We consider a total of 16 HCS crossing events, including 8 observed by PSP and 8 by ACE, as listed in Table 1. All PSP events were reported previously (Phan et al., 2020; Phan et al., 2021; Phan et al., 2022), including 7 with signatures of reconnection exhaust and 1 without. Two ACE events (A1 and A3) with reconnection signatures were reported previously (Gosling et al., 2005c; Gosling et al., 2006). Additional ACE HCS crossing events are identified mainly based on the signature of electron strahl reversal on the two sides of HCS. The suprathermal electron pitch angle data and plots in the solar wind frame can be found at https://izw1.caltech.edu/ACE/ASC/DATA/level3/swepam/index.html. We then visually verify that three of these events show signatures of reconnection exhaust and three do not. We also find several ACE events with less pronounced signatures of reconnection exhaust, and they are excluded from our analysis. A more thorough statistical analysis is planned for the future.

For each HCS event, we determine its duration Δ t by visual inspection of magnetic field and plasma data. Magnetic field data are measured by ACE/MFI and PSP/FIELDS/MAG; and plasma data are measured by ACE/SWEPAM (A1, A3–A8), ACE/SWICS (A2, A7), PSP/SWEAP/SPC (P1–P6), and PSP/SWEAP/SPAN-i (P7, P8). Following Phan et al. (2021), we perform the minimum variance analysis (MVA) of magnetic field to determine the orientation of the observed current sheets (Sonnerup and Cahill, 1967), and the width of a current sheet is calculated by w = u n Δ t , where u n is the spacecraft-frame flow velocity in the minimum variance direction. It has been noted by several previous studies (Knetter et al., 2004; Wang et al., 2024; Eriksson et al., 2024) that MVA sometimes produces highly questionable estimates of the current sheet normal. A more accurate normal direction can be calculated by the cross product between magnetic field before and after the current sheet, i.e., n ̂ = B 1 × B 2 / | B 1 × B 2 | . A hybrid MVA coordinates are then completed by m ̂ = n ̂ × l ̂ MV A / | n ̂ × l ̂ MV A | , and l ̂ = m ̂ × n ̂ . We use the hybrid coordinates for all but four events with large magnetic shear angles (A7, P3, P4, and P7), since the cross product normal becomes less reliable in such cases. The standard LMN coordinates with MVA are used for these four events. The solar wind speed u s w and Alfvén speed V A are calculated separately for periods before and after HCS crossings, giving two numbers for each event in Table 1.

Magnetic reconnection is identified by the signature of outflow jets in the maximum variance direction, i.e., l -direction in the LMN coordinate (e.g., Phan et al., 2010). Figure 1 shows the time series plots for Events A1 and P2 as examples of nonreconnection and reconnection events. The magnetic field and velocity are plotted in the hybrid MVA coordinates. The vertical dashed lines bound the approximate extent of HCS crossing, which is used for the calculation of current sheet width. An outflow jet in the l -direction is seen in the reconnection event P2 but not in the nonreconnection event A2.

For completeness, the LMN unit vectors and magnetic shear angle for all HCS events are listed in Table 2. The LMN coordinates calculated with standard MVA are listed for four events (labeled with an asterisk in the table), three of them having a shear angle 170 ° or larger (A7, P4, and P7). Event P3 has a shear angle of 163 ° , but the hybrid coordinates exhibits more pronounced variations in B n than B m for this event (not shown here). Nevertheless, the difference in the estimated current sheet width due to the two coordinate systems is about 25%, which is not significant. For all other events, the hybrid LMN coordinates with cross product normal are listed.

We have identified the reconnection and nonreconnection events and listed them in Table 1. For most of PSP crossing events, 7 of 8 analyzed events show reconnection whereas only 5 out of 8 ACE crossing events show reconnection. To further understand this result, we propose an approach to analyze the “normalized” thickness of these HCS events, which is usually characterized by the aspect ratio between its width ( w ) and length ( L ) . Thin current sheets with a sufficiently small aspect ratio are subject to MHD tearing instabilities that trigger the onset of magnetic reconnection (e.g., Furth et al., 1963; Uzdensky and Loureiro, 2016; Comisso et al., 2016; Comisso et al., 2017; Huang et al., 2017). Inspired by this scenario, we interpret the observational results in Section 2 as a manifestation of the onset process due to tearing. We now examine the current sheet width and the length scale associated with the propagation of Alfvén waves as key parameters.

The observed width w of HCS can be obtained from observations and is plotted in the top panel of Figure 2 for each event. For PSP events, the width is plotted as a function of the spacecraft location at the time of the observation. ACE events are simply ordered horizontally by the event number in Table 1. The HCS width spans several orders of magnitude and tends to increase with radial distance due to the expansion of the solar wind, consistent with the expectation from previous work (Smith, 2001). Reconnection events are represented by blue colored dots, while non-reconnection events are represented by orange colored crosses.

The relevant lengthscale L of HCS for possible reconnection is defined by the distance traveled by Alfvén waves during the time for the solar wind to propagate from some inner boundary r s to the observer’s location r . This can be calculated by the formula, L = ∫ r s r V A r ′ d r ′ u s w . (1) We consider the super-Alfvénic solar wind, so L is typically much shorter than r . This Alfvén length scale may be regarded as an estimate of the length of the reconnection region within HCS if the onset takes one Alfvén crossing time. This is a reasonable assumption as perturbations to the current sheet is communicated by Alfvén waves in MHD.

For the radial evolution of the Alfvén speed V A ( r ) , we assume Parker’s spiral interplanetary magnetic field (Parker, 1958; Weber and Davis Jr, 1967) and a constant solar wind speed u s w (which implies an r − 2 scaling for the solar wind density), V A = V A s r r s − 1 1 + Ω 2 r − r s 2 / r 2 u s w 2 1 / 2 , (2) where Ω ≃ 2.8 × 1 0 − 6 s − 1 is the angular velocity of solar rotation; and V A s is the Alfvén speed at the solar wind source surface r = r s , where the magnetic field is radial. The lower limit in Equation 1 is taken to be at r s , and we approximate it as the Alfvén critical radius so that it can be determined numerically by the root of the equation V A ( r ) = u s w . The actual location of the source surface is likely to be within the Alfvén critical radius based on observations and models (Wilcox et al., 1980; Burlaga et al., 1981; Kasper et al., 2021; Verscharen et al., 2021; Chhiber et al., 2022), but magnetic field is still predominantly radial at the Alfvén radius, justifying the approximation.

With the observed width w and length L calculated by Equation 1, we define an aspect ratio as w / L and plot it in the bottom panel of Figure 2 for the PSP and ACE events from Table 1. For each event, two values are computed based on parameters measured on two sides of the current sheet, yielding a range for the ratio w / L ; this may be taken as the uncertainty. Most of the uncertainty is quite small so the points from two sides of the current sheets essential overlap.

Figure 2 suggests that reconnection events tend to have a smaller ratio w / L . A threshold ratio may be present around 0.01, as most reconnection events have ratio smaller that (with the exception of PSP events P1 and P8). The feature appears applicable to both PSP and ACE events. Our results suggest that the ratio between these two length scales, which characterize the thickness of HCS, is important in determining whether magnetic reconnection occurs at the HCS. These results are physically sensible because a small current sheet thickness is expected to favor reconnection based on MHD tearing instabilities. And a longer Alfvén length scale means that given a current sheet width, the tearing instability has more time to grow.

Some outstanding uncertainties should be noted regarding our width and length estimates. First, several of the events contain clear signatures of partial HCS crossings: the B r component of magnetic field switches up and down within the identified HCS boundaries, suggesting that the spacecraft likely moves back and forth within the HCS. The partial crossings are most prominent in Events P1, but are also seen in P3, P8, A6, and A8, and will likely lead to overestimating the width. Second, assuming a constant solar wind speed outside the Alfvén surface will underestimate the length L since the solar wind is accelerating. We expect this effect to be strongest for events closest to the Sun (e.g., P7 and P8). For events far away from the Sun, the solar wind speed has likely stabilized throughout most of the propagation time. Incidentally, the two “outliers” in Figure 2 (Events P1 and P8) with higher w / L ratios than other reconnecting events may be explained by the uncertainties. The time series data for these two events are shown in Figure 3. Event P1 is the most impacted event by partial crossings, while P8 is affected by both partial crossings and solar wind acceleration. Both uncertainties are expected to lead to an overestimation of the w / L ratio. The time series plots for these two events are included in the appendix. Third, both MVA and the cross product methods for determining the current sheet normal are subject to errors in certain situations, causing uncertainties in the width estimates. Finally, the visual identification of HCS boundaries is a somewhat subjective matter. More systematic analysis of HCS, possibly with automatic detection algorithm, is beyond the scope of the present work and is deferred for a future study.

In general, our analysis confirms previous results (Gosling et al., 2005a; Phan et al., 2021, etc.) suggesting that the occurrence of reconnection at HCS appears more frequent closer to the Sun. Here, we propose a scenario to explain this observation. Because the solar wind is super-Alfvénic and supersonic for these crossing events, we consider how causality will influence the possible onset of reconnection. Considering a segment of the HCS that may eventually undergo reconnection, as illustrated in Figure 4. The length of the current sheet region is L c s . We can define two timescales, one for the solar wind to propagate to r as τ s w , and the other for Alfvén waves to communicate over L c s as τ A . Both timescales will evolve as the solar wind expands. We suggest that, for a given initial perturbation, the longer the solar wind propagation time is with respect to the Alfvén time, the more likely that the HCS will become unstable and undergo reconnection.

Based on the above argument, a crucial number is the ratio between τ s w and τ A , defined as R A r ≡ τ s w τ A = ∫ r s r V A r ′ d r ′ L c s r u s w r ′ . (3) Using the radial profile of Alfvén speed (Equation 2), a constant solar wind speed u s w , and an expanding box size L c s = L s ( r / r s ) , which corresponds to a linear increase of HCS width with distance suggested by Smith (2001), we can calculate the ratio given by Equation 3, R A = ∫ r s r V A s u s w r ′ r s − 1 1 + Ω 2 r ′ − r s 2 / r ′ 2 u s w 2 1 / 2 1 L s r r s − 1 d r ′ . (4) The integration can be carried out numerically. Similar to the previous section, we assume that the source surface is located at the Alfvén critical point where u s w = V A . Within the critical radius, the acceleration of solar wind bulk flow cannot be neglected. The effect of such acceleration is that the current sheet is stretched in length, which then leads to the increase of the aspect ratio. Indeed, reconnection at HCS within the critical radius is commonly seen at the tip of the streamer belt (e.g., Lavraud et al., 2020; Réville et al., 2020) and this is a separate subject from our discussions.

As a demonstration, Figure 5 plots the radial profiles of Alfvén speed V A (blue curve) and the ratio R A (orange curve) with nominal solar wind parameters r s = 0.04 au and u s w = V A s = 2.7 × 1 0 7 cm/s. The box size at the inner boundary is set as L s = 1.3 × 1 0 11 cm, which is about 1,000 times the width at r s as indicated by Figure 2. The Alfvén speed decreases with increasing distance as ∼ r − 1 near the Sun and approaches a constant asymptotically in the outer heliosphere. Very close to the Sun, R A is small because the short solar wind propagation time τ s w ; far away from the Sun, R A becomes small again because the expansion of box size L and the decreasing Alfvén speed. As a result, the ratio attains a maximum value at some radial distance within 0.3 au. At 1 au, the time available for growth has been reduced in half compared to 0.2 au. This provides a natural explanation to why HCS reconnection is more common at smaller radial distances. At r ≫ 1 au, the ratio R A approaches a finite positive number asymptotically.

These results can be affected by the parameters. For example, a larger r s or V A s would lead to a larger V A at all distances, and thus increases the ratio R A overall; and a larger L s would increase τ A and thus decreases the ratio R A . However, the general behavior of the curves in Figure 5 is independent of the choice of these parameters. Meanwhile, the behavior of R A can be affected by the assumption of L c s ∝ r . If a different scaling is adopted, i.e., L c s ∝ r α , the ratio R A will decrease at large r only if α > 0.5 . A larger index α would lead to a more rapid decrease of the R A at large distances. It does appear from Figure 2 that the width increases with r at least as fast as linearly, but the statistics is limited by the small number of events considered. More realistic models of the solar wind speed and Alfvén speed may be used in the future, which will improve the estimate of r s .

We note that the threshold ratio of ∼ 0.01 from our observational analysis is reminiscent of the critical aspect ratio of plasmoid instability (Loureiro et al., 2007; Bhattacharjee et al., 2009; Huang and Bhattacharjee, 2010). However, this seems to be a coincidence because an aspect ratio of 0.01 would correspond to a very thick current sheet compared to the thickness suggested by the Sweet-Parker or ideal tearing model when the Lundquist number S = V A L c s / η ≫ 1 0 6 ( η is the coefficient of resistivity) (Parker, 1957; Sweet, 1958; Pucci and Velli, 2014; Huang et al., 2017). (The Lundquist number of the weakly collisional solar wind is S ≳ 1 0 12 .) This means that the triggering of magnetic reconnection at HCS is likely not due to tearing instability alone, which is slow in this situation. Although plasmoid instability and kinetic processes can explain fast reconnection rates, the problem of reconnection onset remains an open question. Reconnection exhausts observed at HCS can be as wide as thousands of ion inertial lengths, suggesting that kinetic processes may not be the main triggering mechanism. We speculate that solar wind turbulence may play an important role in reconnection onset at thick current sheets. Numerical simulations may be helpful in illuminating the onset process, though 3D MHD simulations of magnetic reconnection with realistic turbulence injection in the very high- S regime are still challenging.

Next, we briefly discuss the potential consequences of magnetic reconnection at HCS for the solar wind dynamics and particle energization.

A major consequence of magnetic reconnection is plasma heating and particle acceleration (Li et al., 2021). If magnetic reconnection is indeed universally occurring at HCS, it could make a considerable contribution to the heating and acceleration of solar wind, especially in the inner heliosphere. While Gosling et al. (2005a) find an absence of particle acceleration due to magnetic reconnection in the solar wind, more recent observations have reported evidence of energetic particles associated with the HCS (e.g., Khabarova and Zank, 2017; Zhao et al., 2019; Desai et al., 2022; Phan et al., 2022).

The magnetic energy conversion rate per unit area of the current sheet can be estimated as ε m = B 2 4 π V rec , where V rec ≃ ( w / L c s ) V A is the reconnection rate defined as the inflow velocity. Here, we assume a constant reconnection rate V rec = 0.01 V A according to the results from data analysis, assuming L c s ≃ L . The average energy gain per electron-proton pair Δ E p 1 due to magnetic reconnection at HCS as a function of the radial distance is Δ E p 1 r = 1 2 r Δ μ n r 4 π r L c s ∫ r s r 4 π r ′ L c s B 2 r ′ 4 π V rec r ′ d r ′ u s w . Here, Δ μ determines the volume filling factor of the deposited energy. If the converted magnetic energy is deposited to the entire heliosphere, i.e., Δ μ = 1 , it would yield a lower limit for Δ E p 1 . The upper limit is attained when all the converted energy is deposited into the narrow region corresponding to the current sheet thickness, so that Δ μ ∼ cos ( π / 2 − w s / r s ) ≃ 0.002 . The upper and lower limits of Δ E p 1 are plotted in Figure 6. The same typical parameters as in Figure 5 are used. Additionally, the magnetic field at the inner boundary is chosen as B ( r s ) ≃ 1.9 × 1 0 − 2 G according to the nominal value observed by PSP (Badman et al., 2021). This figure shows that a pair of particles may gain ∼ 1 eV–0.6 keV energy on average for our chosen parameters. This is significant compared to the typical solar wind thermal energy ( ∼ 20 eV) or even kinetic energy ( ∼ 1 keV). Indeed, keV-energy proton beams are observed near the reconnecting HCS (Phan et al., 2022). The black dashed line shows the typical thermal energy per electron-proton pair as a comparison, assuming the thermal energy per particle is 10 eV at 1 AU and the thermal energy follows an r − 1 profile based on empirical evidence (Marsch et al., 1982).

The estimated energy release is based on continued magnetic reconnection throughout the entire range of radial distance. In reality, magnetic reconnection may only occurs at a small range of radial distance as suggested by our analysis, which will inevitably decrease the total released magnetic energy. In addition, fast reconnection is known to be bursty, which may also limit the fraction of HCS undergoing reconnection at a time. Therefore, the calculation shown here is likely an overestimation for the conversion of magnetic energy.

Another potential application of reconnection at the HCS is for the outer heliosphere, namely, the heliosheath, which is the region between the heliospheric termination shock and the heliopause. While Figure 5 shows that the solar wind expansion makes reconnection less likely in the distant heliosphere further away from the Sun, it is possible that reconnection switches on again due to the compression at the termination shock. This possibility was discussed previously as an explanation of the increasing intensity of anomalous cosmic rays downstream of the termination shock (e.g., Drake et al., 2010; Zank et al., 2015; Le Roux et al., 2016; Zank et al., 2021). Since the number R A does not fall off to zero but tends to a constant asymptotically at r → ∞ (Figure 5), a compression by a factor of 2.5 at the termination shock as observed by the Voyagers and a deceleration of the solar wind flow could elevate the R A ratio to a level comparable to that of the inner heliosphere, and thus significantly enhance the likelihood of reconnection. It should be noted that the heliosheath is a high-beta subsonic environment that is thermally dominated by interstellar pickup ions, unlike the inner heliosphere where there is significant free magnetic energy. Therefore, if reconnection does occur in the heliosheath, its impact on the solar wind dynamics may be weak.