Statistical Properties of the Distribution and Generation of Kinetic‐Scale Flux Ropes in the Terrestrial Dayside Magnetosheath

The generation of kinetic‐scale flux ropes (KSFRs) is closely related to magnetic reconnection. Both flux ropes and reconnection sites are detected in the magnetosheath and can impact the dynamics upstream of the magnetopause. In this study, using the Magnetospheric Multiscale satellite, 12,623 KSFRs with a scale <20 RCi are statistically studied in the Earth's dayside magnetosheath. It is found that they are mostly generated near the bow shock (BS), and propagate downstream in the magnetosheath. Their quantity significantly increases as the scale decreases, consistent with a flux rope coalescence model. Moreover, the solar wind parameters can control the occurrence rate of KSFRs. They are more easily generated at high Mach number, large proton density, and weak magnetic field strength of the solar wind, similar to the conditions that favor BS reconnection. Our study shows a close connection between KSFR generation and BS reconnection.

• Kinetic-scale flux ropes observed in the magnetosheath are primarily generated near the bow shock (BS) and travel to downstream magnetosheath • The quantity of flux ropes significantly increases as their scale decreases, which is in accordance with the FR coalescence model • The occurrence of flux ropes is influenced by solar wind parameters, and could strongly correlate with BS reconnection

Supporting Information:
Supporting Information may be found in the online version of this article.
Most studies on KSFRs in the dayside by the MMS satellites have primarily been case studies in the magnetosheath and on the magnetopause (e.g., Alm et al., 2018;Dokgo et al., 2021;Hwang et al., 2020;Y Liu et al., 2018;Robertson et al., 2021;Tang et al., 2018aTang et al., , 2018b; S. M. Wang et al., 2019Wang et al., , 2020)).This can be attributed to the MMS satellite's prolonged dwell time in the magnetosheath and many high-precision data for the magnetopause crossing.Since magnetic reconnection can occur in multiple locations, the major source of the KSFRs in the magnetosheath is still unclear.These situations highlight the necessity of conducting statistical studies on KSFRs as single case studies are insufficient.
Previous statistics of FRs in the dayside magnetosheath mainly focused on large-scale (usually larger than 20 ion gyroradius (R Ci ), and R Ci typically on the scale of tens of kilometers in the magnetosheath), investigating plasma acceleration near the magnetopause (Akhavan-Tafti, Slavin, Eastwood, et al., 2019, Akhavan-Tafti, Slavin, Sun, et al., 2019), as well as exploring their relationship with the interplanetary magnetic field (IMF) (Dahani et al., 2022;Kieokaew et al., 2021;Kuo et al., 1995;Y. Wang et al., 2006).Statistical studies on KSFRs mainly focus on Earth's magnetotail (Jiang et al., 2021;Sun et al., 2019), shedding light on the energy dissipation and electron acceleration by KSFRs through secondary reconnection.However, statistics on KSFRs in the dayside magnetosheath are extremely lacking.
In our study, we utilize magnetic field and plasma data from the MMS satellites to identify KSFRs in the magnetosheath in Section 2 and investigate their scale, spatial distributions, propagation characteristics, and upstream conditions in Section 3. In Section 4, we discuss their relationship with reconnection processes.

Data and Events Selection
We use MMS1 magnetic field data from the Flux-Gate Magnetometers (FGM, Russell et al., 2016) and plasma data from the Fast Plasma Investigation (FPI, Pollock et al., 2016) from September 2015 to June 2022 to select KSFRs.The measurement of the FGM instrument is 16 Hz in survey mode and 128 Hz in burst mode.The measurement cadence of FPI is 4.5 s for both electrons and ions in survey mode, 30 ms for electrons and 150 ms for ions in burst mode.The solar wind data with 1 min cadence from the OMNI database are used for the statistical study in Section 3.
To identify candidate dayside KSFRs, we rely on the changes in the total magnetic field and in the helical magnetic field configuration observed in the MMS1 satellite's dayside time-series data.Several steps are used to select KSFRs based on automatic routines using survey mode and burst mode data respectively, as shown below and in Figure 1.
1. Find magnetic peaks from the magnetic field data: We calculate the average value (B ave ) and maximum value (B max ) of the magnetic field data every 10 s   max  ave > 1.3 is used to determine a magnetic peak structure which is regarded as a candidate of significant dayside FR.The start time and end time of magnetic peaks correspond to the moment when the magnetic fields on both sides of the maximum value first reach the average value, and the time between them is considered as the duration of peaks (t peak ). 2. Select FRs that clearly stand out from the surrounding background: We use the criterion of max − ave std > 1.5 .
Here,  std is the standard deviation of the background magnetic field strength.It serves as a measure of the overall disturbance level 3. To prevent events with a large duration from affecting the data calculation of the time window, we require t peak to be less than 10/3 s. 4. To ensure an adequate number of data points within the candidate FR, it should be observed by at least 8 data points.Therefore, candidate events should satisfy the condition t peak ≥ 0.5625 s.This criterion is based on the magnetic field accuracy of the FGM in survey mode resolution (16 Hz). 5. Select the structures with bipolar magnetic signature: The Minimum Variance Analysis (MVA, Sonnerup & Cahill, 1967) is used within t peak to establish a local LMN coordinate system of the FR, and we limit the eigenvalues to ensure the accuracy of the coordinate system ( ).We calculated the angles between the FR's magnetic fields (averaged) on the L-N and M-N planes on either side of the FR's center.The plane containing the larger of these two angles is considered to exhibit significant magnetic field rotation and is deemed the cross-section.That is, the bipolar direction has been identified and used in conjunction with the N-direction to define the cross-section, and the remaining directions (L or M) are considered axial.The magnetic hodogram in the bipolar-axial plane was then fitted to an ellipse using the least squares method (fitting degree r 2 ≥ 0.9) (Elphic et al., 1980;Fitzgibbon et al., 1999) to find a better FR configuration.This method has been widely used to determine the FRs observed near different planets (Y.Chen et al., 2017;Hara et al., 2017;Smith et al., 2017).6.To differentiate between FRs, magnetic bottles, and wave-like fluctuations, which all may show a bipolar magnetic field signature during the satellite crossing (see Yao et al., 2018 and Supporting Information for details), the angle (θ Bv ) between the ion bulk velocity and the magnetic field direction is calculated on the cross-section.Because the trajectory of the satellite or background ion flow will be tangent to the magnetic field line of the FR at some points on the cross-section, θ Bv should be close to 0° or 180° inside the FR.So we set θ Bv to be less than 15° or greater than 165°.7. The averaged plasma parameters of 10 s before and after the event are considered as the background in survey mode (5 s in burst mode).To exclude FRs in the magnetosphere, we reject events with ion density N i < 2 cm −3 and electron temperature T e// > 100 eV (Gao et al., 2022).To exclude FRs in the solar wind, we reject events with the following conditions: N i < 10 cm −3 | T e < 15 eV and N i < 20 cm −3 & T e < 26 eV (Kiran et al., 2020;Tu, 1988), where | and & refers to logical "or" and "and."Furthermore, to address the situations where denser solar wind may resemble the magnetosheath, plasma flow speed V i > 300 km/s is used to reject those solar wind events (e.g., Hunderhausen 1972;Tu, 1988).
Finally, 12,623 KSFRs from survey mode data and 2,644 from burst mode data are used for statistical studies.Figures 1h-1m are KSFR examples selected every 2,500 events from the survey and burst mode events list (Tables S1 and S2 in Supporting Information S1).

Occurrence Rate of KSFRs
The dwell time of MMS satellites (see Figure S2 in Supporting Information S1) from September 2015 to June 2022 is used to calculate the occurrence rate of the FRs. Figure 2a shows the occurrence rate of KSFRs inside the dayside magnetosheath calculated as the quantity of KSFRs divided by the MMS dwell time in each cell.Both Figure 2a and Figure S2 in Supporting Information S1 divide the XY plane into 0.5 × 0.5 R E cells.Since burst mode data has a certain location bias, we only show the result selected from the survey mode data in Figure 2a.For each FR event, its distance from the magnetopause (D MP ) and distance from the bow shock (D BS ) are calculated using the magnetopause and BS model (Farris & Russell, 1994;Shue et al., 1998).The occurrence rate of the FRs as a function of D BS is shown in Figure 2b.However, the magnetosheath thickness varies with the properties of the upstream solar wind.To eliminate the influence caused by the uneven thickness of the magnetosheath, we calculate the relative distance from the FR location to the BS, + , and the corresponding result is shown in Figure 2c.
The results show that the occurrence rate of KSFRs near the BS is significantly higher than that near the magnetopause.The events selected from burst mode data show a similar generation trend (Figure S3 in Supporting Information S1).

Propagation
Using the spatio-temporal difference (STD) method (Shi et al., 2006(Shi et al., , 2019;;and examples in Yao et al., 2016;Yao, Hamrin et al., 2020), the propagation directions of the 2,644 KSFRs observed in the burst mode data were calculated (the errors in the STD results for the survey mode data is too large to be analyzed and those results are not shown here).It is worth noting that the KSFRs selected from the burst mode data are only used here to calculate the direction of the structure's motion.The result shows that 87% of KSFRs propagate downstream of the magnetosheath in the satellite frame (Figure 3a).It can, therefore, be inferred that the higher occurrence rate of KSFRs near the BS is not due to the sunward propagation of the KSFRs generated in the magnetosheath, but due to that the KSFRs are generated near the BS.
Previous studies reported that FRs coalesce during propagation (Alm et al., 2018;Guo et al., 2021;R Wang et al., 2016).Here we find that the relationship between the quantity of KSFRs and their cross-section scale (calculated from the event duration and local plasma flow velocity in the cross-section) is consistent with an FR coalescence model (Figure 3c).This model describes the dynamic evolution of FRs in current layers (Figure 3b), explaining how they grow to a more macroscopic scale by coalescing (Fermo et al., 2010).This model has the numerical steady-state solution   ∞() = ∫ ∞ 0 Ψ∞(Ψ, ) ∝  −∕ 0 , where ψ is magnetic flux (Fermo et al., 2011).Coalescence is considered to be caused by magnetic reconnection if the quantity of the FRs (   ∞ ) is exponential to the FR scale (r) (see examples in Akhavan-Tafti et al. (2018) which studied the coalescence and growth of large-scale FRs near the magnetopause).Our study confirms that this model is applicable to smaller-scale FRs.In Figure 3c, the scale and scale normalized by R Ci are shown as a function of the KSFRs' quantity.The quantity increases exponentially as the scale decreases (the exponential coefficient is >0.95, indicating a very well fitting).

Impaction of Solar Wind Parameters on the Occurrence of KSFRs
The properties of the upstream solar wind can influence the magnetosheath environment, and then potentially control the occurrence rate of KSFRs.To explore this relationship, we investigate the solar wind Alfvén Mach number (M A ), proton density (N i ), velocity (V SW ), and magnetic field strength (B SW ).
The statistics utilized 1 min resolution solar wind parameters obtained from the OMNI database from September 2015 to June 2022 as the background for normalization (Figure 4, left column).The overall KSFRs from survey mode are divided into two groups: one located closer to the BS ( + ∈ [0, 0.5] ) and the other closer to the magnetopause ( + ∈ [0.5, 1] ).Considering that the tracking errors of the upstream solar wind parameters in the second group are large, only the first group is analyzed here.The numbers of the KSFRs as a function of various solar wind parameters are plotted in the middle column of Figure 4, and the right column shows the results normalized concerning the background.The M A , V SW and B SW have been projected into the local BS normal direction.The shaded area represents the credibility interval of the statistical results, and events in each bin falling within this interval account for more than 5% of the total number of events.We consider that the credibility of the results in the interval can not be guaranteed if the number of events is small.It can be found that the occurrence rate of KSFRs exhibits a positive correlation with the solar wind M A (Figure 4c), N i (Figure 4i) and a negative correlation with B SW (Figure 4f).

Discussions and Conclusions
Using the measurement of the MMS satellites, 12,623 KSFRs were found in the dayside magnetosheath.Their occurrence rate, propagations, and upstream solar wind conditions were statistically analyzed.The main conclusions are as follows: 1.The occurrence rate of KSFRs near the BS is higher than that near the magnetopause.2. Most of the KSFRs (87%) move downstream of the BS, and there is an exponential relationship between their scale and quantity, consistent with the FR growth model (Fermo et al., 2010).3. The KSFRs are more easily generated under high solar wind M A , large N i , and weak B SW .
The KSFRs generated by magnetopause and magnetosheath reconnection have been extensively studied previously.Since our study finds that the FR will propagate downstream in the magnetosheath after generation (Figure 3a), neither magnetopause nor magnetosheath reconnection can explain the higher occurrence rate of the KSFRs near the BS.Since the launch of MMS, BS reconnection has gradually entered the field of view (e.g., Bessho et al., 2019;Z. Chen et al., 2019;Gingell et al., 2017Gingell et al., , 2019;;Hamrin et al., 2019;Matsumoto et al., 2015;S. Wang et al., 2019;).Simulation studies have shown that after BS reconnection occurs, a large number of KSFRs will be formed in the transition region and downstream of the shocks (Bessho et al., 2019;Gingell et al., 2017).This is consistent with the high occurrence rate of FRs downstream of the BS, as obtained by our statistics.The location of most BS reconnection current sheets obtained by Gingell et al. (2020) occurs within five R E downstream from the BS, similar to the distribution of KSFRs in our study (Figure 2b).Gingell et al. (2020) statistically studied the relationship between the occurrence of BS reconnection current sheets and the M A , and found that the occurrence of the reconnection current sheet increases with the increase of M A , similar to our results.The higher the M A , the more fully developed turbulence is at the sub-ion scale (Li et al., 2020), which is conducive to the reconnection and KSFRs.Therefore, it can be observed that the occurrence rate of KSFRs near the BS is high (Figure 2a).The conditions that favor the generation of KSFRs in the solar wind and those conducive to the foreshock transients (e.g., T. Liu et al., 2017Liu et al., , 2022;;Zhang et al., 2022) share similarities and differences.Investigating their intrinsic connections and distinctions is critically significant research.Our research shows that BS reconnection should be important in generating KSFRs within the magnetosheath.This is inconsistent with previous points that the KSFRs inside the magnetosheath are often generated by magnetosheath reconnection and turbulence.The energy conversion process of magnetic reconnection and turbulence can affect a wider area within the magnetosheath through the generation and dissipation of FR.Our statistics in the dayside magnetosheath suggest that KSFRs near the BS deserve more in-depth investigation.

Figure 1 .
Figure 1.The selections and examples of dayside kinetic-scale flux ropes.(a) Select magnetic peaks.(b) Select a less disturbed background.(c) Prevent events from affecting the time window.(d) Delete events with less than 8 data points.(e) Select helical magnetic field.(f) Distinguish flux ropes, magnetic bottles, and wave-like fluctuations.(g) Determine the location.The quantity of each step (retained events/dropped events) is indicated below.(h-m) Examples have been chosen based on their numerical order within the chronological sequence of statistical results.

Figure 2 .
Figure 2. The occurrence rate of kinetic-scale flux ropes (KSFRs).(a) The occurrence rate normalized by MMS1 dwell time.The white lines show the bow shock and magnetopause model using parameters obtained as mean values of all KSFRs.Occurrence rate as a function of panel (b) the distance from the bow shock (BS), and (c) the relative distance from the bow shock.

Figure 3 .
Figure 3.The propagations and scale of kinetic-scale flux ropes (KSFRs) in the magnetosheath.(a) The direction of motion of KSFRs in the satellite reference system.(b) The relationship between the scale and quantity of large-scale FRs (from Fermo et al., 2011, Figure7).(c) The quantity of KSFRs as a function of the scale in the unit of km (red) and R Ci (blue).The results are exponential fitted (dashed lines, with R 2 (0-1) the fit coefficient).

Figure 4 .
Figure 4. Relationship between kinetic-scale flux ropes (KSFRs) (using events close to the bow shock) and solar wind parameters.Statistics of the solar wind parameters using the OMNI database from September 2015 to June 2022 (left panels) and of the observed KSFRs (middle panels), and normalized results (right panels, i.e., middle/left) are shown.From top to bottom are the solar wind Alfvén Mach number M A , proton density N i , solar wind velocity V SW , and magnetic field strength B SW .The shaded area in the figure is the credibility interval of the statistical results.