proportionally larger multiples of f ray to increase the histogram bin size and match the 9,076 and 6 hr bin sizes for ease of comparison.
Figure 1 shows the statistics of how many segments were created, how many passed the quality checks and were analyzed, and how many contained at least 1 significant frequency at the 95% confidence threshold.In all cases for protons, between ∼50% and 70% of the solar wind segments pass the quality control checks sufficiently to be analyzed.For α/p, this is closer to ∼30%.This difference is driven by the ability to clearly distinguish the alpha peak in the Wind SWE data, which is easiest during intervals of slower, denser solar wind.Also of note is that the vast majority of the data is from the slow solar wind.Additionally, for short window lengths (6 hr and 9,072 Mm) ∼60% of segments contain at least 1 significant frequency at the 95% level, while for longer window lengths (36 hr and 54,432 Mm) this approaches 100%.

Narrow Band Enhancements of Proton and α/p ODs
We follow Kepko et al. (2020) and Viall et al. (2008) and calculate 4-year ODs, which provides a balance between obtaining sufficient binning statistics while providing sufficient granularity to observe any solar cycle dependencies.The 4-year ODs for proton and α/p for the "slow" segments for all window lengths are shown in Figure 2. A related version of the 9,072 proton OD (upper left in Figure 2) was shown in Kepko et al. (2020).For both the proton and α/p ODs there exists a broad occurrence enhancement between ∼100 and 400 Mm that peaks near 170 Mm.The broad enhancement and location of the central peak are independent of the window length.Narrow, localized occurrence enhancements are evident riding on top of the broad ∼100 and 400 Mm enhancement, including four bands near 130, 165, 210, and 300 Mm, as indicated in the L w = 9,072 Mm OD.These correspond to bands I-IV in Kepko et al. (2020).The remaining ODs in Figure 2 demonstrate that these bands are consistently observed across all window lengths, and quite clearly in the α/p statistics.
The "fast" segment 4-year ODs are shown in Figure 3, in the same format as Figure 2. The ODs for all window lengths exhibit broad occurrence enhancements between 200 and 500 Mm, with a peak generally between 200 and 300 Mm.The strong occurrence enhancement near 200-250 Mm corresponds to band IV, as in the "slow" ODs, while the bands near 400 Mm are bands I and V from Kepko et al. (2020).
Finally, Figure 4 shows the 4-year ODs for segments analyzed in frequency.A pronounced, broad occurrence enhancement occurs between 1 and 3 mHz in all window lengths and in both proton and α/p.The occurrence bands from Kepko et al. (2020) and shown in Figures 2 and 3 were defined in length scale space.Since the frequency ODs in Figure 4 do not take into account the speed of the solar wind segment, both the large 1-3 mHz enhancement and the localized bands are a mixture of fast and slow enhancement bands.Since there are far more slow wind than fast wind segments (see Figure 1), the frequency ODs are dominated primarily by the slow wind bands.Therefore, while we cannot make an explicit one-to-one correlation with length scale bands, recurrent bands are clearly evident in Figure 4 as indicated.

Overall Shape and Broad Enhancement in the Proton and α/p ODs
Figure 5 shows every significant frequency identified for the entire 25-year Wind data set in frequency and length, at the 95% confidence level, for each window length, for proton (top) and α/p (bottom).This provides the probability that a particular frequency was detected within that particular f ray bandwidth.For example, ∼1% of 6-hr segments contained a 2.0 mHz ± 0.5f ray periodicity at the 95% confidence level.Noticeably, the narrow band enhancements are still evident in the summations, as highlighted in Figure 5, and these are persistent regardless of window length.Note in the α/p length ODs two additional bands, near 500 and 800 Mm, that were not identified in the Kepko et al. (2020) study because of the uncertainty in the very long length scales (low frequencies) created by the NW effects.The longer windows studied here clearly pull out these longer length scale enhancements.These enhancements are also not strongly evident in the protons, a point we discuss below.There is a notable broad enhancement between 1 and 3 mHz and a second one below 1 mHz in the α/p distribution in the frequency ODs.Note how the edge effects move outward as the window length increases, as expected based on Di Matteo et al. (2021).This is most noticeable in the low frequency range.

Figure 3.
The "fast" 4-year occurrence distributions of statistically significant frequencies that passed the narrowband test at the 95% level for proton and α/p for different segment lengths, in the same format as Figure 2.

Discussion
Our analysis greatly extends previous studies on solar wind PDSs.For the first time, we have demonstrated the occurrence statistics of periodicities in the α/p ratio between ∼0.1 and 5 mHz.The α/p ODs for both time and length exhibit recurrent narrow bands of enhanced occurrence rates that match the previously calculated bands in Kepko et al. (2020) who used only 9,072-Mm length windows and examined only the proton data.
In addition, we calculated the ODs over the entire 25-year Wind data set using different window lengths to examine the overall distribution shape.Both the proton and α/p ODs exhibit a broad enhancement over 1-3 mHz with a peak occurrence centered near 2.1 mHz.Equivalently, the length scale ODs showed an enhancement between 100 and 400 Mm, peaked near 160 Mm in slow wind, and 130 and 500 Mm, centered near 250 Mm in fast wind.These are roughly equivalent to the 2.1 mHz peak converted into length scale using the median speed for "slow" (0.350 Mm/s/2.1 mHz) and "fast" (0.550 Mm/s/2.1 mHz).
Both the narrow occurrence enhancements and the broad 1-3 mHz enhancement are present in the α/p ODs, which is consistent with a solar origin of PDSs, since solar wind composition is set at the Sun.We note that in fast solar wind, the alphas are observed to stream at a differential velocity relative to the protons, by up to several tens of km/s (e.g., Mostafavi et al., 2022).In the frequency distributions, the vast majority of our data segments with validated alpha data and that contain discrete found frequencies comes from the slow solar wind (see Figure 1).Thus, this effect is unimportant for the "slow" wind distributions where differential streaming is small or zero, which comprise most of the ODs.Furthermore, as we demonstrate below, even for the "fast" wind the effect of differential streaming on the found frequencies is small.
We quantify the effect of differential streaming on the ODs by ODs in time and by calculating length series of n α using the measured v α .These results are shown in Figure 6.For all distributions we use data from 1998 to 2003 (6 years), wide enough to produce good statistics but narrow enough to avoid the previously measured solar cycle shifting of the narrow OD enhancements (Kepko et al., 2020).Figure 6a shows the OD in frequency space for n p , n α /n p , and n α .All three distributions show the same broad 1-3 mHz enhancement, and the same inflections at ∼0.5 and ∼3.5 mHz.The only difference is in the alpha OD, which shows an increase in occurrence between 1 and 2 mHz.
Figures 6b and 6c show ODs for n α in length space, using the proton velocity (blue) and separately the alpha velocity (green) to calculate the length series for "slow" and "fast" wind, respectively.If the differential velocity between protons and alphas is negligible, the ODs calculated from these length series should also be negligible.Conversely, if the differential velocity is significant, we would expect to see differences in the observed length scales.The slow wind ODs for the length series calculated using v p (blue) and v α (green) are nearly identical, confirming that for the "slow" wind differential streaming is unimportant and can be neglected.The "fast" wind OD shows a minor offset between the two ODs (Figure 6c), with the alpha OD calculated with v α (green) shifted to longer length scales by about 10-15 Mm.This is most evident by comparing the minima between the two plots in Figure 6c.The length scale of a 2 mHz oscillation, roughly the peak of the OD in time, for a typical fast wind speed of v = 650 km/s is simply L = v/2 mHz = 325 Mm.The expected difference in the length scale of a 2 mHz oscillation is ΔL = (v p − v α )/f.With ΔL = 15 Mm, the differential velocity between protons and alphas, (v p − v α ) is calculated to be 30 km/s, in good agreement with the average observed differential velocity observed at 1 AU (Mostafavi et al., 2022).
Critically, however, even in the fast wind distributions, where the alpha structures may be "slipping" relative to the protons, the changes in the alpha abundances still must have come from the Sun.For example, if a solar-created periodic alpha structure was injected onto a flat proton density profile, any differential velocity would be irrelevant from the perspective of identifying a periodicity.If both alphas and protons were created with a periodicity, differential velocity could introduce a beating between the frequencies of the two components, but otherwise is irrelevant as to the conclusions that alpha variations must be of solar origin.Finally, recent work by Gershkovich et al. (2023) found 30 and 90 min (the same as identified in this study) PDSs in the ACE alphas and ACE relative abundance ratios for heavy elements, including charge states (e.g., O7 + /O6 + ), which have different average differential streaming than the alphas.They could not examine higher frequencies because of the Nyquist for those data.If differential streaming was important, the found frequencies between the alphas and the heavier elements would show significant differences; they do not.

Calculating the False Positive Rate as a Function of Frequency
The ODs in Figure 5 are a mixture of true quasi-periodic signals and so-called "false positives" that are by definition intrinsic to any test for statistical significance at the significance threshold chosen.For an amplitude test, the overall shape of the OD of false positives is a function of how well the assumed spectral fit, in our case the BPL, models the true background spectra.If the assumed spectral background model perfectly fit the observed spectra, one would expect a flat distribution of "false positives."Similarly, if an assumed spectral model consistently underfits one portion of the spectrum, then spectral power at those frequencies would be tested at a comparatively lower confidence threshold than the neighboring frequencies, and the occurrence rate would appear comparatively higher.This effect is seen at both the low and high edges of the ODs in Figure 5, at the boundaries where constraining the model fit is more difficult.Our fit systematically underfits at the end points, leading to an enhancement in the OD at these boundaries, but is less of an issue away from the end points.Recent work by Di Matteo et al. ( 2021) has calculated the false positive rate for different types of time-series, such as those governed by a first order auto-regressive (AR(1)) power law processes, under different assumptions for the fit, and we apply that technique here, in order to determine the physical significance of the large 1-3 mHz (100-400 Mm) occurrence enhancements.
To test whether our spectral analysis background assumption of a BPL fit could introduce a bias into the significant frequency selection, and therefore the shape of the ODs, we performed a series of Monte Carlo simulations on simulated time-series data.The approach is described in Di Matteo et al. ( 2021) and follows Timmer and König (1995) to generate a random time series with a user-defined spectrum, S(f).In our case, S(f) is the BPL function in Equation 1.We used the BPL fit parameters identified in each analyzed Wind segment to generate an artificial time-series that contained the same spectral background shape of the real Wind data segment.This provided a precisely equivalent number of artificial time-series as analyzed Wind segments, with the same spectral shape as the analyzed Wind segments, but without any physical periodic signal included, so that we could directly compare the OD of "false positives" from the artificial time series to the Wind ODs.The adaptive multitaper method weights the contribution of each PSD estimates (one from each taper) with a recursive approach depending on the data itself.This step is required for the analysis of red noise spectra to avoid misrepresentation of the PSD due to spectral leakage of the high order tapers.It is this last step that cannot be represented analytically, hence requiring the numerical Monte Carlo simulations.
This process is illustrated in Figure 7 for a representative 6-hr time-series of real Wind solar wind density measurements (Figure 7a).To this time-series we applied the multi-tapered spectral analysis to produce a spectrum, which we then fit with the BPL function (Figure 7b).For this representative 6-hr segment there was a significant spectral peak passing at the 95% confidence threshold near 2.7 mHz, marked with a circle.Following the method of Timmer and König (1995) we produced a simulated time series (Figure 7c) using the spectral fit parameters from the Wind segment above.We then calculated the spectra of this simulated time-series and applied a BPL fit to determine the occurrence of statistically significant "false positives" (Figure 7d).
Although we did not add a harmonic signal to the artificial time series of Figure 7c, frequencies pass the amplitude test at the different confidence levels, at a rate proportional to the confidence level used.For example, for a straight FFT using a 95% confidence level one expects each frequency to have a 5% probability of containing sufficient power to pass the threshold.Using the multitaper method, as we do here, lowers this so-called "false positive" rate, by reducing spectral leakage, but the basic principle still applies.For this particular simulated time-series, which contained no added signal, a single "false positive" frequency passed the amplitude test at f = 3.2 mHz at the 95% confidence threshold.Finally, we added a pure harmonic signal at f = 1.2 mHz with an amplitude of 5% of the average to the simulated time-series (Figure 7e), and calculated the spectra of this simulated time series with the added artificial signal (Figure 7f).Since we are approximating this periodic train of structures as a pure sine wave, the 5% amplitude provided an amount of relative spectral power observed in event studies (Di Matteo et al., 2022;Viall et al., 2008Viall et al., , 2010;;Viall, Kepko, & Spence, 2009).We note that the rectangular shape of the spectral peak above the background of the added signal (Figure 7f) is a result of the MTM windowing, and an indication of a real signal, and demonstrates why the "false positive" rate is lower for MTM spectra than straight FFTs.
Using this approach of creating realistic, simulated time-series, we calculated ODs of significant "false positives" for every length and time window analyzed, using the BPL fit parameters from each true Wind segment to generate a simulated time-series.This "false positive" distribution is shown in Figures 8a and 8b, along with the observed distribution observed by Wind from Figure 5, for 6-and 12-hr window lengths.The simulated distributions in Figures 8a and 8b show a distribution of enhanced false positives that slowly rises toward the Nyquist frequency, with a sharp increase near the endpoints where our BPL fits tend to underfit the background.The observed Wind OD also follows this overall trend, but importantly, there is no enhancement between 1 and 3 mHz in the simulated ODs that could explain the peak observed in the ODs calculated from real Wind data.We therefore rule out the possibility of the broad enhancement being the result of "false positives."

Forward Modeling
Relative to the simulated ODs in Figures 8a and 8b, the measured ODs have a broad occurrence enhancement peak between 1 and 3 mHz, centered around 2.1 mHz.We are unable to reproduce this broad enhancement with "false positives" alone.This suggests that there exists a more frequent occurrence of frequencies within that range; that is, the ODs contain within them a distribution of physically meaningful periodicities between 1 and 3 mHz.
To test this hypothesis, we next added randomly distributed harmonic signals to the simulated time series, with the following approach.We define a Gaussian distribution of frequencies with center frequency, f c , and width σ.
We add to a percentage, P, of the simulated segments a pure sine wave with a frequency randomly sampled from the Gaussian distribution.As in the example of Figures 7e and 7f, the sine wave has an amplitude of 5.0% of the average number density of the time series.Although the peak-to-peak time series often exhibit variations closer to 20%-50% in event studies (Di Matteo et al., 2022;Viall et al., 2008Viall et al., , 2010;;Viall, Kepko, & Spence, 2009), the structures themselves are a periodic train, not a wave.Here we are approximating them as a pure sine wave, and the 5.0% amplitude empirically gave the right amount of relative spectral power that has been observed.
With these simulated time-series, we then calculate the OD as before, and the results are shown in Figure 8 for different values of σ (Figures 8c and 8d) and P (Figures 8e and 8f), for 6-and 12-hr window lengths.We find that the amplitude of the added signal makes little difference to the results, as the spectral method easily identifies the pure sine wave, and therefore focus on the effects of σ and P on the ODs.
These simulated ODs match features of the observed ODs in two important ways.First, we can match the observed broad OD enhancement between 1 and 3 mHz in the 6-hr segments by assuming ∼20% and 30% of the simulated solar wind time-series contains a random discrete frequency within that band, dependent on the value of σ chosen.Second, on either side of the broad enhancement the observed OD drops below the expected false positive rate.This feature is also observed in the simulated ODs, and is the result of significance testing.In simplest terms, by adding one true harmonic signal, we have effectively reduced the number of false positives in a given spectra by one, thereby lowering the overall false positive rate slightly.The OD of artificial signals is added on top of this new, slightly reduced, false positive distribution.When we increase the forward modeled time series length to 12 hr, almost 60% of segments are required to have a single frequency to reproduce the observed ODs (Figure 8f).The difference between the 6 and 12 hr results likely indicates that in longer segments there is often more than one frequency of PDS, with none of them lasting the entire duration of the data segment.In situ event studies of PDSs often indicate more than one significant frequency present (e.g., Viall, Kepko, & Spence, 2009).

Occurrence Enhancements Below 1 mHz
The 25-year OD for α/p PDSs shows a slight enhancement in low frequencies (f ∼ <1 mHz) that does not appear in the proton distribution (see Figure 5).Near 0.5 mHz, the occurrence rate for α/p starts to increase, and the location of this inflection point does not change as the window length is increased.This suggests it is not related to edge effects but rather represents a physically meaningful enhancement of occurrences.
Figure 9 shows the ratio of the full 25-year α/p ODs to the proton ODs for the different temporal windows, for both the amplitude criteria (Figure 9a) and the amplitude + F-test criteria (Figure 9b).For the majority of the interval, from 1 to 5 mHz, the ratios are relatively flat, indicating that the shape of the ODs over this interval for both proton and α/p are roughly equal.At low frequencies, both types of ratios show an enhancement of the α/p occurrence rates below 1 mHz.The narrow-band + F-test rises significantly and peaks near 0.2 mHz, indicating that frequencies in this range are more likely to have associated α/p variations.A similar behavior is observed in the narrowband ratios, although less pronounced.c-d), and the effect of a constant width, σ = 0.8 with varying percentages, P. By adjusting the width of the distribution of discrete frequencies and the fraction of segments with the added signal we are able to roughly match the observed OD, with σ = 0.8, P = 20% for the 6-hr windows, and σ = 0.8, P = 60% for the 12-hr windows.
For the large enhancement centered between 1 and 3 mHz, we observe no change in the α/p OD ratio, suggesting that the source of the periodicities produces equal proton and α/p across this frequency regime and suggests a common source.The enhancement at low (<1 mHz) is obviously different, in that these are more likely to exhibit α/p periodicities relative to those between 1 and 3 mHz.This difference in α/p characteristics combined with the clear separation from the broad peak at 2.1 mHz suggests a different solar source mechanism, one that is even more likely to produce alpha variations.
Similar to the broad 1-3 mHz enhancement, the sub 1 mHz region also contains localized enhancements (Figure 10).These bands, labeled "A" and "B" in Figure 10, are evident in the α/p, both in time and length, but not observed in the proton distributions (although a hint of band A appears in the protons of Figure 10).These bands, "A" and "B," are the same bands indicated in Figure 5.

Implications
By comparing the false positive rates and testing different window sizes we have identified at least two broad enhancements in the ODs of statistically significant frequencies observed in the Wind number density measurements: a fairly broad distribution between 1 and 3 mHz, and a second distribution below 1 mHz that appears to peak near 0.2 mHz (∼90 min).While the precise mechanism producing solar wind PDSs is still not solved, the α/p statistics and the examination of different window lengths clarifies how often they come from the Sun, and how many mechanisms and/or sources could be playing a role.
Both the proton and alpha number densities can evolve en route to 1 AU where they are measured.They can squish and expand leading to spatial (temporal as they flow past the spacecraft) variations.As a general rule, the alpha/proton ratio does not change; all PDSs observed in α/p were created at the Sun.Hollweg et al. (2014) developed an analytic solution for the in situ generation of PDS via slow-mode wave growth, but concluded this was not a viable explanation for PDSs.Additionally, multiple event studies have ruled out scenarios such as boundary waves or wavy current sheets (Crooker et al., 1996;Kepko et al., 2016;Viall, Spence, & Kasper, 2009).
The source of PDSs that do not have an α/p change are ambiguous; they could have formed en route from the Sun or been injected without a clear signature in the composition.More of the low frequency (<∼1 mHz) PDSs have associated α/p periodicities than the 1-3 mHz PDSs, suggesting that the low frequency PDSs have a different solar source, one that is especially likely to create composition variations.We therefore believe there are at least two separate distributions: a broad 1-3 mHz distribution that produces relatively equal number of proton and α/p changes, and a low frequency (<0.5 mHz) distribution that regularly has α/p changes.
Our results demonstrate that there are at least two populations of PDSs, and that both come from the Sun, but does not establish what determines the frequencies nor what mechanism(s) produces the plasma variations.However, the body of research on PDSs provides constraints on the generation mechanism(s) and here we speculate on their source(s).
Periodic magnetic reconnection of closed magnetic-fields in the solar corona is one highly likely possibility for creating PDSs.This would release relatively hot, dense, plasma with different composition and alpha to proton abundances from closed magnetic fields onto open magnetic fields (e.g., Moses et al., 2020).This could occur through interchange reconnection, or pinchoff reconnection at the tips of helmet streamers/heliospheric current sheet (Higginson & Lynch, 2018;Higginson et al., 2017).The S-Web represent regions where such reconnection is likely to occur between open and closed solar magnetic field (Antiochos et al., 2011;Linker et al., 2011), and includes blobs at the heliospheric current sheet (Sheeley et al., 1999).Direct evidence of reconnection at S-web arcs has been observed recently in extended EUV images (Chitta et al., 2022).Additionally, jets in the solar corona are locations where reconnection is known to inject hot plasma into the corna and out into the solar wind (Raouafi et al., 2016).Often PDSs are associated with small flux ropes (Di Matteo et al., 2019;Kepko et al., 2016;Lavraud et al., 2020), consistent with reconnection-generation, and small flux ropes of the minutes-to-hours size scales of PDSs were observed in MESSENGER data by Murphy et al. (2020).Similar to our PDSs, they concluded that there were two populations of flux ropes, both formed close to the Sun, with one associated with the HCS and the other away from the HCS.
Switchbacks are an in situ solar wind structure receiving attention because of their possible relation to interchange reconnection at the sun.It is important to note that there is no density signature of magnetic switchbacks when they are observed.Depending on the magnetic topology the magnetic signature of the interchange reconnection can propagate away from the Sun at the Alfvén speed, while the density signature remains behind and embedded in the solar wind flow (Higginson & Lynch, 2018).Thus, it is possible that PDSs have a similar underlying physical mechanisms as switchbacks, but no direct link has been established.Melnikov, 2009).QPPs are regularly observed in other stars, for example, using XMM-Newton (Cho et al., 2016) and GALEX (Doyle et al., 2018;Welsh et al., 2006), pointing to the universality of periodic magnetic reconnection in solar and stellar atmospheres.
Since the α/p distributions suggest the low frequencies have a different source than the higher frequencies, we describe observations and theories specific to each scale size next.

Sources of Low Frequency (<∼1 mHz) PDSs
In the case of the low frequencies, while this is the first statistical result demonstrating sub mHz (∼45-90 min) PDSs in situ at 1 AU, there is ample evidence in previous event studies that reconnection at the Sun is associated with PDSs.For example, Kepko et al. (2016) studied an event with 90-min (0.2 mHz) periodicities that exhibited strong changes in the C/O and C6/C5 ratios and associated magnetic connectivity changes, including electron strahl dropouts, all strongly suggestive of solar magnetic reconnection.Gershkovich et al. ( 2022) studied 4 events and found 0.2-0.6 mHz periodicities in multiple composition ratios, suggesting the release of closed-field coronal plasma into the solar wind.More recently, Gershkovich et al. ( 2023) studied 14 years of 12-min (f ny = 0.65 mHz) ACE composition data and found that the elemental and ionic composition data contained distinct f = 0.4 and f = 0.2 peaks in the occurrence rates, consistent with a solar source.Di Matteo et al. ( 2019), using Helios data, found 0.2-0.6 mHz PDSs associated with temperature changes, and observed a flux rope associated with a train of PDSs.In short, multiple lines of in situ observational evidence, including composition changes, temperature changes, embedded flux ropes, and magnetic connectivity (strahl), are consistent with magnetic reconnection as a source of PDSs.The low frequency (<1 mHz) PDSs in situ event studies tend to be associated with the HCS or S-web arcs (Di Matteo et al., 2019;Gershkovich et al., 2022;Kepko et al., 2016;Lavraud et al., 2020;Rouillard et al., 2020).Using STEREO SECCHI HI1 remote imaging data of density between 15 and 50 solar radii from the Sun, Viall et al. (2010) identified 100-min, 3-and 5-hr PDSs.Viall and Vourlidas (2015) using SECCHI COR2 (between 2.5 and 15 solar radii) observed an a preference for ∼90 min periodicities near helmet streamers and the HCS.
There have been several theoretical studies that examined what process(es) could produce periodic reconnection at helmet streamers/the HCS.Réville et al. (2020), using MHD simulations, was able to explain ∼1-2 hr, or 0.1-0.3mHz, periodic oscillations as the periodic release of flux ropes from the tip of helmet streamers via a flow-modified tearing mode.Schlenker et al. (2021) also found oscillations in this range present in MHD modeling of streamers with thermal nonequilibrium.Allred and MacNeice (2015) in a 2.5d MHD simulation found the helmet streamer to periodically disconnect, with the periodicty determined by the coronal heating rate.A separate study also using MHD simulations found the streamer to produce plasmoids periodically, with associated He variations, albeit on longer timescales than observed here (Endeve et al., 2004(Endeve et al., , 2005)).In a study of interchange reconnection in 3D MHD simulations, Lynch et al. (2014) find a characteristic timescale of reconnection of ∼0.5 mHz, which they comment is similar to the global characteristics Alfvén frequency.Peterson et al. (2021), in experiments with the Big Red Ball plasma confinement device, observed periodic plasmoid ejections from streamers as well as in their supporting MHD simulations of the experiment, and linked the characteristic ∼90-min plasmoid formation to be dependent on pressure gradients and magnetic curvature.Finally, Pylaev et al. (2017) linked this range of periodicities observed in the helmet streamers to the acoustic cut-off frequency and subsequent recurrent shocks.

Sources of 1-3 mHz PDSs
In the case of the higher frequency PDSs, f = 1-3 mHz, a ubiquitous source of mHz pulsations in the solar corona that could be driving periodic reconnection is the solar acoustic pressure (p)-modes.Numerous modeling studies have indicated that the solar p-modes make their way into the corona through a multi-step process of mode conversion into transverse Alfvén-like waves, and contribute substantially to mHz wave activity in the corona (Cally, 2016;Cally & Goossens, 2008;Cally & Hansen, 2011;Jefferies et al., 2006).Rather than a direct transmission of acoustic p-modes into the corona, transmission across the photospheric boundary leads to this energy being mode converted into transverse Alfvén waves.Note that there have been some attempts to link solar p-modes directly to periodicities observed in solar wind magnetic field and velocity measurements (e.g., Thomson et al., 1995Thomson et al., , 2001)), but here we are talking about an entirely different mechanism.We speculate that the p-modes that mode convert into Alfvénic fluctuations in the corona periodically drive magnetic reconnection events in the corona -either pinch-off reconnection or interchange reconnection -which releases mass periodically into the solar wind.This quasi-periodic release of plasma leads to advecting density structures in quasi pressure balance (Di Matteo et al., 2019), rather than waves.
Observationally, several studies using Hinode Coronal Multi-channel Polarimeter (CoMP) measurements have observed an enhancement of spectral power in transverse velocity fluctuations in the solar corona near ∼3 mHz (Morton et al., 2016;Tomczyk et al., 2007).More recent observational work using both CoMP and SDO have demonstrated a spectrum of mHz transverse velocity fluctuations in >95% of coronal spectra (Morton et al., 2019), including coronal holes, quiet Sun, and active regions.Additionally, while (Morton et al., 2019) found a spectral power peak of these coronal velocity fluctuations near 4 mHz, the OD peaked at 2 mHz (see Morton et al., 2019; Figure 1c).This is an important distinction since our study is based on occurrence rates of discrete periodicities, rather than their total power.
In Figure 11 we compare our observed proton OD for 6-hr segments (black), the occurrence rate of periodicities observed by CoMP in the lower corona by Morton et al. (2019) (red), and the simulated OD (blue) that included an added distribution of frequencies (green), for both 95% and 90% confidence levels.The comparison between the added distribution of frequencies (green), which was chosen to match the observed ODs, with the CoMP distribution, shows excellent agreement.This comparision is highly suggestive of a direct link between the 1-3 mHz distribution of PDSs and the CoMP and SDO measured transverse velocity fluctuations in the solar corona.
As these ubiquitous transverse velocity fluctuations propagate through the lower corona they could trigger magnetic reconnection at locations prone to reconnect, such as along S-web corridors (Antiochos et al., 2011).This reconnection would release previously closed plasma onto open field lines, and if the reconnection is periodic, a PDS would result.In the very low corona, jetlets are a well-observed phenomena, known to inject plasma into the solar wind with quasi-periodic recurrence rates of 3-5 min, and thus likely created by magnetic reconnection driven by p-modes from below (Kumar et al., 2022;Raouafi et al., 2023).Indeed, Cattell et al. (2021) used Parker Solar Probe and SDO data and found 5-min EUV periodicities of an active region associated with Type III repetition rate observed by Parker, linking the observations to periodic impulsive heating events.Sterling et al. (2020) suggested that p-modes might force magnetic fields together at an ∼5 min rate, inducing small-scale flux cancellations, build miniature flux ropes or filaments, that then erupt to create jet-like features.
While the multi-step process linking solar acoustic p-mode oscillations to periodically driven reconnection reasonably explains the broad 1-3 mHz OD enhancement, the source of the localized bands in the ODs is not immediately evident.It is possible that a resonance between a transmitted acoustic p-mode and a local coronal loop would occur.Since an individual loop has a narrow range of periodicity for which to find resonance, this coupling would be discrete.For example, though the periods of kink oscillations span a broad range, 1-28 min, they are empirically correlated with loop length and are in the right range to get this discrete resonance matching (Nakariakov et al., 2021).Kepko et al. (2020) noted a solar cycle dependence of the observed localized bands associated with the "terminator" (McIntosh et al., 2022).The terminator is when the extended solar cycle of the prior cycle ends, and next solar cycle takes over, and is also associated with an abrupt increase in the amount of hot plasma in active regions, which varies by more than an order of magnitude over the solar cycle (Schonfeld et al., 2017).It has also been shown that the terminator is associated with a rapid depletion then recovery in the abundance ratio (Alterman et al., 2021).We speculate that the evolution of the localized bands could be due to interchange reconnection with active region loops (Mandrini et al., 2014;Mason et al., 2019;Zanna et al., 2011), such that the change of the plasma properties and magnetic field loop lengths changes with the solar cycle, resulting in different resonances, and different scales of PDSs.
In addition to mode-converted p-mode oscillations, the solar atmosphere contains a plethora of oscillations in the ranges studied here that could drive and/or modulate magnetic reconnection, including compressive waves with frequencies of 10-15 min in polar plumes (Deforest & Gurman, 1998).For a review of oscillations in the solar atmosphere, see review by Roberts (2000).

Conclusion
There now exists a substantial body of remote and in situ observations of solar wind PDSs.These observations span remote imaging of the low frequency PDSs deep in the solar atmosphere (DeForest et al., 2018;Viall & Vourlidas, 2015;Viall et al., 2010), to in situ observations as close as 0.3 AU (Di Matteo et al., 2019), with substantial event and statistical studies at L1 and Earth, where they have been observed by many different spacecraft in the solar wind spacecraft and inside Earth's magnetosphere (Di Matteo et al., 2022;Dyrud et al., 2008;Fenrich & Waters, 2008;Gershkovich et al., 2022Gershkovich et al., , 2023;;Kepko & Viall, 2019;Kepko et al., 2002Kepko et al., , 2020;;Stephenson & Walker, 2002;Viall et al., 2008;Viall, Kepko, & Spence, 2009).The remote observations, coming as near to the sun as 2.5 solar radii, provide compelling evidence that the low frequency (<1 mHz) PDSs come straight from the solar atmosphere, but cannot provide insight into the higher frequency (>1 mHz) PDSs because of Nyquist limitations.The higher cadence in situ measurements have, through observations of flux ropes and disconnections within PDSs (Di Matteo et al., 2019;Kepko et al., 2016), and recent ACE observations of periodicities in heavy ion composition (Gershkovich et al., 2022(Gershkovich et al., , 2023)), provided compelling event and statistical studies that the 1-3 mHz PDSs also have a solar origin.
Using 25 years of Wind solar wind measurements at L1 we have calculated the ODs of PDSs in both the proton and α/p solar wind data, to provide quantitative information on the PDS occurrence rate and also further insight into a possible solar origin.We have demonstrated at least two populations of PDSs exist, and the preponderance of evidence is consistent with a solar origin.We find the same narrow band enhancements in the α/p as were observed in previous studies for the protons, at Lengths of 90, 110, 140, 170, and 320 Mm, and 2.0, 2.4, and 2.9 mHz in time.Since composition cannot change en route, this indicates that the formation mechanism(s) that leads to the occurrence enhancements occur at the Sun.Second, we have demonstrated that a broad distribution of enhanced occurrence rates that occurs in the 1-3 mHz range, peaking near 2.1 mHz, is physically meaningful.We were able to reproduce this enhancement in the OD by assuming an added Gaussian distribution of harmonic signals to ∼30% of the 6-hr time-series.This assumed distribution of periodicities bears strong resemblance to the distribution of transverse velocity fluctuations observed in the solar corona, suggestive of a link between p-mode driven reconnection in the solar corona and the PDSs.Finally, we find that a second, different occurrence enhancement distribution exists below ∼1 mHz, peaking around 0.2 mHz (90 min), with an enhanced likelihood of exhibiting α/p variations relative to other frequencies.
Although previous work has demonstrated that the solar wind contains PDSs at specific frequencies, this is the first time we have been able to demonstrate that: (a) these frequencies were also observed statistically in the alpha/proton ratio, (b) the statistics of the low frequency PDSs, and that (c) the broad enhancement between 1 and 3 mHz is physically meaningful.We speculate that the broad enhancement of PDSs between 1 and 3 mHz is driven in some way by the solar p-modes, perhaps through periodic reconnection driven by the velocity fluctuations once the p-modes have mode-converted their initially acoustic energy into transverse Alfvénic fluctuations in the lower corona.There is observational and simulation evidence that PDSs at low frequency (∼1 mHz) are likely associated with streamers/S-web arcs and the HCS.Importantly, we are not implying that p-modes make it out directly into the solar wind; an intermediate step of plasma release is required, likely through magnetic reconnection.We summarize these different sources of PDSs in Figure 12.
These results have important implications for the release of plasma that becomes the solar wind, and for solar wind-magnetosphere coupling, where PDSs drive coherent, global ULF pulsations.While the low frequency/ larger PDSs have been observed in remote images of the corona, observations at high spatial and temporal resolution and with high signal-to-noise ratio in the middle corona are needed to conclusively determine which high frequency phenomena in the solar atmosphere result in the 1-3 mHz PDSs.

Figure 1 .
Figure1.The number of segments created for all window lengths in time and length (for both "slow" and "fast" wind) are shown in green.The number of proton segments that passed the quality control checks and were analyzed are shown in dark blue, while the number of these analyzed segments that contained ≥1 discrete frequencies, ignoring those within NW bandwidth of f = 0 and f ny , are shown in light blue.The equivalent number of segments for α/p are shown in red and light red.

Figure 2 .
Figure 2. The "slow" 4-year occurrence distributions of statistically significant frequencies that passed the narrowband (amplitude) test for proton and α/p for different segment lengths at the 95% confidence level.Regions of localized enhancements (bands) of discrete frequencies, based on the Kepko et al. (2020) identification, are indicated.

Figure 4 .
Figure 4.The 4-year occurrence distributions of statistically significant frequencies that passed the narrowband test for proton and α/p for different segment window lengths of time.

Figure 5 .
Figure5.Proton (top)  and α/p (bottom) 25-year occurrence distributions for significant frequencies at the 95% confidence threshold for different time and length scale windows.Bin sizes are effectively f ray of the segment length, and the y-axis is the probability of observing a particular frequency within that f ray .Bands identified in the 9,072 Mm window lengths byKepko et al. (2020) are indicated.The two unlabeled bands near 500 and 800 Mm were not observed in that study, because they are near the NW bandwidth of the zero frequency, but are well-resolved in the longer window lengths.

Figure 6 .
Figure6.(a) Six-year occurrence distribution of statistically significant frequencies observed in the alpha (blue) α/p ratio (red) and proton (black).The equivalent distributions for the "slow" (b) and "fast" assuming a length series generated with the proton velocity (blue) and the alpha velocity (green).

Figure 7 .
Figure 7. Panel (a) contains a real 6-hr segment of solar wind proton data from the Wind spacecraft, while the MTM spectra (black), bending power law spectral background fit (red) and 95% confidence threshold (green) are shown in (b).Equivalent plots are shown for a simulated time-series with no added periodicity (c-d) and with a single periodic signal added (e-f).

Figure 8 .
Figure 8. (a and b) Monte Carlo simulation results comparing the "false positive" occurrence distribution (OD) against the observed Wind distribution for (a) 6-hr and (b) 12-hr window lengths, indicating that false positives alone are unable to reproduce the observed ODs.Panels(c-f) show the effect of adding a distribution of discrete frequencies to the simulate time-series with varying widths, σ, and constant percentage (P) of segments with an added frequency (c-d), and the effect of a constant width, σ = 0.8 with varying percentages, P. By adjusting the width of the distribution of discrete frequencies and the fraction of segments with the added signal we are able to roughly match the observed OD, with σ = 0.8, P = 20% for the 6-hr windows, and σ = 0.8, P = 60% for the 12-hr windows.

Figure 9 .
Figure9.Ratios of the α/p and proton full 25-year occurrence distributions (ODs) (i.e., ratios of the ODs shown in Figures5a and 5b) for both narrowband (left) and narrowband + F-test.The low frequency (<1) ratios show significant differences between α/p and proton, indicating different source mechanisms.

Figure 10 .
Figure 10.Zoom in of the low frequency (long length scale) range for proton (top) and α/p occurrence distributions.Two localized enhancements bands are observed, primarily in the α/p ratio, indicated by "A" and "B."These bands were not observed in the Kepko et al. (2020) study, which examined only 9,072 Mm lengths, due to edge effects of the short windows.

Figure 11 .
Figure 11.Comparison of the 25-year proton 6-hr occurrence distribution (OD) (black) compared to the simulated OD (blue) using artificial time-series and an added distribution of discrete periodicities (green).The vertically scaled OD of transverse velocity fluctuations from Morton et al. (2019) is shown in red.

Figure 12 .
Figure 12.Pictorial diagram summarizing our hypothesized distinct distributions in periodic density structures by helmet streamer/HCS reconnection release at the lowest frequencies (longest length scales) and p-mode driven release.