Axial Flux Evolution of Small-scale Magnetic Flux Ropes from 0.06 to 10 au

Small-scale magnetic flux ropes (SMFRs) fill much of the solar wind, but their origin and evolution are debated. We apply our recently developed, improved Grad–Shafranov algorithm for the detection and reconstruction of SMFRs to data from Parker Solar Probe, Solar Orbiter, Wind, and Voyager 1 and 2 to detect events from 0.06 to 10 au. We observe that the axial flux density is the same for SMFRs of all sizes at a fixed heliocentric distance but decreases with distance owing to solar wind expansion. Additionally, using the difference in speed between SMFRs, we find that the vast majority of SMFRs will make contact with others at least once during the 100 hr transit to 1 au. Such contact would allow SMFRs to undergo magnetic reconnection, allowing for processes such as merging via the coalescence instability. Furthermore, we observe that the number of SMFRs with higher axial flux increases significantly with distance from the Sun. Axial flux is conserved under solar wind expansion, but the observation can be explained by a model in which SMFRs undergo turbulent evolution by stochastically merging to produce larger SMFRs. This is supported by the observed log-normal axial flux distribution. Lastly, we derive the global number of SMFRs above 1015 Mx near the Sun to investigate whether SMFRs begin their journey as small-scale solar ejections or are continuously generated within the outer corona and solar wind.


Introduction
Small-scale magnetic flux ropes (SMFRs) are magnetic flux ropes (twisted flux tubes) in the solar wind similar to magnetic clouds in coronal mass ejections (CMEs), but smaller and without other signatures that CMEs have (Moldwin et al. 1995(Moldwin et al. , 2000)).At 1 au, CMEs typically have a duration of more than 10 hr, while SMFRs are rarely more than a few hours in duration.A lower bound for the duration of SMFRs (down to 30 s has been resolved) has not yet been found owing to the limited temporal resolution of spacecraft measurements (Cartwright & Moldwin 2008, 2010;Hu et al. 2018;Farooki et al. 2024b), but they have been observed at ion scales (Eastwood et al. 2021).SMFRs differ significantly from CMEs in plasma properties (Moldwin et al. 1995; for a recent statistical comparison, see Farooki et al. 2024a).SMFRs have been observed abundantly in record proximity to the Sun in Parker Solar Probe (PSP) observations (Zhao et al. 2020(Zhao et al. , 2021;;Chen et al. 2021;Pecora et al. 2021a;Chen & Hu 2022).There is evidence that SMFRs trap, exclude, and accelerate energetic particles (Khabarova & Zank 2017;Zhao et al. 2018;Khabarova et al. 2021;Pecora et al. 2021b;Pezzi et al. 2021;Van Eck et al. 2022;Le Roux 2023).The major open questions regarding SMFRs can be separated as (1) whether SMFRs originate from the Sun or form continuously within the solar wind and (2) whether the SMFRs undergo significant evolution within the solar wind besides the effects of solar wind expansion.
For the possible origins of SMFRs, early studies proposed reconnection across the heliospheric current sheet (HCS; Moldwin et al. 1995Moldwin et al. , 2000;;Cartwright & Moldwin 2008, 2010) or small CMEs and other solar eruptions (Feng et al. 2007(Feng et al. , 2008;;Rouillard et al. 2011;Higginson & Lynch 2018).Such transient generation mechanisms could explain the small numbers of SMFRs found in early studies.But with the abundant number of SMFRs discovered using Grad-Shafranov (GS) based automated detection (Hu et al. 2018), the finding that SMFR properties are controlled by the properties of the "background" solar wind (Zhai et al. 2023;Farooki et al. 2024a), and our recent finding that the SMFR filling factor (∼35%) is independent of solar wind type, solar activity, and distance to the HCS (Farooki et al. 2024b), other mechanisms that predict SMFRs that fill a significant fraction of the solar wind gain appeal.Based on observations of magnetic discontinuities, Borovsky (2008) proposed that the solar wind is filled with flux tubes of photospheric origin.However, recent findings indicate that small-scale solar eruptions may contribute numerous flux ropes to the heliosphere (Sterling & Moore 2020;Huang et al. 2023;Sterling et al. 2024).
Another model that can explain the abundance of SMFRs is one in which the solar wind is dominated by quasi-2D magnetohydrodynamic (MHD) turbulence, which predicts that abundant flux ropes can form in much less than the time it takes for the solar wind to reach 1 au (Matthaeus et al. 2007;Greco et al. 2008Greco et al. , 2009;;Servidio et al. 2008;Wan et al. 2009;Zank et al. 2017).Recent direct observation of bursty and turbulent reconnection in the solar wind (Wang et al. 2022)  turbulence-driven model predicts (Greco et al. 2009) a current density distribution consistent with the distribution derived from GS-based detection (Zheng & Hu 2018), although it is unclear when and where this distribution forms.The merging of flux ropes is an important mechanism that allows more and more large flux ropes to form by turbulence (e.g., Zhou et al. 2019Zhou et al. , 2020)).This falls under the picture of 1/f low-frequency noise observed in the solar wind resulting from multiplicative processes leading to log-normally distributed correlation lengths (Matthaeus & Goldstein 1986).A case study of two flux ropes merging into one measured by two radially separated spacecraft was reported in Hu et al. (2019), but merging has not been shown to have a statistical effect.Although a statistical increase in size has been observed (Cartwright & Moldwin 2010;Chen & Hu 2020), it is difficult to separate the contributions of merging and solar wind expansion.
Merging of flux ropes would produce SMFRs with larger axial flux, unaffected by expansion due to the frozen-in flux theorem.To our knowledge, the evolution of SMFR axial flux has not been investigated in previous studies.In previous GS-based studies, statistical analysis of SMFR axial flux was not possible because the original GS-based detection algorithm did not incorporate GS reconstruction, so axial flux had to be evaluated on a case-by-case basis.However, our improved GS-based algorithm (Farooki et al. 2024b) reconstructs cross sections for all detected events, which can be directly used to calculate axial flux.
In this Letter, we study the evolution of SMFR axial flux using spacecraft measurements between 0.06 and 10 au.

Event Detection
The improved automated GS-based SMFR detection algorithm is detailed in Farooki et al. (2024b).Essentially, the algorithm detects events exhibiting MHD equilibrium and having measurements consistent with a 2.5D (∂/∂z = 0) SMFR with a particular orientation ẑ.The algorithm exhaustively searches each possible event interval, reduces the search space by process of elimination, and outputs a set of nonoverlapping time intervals and 2D reconstructed cross section B(x, y).This cross section can be used to calculate the axial flux Φ ≡ ∫B z dA where the integration is performed over the region of closed field lines.Benchmarks against simulated spacecraft measurements to evaluate the reliability of the algorithm are available in Farooki et al. (2024b).The algorithm takes spacecraft measurements of the magnetic field B(t), proton velocity in the spacecraft reference frame v p (t), proton density n p (t), and gas pressure P gas (t).
Measurements from 0.06 to 0.25 are from the PSP (Fox et al. 2016).Between 0.3 and 1 au, we used Solar Orbiter (SolO; Müller et al. 2020) data.We took advantage of the large volume of data available at 1 au by using the events detected from Wind (Wilson et al. 2021) data in Farooki et al. (2024b).For 1-10 au, we used data from Voyager 1 and 2. For details about the instruments used and how the datasets were processed, see the Appendix.The sliding window durations and other settings used for the algorithm are the same as in Farooki et al. (2024b).For PSP, we used data from the first 16 encounters, from the end of 2018 to 2023.SolO measurements from 2020 to 2023 were used, although the time that measurements were available in 2020 and 2021 was limited.Voyager 1 and 2 data from 1977 to 1980 were used.In total, PSP detected 15,699 events, SolO detected 28,181 events, and Voyager 1 and 2 combined detected 11,841 events.SolO detected more events than PSP because PSP events were only detected during encounters.
Figure 1 shows three examples of the events detected by the algorithm.From left to right, the events were detected by PSP, SolO, and Voyager 1.The events were the largest events from dates near perihelion for PSP and SolO and relatively far from the Sun for Voyager 1, but with relatively good data coverage for Voyager 1 and SolO.Dates in which the largest event appeared to be related to a CME were avoided.The top panel is the reconstructed cross section, where brighter pixels indicate a stronger axial flux density.The black contours represent the field lines projected onto the plane perpendicular to the flux rope axis.The bottom panel displays the magnetic field measurements within the event interval transformed into the flux rope coordinate system (as described in detail in Farooki et al. 2024b).The dashed line overlaid on the measurements is the prediction of the reconstructed model.This figure shows that the magnetic field strength is lower in SMFRs further from the Sun and that the size of a typical large event increases significantly with distance from the Sun (due to some combination of solar wind expansion and flux rope merging).

Axial Flux Density
Figure 2 shows the relationship between the cross-sectional area σ, the axial flux Φ, and the heliocentric distance r.Based on this figure, Φ ∝ σ, with axial flux density Φ/σ dependent on r but not σ.The change in flux density is thus a natural consequence of the solar wind expansion and flux conservation in the flux ropes.The size-independent axial flux density is presumably a consequence of the strong guide field in the solar wind (the Parker spiral).Even though axial flux is conserved under solar wind expansion, the number of SMFRs with higher axial flux can increase if smaller SMFRs merge into larger ones.The relative increase of higher axial flux events will be shown below, but note that even in this figure one can see that PSP does not have many events with as much flux as the larger values observed by Voyager 1 and 2, despite PSP having a larger number of events.

Collision Times
If flux rope merging is to occur, SMFRs must be driven together and reconnect.Each SMFR has a different velocity, and the difference in velocity can be large, sometimes comparable to the solar wind bulk velocity.If the velocity differences are large enough, SMFRs can collide fast enough to overcome the effects of solar wind expansion.Once they are close enough, if their embedded currents are parallel, they will be drawn together and merge owing to the coalescence instability (Finn & Kaw 1977).If they are oppositely oriented, they will be repelled.Thus, SMFRs can merge and bounce depending on the orientation of their currents.
Suppose that a spacecraft observes a quasi-static SMFR (at x = 0) whose observation ends at time t A with speed  x A (in the spacecraft's frame of reference), followed by another one whose observation begins at t B with speed  x B .The first is referred to as A, and the second is referred to as B. Neglecting discrepancies in the direction of the velocity and orientation (since they tend to be small), we can use the following simplistic model to estimate the time for the boundaries of the two flux ropes to collide.The positions of the boundaries at time t are given by Taking t A ≡ 0 and t B ≡ Δt and solving for the collision time τ c at which In the case of equal velocity, , and in the case where the earlier event travels faster than the later event, < , which implies that they will never collide (although A may have been separated or pushed away from B at time −|τ c |, at least from a kinematic perspective).Negative τ c is still relevant because it corresponds to an SMFR pair moving apart.
, where v FR is the flux rope velocity determined by minimizing the electric field included in the output of the detection algorithm (Farooki et al. 2024b), we show the distribution of τ c for pairs of SMFRs observed at difference distances in Figure 3. Here, only events with flux above 10 15 Mx are included in the calculations, since that range is well resolved for all distances (Figure 2; although it is not strictly well resolved for PSP, the orientations are primarily radial near the Sun, so in practice it is reasonably well resolved).
Figure 3 indicates that nearly 50% of adjacent SMFR pairs observed by PSP can collide in <100 hr.Each SMFR is involved in at least two pairs (one with the event before it, one with the event after it).This suggests that the vast majority of SMFRs will make contact with each other at least once during the 100 hr transit time to 1 au.
The collision time tends to increase with heliocentric distance (note that the x-axis is logarithmic).Analysis of the data indicates that this is due in part to the difference in velocity becoming smaller with distance (for reasons that are not entirely clear, but possibly because the process of merging results in the cancellation of momentum), in addition to the spatial separation becoming larger with distance (for a study of the waiting time distribution; see Hu et al. 2018).Interestingly, the collision time appears to be log-normally distributed, suggesting that it may evolve through multiplicative processes Each line represents direct proportionality, with the slope being the average axial flux density for the sample.The shaded region is partially unresolved because SMFRs of diameter below (600 km s −1 )(30 s) (due to the minimum 30 s sliding window) can only be detected when the orientation is sufficiently closely aligned with the velocity in the spacecraft frame of reference (Farooki et al. 2024b).(Matthaeus & Goldstein 1986).Additionally, it appears that, despite the increasing collision time, the collisions do not stop entirely.For example, nearly 50% of pairs at Wind should collide within the 1000 hr transit to 10 au.Thus, continued evolution can be expected, although at a decaying rate.
Another effect visible in Figure 3 is that farther away from the Sun negative τ c is more common than positive τ c .This is especially obvious in the Wind and Voyager data.This, too, can be explained by the merging model: pairs on collision courses will tend to be annihilated by merging, while pairs that are moving farther apart will last longer and thus be more likely to be observed.The typical gap between SMFRs increases with distance from the Sun (Chen & Hu 2020), which is expected from this behavior.

Increase in Flux Limit
Analysis of the data reveals that SMFRs with larger flux become significantly more common with distance from the Sun.As a qualitative visualization of this effect, a map of the detected events is presented in Figure 4. To avoid low-quality but relatively large outliers interfering with the plot, only relatively high quality events with difference residue R diff < 0.2 (Farooki et al. 2024b) are included.Events with flux more than 10 19 Mx are not plotted because they may be CMEs instead of SMFRs.Only Wind events between 2018 and 2022 are included in this figure so that outliers that occur more frequently at different phases of the solar cycle can be avoided.The number of events detected by each spacecraft depends on the period over which it took measurements, which varies significantly between spacecraft, so it is not the focus of this figure.Periods with a reduced number of events, prominent in SolO and PSP data, are primarily due to data gaps.As seen in this figure, some events have much larger axial flux than the majority.This is because SMFR axial flux follows a log-normal distribution with a significant tail.In the SolO data, relatively high flux events are common close to 1 au but rare close to 0.3 au.Comparing PSP events to Wind events similarly shows a major increase in high-flux events.
For a quantitative understanding of the data, Figure 5(a) compares the complementary cumulative distribution of axial flux over different ranges of heliocentric distance (complementary meaning that it starts at 1 instead of 0).The cumulative distribution starts at 10 15 Mx.A convenient way to summarize the cumulative distributions is to define a scalar quantity that is related to the tail of the distribution.We define this "flux limit" as the 90th percentile of the axial flux of the samples having axial flux greater than 10 15 Mx.The flux limit as a function of distance from the Sun is shown in Figure 5(b).We also show the flux limit as a function of plasma age, estimated as the flux rope distance from the Sun divided by its velocity.
Overall, Figure 5 tells us that there is a statistically significant increase in the proportion of SMFRs with increasingly high levels of axial flux as a function of distance from the Sun (or, alternatively, plasma age).This trend is clear both within the data set of individual spacecraft and across all the data sets put together.While there is some slight disagreement between Wind and SolO, it is much smaller than the overall trend considering all the data sets together.Most likely, the small disagreement is due to the large number of data gaps in the SolO data set.From Figure 5(a), it is clear that the proportion of larger-flux events compared to the proportion of smaller-flux events increases significantly with distance from the Sun.In Figure 5(b), we see that there is a statistically significant trend for the flux limit to increase with heliocentric distance.Figure 5(c) shows that the same trend holds for a fixed distance but a different velocity-inferred plasma age.

Global Number of SMFRs
The relationship between the number of SMFRs detected by a spacecraft at a fixed point and the global rate of SMFRs passing through an imaginary sphere with radius equal to the spacecraft heliocentric distance has been derived by Janvier et al. (2014).It depends on the diameter of the flux rope and the length of its projection onto the imaginary sphere.Unfortunately, through single-spacecraft measurements, we do not have any information about the length of SMFRs, e.g., whether it is the same for all SMFRs at a certain distance or a function of diameter.However, near the Sun, the Parker spiral direction is nearly radial.SMFRs tend to be closely aligned with the Parker spiral (Hu et al. 2018;Chen & Hu 2020;Farooki et al. 2024b).The length of their projection onto the sphere is thus simply equal to their diameter.However, there arises another complication: The azimuthal motion of PSP is comparable to the solar wind's radial speed.Presumably, the length of a flux rope is much larger than its diameter.Therefore, it is as if PSP is azimuthally sweeping through SMFRs instead of them being radially advected past PSP.Then, the rate of SMFR observations should be proportional to PSP's azimuthal speed and largely independent of the actual rate of SMFRs passing.
Rather than estimate the rate of SMFRs, we propose that it is possible to estimate the total number of SMFRs passing through a heliocentric shell at any given time.This is similar to estimates of the number of erupting filaments on the Sun at any given time (e.g., Sterling & Moore 2016).This can be done as follows: Consider PSP as passing along the surface of a large sphere of radius r, PSP's heliocentric distance.We take PSP's motion as an azimuthal arc.Assuming an isotropic azimuthal distribution but all flux ropes along the sphere's equator, the fraction of flux ropes observed is equal to the angular length of the arc divided by 360°.Now allowing for a variable distribution of latitude, the probability of observing a flux rope of diameter D during PSP's azimuthal sweep is the probability that its cross section intersects with the sphere's equator.Assuming for simplicity an isotropic orientation distribution, noting that randomly distributed points on a sphere have a nonuniform latitudinal distribution but a uniform distribution of the latitude's sine (Deserno 2004), the probability of the flux rope being observed is D/2r.The global number of flux ropes measured by PSP can then be estimated as follows: where Δf is the total azimuthal distance covered by PSP, the summation is over the flux ropes observed by PSP, D i is the diameter of the ith flux rope, and r i is its distance from the Sun.The main idea is that if, for example, the probability for a flux rope that was observed by PSP is 0.01, then we should count it as 100 flux ropes, assuming that there were 99 other such flux ropes at other latitudes.By doing so, our estimate should approach the true global number on average.
To facilitate comparison with solar observations and to ensure the validity of the azimuthal sweeping approximation, we estimate the global number of SMFRs passing through a spherical shell up to 0.07 au (since the closest event is at around 0.06 au, this range is ∼12-14 R e from the solar surface).To calculate Δf, we use the difference in PSP's longitude when it first enters this range and when it exits.To calculate the longitude relative to the overall initial mass function structure, which corotates with the Sun at such a close distance, we subtracted 360°× t/24.47 days, where t is the time and 24.47 days is the solar rotation period at the equator.PSP only entered this range of distances in the last 7 of the first 16 encounters.Each of these contributed 68°-71°, yielding a total of Δf = 480°.
Finally, to account for differing detection limitations in different data sets, we provide the estimate for different minimum fluxes.The results are listed in Table 1.It must be understood that these are approximate order-of-magnitude estimates.In Section 7, we discuss how these results can be compared with solar observations.

Discussion and Conclusions
The findings of this Letter can be summarized as follows: 1. SMFR axial flux density is independent of its size.As a result, SMFR axial flux is directly proportional to area (Figure 2), consistent with the previous result based on 1 au measurements (Farooki et al. 2024b).The axial flux density decreases with distance based on measurements from 0.06 to 10 au. 2. The relative velocity between individual SMFRs is so large that SMFRs collide with each other frequently enough for significant merging and reconnection to occur (Figure 3). 3. The collision time is log-normally distributed and increases significantly with distance from the Sun owing to decreasing differential velocity and increasing spatial separation (Figure 3). 4. SMFRs with higher flux become more common with increased distance from the Sun (Figures 4 and 5(b)).The number of higher flux SMFRs also increases with plasma age, as inferred from the velocity and distance from the Sun Figure 5(c).We argue that such an evolution of SMFR flux results from merging of SMFRs.
It is interesting to note that even at a fixed distance the flux limit increases as a function of plasma age in Figure 5.It has long been observed that SMFRs tend to be larger in the slow solar wind compared to the fast solar wind (Borovsky 2008;Hu et al. 2018;Farooki et al. 2024b).In Farooki et al. (2024b) we found that this may indicate that SMFRs are more compressed in the fast solar wind.However, if SMFRs undergo a turbulent merging process over time, then the difference in age between fast and slow solar wind is another possible explanation for the difference in SMFR size between the two solar wind types.
Can these observations shed light on the origin of SMFRs?The increase of the axial flux limit of SMFRs with distance or plasma age is strong evidence that SMFRs undergo significant evolution within the solar wind that cannot be explained by expansion alone.Yet, even at its closest approach to the Sun, PSP observes many SMFRs, implying that they originate at less than ∼12 R e from the solar surface.Thus, we are not in a position to tell whether the SMFRs are continuously generated in the solar wind or they all originate from the Sun.However, we can say that at least some SMFRs originate within 12 R e of the Sun.Moreover, that their axial flux is log-normally distributed at this distance may indicate either that they were generated via multiplicative processes, such as magnetic reconnection of smaller tubes (Matthaeus & Goldstein 1986), on the Sun or at least that they underwent significant evolution between the Sun and 12 R e via turbulence.
Indeed, a property of turbulence is the evolution toward states of equilibrium, generally called dynamic alignment and selective decay (Montgomery et al. 1978;Ting et al. 1986;Stribling & Matthaeus 1991).The former represents a dynamical tendency toward states of aligned velocity and magnetic fields (Alfvénic state) (Dobrowolny et al. 1980;Grappin et al. 1982).The latter is obtained when the plasma is dominated by the magnetic energy, and it is possibly organized in a helical configuration (Montgomery et al. 1978;Matthaeus & Montgomery 1980).Together with these long-term asymptotic states, more rapid evolution occurs on patches of quasi-  , c) Flux limit as a function of distance from the Sun and as a function of plasma age, estimated as a flux rope's distance from the Sun divided by its velocity.In panels (b) and (c), each line represents the range of distances used to sample the events.The number of events in each sample is listed below the lines.The 95% confidence level uncertainty estimated with bootstrapping is shown as a shaded region around each line.In panel (b), the distance is virtually constant in the case of Wind, so a single point at 1 au is used instead of a line.The colors of the lines indicate the spacecraft used to detect the events, as labeled in the legend in the lower right corner.Global number 10 5.8 10 4.9 10 3.6 10 2.0 equilibrium via suppression of nonlinearities.The effect is that of a "cellularized" turbulence with localized regions where the nonlinear effects are depleted (Pelz et al. 1985;Montgomery et al. 1992;Matthaeus et al. 2008;Servidio et al. 2008).Such effects have been observed in the solar wind (Osman et al. 2011;Servidio et al. 2014) and the magnetosheath (Pecora et al. 2023).Equilibria include force-free (j × B = 0) and forcebalanced states (j × B = ∇p) to which the GS equation (Grad & Rubin 1958;Hu 2017) is applicable.These dynamical processes support the idea that the turbulent solar wind locally generates more ordered structures that can be identified as flux ropes.The combination of a patchy dynamical "cellularized" turbulence with the reconnection of adjacent flux tubes (that may be one explanation for the log-normally distributed fluxes) and the survival of some flux ropes that originated in the lower solar atmosphere provides several channels for the emergence of these structures throughout the entire heliosphere.If all SMFRs originate from solar eruptions, what mechanisms can release such numerous SMFRs into the heliosphere across such a wide range of scales?Rouillard et al. (2011) traced small CMEs and streamers to 1 au, confirming that they are the source of at least some SMFRs.However, the events were all above 0.05 au in diameter, and SMFRs above ∼0.01au in diameter at 1 au are rare and exhibit different solar activity dependence (Farooki et al. 2024b).The global rate of SMFR occurrence is too abundant by at least an order of magnitude to correspond to CMEs, even narrow ones, although blowout jets or streamers in the corona remain a viable candidate for the origin of SMFRs (Janvier et al. 2014).Some "streamer blob" SMFRs may originate from reconnection in the HCS at the top of the helmet streamer belt (e.g., Crooker et al. 1996;Sanchez-Diaz et al. 2017;Higginson & Lynch 2018).These would be more prevalent in the slow solar wind.Other SMFRs, in the fast solar wind, may originate from coronal blowout jets (e.g., Moore et al. 2010Moore et al. , 2013;;Pariat et al. 2010;Liu et al. 2011;Archontis & Hood 2013;Huang et al. 2023).However, the similarities between SMFRs in the fast and slow solar wind (Farooki et al. 2024b), despite the differences in likely generation mechanisms, may suggest that most of the SMFRs, especially at the smallest scales, are likely to originate from in situ turbulent mechanisms, or at least evolve into a similar state through turbulence.
To test potential candidates for SMFR solar origins, axial flux measurements based on GS reconstruction can be directly compared with solar measurements.Borovsky (2008) previously estimated the magnetic flux of flux tubes based on average field strength and estimated size and found agreement with the photospheric flux distribution reported by Parnell (2002).However, Parnell et al. (2009) found a different distribution that appears to be incompatible with the distribution in Borovsky (2008) and our distribution.On the other hand, since the Parnell et al. (2009) photospheric flux distribution extends below 10 16 Mx, it is plausible that SMFRs even with flux as low as 10 15 Mx come from the Sun.Even if significant evolution occurs in the solar wind, SMFRs may start as a seed population of flux ropes from small-scale solar eruptions.Huang et al. (2023) compared the number of small-scale coronal jet ejections in coronal hole regions expected to be observed by PSP per day.They found a frequency comparable to that of SMFRs identified in previous studies.However, it is important to consider the minimum axial flux that is well resolved by both the solar observation and the in situ observation for a meaningful comparison.Otherwise, the number of events changes by orders of magnitude (Table 1).No information about magnetic flux is currently available for the events in Huang et al. (2023).It may be possible to compare with existing estimates of the number of eruptions on the Sun at any given time as a function of size (e.g., Sterling & Moore 2016;Sterling et al. 2024) if one makes assumptions about the field strength.However, the number of flux ropes on the Sun at a time is not necessarily the number of flux ropes passing through the inner heliosphere at a time, which depends on the SMFR rate of occurrence (unknown in the near-Sun interplanetary space), speed (known through spacecraft measurements), and length (completely unknown as of yet).
Recently, switchbacks (SBs), sudden magnetic field reversals, have received a great deal of attention owing to their prominence in PSP observations (e.g., Bale et al. 2019;Kasper et al. 2019).There is evidence that SBs undergo active evolution within the solar wind (Pecora et al. 2022;Jagarlamudi et al. 2023).SMFRs are not widely considered to be related to SBs.However, some studies have found that the observational features of SBs are similar to what one would see if a spacecraft passes through SMFRs.For example, Drake et al. (2021) demonstrated how simulations of SMFRs can reproduce key aspects of SB observations such as rapid rotation of the radial component into the transverse direction and even a reversal of the radial component.Agapitov et al. (2022) further described how flux ropes affected by expansion and merging can explain observed SB properties, a model that is strengthened by our findings.While these studies proposed coronal origins for the SMFRs, SMFRs of local origin in the solar wind would have similar properties.Additionally, it is important to note that there are likely multiple sources of SBs, some of which may be pure Alfvén waves while others may be SMFRs.
A major reason that SMFRs are not widely believed to be related to SBs is that SMFRs were traditionally believed to be non-Alfvénic (Cartwright & Moldwin 2008, 2010;Marubashi et al. 2010;Yu et al. 2016;Hu et al. 2018) while SBs generally contain Alfvénic flow (Kasper et al. 2019), a signature of Alfvén waves (Belcher & Davis 1971).However, SMFRs can also be Alfvénic and contain torsional Alfvén waves (Gosling et al. 2010).In fact, Alfvénicity increase is a natural consequence of flux rope merging (Agapitov et al. 2022), and Alfvénicity in SBs tends to be lower than in the surrounding solar wind (Agapitov et al. 2023).PSP observations have shown that near the Sun most SMFRs are highly Alfvénic (even though the Alfvénic and non-Alfvénic SMFRs share the same statistical properties) and that there is significant overlap between SMFRs and SB intervals (Chen et al. 2021;Chen & Hu 2022).Recently, we found that even at 1 au most SMFRs are Alfvénic, despite exhibiting signatures inconsistent with Alfvén waves such as nonzero changes in magnetic field strength anticorrelated with density variations (Farooki et al. 2024b).The Alfvénicity is reduced in the slow solar wind and higher in the fast solar wind.This Letter further supports that SMFRs are not just large-amplitude Alfvén waves, since we do not expect them to merge in the way that flux ropes can.Alfvénicity being a consequence of ongoing turbulence is also compatible with our results, since it should decrease with plasma age as merging becomes less frequent, consistent with the well-known property that near the Sun both fast and slow solar wind types are highly Alfvénic, whereas at 1 au only the fast solar wind is highly Alfvénic, and in the outer heliosphere Alfvénicity is highly reduced.Decrease in Alfvénicity as solar wind turbulence ages has been reported previously (Matthaeus et al. 2004;Breech et al. 2005).
A limitation of single-spacecraft SMFR studies is that singlespacecraft measurements cannot provide a unique reconstruction of the magnetic field owing to their inability to directly determine the invariant direction used for reconstruction, as discussed in Farooki et al. (2024b).Initial results from our forthcoming study using multispacecraft data to determine the invariant direction show that a significant portion of the singlespacecraft events can be validated using multispacecraft data and that the statistical properties determined from singlespacecraft data are consistent with those from multispacecraft data.The focus of this Letter is the statistical difference between the results from applying the same algorithm to different data sets, so the single-spacecraft limitations are highly unlikely to affect the conclusions.
reason for the slight disagreement between SolO and Wind, but SolO measurements remain a valuable inclusion owing to its coverage of the range between PSP and Wind observations.

Figure 1 .
Figure 1.Three examples of SMFR events detected by our algorithm from PSP, SolO, and Voyager 1 data.The top panels display the GS reconstruction of each event.The bottom panels show the raw measurements transformed into the flux rope coordinate system.

Figure 2 .
Figure2.Area-flux scatter plot.It is in log-log format, so the slope of each proportionality is represented as the y-intercept.Samples of events from different spacecraft and ranges of heliocentric distance are plotted separately.Each line represents direct proportionality, with the slope being the average axial flux density for the sample.The shaded region is partially unresolved because SMFRs of diameter below (600 km s −1 )(30 s) (due to the minimum 30 s sliding window) can only be detected when the orientation is sufficiently closely aligned with the velocity in the spacecraft frame of reference(Farooki et al. 2024b).

Figure 3 .
Figure 3. Histogram of τ c at different ranges of heliocentric distance.Binned in log space.The y-axis is in linear space (values not shown to save space).

Figure 4 .
Figure4.Map of events detected in the inner heliosphere in heliocentric inertial polar coordinates.Each event is plotted as a circle such that the area of the marker is proportional to the axial flux of the SMFR (not its cross-sectional area, which naturally increases as a function of distance owing to the expansion of the solar wind).

Figure 5 .
Figure 5. (a) Comparison of axial flux for different ranges of distance from the Sun in the form of a (complementary) empirical cumulative distribution function.(b, c) Flux limit as a function of distance from the Sun and as a function of plasma age, estimated as a flux rope's distance from the Sun divided by its velocity.In panels (b) and (c), each line represents the range of distances used to sample the events.The number of events in each sample is listed below the lines.The 95% confidence level uncertainty estimated with bootstrapping is shown as a shaded region around each line.In panel (b), the distance is virtually constant in the case of Wind, so a single point at 1 au is used instead of a line.The colors of the lines indicate the spacecraft used to detect the events, as labeled in the legend in the lower right corner.

Table 1
Global Number of SMFRs