Cluster Observations of Foreshock Bubbles and Their Boundary Shocks

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L. Cai, J. Y. Yang, Y. D. Ma, Tieyan Wang, Y. Y. Liu, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7657123/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 11 Apr, 2026 Read the published version in Earth, Planets and Space → Version 1 posted 5 You are reading this latest preprint version Abstract Foreshock Bubbles (FBs) are transient structures in the Earth's foreshock region, which are diamagnetic cavities formed by hot ion concentration around interplanetary magnetic field discontinuities and have significant compressional boundary shocks contributed to particle acceleration. We present here Cluster observations of FB events from January 2002 to April 2007 that each was encountered by all four spacecraft in order to accurately determine the parameters of its boundary shock. Statistical distributions show that the majorities of the FB boundary shocks are supercritical and steep with large magnetic compression ratios and are in quasi-perpendicular direction to their upstream magnetic field. The magnetic compression ratios of FB boundary shocks are roughly correlated positively with their shock normal angles. Additionally, the magnetic compression ratios enhance with increasing upstream incident velocities, which is interpreted as a manifestation of diamagnetic Hall current generation inside the boundary. These results along with the conclusions given in previous numerical simulations and laboratorial experiments suggest a fast formation of a sharp boundary shock by the Larmor coupling between the super-thermal ions and magnetized ambient plasma in a hot plasma expanding process. Foreshock Bubbles magnetic field compression Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1. Introduction The Earth’s foreshock region is full of back-streaming super-thermal particles reflected from the bow shock (Eastwood et al., 2005 ). The interaction between the super-thermal particles and the solar wind can generate many transient events including foreshock bubble (FB) (Omidi et al., 2010 ; Turner et al., 2013 , Guo et al.,2022, 2023), hot flow anomaly (HFA) (Sibeck et al., 1999 ; Schwartz et al., 2000 ; Lin, 2002 ; Lucek et al., 2004 ; Eastwood et al., 2008 ; Jacobsen et al., 2009 ; Zhang et al., 2010 ), spontaneous HFA (SHFA) (Zhang et al., 2013 ; Omidi et al., 2013 ; Omidi et al., 2016), foreshock caviton (Omidi, 2007; Blanco-Cano et al., 2009 ; Blanco‐Cano et al., 2011; Kajdič et al., 2011 ; Kajdič et al., 2013 ), foreshock cavity (Sibeck et al., 2002 ; Schwartz et al., 2006 ; Billingham et al., 2008 ). All these foreshock transients have a core region surrounded by compressional boundaries or shocks, and both the density and the magnetic field are deeply depressed in the core region. For example, foreshock bubbles are formed by IMF discontinuities interacting with back-streaming ion beams in the foreshock region, whereas HFAs are generated after the interplanetary discontinuities hitting the bow shock. When the back-streaming ion beams pass through the discontinuity, they are deflected due to the rotation of the magnetic field vectors. The deflected beam interacts with the solar wind, resulting in the deceleration of the solar wind and the formation of fast magnetosonic shock wave that expands sunward with time and is followed by the strong sheath plasma and a core region with low magnetic field strength and hot and tenuous plasmas with energetic particles. The whole FB structure convects along with the solar wind in the anti-sunward direction (Omidi et al., 2010 ; Turner et al., 2013 ). FBs are supposed to generate on the upstream sides of interplanetary discontinuities and thus have compressional boundaries on their leading edge, while HFAs have compressional boundaries on both the leading and trailing edges. The FB sizes range from 2 to 15 \(\:{R}_{e}\) (Liu et al 2016 , Turner et al, 2020 ) and with expansion velocity 139 ~ 470 km/s faster than HFA and SHFA (Turner et al., 2020 , Liu et al., 2023 ). FBs have been observed not only in the dayside foreshock but also in the Mid-tail foreshock (Liu.et al, 2021). Foreshock transients not only have significant magnetospheric impacts due to pressure variants in their core region, but also can be effective accelerators of foreshock energetic particles and provide seed populations for further particle acceleration at the parent bow shock (e.g., Cai & Wei 2020 , Wei et al,2022, Omidi et al., 2010 ; T. Z. Liu, Angelopoulos, et al., 2017 ; Wilson et al., 2016; T. Z. Liu, Hietala, et al., 2016 ; T. Z. Liu et al., 2019 , T. Z. Liu, An, et al., 2020 ; T. Z. Liu, Lu, et al., 2017 , 2018 , 2020 ; Turner et al, 2018 ; Omidi et al., 2021 ). MMS observed that HFA can accelerate ions up to hundreds KeV (Turner et al, 2018 ). Electrons in their core can be accelerated to hundreds KeV by all transients (T. Z. Liu, Angelopoulos, et al., 2017 ; Wilson et al., 2016). Besides those acceleration mechanisms such as reconnection which have been suggested to occur inside the structures, Fermi-type mechanisms are main acceleration processes where (Omidi et al., 2021 ; Turner et al., 2018 , Liu, Lu et al., 2017 ) their sharp boundaries play a crucial role. For instance, those earthward high-speed ions in a FB are reflected and accelerated at the earthward moving FBs boundary through partial gyration along the convection electric filed, which means the boundary can act as a magnetic mirror to accelerate ions repeatedly bouncing between the converging compression boundaries and the core. Recent simulations and experiments have focused on the formation of compression boundaries in hot plasma expansions in the MHD or kinetic views. A laboratory investigation shows the collisionless shock can form experimentally within one gyro-period of ion ( \(\:{{\omega\:}}_{\text{c}\text{i}}^{-1}\) ) (Bondarenko.A.S., et al,2017a, 2017b, Hewett, D.W, et al,2011, Clark, S.E, et al, 2013). During one gyro period of ion, the intensification of the magnetic compression due to the accumulation of accelerated ambient ions results in significant expulsion of back-streaming ions from the leading edge of the expansion. The pileup of trapped back-streaming ions leads to density gradients over scales of order the compressed gyro-radius, that is, its formation contributes to Larmor coupling effect. The strong Larmor coupling electric fields created by back-streaming ion current moving across the background magnetic field act to accelerate ambient ions and decelerate back-streaming particles and thus result in collisionless momentum and energy exchange between the two species. Simultaneously, the hot core expansion drives a diamagnetic current that generates a leading magnetic compression and trailing magnetic cavity. Numerical simulation results also show that in the foreshock secondary shock formation the ambient ions gain momentum in the core expanding direction via the induction electric field due to the energetic hot ions moving out of hot layer across the magnetic field in their gyrations. This energy exchange between the fields and the hot ions ceases till the compressional boundary finally detaches from the hot populations. Meanwhile, the compressional boundary moves at supermagnetosonic speed and steepens into a shock with scale of ion cyclotron radius (An et al., 2020 ). In this paper, we present Cluster observations of FB events from January 2002 to April 2007. First, three case events are reported with an emphasis on their magnetic field profiles of the boundary shock. Next, statistical characteristic of selected FB events that each was encountered by all four Cluster spacecraft are presented. Finally, we give a brief summary and discussion of this study. 2. Observations: case examples For this study, we use data from the Cluster mission (Escoubet et al., 2001 ). The Cluster magnetic field and ion measurements are from the FGM experiment (Balogh et al., 2001 ) and the CIS/HIA instrument (Rème et al., 2001 ) respectively. The pristine solar wind data are from the ACE mission (Stone et al., 1998 ). The Cluster tetrahedron enables the accurate estimation of the boundary normal and propagation speed of a FB by the Timing technique: \(\:{\mathbf{r}}_{i4}\bullet\:\) n = \(\:{\text{v}}_{n}\) Δ \(\:{\text{t}}_{i4}\) , i = 1, 2, 3, where n is the normal vector, \(\:{\text{v}}_{n}\) is the normal speed, \(\:{\mathbf{r}}_{i4}\) and Δ \(\:{\text{t}}_{i4}\) are the separation vector and the time difference between spacecraft i = 1, 2, 3 and the reference Spacecraft 4, respectively (Schwartz, 1998 ). 2.1 26 March 2005 event On 26 March 2005, the Cluster spacecraft tetrahedron located in the foreshock region encountered a transient structure. Figure 1 shows the magnetic field and plasma observations of this structure. The ambient solar wind velocity is about 680 km/s, the magnetic field strength is 4.3nT and the density is 2.4 \(\:{cm}^{-3}\) . The solar wind plasma \(\:{\beta\:}\:\) is1.75 and the flow pressure is 2.25 nPa. The Alfven Mach number and magnetosonic Mach number are 13.35 and 7.75 respectively. These parameters are characteristic of fast solar wind streams in solar wind–magnetosphere interactions (Dai et al.2023,2024). As shown the ions energy spectrum in panel A) in Fig. 1 , in the core center detected around 03:57 UT, ion energy flux with energy of KeVs are enhanced, and the thermalized ions of a temperature are up to 1000 eV. The density decreases to about 1.0 \(\:{cm}^{-3}\) as shown in panel C), and the solar wind is strongly deflected with the X-component of the velocity dropping from ~-700km/s to ~-400km/s and the Y and Z components both increasing to 200km/s as shown in panel D). As shown in penal B), the magnetic field strength in the core region is lower than 1nT, but reaches 22 nT within the strong compression boundary. Since only obvious upstream edge but no downstream edge is detected, this transient structure is basically identified as a FB. The compressional boundary of the FB has a sharp mono-peak magnetic field profile. Its maximum magnetic compression ratio \(\:{B}_{\perp\:max}/{B}_{0}\) and maximum density compression ratio \(\:{\text{N}}_{\text{m}\text{a}\text{x}}/{\text{N}}_{0}\) are 3.48 and 3.78 respectively. Its propagating velocity calculated by the timing method is 286km/s in the solar wind rest frame. This boundary is a perpendicular shock with the angle between its normal and the upstream magnetic field vector ( \(\:{\theta\:}_{BN}\) ) of ~ 82.8 o . The shock Alfven Mach number and magnetosonic Mach number are 5.16 and 3.24 respectively. The shock duration is about 16s, ~ 0.14 \(\:{{\omega\:}}_{\text{c}\text{i}}^{-1}\) , where \(\:{\:{\omega\:}}_{\text{c}\text{i}}^{-1}\) is ion gyration period using the minimum magnetic field in the FB core to calculate. The shock transverse dimension is about 4575km ~ 4. 34 \(\:{{\rho\:}}_{\text{i}\text{c}}\) , where \(\:{{\rho\:}}_{\text{i}\text{c}}\) is gyration radius also using the minimum magnetic field and max temperature in the FB core. The FB formation resembles the scenarios described in previous simulations and laboratory experiments of collisionless coupling between explosive hot plasma and magnetized ambient plasma, and the shock boundary also has a similar magnetic profile (see Fig. 1 d in Bondarenko et al 2017 ). Thus, the formation of the FB shock may also contribute to the diamagnetic current (Hall current) generated by the Larmor coupling processes between the super-thermal ions and background ions. The strong coupling electric fields created by back-streaming ion current moving across the background magnetic field results in a net Hall current, which consequently reduces the magnetic field on one side (the cavity) and enhances it on the other side (the compressional boundary). Consequently, a shock forms as fast as one gyro period of ion. 2.2 01 March 2007 event Figure 2 shows magnetic field and plasma observations of a foreshock transient structure on 01 March 2007. Upstream solar wind conditions of this event are high speed velocity (600km/s), low magnetic field strength (3.8nT), low density (3.1 \(\:{cm}^{-3}\) ) and high Mach numbers (Ma ~ 15.5 and Mm ~ 7.6) A core with low magnetic field strength and density, strongly deflected plasma flow and high temperature up to 1800eV is detected about 04:59:05 UT. The time scale of the FBs core is about 23s. Similarly, this structure is identified as FB due to lack of an obvious downstream edge. The upstream edge is encountered at 04:59:10 UT with a density compression ratio of ~ 6.96. This boundary displays a double-peak magnetic field profile, which is different from that of the previous event. The maximum magnetic compression ratio is ~ 7.54 and ~ 5.45, respectively, in the trailing peak at 04:59:10 UT and leading peak at 04:59:17 UT. The orientation and normal velocity of the two peaks are determined separately by the timing method. The trailing shock propagation velocity is 523.652 km/s, and the angle \(\:{\theta\:}_{BN}\) is 66.7 o . For the leading one they are 448.095 km/s and 74.7 o respectively. The leading and trailing shock are all quasi-perpendicular shock. The Alfven Mach and magnetosonic Mach number of the trailing shock are accordingly ~ 11.71 and ~ 5.71 and the leading ones are ~ 3.0 and ~ 1.46 respectively. The two shocks time scale are 9s and 9s ~ 0.048 \(\:{{\omega\:}}_{\text{c}\text{i}}^{-1}\) , and the transverse dimension are 4712km ~ 2.05ρ ic and 4032.4km ~ 1.75ρ ic respectively. The double shocks magnetic profile in this event is highly similar to that of MMS observations reported in Turner, et al., 2021 (see Figure A1 in this reference). In MMS observations, a new shock generates out of the foot region of the original shock of a foreshock transient structure. The high correlations of multi-spacecraft measurements during their crossings of this new shock suggest that it is unlikely random fluctuations. It is also not the ripple wave along shock surface since the presumed time lags among the MMS tetrahedron vertexes don’t match the observations. While it is explained as reformation phenomenon, that is, the enhanced total B at the original ramp reflects a significant fraction of incident solar wind ions back into the upstream region, resulting in a new shock ramp. In the present event, considering the Cluster tetrahedron size is of thousands of kilometers which is much larger than the shock ripple wavelength of ~ 100 to 200 kilometers, it excludes the possibility of shock surface wave explanation but supports a shock reformation interpretation. 2.3 30 March 2004 event The observations of 30 March 2004 foreshock bubble event are shown in Fig. 3 . Around 15:07:20UT, a low plasma density and high temperature up to 800eV core with strong plasma flow deflections and thermalized ions, is observed. The FBs core duration approximately 27s. A sharp shock is observed in the period of 15:07:28 − 15:07:33UT, and another upstream shock encountered during the period of 5:07:34 − 15:07:49UT is much gentle. The time scale of the trailing and leading shocks are 5s ~ 0.04 \(\:{{\omega\:}}_{\text{c}\text{i}}^{-1}\) and 15s ~ 0.12 \(\:{{\omega\:}}_{\text{c}\text{i}}^{-1}\) respectively. The maximum magnetic compression ratios of trailing and leading shocks are 3.51 and 1.28 respectively. The Ma and Mm of the leading shock are 4.90 and 3.82, while those of the trailing shock are much lower as 0.41 and 0.31 respectively. The leading and trailing shock of FBs propagation velocity are 102km/s and 118.4 km/s. The transverse dimension of the leading shock accordingly is 1530.0km ~ 1.55ρ ic significantly longer than that of the trailing shock of 592km ~ 0.60ρ ic . Particularly, the trailing shock is a quasi-perpendicular shock with a normal angle \(\:{\theta\:}_{BN}\) of 65.0 o , while the leading shock is a quasi-parallel one with the angle \(\:\:{\theta\:}_{BN}\) of 31.7 o . The magnetic profile of the trailing shock feature, like that of 01March 2007 event, is steep and short duration. However, the leading shock has much longer duration and flatter magnetic field strength profile. In contract to those steep shocks generated by strong diamagnetic currents, this shock more likely evolve from steepening quasi-parallel propagating kinetic magnetosonic waves that are also excited by the hot core ions interacting with ambient plasmas (Omidi.N and D. Winske, 1990). 3. Statistical analyses To conduct statistical analyses for the FB shock boundary, we pick up foreshock transient events without an obvious downstream compressional boundary or with a downstream compressional boundary whose magnetic field strength is much lower than that of the upstream boundary (the magnetic field strength ratio of the upstream to the downstream boundary is at least 3). From January 2002 to April 2007, totally 44 FB events that each was encountered by all four Cluster spacecraft are selected. 3.1 Solar Wind conditions The OMNI data with 1 min time resolution are used to analyze the upstream solar wind conditions of the FBs. The distribution of total solar wind magnetic field strength as shown in Fig. 4 A peaks at approximately 4 nT. The solar wind density mainly ranges in 2.0–4.0 \(\:{cm}^{-3}\) , and the distribution of their temperature is within 10–20 eV as shown in Fig. 4 B and 4 C respectively. The solar wind speed is roughly from 400km/s to 800km/s with a peak around 600km/s as displayed in Fig. 4 D. Furthermore, Fig. 4 E and 4 F show the distributions of the Alfvenic and magnetosonic Mach numbers respectively. Most FB events occur at Alfvenic Mach numbers between 8.0 and15.0 and accordingly at magnetosonic Mach numbers between 2.0 and 6.0. As depicted in Fig. 4 G, the distribution of plasma beta is concentrated around 1.0 and over half of FBs solar wind beta are larger than 1.0, which means the upstream solar wind are mainly warm plasmas. Figure 4 H shows that the solar wind flowing pressure is between 1 and 4 Pa. Generally, the favorable solar wind conditions for these FB events are high flow speed, high Mach numbers, low density, low magnetic field strength and high plasma beta β. These results are consistent with previous statistical studies (Wang et al 2021 , Vu et al., 2022 ). 3.2 Location of FBs The Solar Foreshock Coordinate (SFC) system initially developed by Greenstadt and Baum ( 1986 ) to map the onset of ULF waves in the foreshock, has been widely used (Meziane and d’Uston ,1998; André.N, et al, 2015) to locate the boundaries of foreshock structures and to compare foreshock transient structures (Billingham. l. et al, 2008, Kajdie. P. et al., 2013,2017). In the SFC system within a plane containing the IMF vector and the solar wind velocity, the horizontal axis Dbt represents the distance along the tangent field line from the point of tangency to a point immediately upstream of the event, and the vertical axis XF is the distance along the solar wind flow direction from that point to the event. This system effectively normalizes upstream coordinates relative to the bow shock, allowing comparison of upstream phenomena independent of variations in solar wind properties and IMF orientation. Calculating distances XF and Dbt in the SFC system follows the method by Greenstadt and Baum ( 1986 ), and the Slavin bow shock model (1984) is used here. The bow shock scale is calculated by time-shifted dynamic pressure. The location distribution of the selected FB events in the SFC system is displayed in Fig. 5 . Those FB events occur in XF lower than 30R e and in D bt lower than 40R e . The horizontal dashed green line represents a nominal tangent line that magnetically connects to the bow shock. Notably, no events were outside the tangent line since variable shock scales are used here, whereas it would improperly happen when a single shock scale is used. In previous studies, a fitting line of transient structure locations in the SFC system gives the critical boundaries in the foreshock region, for example, fittings show the ULF wave boundary by Greenstadt and Baum ( 1986 ) and the intermediate ion boundary by Meziane and d’Uston ( 1998 ). Here, we use a fitting to give the average location of those FB events in the foreshock region. The linear fitting approximation is XF = 0.54 + 0.75D bt and the coefficient is 0.88. This fitting line slope is larger than that of particular boundary fittings given in previous reports. 3.3 Statistical Characteristics of FB Structures The distributions of the FB shock and hot FB core features are shown in Fig. 6 . Most of the FBs shock durations are less than 0.5 ion cyclotron period, with very few exceeding 1.0 as shown in Fig. 6 A. The transverse scales of FBs shock are typically longer than or equivalent to one ion gyro radius, ranging between 2 and 4 ion gyro radiuses, as displayed by Fig. 6 B. The core magnetic field strengths are lower than half of the background magnetic field, and most of the core density are also lower than half of the background density, as shown in Fig. 6 C and 6 D. The strong coupling between explosive core plasma and cold ambient plasma consequently results in the formation of FB compressional boundary. Figure 7 shows the scatter plot of FB maximum magnetic field compression ratios versus their shock normal angles \(\:{\theta\:}_{BN}\) . The shocks are mainly qusi-perpendicular with angles \(\:{\theta\:}_{BN}\) between 60° and 90°. Moreover, in a full view there is roughly a tendency that FB shocks with larger compression ratios have larger normal angles. The feature can be interpreted as a result of magnetic field Gauss’s Law \(\:\nabla\:\bullet\:\overrightarrow{B}\) =0. When a FB core continuously expands outward, the magnetic field boundary is compressed more strongly and the magnetic field strength becomes larger, consequently magnetic field lines tend to become perpendicular to the boundary normal direction so that the magnetic field divergence across the boundary keeps to a minimum. Figure 8 A and 8 C show the distributions of maximum magnetic compression ratios of FB shock versus their Alfvén and magnetosonic Mach numbers respectively. The green diamonds, red triangles and blue crosses represent shocks with a single peak, and the leading and trailing shocks of the double-peak ones respectively. The FB shocks speeds are calculated in solar wind rest frame. Their Alfvén and magnetosonic Mach numbers are mainly within 1 ~ 10 and 1 ~ 6 respectively as shown in Fig. 8 A and 8 C, that is, most of the FB shocks are fast and supercritical shock. Accordingly, most of the FB shocks exhibit strong magnetic compression and their ratios can reach up to 10. As reported in the previous laboratory studies of supercritical shock as well as subcritical shock formation, the shock magnetic compression tends to be stronger with a larger magnetosonic or Alfvén Mach number (Schaeffer. D.B. et al, 2017; Schaeffer. D.B. et al, 2015). In general, the observational results presented here show similar features to these laboratory investigations, that is, the FB shock magnetic field amplitudes enhance with increasing upstream incident velocities as implied by the linear fittings in Fig. 8 . Figure 8 A shows the correlation between maximum magnetic compression ratios and Alfvén Mach numbers for all the FB shocks, its correlation coefficient is 0.41 and slope is 0.20. While the magnetic compression ratios and Alfvén Mach number for the trailing shocks separately is more relevant with a coefficient 0.74and a slope 0.30 as shown in Fig. 8 B. Similarly, there are correlations between magnetic compression ratios and magnetosonic Mach numbers. Their correlation coefficient for all the shocks and the sole trailing shocks are 0.37 and 0.70 respectively, and their corresponding slope are 0.38 and 0.52 as shown in Fig. 8 C and 8 D. 4. Summary and Discussion In the case analysis section, three FB events are reported with an emphasis on their magnetic field profiles of the boundary shock. The first one has a boundary with a sharp mono-peak magnetic field compression, which is supercritical and quasi-perpendicular to the upstream solar wind magnetic field direction. The formation of this FB shock may contribute to the diamagnetic current generated by the Larmor coupling processes between the super-thermal ions and background ions. Consequently, a shock forms as fast as one gyro period of ion. The second event however has a boundary with double-peak magnetic field strength which are also sharp and comparable. These two shocks are both supercritical and quasi-perpendicular. This double-peak magnetic profile can be interpreted as the shock reformation, that is, a new shock generates out of the foot region of the original shock. The third event also has double shocks but with distinct features in their magnetic field profiles. One shock is sharp and is the same as those mentioned above, while the other has a much larger spatial size and lower magnetic field compression ratio. In addition, their shock normal directions are distinct by quasi-perpendicular and quasi-parallel to the solar wind magnetic field respectively. Accordingly, this broad shock may be explained as a steepening parallel-propagating kinetic magnetosonic shock instead of a reformatting one. The statistical results show that FB events are more likely detected in the quasi-parallel foreshock region under solar wind conditions of high flow speed, high Mach numbers, low density, low magnetic field strength and high plasma beta β. The FB core expansion periods are shorter than one ion cyclotron period, within this time scale, the core expanding quickly and generate a steep magnetic compression boundary with transverse scales larger than one ion gyro radius. These results are consistent with conclusions given in previous simulations (An et al., 2020 ) and laboratorial experiments (Schaeffer et al., 2017 a, b), which predict that magnetic field pile-up steepens into a shock as at super magnetosonic speeds. Moreover, the majorities of the FB boundary shocks are supercritical and steep with large magnetic compression ratios and are in quasi-perpendicular direction to their upstream magnetic field. Besides the feature that there is roughly a positive correlation between the magnetic compression ratios of the FB boundary shocks and their shock normal angles, another noteworthy fact revealed in the statistical analyses is that the shock magnetic compression ratios enhance with increasing upstream incident velocities. This point can be interpreted as a manifestation of diamagnetic current generation inside the boundary. Generally we have \(\:{\text{j}}_{\text{r}}=\text{n}\text{e}\left({\text{v}}_{\text{i}}+{\text{v}}_{\text{e}}\right)=(\nabla\:\times\:\text{B}{)}_{r}\approx\:\frac{\varDelta\:{\text{B}}_{\perp\:\text{m}\text{a}\text{x}}}{\varDelta\:{\text{L}}_{\text{r}}}\) , where \(\:{\text{j}}_{\text{r}}\) , \(\:\:\text{n}\) , \(\:{\text{v}}_{\text{e}}\) and \(\:{\text{v}}_{\text{i}}\) are the radial electric current, density, electrons velocity and ions velocity respectively, and \(\:\varDelta\:{\text{B}}_{\perp\:\text{m}\text{a}\text{x}}\) and \(\:\:\varDelta\:{\text{L}}_{\text{r}}\) are the maximum magnetic field perpendicular to FB normal and expanding boundary radial scale respectively. During the interaction between hot core plasma and cold ambient plasma, the hot core expansion leads to a portion of electrons and ions to be decoupled inside the boundary, thus there are approximately \(\:\:{\varDelta\:\text{L}}_{\text{r}}\sim{{\lambda\:}}_{\text{i}}\) , \(\:{\text{v}}_{\text{i}}\sim\) 0, \(\:{\text{v}}_{\text{e}}\sim{\text{v}}_{\text{e}\text{x}\text{p}}\) , where \(\:{\text{v}}_{\text{e}\text{x}\text{p}}\) is the boundary expanding velocity, and \(\:{{\lambda\:}}_{\text{i}}\) is the ion inertial length. Then we get \(\:{\alpha\:}{\text{n}}_{0}\text{e}{\text{v}}_{\text{e}\text{x}\text{p}}=\frac{{\varDelta\:\text{B}}_{\perp\:\text{m}\text{a}\text{x}}}{\varDelta\:{\text{L}}_{\text{r}}}\) , or \(\:\frac{\nabla\:{\text{B}}_{\perp\:\text{m}\text{a}\text{x}}}{{\text{B}}_{0}}\) = \(\:{\alpha\:}{\text{n}}_{0}\text{e}{\text{v}}_{\text{e}\text{x}\text{p}}·{{\lambda\:}}_{\text{i}}\) / \(\:{\text{B}}_{0}\propto\:{\text{v}}_{\text{e}\text{x}\text{p}}\) / \(\:{\text{v}}_{\text{A}}\) , where \(\:{\alpha\:}\) is the ration of decoupled particles. Finally, we obtain \(\:\frac{\nabla\:{\text{B}}_{\perp\:\text{m}\text{a}\text{x}}}{{\text{B}}_{0}}\propto\:{\text{M}}_{\text{A}}\) , i.e. magnetic field strength changes across the boundary generated by the diamagnetic Hall current enhance with increasing upstream incident velocities. Simultaneously, the strong Larmor coupling electric fields created by energetic ions moving across the background magnetic field act to accelerate ambient ions and decelerate energetic particles and thus result in collisionless momentum and energy exchange between these two species. Declarations Competing interests The authors declare that they have no competing interests. Author details 1 State Key Laboratory of Solar Activity and Space Weather, National Space Science Center, Chinese Academy of Sciences,Beijing 100190 2 School of Space and Earth Sciences, Beihang University, Beijing, 100191, People’s Republic of China; 3 Key Laboratory of Space Environment Monitoring and Information Processing, Ministry of Industry and Information Technology, Beijing, People’s Republic of China 4 Department of Geophysics, School of Earth Sciences, Yunnan University, Kunming, 650091, China Funding This research is partly by National Natural Science Foundation of China (NSFC) grants: 42374202, 42350710793, 42188101, 42174207), the Specialized Research Fund for State Key Laboratories of China, and the Strategic Pioneer Program on Space Science II, Chinese Academy of Sciences, Grants XDA15350201, XDA15052500. Author Contributions: Data and dynamics analysis, review & editing, Xinhua Wei and ChunLin. Cai; Discussion and review & editing, JunYing Yang, YuDuan Ma, TieYan Wang, YangYang Liu and Lei Dai. All authors have read and agreed to the published version of the manuscript. Acknowledgments We sincerely thank anonymous reviewers and the editors of EPS for their valuable and constructive comments for improving this manuscript. Data Availability Statement: The Custer data are available from Cluster Science Archive. The OMNI data were obtained from GSFC/ SPDF OMNIWeb ( http://omniweb.gsfc.nasa.gov ). References An X, Liu TZ, Bortnik J, Osmane A, Angelopoulos V (2020) Formation of foreshock transients and associated secondary shocks. Acta Pathol Japonica 901:73. https://doi.org/10.3847/1538-4357/abaf03 Andrés N, Meziane K, Mazelle C, Bertucci C, Gómez D (2015) The ULF wave foreshock boundary: Cluster observations. 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J Phys Res 115:A12235. https://doi.org/10.1029/2009JA015180 Zhang H, Sibeck D, Zong QG, Omidi N, Turner D, Clausen LN (2013) Spontaneous hot flow anomalies at quasi-parallel shocks:1. Observations. J Geophys Research: Space Phys 118(6):3357–3363. https://doi.org/10.1002/jgra.50376 Supplementary Files GA.png Cite Share Download PDF Status: Published Journal Publication published 11 Apr, 2026 Read the published version in Earth, Planets and Space → Version 1 posted Reviewers agreed at journal 26 Oct, 2025 Reviewers invited by journal 06 Oct, 2025 Editor assigned by journal 06 Oct, 2025 First submitted to journal 28 Sep, 2025 Editorial decision: Major Revision 27 Sep, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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From top to bottom the panels are ion energy spectrogram, total magnetic field strength, ion density, velocities and temperature respectively.\u003c/p\u003e","description":"","filename":"FBfig1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/8e810fc844e5ea9dc2263e91.jpg"},{"id":93806588,"identity":"3906b230-134e-4e9e-8f18-e82944f29fa4","added_by":"auto","created_at":"2025-10-17 18:19:18","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":496453,"visible":true,"origin":"","legend":"\u003cp\u003eObservations of 01 March 2007 event. The figure is in the same format as Figure 1.\u003c/p\u003e","description":"","filename":"FBfig2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/a2111dae54fe4fd7ebffd8e9.jpg"},{"id":93805814,"identity":"f65fa307-4b57-4625-a0a5-6a2603cb7a9c","added_by":"auto","created_at":"2025-10-17 18:03:18","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":505697,"visible":true,"origin":"","legend":"\u003cp\u003eObservations of 30 March 2004 event. The figure is in the same format as Figure 1.\u003c/p\u003e","description":"","filename":"FBfig3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/231f25afb1a9abfaae2c30ee.jpg"},{"id":93806419,"identity":"d62ae363-1e92-4be3-9822-0108c4130620","added_by":"auto","created_at":"2025-10-17 18:11:18","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":414183,"visible":true,"origin":"","legend":"\u003cp\u003eDistributions of FBs occurrence as a function of background Solar wind conditions normalized to average occurrence in January 2002 and April 2007. (A) Solar wind total Magnetic field; (B) Density; (C) Temperature, (D) Solar wind velocity, (E-F) Alfvenic Mach number and magnetosonic Mach number respectively, (G) Solar wind pressure, (F) Beta.\u003c/p\u003e","description":"","filename":"FBfig4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/bce4e1c0b180a161db0c9392.jpg"},{"id":93805812,"identity":"3ef08278-7ff6-470a-9071-1fa652b68226","added_by":"auto","created_at":"2025-10-17 18:03:18","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":282276,"visible":true,"origin":"","legend":"\u003cp\u003eFB locations in the SFC system. The horizontal dash green line represents a nominal tangent line. The diagonal black line is a fit to FB events.\u003c/p\u003e","description":"","filename":"FBfig5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/2378e36510899021a6c450e6.jpg"},{"id":93806421,"identity":"f4a789dd-3aa4-4880-95ba-01b37e08c5d1","added_by":"auto","created_at":"2025-10-17 18:11:18","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":155493,"visible":true,"origin":"","legend":"\u003cp\u003eDistributions of the FB shock and core features. (A) and (B) are FB shock duration and scale size respectively. (C) and (D) magnetic field and density depletions in the FB cores.\u003c/p\u003e","description":"","filename":"FBfig6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/5b92803022b2a997c4c77b69.jpg"},{"id":93805818,"identity":"ab4a1503-87f8-41ef-bfe7-cd833b6e30dd","added_by":"auto","created_at":"2025-10-17 18:03:18","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":315382,"visible":true,"origin":"","legend":"\u003cp\u003eScattering plots of shock Alfvén and magnetosonic Mach numbers versus their magnetic compression ratios respectively, where the green diamonds, red triangles and blue crosses represent shocks with a single peak, and the leading and trailing shocks of the double-peak ones respectively.\u003c/p\u003e","description":"","filename":"FBfig7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/13caa045c0c030d138bc38de.jpg"},{"id":93806423,"identity":"d86aeef0-9407-441a-8142-7f3ebc9d2dea","added_by":"auto","created_at":"2025-10-17 18:11:18","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":182911,"visible":true,"origin":"","legend":"\u003cp\u003eScattering plot of the shock magnetic field compression ratios versus their normal angles.\u003c/p\u003e","description":"","filename":"FBfig8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/ffd67dde09ef829c3cd59d35.jpg"},{"id":106810876,"identity":"b0005a73-13e0-466c-b9fb-2ae159524cc1","added_by":"auto","created_at":"2026-04-13 16:17:13","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3454568,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/6404ae58-6a77-48b2-8050-4d54276d755c.pdf"},{"id":93805823,"identity":"39a41421-cdb2-4da3-9eeb-59c60d89a811","added_by":"auto","created_at":"2025-10-17 18:03:18","extension":"png","order_by":5,"title":"","display":"","copyAsset":false,"role":"supplement","size":1047914,"visible":true,"origin":"","legend":"","description":"","filename":"GA.png","url":"https://assets-eu.researchsquare.com/files/rs-7657123/v1/c0dc145b003d11c8bcac17b3.png"}],"financialInterests":"","formattedTitle":"Cluster Observations of Foreshock Bubbles and Their Boundary Shocks","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eThe Earth\u0026rsquo;s foreshock region is full of back-streaming super-thermal particles reflected from the bow shock (Eastwood et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2005\u003c/span\u003e). The interaction between the super-thermal particles and the solar wind can generate many transient events including foreshock bubble (FB) (Omidi et al., \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Turner et al., \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2013\u003c/span\u003e, Guo et al.,2022, 2023), hot flow anomaly (HFA) (Sibeck et al., \u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Schwartz et al., \u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Lin, \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Lucek et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Eastwood et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Jacobsen et al., \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Zhang et al., \u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e2010\u003c/span\u003e), spontaneous HFA (SHFA) (Zhang et al., \u003cspan citationid=\"CR71\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Omidi et al., \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Omidi et al., 2016), foreshock caviton (Omidi, 2007; Blanco-Cano et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Blanco‐Cano et al., 2011; Kajdič et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Kajdič et al., \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), foreshock cavity (Sibeck et al., \u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Schwartz et al., \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2006\u003c/span\u003e; Billingham et al., \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). All these foreshock transients have a core region surrounded by compressional boundaries or shocks, and both the density and the magnetic field are deeply depressed in the core region. For example, foreshock bubbles are formed by IMF discontinuities interacting with back-streaming ion beams in the foreshock region, whereas HFAs are generated after the interplanetary discontinuities hitting the bow shock. When the back-streaming ion beams pass through the discontinuity, they are deflected due to the rotation of the magnetic field vectors. The deflected beam interacts with the solar wind, resulting in the deceleration of the solar wind and the formation of fast magnetosonic shock wave that expands sunward with time and is followed by the strong sheath plasma and a core region with low magnetic field strength and hot and tenuous plasmas with energetic particles. The whole FB structure convects along with the solar wind in the anti-sunward direction (Omidi et al., \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Turner et al., \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). FBs are supposed to generate on the upstream sides of interplanetary discontinuities and thus have compressional boundaries on their leading edge, while HFAs have compressional boundaries on both the leading and trailing edges. The FB sizes range from 2 to 15\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{R}_{e}\\)\u003c/span\u003e\u003c/span\u003e (Liu et al \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2016\u003c/span\u003e, Turner et al, \u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) and with expansion velocity 139\u0026thinsp;~\u0026thinsp;470 km/s faster than HFA and SHFA (Turner et al., \u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e2020\u003c/span\u003e, Liu et al., \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). FBs have been observed not only in the dayside foreshock but also in the Mid-tail foreshock (Liu.et al, 2021).\u003c/p\u003e\u003cp\u003eForeshock transients not only have significant magnetospheric impacts due to pressure variants in their core region, but also can be effective accelerators of foreshock energetic particles and provide seed populations for further particle acceleration at the parent bow shock (e.g., Cai \u0026amp; Wei \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2020\u003c/span\u003e, Wei et al,2022, Omidi et al., \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; T. Z. Liu, Angelopoulos, et al., \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Wilson et al., 2016; T. Z. Liu, Hietala, et al., \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; T. Z. Liu et al., \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2019\u003c/span\u003e, T. Z. Liu, An, et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; T. Z. Liu, Lu, et al., \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2017\u003c/span\u003e, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2018\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Turner et al, \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Omidi et al., \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). MMS observed that HFA can accelerate ions up to hundreds KeV (Turner et al, \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Electrons in their core can be accelerated to hundreds KeV by all transients (T. Z. Liu, Angelopoulos, et al., \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Wilson et al., 2016). Besides those acceleration mechanisms such as reconnection which have been suggested to occur inside the structures, Fermi-type mechanisms are main acceleration processes where (Omidi et al., \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Turner et al., \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2018\u003c/span\u003e, Liu, Lu et al., \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) their sharp boundaries play a crucial role. For instance, those earthward high-speed ions in a FB are reflected and accelerated at the earthward moving FBs boundary through partial gyration along the convection electric filed, which means the boundary can act as a magnetic mirror to accelerate ions repeatedly bouncing between the converging compression boundaries and the core.\u003c/p\u003e\u003cp\u003eRecent simulations and experiments have focused on the formation of compression boundaries in hot plasma expansions in the MHD or kinetic views. A laboratory investigation shows the collisionless shock can form experimentally within one gyro-period of ion (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\omega\\:}}_{\\text{c}\\text{i}}^{-1}\\)\u003c/span\u003e\u003c/span\u003e) (Bondarenko.A.S., et al,2017a, 2017b, Hewett, D.W, et al,2011, Clark, S.E, et al, 2013). During one gyro period of ion, the intensification of the magnetic compression due to the accumulation of accelerated ambient ions results in significant expulsion of back-streaming ions from the leading edge of the expansion. The pileup of trapped back-streaming ions leads to density gradients over scales of order the compressed gyro-radius, that is, its formation contributes to Larmor coupling effect. The strong Larmor coupling electric fields created by back-streaming ion current moving across the background magnetic field act to accelerate ambient ions and decelerate back-streaming particles and thus result in collisionless momentum and energy exchange between the two species. Simultaneously, the hot core expansion drives a diamagnetic current that generates a leading magnetic compression and trailing magnetic cavity. Numerical simulation results also show that in the foreshock secondary shock formation the ambient ions gain momentum in the core expanding direction via the induction electric field due to the energetic hot ions moving out of hot layer across the magnetic field in their gyrations. This energy exchange between the fields and the hot ions ceases till the compressional boundary finally detaches from the hot populations. Meanwhile, the compressional boundary moves at supermagnetosonic speed and steepens into a shock with scale of ion cyclotron radius (An et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eIn this paper, we present Cluster observations of FB events from January 2002 to April 2007. First, three case events are reported with an emphasis on their magnetic field profiles of the boundary shock. Next, statistical characteristic of selected FB events that each was encountered by all four Cluster spacecraft are presented. Finally, we give a brief summary and discussion of this study.\u003c/p\u003e"},{"header":"2. Observations: case examples","content":"\u003cp\u003eFor this study, we use data from the Cluster mission (Escoubet et al., \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). The Cluster magnetic field and ion measurements are from the FGM experiment (Balogh et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2001\u003c/span\u003e) and the CIS/HIA instrument (R\u0026egrave;me et al., \u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e2001\u003c/span\u003e) respectively. The pristine solar wind data are from the ACE mission (Stone et al., \u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e1998\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eThe Cluster tetrahedron enables the accurate estimation of the boundary normal and propagation speed of a FB by the Timing technique: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{r}}_{i4}\\bullet\\:\\)\u003c/span\u003e\u003c/span\u003e\u003cb\u003en\u003c/b\u003e\u0026thinsp;=\u0026thinsp;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{v}}_{n}\\)\u003c/span\u003e\u003c/span\u003eΔ\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{t}}_{i4}\\)\u003c/span\u003e\u003c/span\u003e, i\u0026thinsp;=\u0026thinsp;1, 2, 3, where \u003cb\u003en\u003c/b\u003e is the normal vector, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{v}}_{n}\\)\u003c/span\u003e\u003c/span\u003e is the normal speed, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{r}}_{i4}\\)\u003c/span\u003e\u003c/span\u003e and Δ\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{t}}_{i4}\\)\u003c/span\u003e\u003c/span\u003e are the separation vector and the time difference between spacecraft i\u0026thinsp;=\u0026thinsp;1, 2, 3 and the reference Spacecraft 4, respectively (Schwartz, \u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e1998\u003c/span\u003e).\u003c/p\u003e\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e2.1 26 March 2005 event\u003c/h2\u003e\u003cp\u003eOn 26 March 2005, the Cluster spacecraft tetrahedron located in the foreshock region encountered a transient structure. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the magnetic field and plasma observations of this structure. The ambient solar wind velocity is about 680 km/s, the magnetic field strength is 4.3nT and the density is 2.4\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{cm}^{-3}\\)\u003c/span\u003e\u003c/span\u003e. The solar wind plasma \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}\\:\\)\u003c/span\u003e\u003c/span\u003eis1.75 and the flow pressure is 2.25 nPa. The Alfven Mach number and magnetosonic Mach number are 13.35 and 7.75 respectively. These parameters are characteristic of fast solar wind streams in solar wind\u0026ndash;magnetosphere interactions (Dai et al.2023,2024). As shown the ions energy spectrum in panel A) in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, in the core center detected around 03:57 UT, ion energy flux with energy of KeVs are enhanced, and the thermalized ions of a temperature are up to 1000 eV. The density decreases to about 1.0\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{cm}^{-3}\\)\u003c/span\u003e\u003c/span\u003eas shown in panel C), and the solar wind is strongly deflected with the X-component of the velocity dropping from ~-700km/s to ~-400km/s and the Y and Z components both increasing to 200km/s as shown in panel D). As shown in penal B), the magnetic field strength in the core region is lower than 1nT, but reaches 22 nT within the strong compression boundary. Since only obvious upstream edge but no downstream edge is detected, this transient structure is basically identified as a FB.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe compressional boundary of the FB has a sharp mono-peak magnetic field profile. Its maximum magnetic compression ratio \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{B}_{\\perp\\:max}/{B}_{0}\\)\u003c/span\u003e\u003c/span\u003e and maximum density compression ratio \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{N}}_{\\text{m}\\text{a}\\text{x}}/{\\text{N}}_{0}\\)\u003c/span\u003e\u003c/span\u003e are 3.48 and 3.78 respectively. Its propagating velocity calculated by the timing method is 286km/s in the solar wind rest frame. This boundary is a perpendicular shock with the angle between its normal and the upstream magnetic field vector (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\theta\\:}_{BN}\\)\u003c/span\u003e\u003c/span\u003e) of ~\u0026thinsp;82.8\u003csup\u003eo\u003c/sup\u003e. The shock Alfven Mach number and magnetosonic Mach number are 5.16 and 3.24 respectively. The shock duration is about 16s, ~\u0026thinsp;0.14 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\omega\\:}}_{\\text{c}\\text{i}}^{-1}\\)\u003c/span\u003e\u003c/span\u003e, where\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\:{\\omega\\:}}_{\\text{c}\\text{i}}^{-1}\\)\u003c/span\u003e\u003c/span\u003e is ion gyration period using the minimum magnetic field in the FB core to calculate. The shock transverse dimension is about 4575km\u0026thinsp;~\u0026thinsp;4. 34 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\rho\\:}}_{\\text{i}\\text{c}}\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\rho\\:}}_{\\text{i}\\text{c}}\\)\u003c/span\u003e\u003c/span\u003e is gyration radius also using the minimum magnetic field and max temperature in the FB core.\u003c/p\u003e\u003cp\u003eThe FB formation resembles the scenarios described in previous simulations and laboratory experiments of collisionless coupling between explosive hot plasma and magnetized ambient plasma, and the shock boundary also has a similar magnetic profile (see Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ed in Bondarenko et al \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Thus, the formation of the FB shock may also contribute to the diamagnetic current (Hall current) generated by the Larmor coupling processes between the super-thermal ions and background ions. The strong coupling electric fields created by back-streaming ion current moving across the background magnetic field results in a net Hall current, which consequently reduces the magnetic field on one side (the cavity) and enhances it on the other side (the compressional boundary). Consequently, a shock forms as fast as one gyro period of ion.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e2.2 01 March 2007 event\u003c/h2\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows magnetic field and plasma observations of a foreshock transient structure on 01 March 2007. Upstream solar wind conditions of this event are high speed velocity (600km/s), low magnetic field strength (3.8nT), low density (3.1\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{cm}^{-3}\\)\u003c/span\u003e\u003c/span\u003e) and high Mach numbers (Ma\u0026thinsp;~\u0026thinsp;15.5 and Mm\u0026thinsp;~\u0026thinsp;7.6) A core with low magnetic field strength and density, strongly deflected plasma flow and high temperature up to 1800eV is detected about 04:59:05 UT. The time scale of the FBs core is about 23s. Similarly, this structure is identified as FB due to lack of an obvious downstream edge.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe upstream edge is encountered at 04:59:10 UT with a density compression ratio of ~\u0026thinsp;6.96. This boundary displays a double-peak magnetic field profile, which is different from that of the previous event. The maximum magnetic compression ratio is ~\u0026thinsp;7.54 and ~\u0026thinsp;5.45, respectively, in the trailing peak at 04:59:10 UT and leading peak at 04:59:17 UT. The orientation and normal velocity of the two peaks are determined separately by the timing method. The trailing shock propagation velocity is 523.652 km/s, and the angle \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\theta\\:}_{BN}\\)\u003c/span\u003e\u003c/span\u003e is 66.7\u003csup\u003eo\u003c/sup\u003e. For the leading one they are 448.095 km/s and 74.7\u003csup\u003eo\u003c/sup\u003e respectively. The leading and trailing shock are all quasi-perpendicular shock. The Alfven Mach and magnetosonic Mach number of the trailing shock are accordingly\u0026thinsp;~\u0026thinsp;11.71 and ~\u0026thinsp;5.71 and the leading ones are ~\u0026thinsp;3.0 and ~\u0026thinsp;1.46 respectively. The two shocks time scale are 9s and 9s\u0026thinsp;~\u0026thinsp;0.048\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\omega\\:}}_{\\text{c}\\text{i}}^{-1}\\)\u003c/span\u003e\u003c/span\u003e, and the transverse dimension are 4712km\u0026thinsp;~\u0026thinsp;2.05ρ\u003csub\u003eic\u003c/sub\u003e and 4032.4km\u0026thinsp;~\u0026thinsp;1.75ρ\u003csub\u003eic\u003c/sub\u003e respectively.\u003c/p\u003e\u003cp\u003eThe double shocks magnetic profile in this event is highly similar to that of MMS observations reported in Turner, et al., \u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e2021\u003c/span\u003e(see Figure A1 in this reference). In MMS observations, a new shock generates out of the foot region of the original shock of a foreshock transient structure. The high correlations of multi-spacecraft measurements during their crossings of this new shock suggest that it is unlikely random fluctuations. It is also not the ripple wave along shock surface since the presumed time lags among the MMS tetrahedron vertexes don\u0026rsquo;t match the observations. While it is explained as reformation phenomenon, that is, the enhanced total B at the original ramp reflects a significant fraction of incident solar wind ions back into the upstream region, resulting in a new shock ramp. In the present event, considering the Cluster tetrahedron size is of thousands of kilometers which is much larger than the shock ripple wavelength of ~\u0026thinsp;100 to 200 kilometers, it excludes the possibility of shock surface wave explanation but supports a shock reformation interpretation.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003e2.3 30 March 2004 event\u003c/h2\u003e\u003cp\u003eThe observations of 30 March 2004 foreshock bubble event are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Around 15:07:20UT, a low plasma density and high temperature up to 800eV core with strong plasma flow deflections and thermalized ions, is observed. The FBs core duration approximately 27s.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eA sharp shock is observed in the period of 15:07:28\u0026thinsp;\u0026minus;\u0026thinsp;15:07:33UT, and another upstream shock encountered during the period of 5:07:34\u0026thinsp;\u0026minus;\u0026thinsp;15:07:49UT is much gentle. The time scale of the trailing and leading shocks are 5s\u0026thinsp;~\u0026thinsp;0.04\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\omega\\:}}_{\\text{c}\\text{i}}^{-1}\\)\u003c/span\u003e\u003c/span\u003e and 15s\u0026thinsp;~\u0026thinsp;0.12\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\omega\\:}}_{\\text{c}\\text{i}}^{-1}\\)\u003c/span\u003e\u003c/span\u003e respectively. The maximum magnetic compression ratios of trailing and leading shocks are 3.51 and 1.28 respectively. The Ma and Mm of the leading shock are 4.90 and 3.82, while those of the trailing shock are much lower as 0.41 and 0.31 respectively. The leading and trailing shock of FBs propagation velocity are 102km/s and 118.4 km/s. The transverse dimension of the leading shock accordingly is 1530.0km\u0026thinsp;~\u0026thinsp;1.55ρ\u003csub\u003eic\u003c/sub\u003e significantly longer than that of the trailing shock of 592km\u0026thinsp;~\u0026thinsp;0.60ρ\u003csub\u003eic\u003c/sub\u003e. Particularly, the trailing shock is a quasi-perpendicular shock with a normal angle \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\theta\\:}_{BN}\\)\u003c/span\u003e\u003c/span\u003e of 65.0\u003csup\u003eo\u003c/sup\u003e, while the leading shock is a quasi-parallel one with the angle\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:{\\theta\\:}_{BN}\\)\u003c/span\u003e\u003c/span\u003e of 31.7\u003csup\u003eo\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eThe magnetic profile of the trailing shock feature, like that of 01March 2007 event, is steep and short duration. However, the leading shock has much longer duration and flatter magnetic field strength profile. In contract to those steep shocks generated by strong diamagnetic currents, this shock more likely evolve from steepening quasi-parallel propagating kinetic magnetosonic waves that are also excited by the hot core ions interacting with ambient plasmas (Omidi.N and D. Winske, 1990).\u003c/p\u003e\u003c/div\u003e"},{"header":"3. Statistical analyses","content":"\u003cp\u003eTo conduct statistical analyses for the FB shock boundary, we pick up foreshock transient events without an obvious downstream compressional boundary or with a downstream compressional boundary whose magnetic field strength is much lower than that of the upstream boundary (the magnetic field strength ratio of the upstream to the downstream boundary is at least 3). From January 2002 to April 2007, totally 44 FB events that each was encountered by all four Cluster spacecraft are selected.\u003c/p\u003e\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003e3.1 Solar Wind conditions\u003c/h2\u003e\u003cp\u003eThe OMNI data with 1 min time resolution are used to analyze the upstream solar wind conditions of the FBs. The distribution of total solar wind magnetic field strength as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eA peaks at approximately 4 nT. The solar wind density mainly ranges in 2.0\u0026ndash;4.0\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{cm}^{-3}\\)\u003c/span\u003e\u003c/span\u003e, and the distribution of their temperature is within 10\u0026ndash;20 eV as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eB and \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eC respectively. The solar wind speed is roughly from 400km/s to 800km/s with a peak around 600km/s as displayed in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eD. Furthermore, Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eE and \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eF show the distributions of the Alfvenic and magnetosonic Mach numbers respectively. Most FB events occur at Alfvenic Mach numbers between 8.0 and15.0 and accordingly at magnetosonic Mach numbers between 2.0 and 6.0. As depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eG, the distribution of plasma beta is concentrated around 1.0 and over half of FBs solar wind beta are larger than 1.0, which means the upstream solar wind are mainly warm plasmas. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eH shows that the solar wind flowing pressure is between 1 and 4 Pa.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eGenerally, the favorable solar wind conditions for these FB events are high flow speed, high Mach numbers, low density, low magnetic field strength and high plasma beta β. These results are consistent with previous statistical studies (Wang et al \u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e2021\u003c/span\u003e, Vu et al., \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e3.2 Location of FBs\u003c/h2\u003e\u003cp\u003eThe Solar Foreshock Coordinate (SFC) system initially developed by Greenstadt and Baum (\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e1986\u003c/span\u003e) to map the onset of ULF waves in the foreshock, has been widely used (Meziane and d\u0026rsquo;Uston ,1998; Andr\u0026eacute;.N, et al, 2015) to locate the boundaries of foreshock structures and to compare foreshock transient structures (Billingham. l. et al, 2008, Kajdie. P. et al., 2013,2017). In the SFC system within a plane containing the IMF vector and the solar wind velocity, the horizontal axis Dbt represents the distance along the tangent field line from the point of tangency to a point immediately upstream of the event, and the vertical axis XF is the distance along the solar wind flow direction from that point to the event. This system effectively normalizes upstream coordinates relative to the bow shock, allowing comparison of upstream phenomena independent of variations in solar wind properties and IMF orientation.\u003c/p\u003e\u003cp\u003eCalculating distances XF and Dbt in the SFC system follows the method by Greenstadt and Baum (\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e1986\u003c/span\u003e), and the Slavin bow shock model (1984) is used here. The bow shock scale is calculated by time-shifted dynamic pressure.\u003c/p\u003e\u003cp\u003eThe location distribution of the selected FB events in the SFC system is displayed in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. Those FB events occur in XF lower than 30R\u003csub\u003ee\u003c/sub\u003e and in D\u003csub\u003ebt\u003c/sub\u003e lower than 40R\u003csub\u003ee\u003c/sub\u003e. The horizontal dashed green line represents a nominal tangent line that magnetically connects to the bow shock. Notably, no events were outside the tangent line since variable shock scales are used here, whereas it would improperly happen when a single shock scale is used. In previous studies, a fitting line of transient structure locations in the SFC system gives the critical boundaries in the foreshock region, for example, fittings show the ULF wave boundary by Greenstadt and Baum (\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e1986\u003c/span\u003e) and the intermediate ion boundary by Meziane and d\u0026rsquo;Uston (\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e1998\u003c/span\u003e). Here, we use a fitting to give the average location of those FB events in the foreshock region. The linear fitting approximation is XF\u0026thinsp;=\u0026thinsp;0.54\u0026thinsp;+\u0026thinsp;0.75D\u003csub\u003ebt\u003c/sub\u003e and the coefficient is 0.88. This fitting line slope is larger than that of particular boundary fittings given in previous reports.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\u003ch2\u003e3.3 Statistical Characteristics of FB Structures\u003c/h2\u003e\u003cp\u003eThe distributions of the FB shock and hot FB core features are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. Most of the FBs shock durations are less than 0.5 ion cyclotron period, with very few exceeding 1.0 as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eA. The transverse scales of FBs shock are typically longer than or equivalent to one ion gyro radius, ranging between 2 and 4 ion gyro radiuses, as displayed by Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eB. The core magnetic field strengths are lower than half of the background magnetic field, and most of the core density are also lower than half of the background density, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eC and \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eD.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe strong coupling between explosive core plasma and cold ambient plasma consequently results in the formation of FB compressional boundary. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e shows the scatter plot of FB maximum magnetic field compression ratios versus their shock normal angles \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\theta\\:}_{BN}\\)\u003c/span\u003e\u003c/span\u003e. The shocks are mainly qusi-perpendicular with angles \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\theta\\:}_{BN}\\)\u003c/span\u003e\u003c/span\u003e between 60\u0026deg; and 90\u0026deg;. Moreover, in a full view there is roughly a tendency that FB shocks with larger compression ratios have larger normal angles. The feature can be interpreted as a result of magnetic field Gauss\u0026rsquo;s Law \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\nabla\\:\\bullet\\:\\overrightarrow{B}\\)\u003c/span\u003e\u003c/span\u003e=0. When a FB core continuously expands outward, the magnetic field boundary is compressed more strongly and the magnetic field strength becomes larger, consequently magnetic field lines tend to become perpendicular to the boundary normal direction so that the magnetic field divergence across the boundary keeps to a minimum.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eA and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eC show the distributions of maximum magnetic compression ratios of FB shock versus their Alfv\u0026eacute;n and magnetosonic Mach numbers respectively. The green diamonds, red triangles and blue crosses represent shocks with a single peak, and the leading and trailing shocks of the double-peak ones respectively. The FB shocks speeds are calculated in solar wind rest frame. Their Alfv\u0026eacute;n and magnetosonic Mach numbers are mainly within 1\u0026thinsp;~\u0026thinsp;10 and 1\u0026thinsp;~\u0026thinsp;6 respectively as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eA and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eC, that is, most of the FB shocks are fast and supercritical shock. Accordingly, most of the FB shocks exhibit strong magnetic compression and their ratios can reach up to 10.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAs reported in the previous laboratory studies of supercritical shock as well as subcritical shock formation, the shock magnetic compression tends to be stronger with a larger magnetosonic or Alfv\u0026eacute;n Mach number (Schaeffer. D.B. et al, 2017; Schaeffer. D.B. et al, 2015). In general, the observational results presented here show similar features to these laboratory investigations, that is, the FB shock magnetic field amplitudes enhance with increasing upstream incident velocities as implied by the linear fittings in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eA shows the correlation between maximum magnetic compression ratios and Alfv\u0026eacute;n Mach numbers for all the FB shocks, its correlation coefficient is 0.41 and slope is 0.20. While the magnetic compression ratios and Alfv\u0026eacute;n Mach number for the trailing shocks separately is more relevant with a coefficient 0.74and a slope 0.30 as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eB. Similarly, there are correlations between magnetic compression ratios and magnetosonic Mach numbers. Their correlation coefficient for all the shocks and the sole trailing shocks are 0.37 and 0.70 respectively, and their corresponding slope are 0.38 and 0.52 as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eC and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003eD.\u003c/p\u003e\u003c/div\u003e"},{"header":"4. Summary and Discussion","content":"\u003cp\u003eIn the case analysis section, three FB events are reported with an emphasis on their magnetic field profiles of the boundary shock. The first one has a boundary with a sharp mono-peak magnetic field compression, which is supercritical and quasi-perpendicular to the upstream solar wind magnetic field direction. The formation of this FB shock may contribute to the diamagnetic current generated by the Larmor coupling processes between the super-thermal ions and background ions. Consequently, a shock forms as fast as one gyro period of ion. The second event however has a boundary with double-peak magnetic field strength which are also sharp and comparable. These two shocks are both supercritical and quasi-perpendicular. This double-peak magnetic profile can be interpreted as the shock reformation, that is, a new shock generates out of the foot region of the original shock. The third event also has double shocks but with distinct features in their magnetic field profiles. One shock is sharp and is the same as those mentioned above, while the other has a much larger spatial size and lower magnetic field compression ratio. In addition, their shock normal directions are distinct by quasi-perpendicular and quasi-parallel to the solar wind magnetic field respectively. Accordingly, this broad shock may be explained as a steepening parallel-propagating kinetic magnetosonic shock instead of a reformatting one.\u003c/p\u003e\u003cp\u003eThe statistical results show that FB events are more likely detected in the quasi-parallel foreshock region under solar wind conditions of high flow speed, high Mach numbers, low density, low magnetic field strength and high plasma beta β. The FB core expansion periods are shorter than one ion cyclotron period, within this time scale, the core expanding quickly and generate a steep magnetic compression boundary with transverse scales larger than one ion gyro radius. These results are consistent with conclusions given in previous simulations (An et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) and laboratorial experiments (Schaeffer et al., \u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e2017\u003c/span\u003ea, b), which predict that magnetic field pile-up steepens into a shock as at super magnetosonic speeds. Moreover, the majorities of the FB boundary shocks are supercritical and steep with large magnetic compression ratios and are in quasi-perpendicular direction to their upstream magnetic field.\u003c/p\u003e\u003cp\u003eBesides the feature that there is roughly a positive correlation between the magnetic compression ratios of the FB boundary shocks and their shock normal angles, another noteworthy fact revealed in the statistical analyses is that the shock magnetic compression ratios enhance with increasing upstream incident velocities. This point can be interpreted as a manifestation of diamagnetic current generation inside the boundary. Generally we have \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{j}}_{\\text{r}}=\\text{n}\\text{e}\\left({\\text{v}}_{\\text{i}}+{\\text{v}}_{\\text{e}}\\right)=(\\nabla\\:\\times\\:\\text{B}{)}_{r}\\approx\\:\\frac{\\varDelta\\:{\\text{B}}_{\\perp\\:\\text{m}\\text{a}\\text{x}}}{\\varDelta\\:{\\text{L}}_{\\text{r}}}\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{j}}_{\\text{r}}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\text{n}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{v}}_{\\text{e}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{v}}_{\\text{i}}\\)\u003c/span\u003e\u003c/span\u003e are the radial electric current, density, electrons velocity and ions velocity respectively, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:{\\text{B}}_{\\perp\\:\\text{m}\\text{a}\\text{x}}\\)\u003c/span\u003e\u003c/span\u003eand\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\varDelta\\:{\\text{L}}_{\\text{r}}\\)\u003c/span\u003e\u003c/span\u003e are the maximum magnetic field perpendicular to FB normal and expanding boundary radial scale respectively. During the interaction between hot core plasma and cold ambient plasma, the hot core expansion leads to a portion of electrons and ions to be decoupled inside the boundary, thus there are approximately\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:{\\varDelta\\:\\text{L}}_{\\text{r}}\\sim{{\\lambda\\:}}_{\\text{i}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{v}}_{\\text{i}}\\sim\\)\u003c/span\u003e\u003c/span\u003e0, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{v}}_{\\text{e}}\\sim{\\text{v}}_{\\text{e}\\text{x}\\text{p}}\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{v}}_{\\text{e}\\text{x}\\text{p}}\\)\u003c/span\u003e\u003c/span\u003e is the boundary expanding velocity, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\lambda\\:}}_{\\text{i}}\\)\u003c/span\u003e\u003c/span\u003e is the ion inertial length. Then we get \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}{\\text{n}}_{0}\\text{e}{\\text{v}}_{\\text{e}\\text{x}\\text{p}}=\\frac{{\\varDelta\\:\\text{B}}_{\\perp\\:\\text{m}\\text{a}\\text{x}}}{\\varDelta\\:{\\text{L}}_{\\text{r}}}\\)\u003c/span\u003e\u003c/span\u003e, or \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\nabla\\:{\\text{B}}_{\\perp\\:\\text{m}\\text{a}\\text{x}}}{{\\text{B}}_{0}}\\)\u003c/span\u003e\u003c/span\u003e=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}{\\text{n}}_{0}\\text{e}{\\text{v}}_{\\text{e}\\text{x}\\text{p}}\u0026middot;{{\\lambda\\:}}_{\\text{i}}\\)\u003c/span\u003e\u003c/span\u003e/\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{B}}_{0}\\propto\\:{\\text{v}}_{\\text{e}\\text{x}\\text{p}}\\)\u003c/span\u003e\u003c/span\u003e/\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{v}}_{\\text{A}}\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}\\)\u003c/span\u003e\u003c/span\u003e is the ration of decoupled particles. Finally, we obtain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\nabla\\:{\\text{B}}_{\\perp\\:\\text{m}\\text{a}\\text{x}}}{{\\text{B}}_{0}}\\propto\\:{\\text{M}}_{\\text{A}}\\)\u003c/span\u003e\u003c/span\u003e, i.e. magnetic field strength changes across the boundary generated by the diamagnetic Hall current enhance with increasing upstream incident velocities. Simultaneously, the strong Larmor coupling electric fields created by energetic ions moving across the background magnetic field act to accelerate ambient ions and decelerate energetic particles and thus result in collisionless momentum and energy exchange between these two species.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eCompeting interests\u003c/h2\u003e\u003cp\u003eThe authors declare that they have no competing interests.\u003c/p\u003e\u003ch2\u003eAuthor details\u003c/h2\u003e\u003cp\u003e\u003csup\u003e1\u003c/sup\u003e State Key Laboratory of Solar Activity and Space Weather, National Space Science Center, Chinese Academy of Sciences,Beijing 100190\u003c/p\u003e\u003cp\u003e\u003csup\u003e2\u003c/sup\u003e School of Space and Earth Sciences, Beihang University, Beijing, 100191, People\u0026rsquo;s Republic of China;\u003c/p\u003e\u003cp\u003e\u003csup\u003e3\u003c/sup\u003eKey Laboratory of Space Environment Monitoring and Information Processing, Ministry of Industry and Information Technology, Beijing, People\u0026rsquo;s Republic of China\u003c/p\u003e\u003cp\u003e\u003csup\u003e4\u003c/sup\u003eDepartment of Geophysics, School of Earth Sciences, Yunnan University, Kunming, 650091, China\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e\u003cp\u003eThis research is partly by National Natural Science Foundation of China (NSFC) grants: 42374202, 42350710793, 42188101, 42174207), the Specialized Research Fund for State Key Laboratories of China, and the Strategic Pioneer Program on Space Science II, Chinese Academy of Sciences, Grants XDA15350201, XDA15052500.\u003c/p\u003e\u003ch2\u003eAuthor Contributions:\u003c/h2\u003e\u003cp\u003eData and dynamics analysis, review \u0026amp; editing, Xinhua Wei and ChunLin. Cai; Discussion and review \u0026amp; editing, JunYing Yang, YuDuan Ma, TieYan Wang, YangYang Liu and Lei Dai. All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgments\u003c/h2\u003e\u003cp\u003eWe sincerely thank anonymous reviewers and the editors of EPS for their valuable and constructive comments for improving this manuscript.\u003c/p\u003e\u003ch2\u003eData Availability Statement:\u003c/h2\u003e\u003cp\u003eThe Custer data are available from Cluster Science Archive. The OMNI data were obtained from GSFC/ SPDF OMNIWeb (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttp://omniweb.gsfc.nasa.gov\u003c/span\u003e\u003cspan address=\"http://omniweb.gsfc.nasa.gov\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003e\u003cspan\u003eAn X, Liu TZ, Bortnik J, Osmane A, Angelopoulos V (2020) Formation of foreshock transients and associated secondary shocks. Acta Pathol Japonica 901:73. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.3847/1538-4357/abaf03\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eAndr\u0026eacute;s N, Meziane K, Mazelle C, Bertucci C, G\u0026oacute;mez D (2015) The ULF wave foreshock boundary: Cluster observations. 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J Geophys Research: Space Phys 118(6):3357\u0026ndash;3363. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1002/jgra.50376\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"earth-planets-and-space","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"epsp","sideBox":"Learn more about [Earth, Planets and Space](http://earth-planets-space.springeropen.com)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/epsp/default.aspx","title":"Earth, Planets and Space","twitterHandle":"@SpringerOpen","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"BMC/SO AJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Foreshock Bubbles, magnetic field compression","lastPublishedDoi":"10.21203/rs.3.rs-7657123/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7657123/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eForeshock Bubbles (FBs) are transient structures in the Earth's foreshock region, which are diamagnetic cavities formed by hot ion concentration around interplanetary magnetic field discontinuities and have significant compressional boundary shocks contributed to particle acceleration. We present here Cluster observations of FB events from January 2002 to April 2007 that each was encountered by all four spacecraft in order to accurately determine the parameters of its boundary shock. Statistical distributions show that the majorities of the FB boundary shocks are supercritical and steep with large magnetic compression ratios and are in quasi-perpendicular direction to their upstream magnetic field. The magnetic compression ratios of FB boundary shocks are roughly correlated positively with their shock normal angles. Additionally, the magnetic compression ratios enhance with increasing upstream incident velocities, which is interpreted as a manifestation of diamagnetic Hall current generation inside the boundary. These results along with the conclusions given in previous numerical simulations and laboratorial experiments suggest a fast formation of a sharp boundary shock by the Larmor coupling between the super-thermal ions and magnetized ambient plasma in a hot plasma expanding process.\u003c/p\u003e","manuscriptTitle":"Cluster Observations of Foreshock Bubbles and Their Boundary Shocks","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-17 18:03:13","doi":"10.21203/rs.3.rs-7657123/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"","date":"2025-10-26T21:54:16+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-10-07T00:21:36+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-10-06T10:54:57+00:00","index":"","fulltext":""},{"type":"submitted","content":"Earth, Planets and Space","date":"2025-09-28T22:54:31+00:00","index":"","fulltext":""},{"type":"decision","content":"Major Revision","date":"2025-09-27T05:20:58+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"earth-planets-and-space","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"epsp","sideBox":"Learn more about [Earth, Planets and Space](http://earth-planets-space.springeropen.com)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/epsp/default.aspx","title":"Earth, Planets and Space","twitterHandle":"@SpringerOpen","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"BMC/SO AJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"dae4aa87-d6c6-4836-a659-6b0703394229","owner":[],"postedDate":"October 17th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2026-04-13T16:15:52+00:00","versionOfRecord":{"articleIdentity":"rs-7657123","link":"https://doi.org/10.1186/s40623-026-02425-8","journal":{"identity":"earth-planets-and-space","isVorOnly":false,"title":"Earth, Planets and Space"},"publishedOn":"2026-04-11 15:58:59","publishedOnDateReadable":"April 11th, 2026"},"versionCreatedAt":"2025-10-17 18:03:13","video":"","vorDoi":"10.1186/s40623-026-02425-8","vorDoiUrl":"https://doi.org/10.1186/s40623-026-02425-8","workflowStages":[]},"version":"v1","identity":"rs-7657123","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7657123","identity":"rs-7657123","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}