Shock Reformation Induced by Ion-scale Whistler Waves in Quasi-perpendicular Bow Shock

Shock Reformation Induced by Ion-scale Whistler Waves in Quasi-perpendicular Bow Shock | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Shock Reformation Induced by Ion-scale Whistler Waves in Quasi-perpendicular Bow Shock Sibo Xu, Jia-Ji Sun, Shan Wang, Jing-Huan Li, Xu-Zhi Zhou, Daniel B. Graham, and 5 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6012766/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Studies have long suggested that shocks can undergo cyclical self-reformation as a type of shock nonstationarity. Until now, providing solid evidence for shock reformation in spacecraft observation and identifying its generating mechanisms remain challenging. In this work, by analyzing Magnetospheric Multiscale (MMS) spacecraft observations, we unambiguously identified shock reformation occurring in a quasi-perpendicular shock. A 2-D particle-in-cell simulation reproduces and explains the observed shock reformation. The simulation reveals that shock dynamics in the foot and ramp region generate two types of ion-scale whistler waves, respectively, each of which can drive shock reformation. Within one single wave period, the wave induces the magnetic field pile-up, ion accumulation and reflection, and upstream-pointing electric field, finally evolving into a new shock front. An interesting finding is that different shock dynamics compete and dominate the reformation at different stages. Our results not only provide evidence that the shock reformation in the present regime can be driven by ion-scale whistler waves, but also demonstrate the detailed kinetic processes how it happens, offering valuable insights into shock dynamics. Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction Collisionless shocks are ubiquitous phenomena in space and astrophysical plasmas 1 – 5 where significant energy conversion occurs between bulk, thermal, and electromagnetic energies. Shocks exhibit intrinsic nonstationarity, arising from microscale physics and influencing shock dynamics 6 – 9 . One form of nonstationarity is shock reformation 10 . For supercritical shocks with the Alfvén Mach number ( M A ) greater than ~ 3, satisfied by typical planetary bow shocks in the solar system, the shock produces reflected ions, which can contribute to shock reformation by accumulating in front of the shock ramp and inducing growing magnetic fields 11 , 12 . As time goes on, the enhanced foot evolves into a new shock front and then replaces the old one 13 , 14 . This entire process is repeated periodically and named shock reformation 15 – 18 . Dynamics and structures in the quasi-perpendicular shock transition region vary with shock parameters like M A , which may be related to different reformation processes. Krasnoselskikh et al. 9 showed that the gradient catastrophe instability at shock ramp induces the whistler precursor, in order to prevent the gradient from growing to infinity. The model predicts that the precursor wavelength is on the ion scale when M A < M w and on the electron scale when M w < M A < M cwn , where, \(\:{M}_{w}=\:\frac{\text{c}\text{o}\text{s}\left({\theta\:}_{Bn}\right)}{2\sqrt{{m}_{e}/{m}_{i}}}\) is the whistler Mach number, and \(\:{M}_{cwn}=\:\frac{\text{c}\text{o}\text{s}\left({\theta\:}_{Bn}\right)}{\sqrt{2{m}_{e}/{m}_{i}}}\) is the critical whistler nonlinear Mach number 9 . Simulations have suggested that this whistler precursor contributes to shock reformation 9 , 19 . Moreover, modified two-stream instability (MTSI), arising from interactions between reflected/incoming ions and electrons, generates whistler waves in the shock foot region 12 , 19 – 22 . When an ion beam drifting relative to electrons exists, MTSI can grow at the intersection of two dispersion curves between whistler waves and beam modes 23 , where the resonant ions satisfy ω = kV , corresponding to Landau resonance. 1-D PIC simulations imply the contribution of such MTSI-whistlers to shock reformation 14 , 19 , 24 . Furthermore, a type of large-scale shock reformation has also been reported in PIC simulations at large M A > M cwn , associated with a phase space vortex between the ramp and the upstream region 14 . The contributions of different mechanisms remain controversial, especially in parameter regimes when multiple mechanisms can co-exist and physics of multiple dimensions needs to be included. Until now, most evidence for the reformation of the quasi-perpendicular bow shock has been provided by numerical simulations. Detection of shock reformation in space observations requires high time resolutions, best aided with multi-spacecraft measurements. Lobzin etl al. 25 is one of the few successful trials to identify shock reformation using Cluster satellites. In addition, other nonstationary phenomena at shocks, e.g., surface waves 26 – 29 , may appear with similar fluctuations as during the shock reformation 30 , 31 , introducing challenges in accurately identifying reformation, not to mention its mechanisms. In this study, we report an Magnetospheric Multiscale (MMS) observation where the reformation in the Earth’s quasi-perpendicular bow shock can be unambiguously identified with clearer details compared to Cluster observations. In addition, a 2-D particle-in-cell (PIC) simulation with m i / m e = 1836 is employed, which reproduces the shock reformation observed by MMS and reveals the driving mechanisms. Results MMS observations MMS crossed a quasi-perpendicular bow shock on 2 February, 2021, at 07:18 UT. At approximately 07:18:15 UT, MMS1 observed a significant increase in the magnetic field amplitude from 12 nT to an overshoot of 70 nT, along with an increase in electron density from 14 cm − 3 to 150 cm − 3 (Fig. 1 a). They are typical features of a shock as the spacecraft transitioned from the upstream solar wind toward the magnetosheath. The shock normal vector n is estimated as [0.928, 0.257, -0.269] in GSE coordinates based on the bow shock model 32 , which is close to that from the mixed mode coplanarity method of magnetic field and ion velocity 33 upstream and downstream of the shock, with a difference of 12.6 degrees. Figure 1 b shows the magnetic field vector in a shock coordinate system n - t 1 - t 2 , where t 2 = n × B u / | B u |, t 1 = t 2 × n , and B u is the upstream magnetic field (average during 07:17:20 to 07:17:40 UT). Thus, we determine the angle ( θ Bn ) between B u and n as 67°, i.e., a quasi-perpendicular bow shock. In the ion normal velocity ( v n ) spectrum (Fig. 1 c), ion reflections are detected in the foot and ramp regions, where the intense phase-space density forms an arc that connects between incoming solar wind ions at v n 0. Key shock parameters are listed in Table 1 . Table 1 Shock parameters in MMS observation and PIC simulation Shock parameters In MMS observations PIC simulation θ Bn 67° 67° Upstream Alfvén speed ( V A ) 67.94 km / s - Alfvén Mach number ( M A ) 6.43 8.74 Ion beta ( β i ) 0.63 0.63 Electron beta ( β e ) 0.87 0.87 Critical whistler nonlinear Mach number ( M cwn ) 11.79 11.79 Whistler Mach number ( M w ) 8.34 8.34 Upstream ion inertial length ( d i ) 60 km - Figures 1 d-f present measurements from MMS2. It registers two primary magnetic peaks around 07:18:13 UT and 07:18:17 UT, respectively (labeled as B and C), in contrast to only one primary magnetic peak observed by MMS1 around 07:18:16 UT (A). Incoming and reflected ion populations separated in v n are detected prior to peak B at MMS2 and A at MMS1 but not between peaks B and C at MMS2. Figures 1 i-k present the reduced 2-D velocity distribution functions (VDFs) of ions at A, B, and C in the v n – v t2 plane. A and B show similar distributions, where reflected ions can be distinguished from the incoming solar wind, forming an arc as they gyro-turn around the magnetic field during reflection 30 , 34 . The similarity indicates that they correspond to the same main shock front. Ions at C exhibit a more isotropic distribution (compared with ion distribution at A and B), where incoming and reflected populations are indistinguishable, indicating more complete ion thermalizations downstream of the shock ramp. Figure 1 m displays the relative positions of the four spacecraft, which are grouped into two pairs: MMS1 & MMS4 and MMS2 & MMS3, primarily separated along n by ~ 50 km. The downstream pair (MMS2 & MMS3) detects the shock earlier, consistent with that the shock passed by the spacecraft from downstream to upstream directions. Measurements within each pair are nearly identical (not shown), suggesting that the different profiles between MMS2 and MMS1 are likely due to the temporal evolution as the shock propagates in n direction, hereafter referred to as “early shock” and “later shock”, respectively. Using the method that considers the thickness of the shock foot with the presence of reflected ions as their gyro-radius 35 , we estimate the shock propagation speed as 17 km/s, suggesting a time lag of approximately 3s between the two spacecraft pairs. Applying a 2.7 s (~ 3 s) forward shift for B t 1 in the early shock (Fig. 1 g), the first magnetic peaks from all four spacecraft well align with each other, further validating that they correspond to the same primary shock front. The second peak present in the early shock vanishes in the later shock. We interpret that it represents an old shock, which gradually damps as a new shock grows at the first peak during reformation. Based on the estimated shock speed, the distance between the old and new shocks is approximately 58 km (~ 1 d i ). Next, we highlight another potential shock reformation cycle in the later shock. At approximately 07:18:12.7 UT, a magnetic field pile-up region is observed (marked by the red shadow in Figs. 1 a-c) in the foot region. All three magnetic field components exhibit large-amplitude oscillations, leading to a rise in total magnetic field strength with a maximum around 40 nT—roughly half of that at the main shock front. The distance between this pile-up region and the primary shock front is also close to 1 d i , suggesting a new reformation cycle. Additionally, within this region, the density shows spikes with a maximum reaching 30 cm − 3 , and the v n spectrum displays a distinct concentration at v n > 0 (highlighted by the circle in Fig. 1 c). These features indicate that ions have been reflected and accumulated in the pile-up region, as in a shock ramp. Figure 1 h presents the normal component of the electric field ( E n )recorded by four spacecraft, with the same time shift applied to MMS2 and MMS3 as in B t1 . Around 07:18:13 UT, positive E n spikes are detected in the later shock within the magnetic field pile-up region, while E n remains around zero in the early shock. Such upstream-pointing electric field spikes, similar to those reported in previous studies 35 , 36 , introduce an electrostatic potential resembling the cross-shock potential at the shock ramp 14 , and they seem to appear along with a growing shock during reformation. Collectively, these observations strongly suggest that the pile-up region in the foot is evolving into a new shock front, inducing the shock reformation. We further investigate the properties of magnetic perturbations responsible to the magnetic pile-up. The wave analysis (see detailed description in Methods, subsection Wave analysis) reveals that the angle between wavevector and magnetic field ( θ kB ) ~ 147°, wavelength (λ) ~ 1.2 d i , the phase speed ( V ph ) is ~ 2.5 V A in the spacecraft frame, and ~ 5.5 V A in the plasma rest frame (PRF), where V A is the upstream Alfvén speed. The waves are propagating upstream, both in the spacecraft frame and the PRF. The ellipticity in both frames are right-handed. The waves are thus classified as an ion-scale whistler mode. The ion VDF in the wave interval (Fig. 1 l) further shows that shock reflected ions are located near V ph , and hence, can be in Landau resonance with these waves. Both the wave properties and resonance conditions are similar to those analyzed in Hull et al. 21 and Lalti et al. 20 , consistent with waves excited by MTSI resonant with reflected ions. PIC simulation In order to verify the shock reformation in MMS observations and understand the underlined mechanisms, we perform a 2-D particle-in-cell (PIC) simulation (see detailed description in Methods, subsection PIC simulation) in the x - y plane (i.e., n - t 1 plane in observations). Thus, this simulation is conducted to investigate the shock dynamics along n and t 1 , while it should be noted that other reformation related processes can occur along the t 2 direction 37 . The parameters are selected to represent the observation regime: super-critical with ion reflections, while \(\:{M}_{A}\lesssim\:{M}_{w}\) to allow for ion-scale precursors. Figure 2 illustrates the process of shock reformation in PIC. Figures 2 a-d present the 2-D pictures of B t at 5.8, 6.0, 6.2 and 6.4 ω ci −1 , where ω ci is the upstream ion cyclotron frequency. Waves are present in the foot, propagating oblique to the shock normal and quasi-parallel to the magnetic field, identified as right-hand ion-scale MTSI-whistler waves Landau resonant with reflected ions, similar to those observed in MMS (see details in Methods, subsection Wave analysis). The black ovals circle a wave peak that will evolve into the new shock. At 5.8 ω ci −1 , it exhibits the same properties with the other waveforms in the foot, with consistent amplitudes, wavefront directions, and propagation velocities. As time progresses to 6.0 ω ci −1 , its amplitude significantly increases; at 6.2 ω ci −1 , its wavefront changes and rotates toward the y -axis, with the orientation gradually aligning with the shock front. Between 6.2 and 6.4 ω ci −1 , the propagation velocity in the y -direction significantly decreases. Between 5.8 and 6.4 ω ci −1 , the wavelength is the only property that does not change. After 6.4 ω ci −1 , the wave peak merges into the shock, losing properties as a propagating wave. Figure 2 e displays 1-D profiles of the total B t between 5.6 ω ci −1 and 6.6 ω ci −1 , cut at the center of the circled patch (marked by white dotted lines in Figs. 2 a-d). Initially, the shock consists of a ramp and a foot with several small magnetic peaks, corresponding to the waves seen in 2-D plots. The amplitude of the wave peak closest to the ramp increases with time. At 6.2 ω ci −1 , the magnetic field magnitude of wave peak exceeds that of the old front (labeled as ‘ O ’) for the first time, marking the evolution into a new shock front (labeled as ‘ N ’). After 6.4 ω ci −1 , the old and the new fronts merge into a single magnetic peak, signaling the completion of shock reformation. From 5.6 to 6.6 ω ci −1 , the timescale of reformation is approximately 1 ω ci −1 , consistent with previous studies 9 , 14 . We note that the time interval from the early shock at MMS2 to the later shock at MMS1 is ~ 2.7s (~ 0.48 ω ci −1 ), during which the shock exhibits obvious evolution but has not yet completed a full reformation cycle, also indicating the reformation period on the order of ion cyclotron period. Figures 2 f-k present a 1-D cut of shock along x -axis at 6.0 ω ci −1 at y = 0.5 d i (along the dotted line in Fig. 2 b). The shaded region highlights the new shock forming ahead of the old shock. The new shock shows ion density enhancements and a significant upstream-pointing electric field, and locally reflected ions start to show up with v n changing from negative towards positive values. These characteristics resemble the magnetic field pile-up region observed by MMS1, supporting the notion that the pile-up region will evolve into a new shock. For the present regime of quasi-perpendicular shocks, two types of ion-scale whistlers may develop, and our simulation reveals their co-existence and competition. Figure 3 displays the magnetic fields at 3 ω ci −1 and 6 ω ci −1 . At 3 ω ci −1 , two kinds of waves are observed in different regions of the shock. In the shock ramp between the first two vertical dashed lines, a stripe of wave front along y can be seen in the 2-D plot (marked by the black oval in Fig. 3 a top), mostly phase standing with the shock. In the 1-D cut (Fig. 3 a bottom), B y and B z exhibit obvious perturbations, while B x does not, further supporting that k is along the shock normal. The wave is thus identified as whistler precursors, with their generation attributed to shock dynamics. The whistler precursor is produced by the shock itself, resulting from the gradient catastrophe instability predicted by 1-D shock models 9 . The region between the second and third dashed lines corresponds to the shock upstream region. The waves propagate obliquely relative to the shock normal ( k marked by the magenta arrow in Fig. 3 a top) already identified as MTSI – whistlers, with their generation attributed to upstream dynamics. At 6 ω ci −1 , the whistler precursors at the ramp disappear, while the MTSI – whistlers remain. Compared to the earlier time, MTSI – whistlers extend closer to the shock ramp and exhibit a larger amplitude. It suggests that the MTSI – whistlers compensate for the magnetic field gradient, so that no additional precursor waves need to be developed in order to avoid the gradient catastrophe, which may explain the disappearance of the precursor. Both the whistler precursors associated with shock dynamics and the MTSI - whistlers associated with upstream dynamics can contribute to shock reformation. At 3 ω ci −1 , the old ramp is located at x ~ 6.2 d i , while the wave peak of the whistler precursor at x ~ 6.8 d i evolves into the new ramp, corresponding to precursor – reformation. At 6 ω ci −1 , the reformed shock front originates from a single wave peak of MTSI – whistlers in the upstream region, indicating MTSI – reformation. Interestingly, both reformation scenarios can explain MMS observations. If the entire pile-up region observed by MMS1 is treated as a single wave peak of the whistler precursor, the shock reformation can be attributed to the whistler precursor, as in the simulation at t = 3 ω ci −1 . The identified MTSI-whistlers in the pile-up region can be regarded as high-frequency magnetic fluctuations superimposed on the low-frequency whistler precursor. On the other hand, if we consider that the magnetic pile-up is purely caused by accumulating 4–5 periods of the MTSI-whistler wave, the shock reformation occurs when a single period of MTSI-whistler enhances and its propagation velocity gradually decreases and finally aligns with the shock. The MITSI-whistler wavelength is close to the distance between old and new shocks, both ~ 1 d i . The consistency supports the possibility that the reformed shock front can evolve from a single MTSI-whistler peak, like the simulation around t = 6 ω ci −1 . In the simulation, the new shock front consists of patches (marked by red circles in Fig. 3 b) with a finite length along y. It originates from the regular MTSI-whistler waveforms but has lost the whistler propagation velocity. Ambient ordinary MTSI-whistler wave fields can propagate through the new shock front, which appears to be a superposition of high-frequency fluctuations on top of low-frequency signals, similar to the features in the magnetic pile-up region observed by MMS1. Conclusion Using MMS observations, combining the magnetic field and ion measurements from two pairs of spacecraft with a ~ d i scale separation, we unambiguously identified the shock reformation process in a quasi-perpendicular shock. The shock is super-critical with presence of reflected ions and M A < M w . These conditions on M A ​satisfy the generation criteria for ion-scale whistler precursors. The shock observed slightly earlier exhibits two magnetic peaks, while the later crossing observed one, due to the reformed new shock front and a damping old shock. A magnetic field pile-up associated with ion-scale whistler waves is observed in the foot region, exhibiting features of another growing shock front, including plasma density spikes, ion reflection and accumulation, and upstream-pointing normal electric field spikes. These results provide the first observational evidence that ion-scale whistlers drive the shock reformation in a quasi-perpendicular bow shock. A 2-D PIC simulation applicable to the observation regime reproduces the reformation and reveals two scenarios (illustrated by Fig. 4 ) for the wave dynamics and shock reformation mechanisms. In the earlier stage (Fig. 4 left), two types of ion-scale whistler waves, associated with shock dynamics in the ramp and foot regions, respectively, co-exist in the shock transition region. Near the shock front, shock dynamics produces whistler precursors that grows out of the magnetic field gradient, labeled as “Shock Gradient-Whistler”. In the shock foot region, whistlers are generated by the kinetic instability MTSI resonant with reflected ions, labelled as “Reflected Ion-Whistler”. The reformation is caused by “Shock Gradient-Whistler” in the early time (Fig. 4 left), and by “Reflected Ion-Whistler” in the later time (Fig. 4 right) as its residing region extends closer to the shock ramp and it replaces the former wave. The novel understanding of competitions between different shock dynamics demonstrates the complicated and dynamic nature of shock processes. Our results provide evidence for shock reformation and its link to ion-scale whistler waves. The conclusions can be applied to collisionless shocks in the same regime that commonly exist in the solar system, advancing our understanding of shock physics and encouraging extensions of relevant studies to additional regimes. Methods Data utilization The displayed space observation data in this manuscript are taken from MMS and OMNI. For MMS data, magnetic field measurements are from the Fluxgate Magnetometer (FGM) 38 in the burst mode at 128 samples/s and the electric fields are from the double probes in the fast mode at 32 samples/s 39–41 . Ion data are provided by the Dual Ion Spectrometer (DIS) onboard the Fast Plasma Investigation (FPI) 42 , with a time cadence of 0.15 s in the burst mode. The electron data are from the Dual Electron Spectrometer (DES) onboard FPI at 0.03 s in the burst mode. Since FPI-DIS is not designed to monitor the solar wind, we use OMNI data to obtain the density and temperature of upstream ions. Excluding Ripples as the Cause of Multiple Magnetic Peaks in MMS observations Multiple magnetic peaks observed along the spacecraft trajectory may result from other types of shock nonstationarities like the surface wave ripples, which can be excluded in the present case for three reasons. Firstly, when multiple spacecraft with separations along n cross a rippled shock front, the spacecraft entering the downstream region first will transition back into the upstream region later 30 , 31 . However, in our case, the magnetic field measurements at the primary shock front from MMS1 (peak A) and MMS2 (peak B) illustrate a first-in-first-out scenario. Secondly, magnetic field profiles from four spacecraft get aligned with a time shift of 2.7 s. This time shift is in close agreement with the shock’s propagation time (~ 3 s) between the two pairs of spacecraft along n . Thirdly, the ion VDF at peak C shows a significant difference from that observed at the main shock front (peaks A and B), suggesting that magnetic peak C is not caused by spacecraft repeatedly crossing the same main shock front due to a rippled shock. Wave analysis The wave analysis of MMS observations is based on the Singular Value Decomposition (SVD) method 43 and Minimum Variance Analysis (MVA) method 44 . Fig. S1a presents the magnetic field components in field-aligned coordinates from MMS1, and Fig. S1b-e display the results of wave analysis in the spacecraft frame obtained from SVD method. Focusing on the pile-up region, the power spectral densities of the magnetic field reveal a marked increase in 1–3 Hz range (highlighted by the circles in Fig. S1b-e). The corresponding ellipticities exhibit the positive values close to 1, suggesting right-hand polarized waves in the spacecraft frame. θ kB within the 1–3 Hz range is quasi-antiparallel, with the phase velocity ( V ph ) ranging from [100, 200] km/s. To further verify the wave properties in the pile-up region, we conduct the MVA of the 1–3 Hz filtered magnetic field and analyze the k (wave vector) component of the Poynting flux to remove the 180 o ambiguity of the MVA result. The results include k = (0.59, -0.76, 0.27) in the shock coordinate n - t 1 - t 2 , θ kB = 147°, and V ph = 148 km/s (estimated using |δ E |/|δ B |) in the spacecraft frame, all close to the SVD results. Using a mean frequency of 2 Hz, the wavelength λ is calculated to be approximately 74 km, close to the upstream d i = 60 km). Considering that the bulk ion velocity along k is -177 km/s, the phase speed in the plasma rest frame (PRF) is 325 km/s ~ 5.5 V A , so the ellipticity in PRF remains right-handed. Thus, we conclude that the wave at 1–3 Hz in the pile-up region belongs to an ion scale whistler mode. Figure 1 l presents a 2-D ion VDF at 07:18:11 UT observed by MMS1 (labeled as ‘D’ in Figs. 1 a-c) in k - B plane, where the horizontal axis is along the wavevector k of the whistlers in the pile-up region and the background magnetic field B is within the plane shown with a white line. The red line marks the wave phase velocity V ph ., It falls within the range of the reflected ion population, suggesting that reflected ions can be in Landau resonance with the waves, satisfying ω = kV ph . Figure S1f-j present the wave analysis based on MMS2 data with a 2.7 s forward shift as in Fig. 1 g. The analysis reveals whistler waves observed by MMS2 in the shock foot and near the ramp (highlighted by the circles in Fig. S1g-j), which exhibit properties closely matching those observed by MMS1. Figure S2 illustrates the wave properties in the shock foot based on the PIC simulation results. Figure S2a displays the 2-D representation of B z at 4.0 ω ci −1 , and we focus on the wave stripes at x ~ [11, 13] d i . Based on the wave vector k perpendicular to the wavefronts as marked by the dashed arrow in Figure S2a, we identify θ k n as 41.6°, θ k B as 30.4°, and the wavelength λ as 0.85 d i . From the displacement of the waveforms between 3.9 and 4.1 ω ci −1 , V ph is determined to be 3.39 V A in the simulation frame, 1.5 V A in the shock frame, and 6.3 V A in PRF. Figure S2b presents the hodogram of the magnetic field in the perpendicular plane taken at ( x , y ) = (12.8, -4.1) d i from 4.0 ω ci −1 to 4.5 ω ci −1 , which suggests a right-handed polarization in all frames. Therefore, the waves in the shock foot are identified as ion-scale whistler waves. Figure S2c-d present the ion (VDF) at ( x , y ) = (12, -4.5) d i in the coordinate system defined with k and background B . The red lines, which mark the velocity component V k = 3.39 V A fall within the range of the reflected ion population, suggesting that the reflected ion component of the VDF is in Landau resonance with the waves. We further use three plasma components represented by Maxwellian distributions: electrons, incoming ions, and reflected ions, to fit the VDF shown in Figure S2c-d, and study the linear instability growth using the Bo kinetic dispersion solver 45 . Table S1 displays the fitting parameters for three components. The result (Fig. S2e) shows that the growth rate reaches a maximum of γ max ~ 1.19 ω ci at ω ~ 3.83 ω ci , with λ ~ 0.67 d i and V ph ~ 2.55 V A in the simulation frame. The ratio E y / iE x is approximately 0.67-0.24 i (not shown here), suggesting a right-handed polarization. The wave properties obtained from the kinetic dispersion solver closely match those in the PIC simulation and MMS observations, providing further evidence that the waves are whistlers and grow from MTSI between Landau-resonant reflected ions and electrons. PIC simulation The numerical part of this work employs the two-dimension particle-in-cell simulation using the VPIC code 46 . The shock is produced by the injection method: plasmas are injected from the right boundary in a uniform upstream magnetic field condition with + x and + y components, and particles are specularly reflected at the left boundary, leading to the formation of a shock that propagates along the + x direction. The simulation uses periodic boundary conditions along y . A real mass ratio m i / m e = 1836 is used to achieve a small M A compared to M w and the 2-D setup allows for the wave propagations both along and oblique to the shock normal. A combination of these two conditions leads to high cost of the computation resources, which is partly why few similar studies have been conducted, but it is crucial for resolving the necessary kinetic physics and developing ion-scale waves critical for the reformation process we are interested in. The injection speed is set as 6.18 V A , and the shock propagates upstream at a speed of 2.56 V A , resulting in a M A of 8.74 ~ M w = 8.34. The 1-D theory predicts that the whistler precursors transition from ion to electron scales as M A increases above M w . It turns out that our 2-D simulation still develops ion-scale precursors close to the transition threshold, and hence present the same features as in observations with M A <M w , justifying the usage of our simulation to conduct comparative analyses with observations. Other shock parameters are configured to match the MMS observation as shown in Table 1 . The system size is L x × L y = 36 d i × 12 d i with 6600 × 2200 grids, the electron plasma to cyclotron frequency ratio based on upstream conditions is ω pe / ω ce = 5, and 200 particles per species are initialized in each computational cell with a uniform distribution. In simulation results, time will be given in units of the inverse of the ion cyclotron frequency ω ci −1 , distances are in units of the ion inertial length d i ( d i = V A / ω ci ), the velocity is in units of the upstream Alfvén speed V A , and the density is in upstream density N 0 . Declarations Data availability MMS data are available at https://lasp.colorado.edu/mms/sdc/public/data following the directories: mms#/fgm/brst/l2 for FGM data, mms#/edp/brst/l2/dce/ for electric field data, mms#/fpi/brst/l2/dis-dist for FPI ion distributions, mms#/fpi/brst/l2/dis-moms for FPI ion moments, and mms#/fpi/brst/l2/des-moms for FPI electron moments. OMNI data used are available at https://omniweb.gsfc.nasa.gov/ . Simulation data are available at https://zenodo.org/records/14625758 . Code availability MMS data analysis is conducted using SPEADAS software ( http://themis.ssl.berkeley.edu ). The Bo solver can be found at http://dx.doi.org/10.17632/cvbrftzfy5.1 . VPIC codes are available at https://github.com/lanl/vpic . Competing interests The authors declare no competing interests. Acknowledgements We thank the MMS team (PI: James Burch) for providing the high-quality data. SBX and SW acknowledge the helpful discussions with Yong-Fu Wang, Yi-Xin Sun, Zhi-Yang Liu, and the ISSI team 555 of “Impact of upstream mesoscale transients on the near-Earth environment” led by Primoz Kajdic and Xochitl Blanco-Cano. Work at PKU is supported by the National Natural Science Foundation of China Grants 42374188 and 42330202. The simulation work was carried out at National Supercomputer Center in Tianjin, China, and the calculations were performed on TianHe-HPC. References Bale SD et al (2005) Quasi-perpendicular shock structure and processes. Space Sci Rev 118:161–203 Richardson JD, Kasper JC, Wang C, Belcher JW, Lazarus AJ (2008) Cool heliosheath plasma and deceleration of the upstream solar wind at the termination shock. 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Phys Plasmas 9:1192–1209 Lembège B, Savoini P (2002) Formation of reflected electron bursts by the nonstationarity and nonuniformity of a collisionless shock front. J Geophys Research: Space Phys 107:SMP–X Treumann RA (2009) Fundamentals of collisionless shocks for astrophysical application, 1. Non-relativistic shocks. Astron Astrophys Rev 17:409 Laming JM (2022) Critical Mach Numbers for Magnetohydrodynamic Shocks with Accelerated Particles and Waves. Astrophys J 940:98 Hada T, Oonishi M, Lembège B, Savoini P (2003) Shock front nonstationarity of supercritical perpendicular shocks. J Geophys Research: Space Phys, 108 Scholer M, Shinohara I, Matsukiyo S (2003) Quas-perpendicular shocks: Length scale of the cross-shock potential, shock reformation, and implication for shock surfing. J Geophys Research: Space Phys 108:SSH–4 Lembege B, Savoini P (1992) Nonstationarity of a two-dimensional quasiperpendicular supercritical collisionless shock by self-reformation. Phys Fluids B 4:3533–3548 Chapman SC, Lee RE, Dendy R (2005) O. Perpendicular shock reformation and ion acceleration. Space Sci Rev 121:5–19 Umeda T, Yamazaki R (2006) Full particle simulation of a perpendicular collisionless shock: A shock-rest-frame model. Earth Planets and Space 58:e41–e44 Caprioli D, Pop AR, Spitkovsky A (2014) Simulations and theory of ion injection at non-relativistic collisionless shocks. Astrophys J Lett 798:L28 Scholer M, Burgess D (2007) Whistler waves, core ion heating, and nonstationarity in oblique collisionless shocks. Phys Plasmas, 14 Lalti A et al (2022) Whistler waves in the foot of quasi-perpendicular supercritical shocks. J Geophys Research: Space Phys, 127, e2021JA029969 Hull AJ et al (2020) MMS observations of intense whistler waves within Earth's supercritical bow shock: Source mechanism and impact on shock structure and plasma transport. J Geophys Research: Space Phys, 125, e2019JA027290 Umeda T, Kidani Y, Matsukiyo S, Yamazaki R (2012) Modified two-stream instability at perpendicular collisionless shocks: Full particle simulations. J Geophys Research: Space Phys, 117 Muschietti L, Lembège B (2017) Two-stream instabilities from the lower-hybrid frequency to the electron cyclotron frequency: application to the front of quasi-perpendicular shocks. Ann Geophys 35:5 Matsukiyo S, Scholer M (2003) Modified two-stream instability in the foot of high Mach number quasi‐perpendicular shocks. J Geophys Research: Space Phys, 108 Lobzin VV et al (2007) Nonstationarity and reformation of high-Mach‐number quasiperpendicular shocks: Cluster observations. Geophys Res Lett, 34 Lembège B et al (2009) Nonstationarity of a two-dimensional perpendicular shock: Competing mechanisms. J Geophys Research: Space Phys, 114 Hao Y et al (2017) Reformation of rippled quasi-parallel shocks: 2-D hybrid simulations. J Geophys Research: Space Phys 122:6385–6396 Umeda T, Daicho Y (2018) Periodic self-reformation of rippled perpendicular collisionless shocks in two dimensions. Ann Geophys 36:1047–1055 Yang Z, Lu Q, Liu YD, Wang R (2018) Impact of shock front rippling and self-reformation on the electron dynamics at low-mach-number shocks. Astrophys J 857:36 Johlander A et al (2016) Rippled quasiperpendicular shock observed by the magnetospheric multiscale spacecraft. Phys Rev Lett 117:165101 Li JH et al (2024) Bow shock ripples and their modulation of whistler wave packets: MMS observations. Geophys Res Lett 51:e2024GL111590 Farris MH, Petrinec SM, Russell CT (1991) The thickness of the magnetosheath: Constraints on the polytropic index. Geophys Res Lett 18:1821–1824 Schwartz SJ (1998) Shock and discontinuity normals, Mach numbers, and related parameters. ISSI Sci Rep Ser 1:249–270 Sckopke N, Paschmann G, Bame SJ, Gosling JT, Russell CT (1983) Evolution of ion distributions across the nearly perpendicular bow shock: Specularly and non-specularly reflected-gyrating ions. J Geophys Research: Space Phys 88:6121–6136 Gosling JT, Baker DN, Bame SJ, Feldman WC, Zwickl RD, Smith EJ (1985) North-south and dawn‐dusk plasma asymmetries in the distant tail lobes: ISEE 3. J Geophys Research: Space Phys 90(A7):6354–6360 Chen LJ et al (2018) Electron bulk acceleration and thermalization at Earth’s quasiperpendicular bow shock. Phys Rev Lett 120:225101 Burgess D et al (2016) Microstructure in two-and three-dimensional hybrid simulations of perpendicular collisionless shocks. J Plasma Phys, 82 Russell CT et al (2016) The magnetospheric multiscale magnetometers. Space Sci Rev 199:189–256 Ergun R et al (2016) The axial double probe and fields signal processing for the MMS mission. Space Sci Rev 199:167–188 Lindqvist P-A et al (2016) The spin-plane double probe electric field instrument for MMS. Space Sci Rev 199:137–165 Torbert RB et al (2016) The FIELDS instrument suite on MMS: Scientific objectives, measurements, and data products. Space Sci Rev 199:105–135 Pollock C et al (2016) Fast plasma investigation for magnetospheric multiscale. Space Sci Rev 199:331–406 Santolík O, Parrot M, Lefeuvre F (2003) Singular value decomposition methods for wave propagation analysis. Radio Sci 38:10–11 Sonnerup BU, Scheible M (1998) Minimum and maximum variance analysis. Anal Methods Multi-Spacecr Data 1:185–220 Xie HS, BO (2019) A unified tool for plasma waves and instabilities analysis. Comput Phys Commun 244:343–371 Bowers K et al (2008) VPIC v. 1.2 (No. vpic). Los Alamos National Laboratory (LANL), Los Alamos, NM (United States) Additional Declarations The authors declare no competing interests. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6012766","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":414615968,"identity":"e630106c-47d6-4aa4-98d8-acd62ccb9470","order_by":0,"name":"Sibo Xu","email":"","orcid":"","institution":"Peking University","correspondingAuthor":false,"prefix":"","firstName":"Sibo","middleName":"","lastName":"Xu","suffix":""},{"id":414615969,"identity":"2d8ed385-c1bc-4137-b601-7f96e8fdae32","order_by":1,"name":"Jia-Ji Sun","email":"","orcid":"","institution":"Peking University","correspondingAuthor":false,"prefix":"","firstName":"Jia-Ji","middleName":"","lastName":"Sun","suffix":""},{"id":414615970,"identity":"e0e07cf2-82e3-4abf-a80c-23cec788f053","order_by":2,"name":"Shan Wang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVRIiWNgGAWjYLCCByCCvbERRPPwEaUlAUTwHD5sAKLYiNcikZYmAaIJauFv7334IbHNLk8+Ises8muOnQwbA/PDRzfwaJE4c9xYIrEtudjwzBuz27LbkoEOYzM2zsGjxUAijUEicRtz4sb2HLPbktuYgVp42KQJaGH+kbitPnFjQ45ZseS2eqK0sAFtOZw4nyMtjfHjtsOEtUicOcZmkfjveOIGYCBLM247zsPGTMAv/O1tzDc+nKlOnN/e2Pjx57Zqe3725oeP8WlBuPAAAwMzD4jFTIxyEJBvYGBg/EGs6lEwCkbBKBhRAACW9UhMKkkL+wAAAABJRU5ErkJggg==","orcid":"","institution":"Peking University","correspondingAuthor":true,"prefix":"","firstName":"Shan","middleName":"","lastName":"Wang","suffix":""},{"id":414615971,"identity":"dd7b9b2e-f4ce-43d3-8703-34d94af7a1c3","order_by":3,"name":"Jing-Huan Li","email":"","orcid":"","institution":"Swedish Institute of Space Physics, Box 537, 75121, Uppsala, Sweden.","correspondingAuthor":false,"prefix":"","firstName":"Jing-Huan","middleName":"","lastName":"Li","suffix":""},{"id":414615972,"identity":"b03120cb-103e-4606-816f-5805a5446006","order_by":4,"name":"Xu-Zhi Zhou","email":"","orcid":"","institution":"Peking University","correspondingAuthor":false,"prefix":"","firstName":"Xu-Zhi","middleName":"","lastName":"Zhou","suffix":""},{"id":414615973,"identity":"3a89832e-06e0-4beb-9dc3-51010ceeb29d","order_by":5,"name":"Daniel B. Graham","email":"","orcid":"","institution":"Swedish Institute of Space Physics, Box 537, 75121, Uppsala, Sweden.","correspondingAuthor":false,"prefix":"","firstName":"Daniel","middleName":"B.","lastName":"Graham","suffix":""},{"id":414615974,"identity":"46e360d4-987d-4a55-85fc-60d6eaf3d209","order_by":6,"name":"Yu-Fei Hao","email":"","orcid":"","institution":"Key Laboratory of Planetary Sciences, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, China.","correspondingAuthor":false,"prefix":"","firstName":"Yu-Fei","middleName":"","lastName":"Hao","suffix":""},{"id":414615975,"identity":"06bb9481-7a40-45a3-9810-90b7b8d3977b","order_by":7,"name":"Qiu-Gang Zong","email":"","orcid":"","institution":"Peking University","correspondingAuthor":false,"prefix":"","firstName":"Qiu-Gang","middleName":"","lastName":"Zong","suffix":""},{"id":414615976,"identity":"a1fe7031-cc66-44ac-8742-3e8a063943ec","order_by":8,"name":"Chao Yue","email":"","orcid":"","institution":"Peking University","correspondingAuthor":false,"prefix":"","firstName":"Chao","middleName":"","lastName":"Yue","suffix":""},{"id":414615977,"identity":"87acb425-8a55-4fa2-9a82-374b862838c9","order_by":9,"name":"Yoshiharu Omura","email":"","orcid":"","institution":"State Key Laboratory of Lunar and Planetary Sciences, Macau University of Science and Technology (MUST), Macau, China.","correspondingAuthor":false,"prefix":"","firstName":"Yoshiharu","middleName":"","lastName":"Omura","suffix":""},{"id":414615978,"identity":"b953f2c4-ac1e-4375-bd68-466150ad7b60","order_by":10,"name":"Yuri V. Khotyaintsev","email":"","orcid":"","institution":"Swedish Institute of Space Physics, Box 537, 75121, Uppsala, Sweden.","correspondingAuthor":false,"prefix":"","firstName":"Yuri","middleName":"V.","lastName":"Khotyaintsev","suffix":""}],"badges":[],"createdAt":"2025-02-12 07:27:13","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-6012766/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6012766/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":76181143,"identity":"57bd2694-1aa1-47d3-9005-b9f43cc19851","added_by":"auto","created_at":"2025-02-13 07:25:05","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":499446,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eOverview of the shock encountered by multiple spacecraft of MMS.\u003c/strong\u003e (a, d) the magnetic field and electron number density in MMS1 and MMS2 respectively. (b, e) The magnetic field vector in MMS1 and MMS2 respectively. (c, f) The ion \u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e spectrum in MMS1 and MMS2 respectively. (g) \u003cem\u003eB\u003c/em\u003e\u003csub\u003et1\u003c/sub\u003e measurement by four spacecraft, where MMS2 and MMS3 are shifted 2.7s forward. (h) \u003cem\u003eE\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e measurement by four spacecraft with the same shift as (g). (i-k) The ion velocity distributions in \u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e-\u003cem\u003ev\u003c/em\u003e\u003csub\u003et2\u003c/sub\u003e plane at the magnetic peaks A, B and C. (l) The ion velocity distributions in \u003cem\u003ek\u003c/em\u003e-\u003cem\u003eB\u003c/em\u003e plane at D. (m) The relative location of the four spacecraft of MMS.\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-6012766/v1/b044fd26402246292a0a6e14.png"},{"id":76181145,"identity":"8006ec1e-9fb7-4918-a086-7e22c2232da1","added_by":"auto","created_at":"2025-02-13 07:25:05","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":1115216,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003ePIC simulation results demonstrating the reformation process.\u003c/strong\u003e (a-d) 2-D pictures of \u003cem\u003eB\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e at 5.8, 6.0, 6.2 and 6.4 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e-1\u003c/sup\u003e. (e) 1-D profiles of the total magnetic field \u003cem\u003eB\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e between 5.6 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e-1\u003c/sup\u003e and 6.6 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e-1\u003c/sup\u003e. ‘O’ and ‘N’ represent the old and new shocks, respectively. (f-k) 1-D cut of shock along \u003cem\u003ex\u003c/em\u003e-axis at \u003cem\u003ey\u003c/em\u003e = 0.5 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e at 6.0 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e-1\u003c/sup\u003e.\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-6012766/v1/869d7dfee03c94aa9356b7f6.png"},{"id":76181147,"identity":"8a5773f2-d67b-410e-a2e9-c8145a61a577","added_by":"auto","created_at":"2025-02-13 07:25:05","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":1784716,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eMagnetic field components in PIC simulation results at 3 \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eω\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cstrong\u003eci\u003c/strong\u003e\u003c/sub\u003e\u003csup\u003e\u003cstrong\u003e-1\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e and 6 \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eω\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cstrong\u003eci\u003c/strong\u003e\u003c/sub\u003e\u003csup\u003e\u003cstrong\u003e-1\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e, illustrating shock reformation induced by the whistler precursor and by MTSI whistlers, respectively. \u003c/strong\u003eThe whistler precursor propagates along the shock normal direction (\u003cem\u003ex\u003c/em\u003e), while the MTSI whistler propagates oblique with respect to \u003cem\u003ex\u003c/em\u003e.\u003cstrong\u003e \u003c/strong\u003eThe 1-D profiles of the magnetic field components are taken along the horizontal dashed lines in the 2-D plots. In (a), reformation is due to precursors while MTSI exists in foot; in (b) MTSI takes over precursors at ramp and leads to reformation.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-6012766/v1/b9cbce8e331241f3e60c84e1.png"},{"id":76181155,"identity":"a70eca41-eced-4af2-86dc-c09fea2ec8dc","added_by":"auto","created_at":"2025-02-13 07:25:06","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":904097,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSchematic of the waves at shock and their roles in shock reformation. \u003c/strong\u003eColor shades represent the magnetic field strength. The black curves represent wave modes and the arrows with label “\u003cem\u003ek\u003c/em\u003e” represent the directions of wavevectors. Label “\u003cem\u003eOld→New\u003c/em\u003e” represents the shock reformation between the old and new shocks (ramps marked by nearly-vertical thick colored curves). Reformation mechanisms vary over time, “Shock Gradient-Whistler” at the earlier time and “Reflected Ion-Whistler” at the later time.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-6012766/v1/750c48cc06edb92e83082b3d.png"},{"id":76182426,"identity":"90be99f7-fc04-4d3e-ab70-dd417eef67ea","added_by":"auto","created_at":"2025-02-13 07:41:08","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5411877,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6012766/v1/2d18fc16-593f-4b7f-81ae-814815638c08.pdf"},{"id":76181363,"identity":"71948034-f269-4776-a1ff-e7916d704885","added_by":"auto","created_at":"2025-02-13 07:33:05","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":938210,"visible":true,"origin":"","legend":"","description":"","filename":"TableSFigS.docx","url":"https://assets-eu.researchsquare.com/files/rs-6012766/v1/c256e54b3b6849696c251137.docx"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eShock Reformation Induced by Ion-scale Whistler Waves in Quasi-perpendicular Bow Shock\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eCollisionless shocks are ubiquitous phenomena in space and astrophysical plasmas\u003csup\u003e\u003cspan additionalcitationids=\"CR2 CR3 CR4\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e where significant energy conversion occurs between bulk, thermal, and electromagnetic energies. Shocks exhibit intrinsic nonstationarity, arising from microscale physics and influencing shock dynamics\u003csup\u003e\u003cspan additionalcitationids=\"CR7 CR8\" citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e. One form of nonstationarity is shock reformation\u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e. For supercritical shocks with the Alfv\u0026eacute;n Mach number (\u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e) greater than ~\u0026thinsp;3, satisfied by typical planetary bow shocks in the solar system, the shock produces reflected ions, which can contribute to shock reformation by accumulating in front of the shock ramp and inducing growing magnetic fields\u003csup\u003e\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e,\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e. As time goes on, the enhanced foot evolves into a new shock front and then replaces the old one\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e,\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e. This entire process is repeated periodically and named shock reformation\u003csup\u003e\u003cspan additionalcitationids=\"CR16 CR17\" citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eDynamics and structures in the quasi-perpendicular shock transition region vary with shock parameters like \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e, which may be related to different reformation processes. Krasnoselskikh et al.\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e showed that the gradient catastrophe instability at shock ramp induces the whistler precursor, in order to prevent the gradient from growing to infinity. The model predicts that the precursor wavelength is on the ion scale when \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e \u0026lt; \u003cem\u003eM\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e and on the electron scale when \u003cem\u003eM\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e \u0026lt; \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e \u0026lt; \u003cem\u003eM\u003c/em\u003e\u003csub\u003ecwn\u003c/sub\u003e, where, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{M}_{w}=\\:\\frac{\\text{c}\\text{o}\\text{s}\\left({\\theta\\:}_{Bn}\\right)}{2\\sqrt{{m}_{e}/{m}_{i}}}\\)\u003c/span\u003e\u003c/span\u003e is the whistler Mach number, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{M}_{cwn}=\\:\\frac{\\text{c}\\text{o}\\text{s}\\left({\\theta\\:}_{Bn}\\right)}{\\sqrt{2{m}_{e}/{m}_{i}}}\\)\u003c/span\u003e\u003c/span\u003e is the critical whistler nonlinear Mach number\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e. Simulations have suggested that this whistler precursor contributes to shock reformation\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e,\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u003c/sup\u003e. Moreover, modified two-stream instability (MTSI), arising from interactions between reflected/incoming ions and electrons, generates whistler waves in the shock foot region\u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e,\u003cspan additionalcitationids=\"CR20 CR21\" citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/sup\u003e. When an ion beam drifting relative to electrons exists, MTSI can grow at the intersection of two dispersion curves between whistler waves and beam modes\u003csup\u003e\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e, where the resonant ions satisfy \u003cem\u003eω\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003ekV\u003c/em\u003e, corresponding to Landau resonance. 1-D PIC simulations imply the contribution of such MTSI-whistlers to shock reformation\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e,\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e,\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e\u003c/sup\u003e. Furthermore, a type of large-scale shock reformation has also been reported in PIC simulations at large \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e \u0026gt; \u003cem\u003eM\u003c/em\u003e\u003csub\u003ecwn\u003c/sub\u003e, associated with a phase space vortex between the ramp and the upstream region\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e. The contributions of different mechanisms remain controversial, especially in parameter regimes when multiple mechanisms can co-exist and physics of multiple dimensions needs to be included.\u003c/p\u003e \u003cp\u003eUntil now, most evidence for the reformation of the quasi-perpendicular bow shock has been provided by numerical simulations. Detection of shock reformation in space observations requires high time resolutions, best aided with multi-spacecraft measurements. Lobzin etl al.\u003csup\u003e\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e is one of the few successful trials to identify shock reformation using Cluster satellites. In addition, other nonstationary phenomena at shocks, e.g., surface waves\u003csup\u003e\u003cspan additionalcitationids=\"CR27 CR28\" citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/sup\u003e, may appear with similar fluctuations as during the shock reformation\u003csup\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e,\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u003c/sup\u003e, introducing challenges in accurately identifying reformation, not to mention its mechanisms. In this study, we report an Magnetospheric Multiscale (MMS) observation where the reformation in the Earth\u0026rsquo;s quasi-perpendicular bow shock can be unambiguously identified with clearer details compared to Cluster observations. In addition, a 2-D particle-in-cell (PIC) simulation with \u003cem\u003em\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e/\u003cem\u003em\u003c/em\u003e\u003csub\u003ee\u003c/sub\u003e = 1836 is employed, which reproduces the shock reformation observed by MMS and reveals the driving mechanisms.\u003c/p\u003e"},{"header":"Results","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n \u003ch2\u003eMMS observations\u003c/h2\u003e\n \u003cp\u003eMMS crossed a quasi-perpendicular bow shock on 2 February, 2021, at 07:18 UT. At approximately 07:18:15 UT, MMS1 observed a significant increase in the magnetic field amplitude from 12 nT to an overshoot of 70 nT, along with an increase in electron density from 14 cm\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e to 150 cm\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ea). They are typical features of a shock as the spacecraft transitioned from the upstream solar wind toward the magnetosheath. The shock normal vector \u003cem\u003en\u003c/em\u003e is estimated as [0.928, 0.257, -0.269] in GSE coordinates based on the bow shock model\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e32\u003c/span\u003e\u003c/sup\u003e, which is close to that from the mixed mode coplanarity method of magnetic field and ion velocity\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e33\u003c/span\u003e\u003c/sup\u003e upstream and downstream of the shock, with a difference of 12.6 degrees. Figure \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003eb shows the magnetic field vector in a shock coordinate system \u003cem\u003en\u003c/em\u003e-\u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e-\u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, where \u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003en\u003c/em\u003e \u0026times; \u003cem\u003eB\u003c/em\u003e\u003csub\u003eu\u003c/sub\u003e / |\u003cem\u003eB\u003c/em\u003e\u003csub\u003eu\u003c/sub\u003e|, \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e \u0026times; \u003cem\u003en\u003c/em\u003e, and \u003cem\u003eB\u003c/em\u003e\u003csub\u003eu\u003c/sub\u003e is the upstream magnetic field (average during 07:17:20 to 07:17:40 UT). Thus, we determine the angle (\u003cem\u003e\u0026theta;\u003c/em\u003e\u003csub\u003eBn\u003c/sub\u003e) between \u003cstrong\u003eB\u003c/strong\u003e\u003csub\u003eu\u003c/sub\u003e and \u003cem\u003en\u003c/em\u003e as 67\u0026deg;, i.e., a quasi-perpendicular bow shock. In the ion normal velocity (\u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e) spectrum (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ec), ion reflections are detected in the foot and ramp regions, where the intense phase-space density forms an arc that connects between incoming solar wind ions at \u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e \u0026lt; 0 and reflected ions at \u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e \u0026gt; 0. Key shock parameters are listed in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\u0026nbsp;\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eShock parameters in MMS observation and PIC simulation\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"3\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eShock parameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eIn MMS observations\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePIC simulation\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003e\u0026theta;\u003c/em\u003e\u003csub\u003eBn\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e67\u0026deg;\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e67\u0026deg;\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eUpstream Alfv\u0026eacute;n speed (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e67.94\u003cem\u003ekm\u003c/em\u003e/\u003cem\u003es\u003c/em\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eAlfv\u0026eacute;n Mach number (\u003c/strong\u003e\u003cstrong\u003eM\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003eA\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e6.43\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e8.74\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eIon beta (\u003c/strong\u003e\u003cstrong\u003e\u0026beta;\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003ei\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.63\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.63\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eElectron beta (\u003c/strong\u003e\u003cstrong\u003e\u0026beta;\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003ee\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.87\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.87\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eCritical whistler nonlinear Mach number (\u003c/strong\u003e\u003cstrong\u003eM\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003ecwn\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e11.79\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e11.79\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eWhistler Mach number (\u003c/strong\u003e\u003cstrong\u003eM\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003ew\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e8.34\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e8.34\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eUpstream ion inertial length (\u003c/strong\u003e\u003cstrong\u003ed\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003ei\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e60\u003c/strong\u003e\u003cstrong\u003ekm\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cdiv class=\"gridtable\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cdiv align=\"char\" class=\"colspec\"\u003eFigures \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ed-f present measurements from MMS2. It registers two primary magnetic peaks around 07:18:13 UT and 07:18:17 UT, respectively (labeled as B and C), in contrast to only one primary magnetic peak observed by MMS1 around 07:18:16 UT (A). Incoming and reflected ion populations separated in \u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e are detected prior to peak B at MMS2 and A at MMS1 but not between peaks B and C at MMS2. Figures \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ei-k present the reduced 2-D velocity distribution functions (VDFs) of ions at A, B, and C in the \u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e \u0026ndash; \u003cem\u003ev\u003c/em\u003e\u003csub\u003et2\u003c/sub\u003e plane. A and B show similar distributions, where reflected ions can be distinguished from the incoming solar wind, forming an arc as they gyro-turn around the magnetic field during reflection\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e30\u003c/span\u003e,\u003cspan class=\"CitationRef\"\u003e34\u003c/span\u003e\u003c/sup\u003e. The similarity indicates that they correspond to the same main shock front. Ions at C exhibit a more isotropic distribution (compared with ion distribution at A and B), where incoming and reflected populations are indistinguishable, indicating more complete ion thermalizations downstream of the shock ramp.\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003em displays the relative positions of the four spacecraft, which are grouped into two pairs: MMS1 \u0026amp; MMS4 and MMS2 \u0026amp; MMS3, primarily separated along \u003cem\u003en\u003c/em\u003e by ~\u0026thinsp;50 km. The downstream pair (MMS2 \u0026amp; MMS3) detects the shock earlier, consistent with that the shock passed by the spacecraft from downstream to upstream directions. Measurements within each pair are nearly identical (not shown), suggesting that the different profiles between MMS2 and MMS1 are likely due to the temporal evolution as the shock propagates in \u003cem\u003en\u003c/em\u003e direction, hereafter referred to as \u0026ldquo;early shock\u0026rdquo; and \u0026ldquo;later shock\u0026rdquo;, respectively. Using the method that considers the thickness of the shock foot with the presence of reflected ions as their gyro-radius\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/sup\u003e, we estimate the shock propagation speed as 17 km/s, suggesting a time lag of approximately 3s between the two spacecraft pairs. Applying a 2.7 s (~\u0026thinsp;3 s) forward shift for \u003cem\u003eB\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e1\u003c/sub\u003e in the early shock (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003eg), the first magnetic peaks from all four spacecraft well align with each other, further validating that they correspond to the same primary shock front. The second peak present in the early shock vanishes in the later shock. We interpret that it represents an old shock, which gradually damps as a new shock grows at the first peak during reformation. Based on the estimated shock speed, the distance between the old and new shocks is approximately 58 km (~\u0026thinsp;1 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e).\u003c/p\u003e\n \u003cp\u003eNext, we highlight another potential shock reformation cycle in the later shock. At approximately 07:18:12.7 UT, a magnetic field pile-up region is observed (marked by the red shadow in Figs. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ea-c) in the foot region. All three magnetic field components exhibit large-amplitude oscillations, leading to a rise in total magnetic field strength with a maximum around 40 nT\u0026mdash;roughly half of that at the main shock front. The distance between this pile-up region and the primary shock front is also close to 1 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e, suggesting a new reformation cycle. Additionally, within this region, the density shows spikes with a maximum reaching 30 cm\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e, and the \u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e spectrum displays a distinct concentration at \u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e \u0026gt; 0 (highlighted by the circle in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ec). These features indicate that ions have been reflected and accumulated in the pile-up region, as in a shock ramp. Figure \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003eh presents the normal component of the electric field (\u003cem\u003eE\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e)recorded by four spacecraft, with the same time shift applied to MMS2 and MMS3 as in \u003cem\u003eB\u003c/em\u003e\u003csub\u003et1\u003c/sub\u003e. Around 07:18:13 UT, positive \u003cem\u003eE\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e spikes are detected in the later shock within the magnetic field pile-up region, while \u003cem\u003eE\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e remains around zero in the early shock. Such upstream-pointing electric field spikes, similar to those reported in previous studies\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e35\u003c/span\u003e,\u003cspan class=\"CitationRef\"\u003e36\u003c/span\u003e\u003c/sup\u003e, introduce an electrostatic potential resembling the cross-shock potential at the shock ramp\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e, and they seem to appear along with a growing shock during reformation. Collectively, these observations strongly suggest that the pile-up region in the foot is evolving into a new shock front, inducing the shock reformation.\u003c/p\u003e\n \u003cp\u003eWe further investigate the properties of magnetic perturbations responsible to the magnetic pile-up. The wave analysis (see detailed description in Methods, subsection Wave analysis) reveals that the angle between wavevector and magnetic field (\u003cem\u003e\u0026theta;\u003c/em\u003e\u003csub\u003ekB\u003c/sub\u003e)\u0026thinsp;~\u0026thinsp;147\u0026deg;, wavelength (\u0026lambda;)\u0026thinsp;~\u0026thinsp;1.2 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e, the phase speed (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eph\u003c/sub\u003e) is ~\u0026thinsp;2.5 \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e in the spacecraft frame, and ~\u0026thinsp;5.5 \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e in the plasma rest frame (PRF), where \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e is the upstream Alfv\u0026eacute;n speed. The waves are propagating upstream, both in the spacecraft frame and the PRF. The ellipticity in both frames are right-handed. The waves are thus classified as an ion-scale whistler mode. The ion VDF in the wave interval (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003el) further shows that shock reflected ions are located near \u003cem\u003eV\u003c/em\u003e\u003csub\u003eph\u003c/sub\u003e, and hence, can be in Landau resonance with these waves. Both the wave properties and resonance conditions are similar to those analyzed in Hull et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e\u003c/sup\u003e and Lalti et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e\u003c/sup\u003e, consistent with waves excited by MTSI resonant with reflected ions.\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003ePIC simulation\u003c/h3\u003e\n\u003cp\u003eIn order to verify the shock reformation in MMS observations and understand the underlined mechanisms, we perform a 2-D particle-in-cell (PIC) simulation (see detailed description in Methods, subsection PIC simulation) in the \u003cem\u003ex\u003c/em\u003e-\u003cem\u003ey\u003c/em\u003e plane (i.e., \u003cem\u003en\u003c/em\u003e-\u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e plane in observations). Thus, this simulation is conducted to investigate the shock dynamics along \u003cem\u003en\u003c/em\u003e and \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, while it should be noted that other reformation related processes can occur along the \u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e direction\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e37\u003c/span\u003e\u003c/sup\u003e. The parameters are selected to represent the observation regime: super-critical with ion reflections, while \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{M}_{A}\\lesssim\\:{M}_{w}\\)\u003c/span\u003e\u003c/span\u003e to allow for ion-scale precursors.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e illustrates the process of shock reformation in PIC. Figures \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ea-d present the 2-D pictures of \u003cem\u003eB\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e at 5.8, 6.0, 6.2 and 6.4 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, where \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e is the upstream ion cyclotron frequency. Waves are present in the foot, propagating oblique to the shock normal and quasi-parallel to the magnetic field, identified as right-hand ion-scale MTSI-whistler waves Landau resonant with reflected ions, similar to those observed in MMS (see details in Methods, subsection Wave analysis). The black ovals circle a wave peak that will evolve into the new shock. At 5.8 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, it exhibits the same properties with the other waveforms in the foot, with consistent amplitudes, wavefront directions, and propagation velocities. As time progresses to 6.0 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, its amplitude significantly increases; at 6.2 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, its wavefront changes and rotates toward the \u003cem\u003ey\u003c/em\u003e-axis, with the orientation gradually aligning with the shock front. Between 6.2 and 6.4 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, the propagation velocity in the \u003cem\u003ey\u003c/em\u003e-direction significantly decreases. Between 5.8 and 6.4 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, the wavelength is the only property that does not change. After 6.4 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, the wave peak merges into the shock, losing properties as a propagating wave.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ee displays 1-D profiles of the total \u003cem\u003eB\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e between 5.6 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e and 6.6 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, cut at the center of the circled patch (marked by white dotted lines in Figs. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ea-d). Initially, the shock consists of a ramp and a foot with several small magnetic peaks, corresponding to the waves seen in 2-D plots. The amplitude of the wave peak closest to the ramp increases with time. At 6.2 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, the magnetic field magnitude of wave peak exceeds that of the old front (labeled as \u0026lsquo;\u003cem\u003eO\u003c/em\u003e\u0026rsquo;) for the first time, marking the evolution into a new shock front (labeled as \u0026lsquo;\u003cem\u003eN\u003c/em\u003e\u0026rsquo;). After 6.4 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, the old and the new fronts merge into a single magnetic peak, signaling the completion of shock reformation. From 5.6 to 6.6 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, the timescale of reformation is approximately 1 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, consistent with previous studies\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e,\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e. We note that the time interval from the early shock at MMS2 to the later shock at MMS1 is ~\u0026thinsp;2.7s (~\u0026thinsp;0.48 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e), during which the shock exhibits obvious evolution but has not yet completed a full reformation cycle, also indicating the reformation period on the order of ion cyclotron period.\u003c/p\u003e\n\u003cp\u003eFigures \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ef-k present a 1-D cut of shock along \u003cem\u003ex\u003c/em\u003e-axis at 6.0 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e at \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.5 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e (along the dotted line in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003eb). The shaded region highlights the new shock forming ahead of the old shock. The new shock shows ion density enhancements and a significant upstream-pointing electric field, and locally reflected ions start to show up with \u003cem\u003ev\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e changing from negative towards positive values. These characteristics resemble the magnetic field pile-up region observed by MMS1, supporting the notion that the pile-up region will evolve into a new shock.\u003c/p\u003e\n\u003cp\u003eFor the present regime of quasi-perpendicular shocks, two types of ion-scale whistlers may develop, and our simulation reveals their co-existence and competition. Figure \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e displays the magnetic fields at 3 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e and 6 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e. At 3 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, two kinds of waves are observed in different regions of the shock. In the shock ramp between the first two vertical dashed lines, a stripe of wave front along y can be seen in the 2-D plot (marked by the black oval in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ea top), mostly phase standing with the shock. In the 1-D cut (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ea bottom), \u003cem\u003eB\u003c/em\u003e\u003csub\u003ey\u003c/sub\u003e and \u003cem\u003eB\u003c/em\u003e\u003csub\u003ez\u003c/sub\u003e exhibit obvious perturbations, while \u003cem\u003eB\u003c/em\u003e\u003csub\u003ex\u003c/sub\u003e does not, further supporting that \u003cstrong\u003ek\u003c/strong\u003e is along the shock normal. The wave is thus identified as whistler precursors, with their generation attributed to shock dynamics. The whistler precursor is produced by the shock itself, resulting from the gradient catastrophe instability predicted by 1-D shock models\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e. The region between the second and third dashed lines corresponds to the shock upstream region. The waves propagate obliquely relative to the shock normal (\u003cstrong\u003ek\u003c/strong\u003e marked by the magenta arrow in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ea top) already identified as MTSI \u0026ndash; whistlers, with their generation attributed to upstream dynamics. At 6 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, the whistler precursors at the ramp disappear, while the MTSI \u0026ndash; whistlers remain. Compared to the earlier time, MTSI \u0026ndash; whistlers extend closer to the shock ramp and exhibit a larger amplitude. It suggests that the MTSI \u0026ndash; whistlers compensate for the magnetic field gradient, so that no additional precursor waves need to be developed in order to avoid the gradient catastrophe, which may explain the disappearance of the precursor.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eBoth the whistler precursors associated with shock dynamics and the MTSI - whistlers associated with upstream dynamics can contribute to shock reformation. At 3 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, the old ramp is located at \u003cem\u003ex\u003c/em\u003e\u0026thinsp;~\u0026thinsp;6.2 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e, while the wave peak of the whistler precursor at \u003cem\u003ex\u003c/em\u003e\u0026thinsp;~\u0026thinsp;6.8 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e evolves into the new ramp, corresponding to precursor \u0026ndash; reformation. At 6 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, the reformed shock front originates from a single wave peak of MTSI \u0026ndash; whistlers in the upstream region, indicating MTSI \u0026ndash; reformation.\u003c/p\u003e\n\u003cp\u003eInterestingly, both reformation scenarios can explain MMS observations. If the entire pile-up region observed by MMS1 is treated as a single wave peak of the whistler precursor, the shock reformation can be attributed to the whistler precursor, as in the simulation at \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;3 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e. The identified MTSI-whistlers in the pile-up region can be regarded as high-frequency magnetic fluctuations superimposed on the low-frequency whistler precursor. On the other hand, if we consider that the magnetic pile-up is purely caused by accumulating 4\u0026ndash;5 periods of the MTSI-whistler wave, the shock reformation occurs when a single period of MTSI-whistler enhances and its propagation velocity gradually decreases and finally aligns with the shock. The MITSI-whistler wavelength is close to the distance between old and new shocks, both ~\u0026thinsp;1 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e. The consistency supports the possibility that the reformed shock front can evolve from a single MTSI-whistler peak, like the simulation around \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;6 \u003cem\u003e\u0026omega;\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e. In the simulation, the new shock front consists of patches (marked by red circles in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eb) with a finite length along y. It originates from the regular MTSI-whistler waveforms but has lost the whistler propagation velocity. Ambient ordinary MTSI-whistler wave fields can propagate through the new shock front, which appears to be a superposition of high-frequency fluctuations on top of low-frequency signals, similar to the features in the magnetic pile-up region observed by MMS1.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eUsing MMS observations, combining the magnetic field and ion measurements from two pairs of spacecraft with a\u0026thinsp;~\u0026thinsp;\u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e scale separation, we unambiguously identified the shock reformation process in a quasi-perpendicular shock. The shock is super-critical with presence of reflected ions and \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e \u0026lt; \u003cem\u003eM\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e. These conditions on \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e ​satisfy the generation criteria for ion-scale whistler precursors. The shock observed slightly earlier exhibits two magnetic peaks, while the later crossing observed one, due to the reformed new shock front and a damping old shock. A magnetic field pile-up associated with ion-scale whistler waves is observed in the foot region, exhibiting features of another growing shock front, including plasma density spikes, ion reflection and accumulation, and upstream-pointing normal electric field spikes. These results provide the first observational evidence that ion-scale whistlers drive the shock reformation in a quasi-perpendicular bow shock.\u003c/p\u003e \u003cp\u003eA 2-D PIC simulation applicable to the observation regime reproduces the reformation and reveals two scenarios (illustrated by Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) for the wave dynamics and shock reformation mechanisms. In the earlier stage (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e left), two types of ion-scale whistler waves, associated with shock dynamics in the ramp and foot regions, respectively, co-exist in the shock transition region. Near the shock front, shock dynamics produces whistler precursors that grows out of the magnetic field gradient, labeled as \u0026ldquo;Shock Gradient-Whistler\u0026rdquo;. In the shock foot region, whistlers are generated by the kinetic instability MTSI resonant with reflected ions, labelled as \u0026ldquo;Reflected Ion-Whistler\u0026rdquo;. The reformation is caused by \u0026ldquo;Shock Gradient-Whistler\u0026rdquo; in the early time (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e left), and by \u0026ldquo;Reflected Ion-Whistler\u0026rdquo; in the later time (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e right) as its residing region extends closer to the shock ramp and it replaces the former wave. The novel understanding of competitions between different shock dynamics demonstrates the complicated and dynamic nature of shock processes.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eOur results provide evidence for shock reformation and its link to ion-scale whistler waves. The conclusions can be applied to collisionless shocks in the same regime that commonly exist in the solar system, advancing our understanding of shock physics and encouraging extensions of relevant studies to additional regimes.\u003c/p\u003e"},{"header":"Methods","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003eData utilization\u003c/h2\u003e \u003cp\u003eThe displayed space observation data in this manuscript are taken from MMS and OMNI. For MMS data, magnetic field measurements are from the Fluxgate Magnetometer (FGM)\u003csup\u003e\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u003c/sup\u003e in the burst mode at 128 samples/s and the electric fields are from the double probes in the fast mode at 32 samples/s\u003csup\u003e39\u0026ndash;41\u003c/sup\u003e. Ion data are provided by the Dual Ion Spectrometer (DIS) onboard the Fast Plasma Investigation (FPI)\u003csup\u003e\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u003c/sup\u003e, with a time cadence of 0.15 s in the burst mode. The electron data are from the Dual Electron Spectrometer (DES) onboard FPI at 0.03 s in the burst mode. Since FPI-DIS is not designed to monitor the solar wind, we use OMNI data to obtain the density and temperature of upstream ions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003eExcluding Ripples as the Cause of Multiple Magnetic Peaks in MMS observations\u003c/h2\u003e \u003cp\u003eMultiple magnetic peaks observed along the spacecraft trajectory may result from other types of shock nonstationarities like the surface wave ripples, which can be excluded in the present case for three reasons. Firstly, when multiple spacecraft with separations along \u003cem\u003en\u003c/em\u003e cross a rippled shock front, the spacecraft entering the downstream region first will transition back into the upstream region later\u003csup\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e,\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u003c/sup\u003e. However, in our case, the magnetic field measurements at the primary shock front from MMS1 (peak A) and MMS2 (peak B) illustrate a first-in-first-out scenario. Secondly, magnetic field profiles from four spacecraft get aligned with a time shift of 2.7 s. This time shift is in close agreement with the shock\u0026rsquo;s propagation time (~\u0026thinsp;3 s) between the two pairs of spacecraft along \u003cem\u003en\u003c/em\u003e. Thirdly, the ion VDF at peak C shows a significant difference from that observed at the main shock front (peaks A and B), suggesting that magnetic peak C is not caused by spacecraft repeatedly crossing the same main shock front due to a rippled shock.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eWave analysis\u003c/h3\u003e\n\u003cp\u003eThe wave analysis of MMS observations is based on the Singular Value Decomposition (SVD) method\u003csup\u003e\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e\u003c/sup\u003e and Minimum Variance Analysis (MVA) method\u003csup\u003e\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e\u003c/sup\u003e. Fig. S1a presents the magnetic field components in field-aligned coordinates from MMS1, and Fig. S1b-e display the results of wave analysis in the spacecraft frame obtained from SVD method. Focusing on the pile-up region, the power spectral densities of the magnetic field reveal a marked increase in 1\u0026ndash;3 Hz range (highlighted by the circles in Fig. S1b-e). The corresponding ellipticities exhibit the positive values close to 1, suggesting right-hand polarized waves in the spacecraft frame. \u003cem\u003eθ\u003c/em\u003e\u003csub\u003ekB\u003c/sub\u003e within the 1\u0026ndash;3 Hz range is quasi-antiparallel, with the phase velocity (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eph\u003c/sub\u003e) ranging from [100, 200] km/s. To further verify the wave properties in the pile-up region, we conduct the MVA of the 1\u0026ndash;3 Hz filtered magnetic field and analyze the \u003cem\u003ek\u003c/em\u003e (wave vector) component of the Poynting flux to remove the 180\u003csup\u003eo\u003c/sup\u003e ambiguity of the MVA result. The results include \u003cem\u003ek\u003c/em\u003e = (0.59, -0.76, 0.27) in the shock coordinate \u003cem\u003en\u003c/em\u003e - \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e - \u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u003cem\u003eθ\u003c/em\u003e\u003csub\u003ekB\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;147\u0026deg;, and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eph\u003c/sub\u003e = 148 km/s (estimated using |δ\u003cem\u003eE\u003c/em\u003e|/|δ\u003cem\u003eB\u003c/em\u003e|) in the spacecraft frame, all close to the SVD results. Using a mean frequency of 2 Hz, the wavelength λ is calculated to be approximately 74 km, close to the upstream \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e = 60 km). Considering that the bulk ion velocity along \u003cem\u003ek\u003c/em\u003e is -177 km/s, the phase speed in the plasma rest frame (PRF) is 325 km/s\u0026thinsp;~\u0026thinsp;5.5 V\u003csub\u003eA\u003c/sub\u003e, so the ellipticity in PRF remains right-handed. Thus, we conclude that the wave at 1\u0026ndash;3 Hz in the pile-up region belongs to an ion scale whistler mode.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003el presents a 2-D ion VDF at 07:18:11 UT observed by MMS1 (labeled as \u0026lsquo;D\u0026rsquo; in Figs.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea-c) in \u003cem\u003ek\u003c/em\u003e-\u003cem\u003eB\u003c/em\u003e plane, where the horizontal axis is along the wavevector \u003cem\u003ek\u003c/em\u003e of the whistlers in the pile-up region and the background magnetic field \u003cb\u003eB\u003c/b\u003e is within the plane shown with a white line. The red line marks the wave phase velocity \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003eph\u003c/em\u003e\u003c/sub\u003e., It falls within the range of the reflected ion population, suggesting that reflected ions can be in Landau resonance with the waves, satisfying \u003cem\u003eω\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003ekV\u003c/em\u003e\u003csub\u003eph\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003eFigure S1f-j present the wave analysis based on MMS2 data with a 2.7 s forward shift as in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eg. The analysis reveals whistler waves observed by MMS2 in the shock foot and near the ramp (highlighted by the circles in Fig. S1g-j), which exhibit properties closely matching those observed by MMS1.\u003c/p\u003e \u003cp\u003eFigure S2 illustrates the wave properties in the shock foot based on the PIC simulation results. Figure S2a displays the 2-D representation of \u003cem\u003eB\u003c/em\u003e\u003csub\u003ez\u003c/sub\u003e at 4.0 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, and we focus on the wave stripes at \u003cem\u003ex\u003c/em\u003e ~ [11, 13] \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e. Based on the wave vector \u003cb\u003ek\u003c/b\u003e perpendicular to the wavefronts as marked by the dashed arrow in Figure S2a, we identify \u003cem\u003eθ\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003en\u003c/sub\u003e as 41.6\u0026deg;, \u003cem\u003eθ\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003eB\u003c/sub\u003e as 30.4\u0026deg;, and the wavelength λ as 0.85 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e. From the displacement of the waveforms between 3.9 and 4.1 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, \u003cem\u003eV\u003c/em\u003e\u003csub\u003eph\u003c/sub\u003e is determined to be 3.39 \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e in the simulation frame, 1.5 \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e in the shock frame, and 6.3 \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e in PRF. Figure S2b presents the hodogram of the magnetic field in the perpendicular plane taken at (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e) = (12.8, -4.1) \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e from 4.0 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e to 4.5 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, which suggests a right-handed polarization in all frames. Therefore, the waves in the shock foot are identified as ion-scale whistler waves.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure S2c-d present the ion (VDF) at (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e) = (12, -4.5) \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e in the coordinate system defined with \u003cb\u003ek\u003c/b\u003e and background \u003cb\u003eB\u003c/b\u003e. The red lines, which mark the velocity component \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e = 3.39 \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e fall within the range of the reflected ion population, suggesting that the reflected ion component of the VDF is in Landau resonance with the waves. We further use three plasma components represented by Maxwellian distributions: electrons, incoming ions, and reflected ions, to fit the VDF shown in Figure S2c-d, and study the linear instability growth using the Bo kinetic dispersion solver\u003csup\u003e\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/sup\u003e. Table S1 displays the fitting parameters for three components. The result (Fig. S2e) shows that the growth rate reaches a maximum of \u003cem\u003eγ\u003c/em\u003e\u003csub\u003emax\u003c/sub\u003e\u0026thinsp;~\u0026thinsp;1.19 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e at \u003cem\u003eω\u0026thinsp;~\u003c/em\u003e\u0026thinsp;3.83 \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e, with λ\u0026thinsp;~\u0026thinsp;0.67 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eph\u003c/sub\u003e ~ 2.55 \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e in the simulation frame. The ratio \u003cem\u003eE\u003c/em\u003e\u003csub\u003ey\u003c/sub\u003e/\u003cem\u003eiE\u003c/em\u003e\u003csub\u003ex\u003c/sub\u003e is approximately 0.67-0.24\u003cem\u003ei\u003c/em\u003e (not shown here), suggesting a right-handed polarization. The wave properties obtained from the kinetic dispersion solver closely match those in the PIC simulation and MMS observations, providing further evidence that the waves are whistlers and grow from MTSI between Landau-resonant reflected ions and electrons.\u003c/p\u003e\n\u003ch3\u003ePIC simulation\u003c/h3\u003e\n\u003cp\u003eThe numerical part of this work employs the two-dimension particle-in-cell simulation using the VPIC code\u003csup\u003e\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e\u003c/sup\u003e. The shock is produced by the injection method: plasmas are injected from the right boundary in a uniform upstream magnetic field condition with +\u0026thinsp;\u003cem\u003ex\u003c/em\u003e and +\u0026thinsp;\u003cem\u003ey\u003c/em\u003e components, and particles are specularly reflected at the left boundary, leading to the formation of a shock that propagates along the +\u0026thinsp;\u003cem\u003ex\u003c/em\u003e direction. The simulation uses periodic boundary conditions along \u003cem\u003ey\u003c/em\u003e. A real mass ratio \u003cem\u003em\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e/\u003cem\u003em\u003c/em\u003e\u003csub\u003ee\u003c/sub\u003e = 1836 is used to achieve a small \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e compared to \u003cem\u003eM\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e and the 2-D setup allows for the wave propagations both along and oblique to the shock normal. A combination of these two conditions leads to high cost of the computation resources, which is partly why few similar studies have been conducted, but it is crucial for resolving the necessary kinetic physics and developing ion-scale waves critical for the reformation process we are interested in. The injection speed is set as 6.18 \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e, and the shock propagates upstream at a speed of 2.56 \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e, resulting in a \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e of 8.74\u0026thinsp;~\u0026thinsp;\u003cem\u003eM\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e = 8.34. The 1-D theory predicts that the whistler precursors transition from ion to electron scales as \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e increases above \u003cem\u003eM\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e. It turns out that our 2-D simulation still develops ion-scale precursors close to the transition threshold, and hence present the same features as in observations with \u003cem\u003eM\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e \u003cem\u003e\u0026lt;M\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e, justifying the usage of our simulation to conduct comparative analyses with observations. Other shock parameters are configured to match the MMS observation as shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The system size is \u003cem\u003eL\u003c/em\u003e\u003csub\u003ex\u003c/sub\u003e \u0026times; \u003cem\u003eL\u003c/em\u003e\u003csub\u003ey\u003c/sub\u003e = 36 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e \u0026times; 12 \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e with 6600 \u0026times; 2200 grids, the electron plasma to cyclotron frequency ratio based on upstream conditions is \u003cem\u003eω\u003c/em\u003e\u003csub\u003epe\u003c/sub\u003e/\u003cem\u003eω\u003c/em\u003e\u003csub\u003ece\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;5, and 200 particles per species are initialized in each computational cell with a uniform distribution. In simulation results, time will be given in units of the inverse of the ion cyclotron frequency \u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e, distances are in units of the ion inertial length \u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e (\u003cem\u003ed\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e = \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e/\u003cem\u003eω\u003c/em\u003e\u003csub\u003eci\u003c/sub\u003e), the velocity is in units of the upstream Alfv\u0026eacute;n speed \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e, and the density is in upstream density \u003cem\u003eN\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e.\u003c/p\u003e"},{"header":"Declarations","content":" \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003eData availability\u003c/h2\u003e \u003cp\u003eMMS data are available at \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://lasp.colorado.edu/mms/sdc/public/data\u003c/span\u003e\u003cspan address=\"https://lasp.colorado.edu/mms/sdc/public/data\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e following the directories: mms#/fgm/brst/l2 for FGM data, mms#/edp/brst/l2/dce/ for electric field data, mms#/fpi/brst/l2/dis-dist for FPI ion distributions, mms#/fpi/brst/l2/dis-moms for FPI ion moments, and mms#/fpi/brst/l2/des-moms for FPI electron moments. OMNI data used are available at \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://omniweb.gsfc.nasa.gov/\u003c/span\u003e\u003cspan address=\"https://omniweb.gsfc.nasa.gov/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Simulation data are available at \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://zenodo.org/records/14625758\u003c/span\u003e\u003cspan address=\"https://zenodo.org/records/14625758\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003eCode availability\u003c/h2\u003e \u003cp\u003eMMS data analysis is conducted using SPEADAS software (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttp://themis.ssl.berkeley.edu\u003c/span\u003e\u003cspan address=\"http://themis.ssl.berkeley.edu\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e). The Bo solver can be found at \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttp://dx.doi.org/10.17632/cvbrftzfy5.1\u003c/span\u003e\u003cspan address=\"10.17632/cvbrftzfy5.1\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. VPIC codes are available at \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://github.com/lanl/vpic\u003c/span\u003e\u003cspan address=\"https://github.com/lanl/vpic\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e\u003cp\u003e \u003ch2\u003eCompeting interests\u003c/h2\u003e \u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eWe thank the MMS team (PI: James Burch) for providing the high-quality data. SBX and SW acknowledge the helpful discussions with Yong-Fu Wang, Yi-Xin Sun, Zhi-Yang Liu, and the ISSI team 555 of \u0026ldquo;Impact of upstream mesoscale transients on the near-Earth environment\u0026rdquo; led by Primoz Kajdic and Xochitl Blanco-Cano. Work at PKU is supported by the National Natural Science Foundation of China Grants 42374188 and 42330202. The simulation work was carried out at National Supercomputer Center in Tianjin, China, and the calculations were performed on TianHe-HPC.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eBale SD et al (2005) Quasi-perpendicular shock structure and processes. Space Sci Rev 118:161\u0026ndash;203\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRichardson JD, Kasper JC, Wang C, Belcher JW, Lazarus AJ (2008) Cool heliosheath plasma and deceleration of the upstream solar wind at the termination shock. 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Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Peking University","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-6012766/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6012766/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eStudies have long suggested that shocks can undergo cyclical self-reformation as a type of shock nonstationarity. Until now, providing solid evidence for shock reformation in spacecraft observation and identifying its generating mechanisms remain challenging. In this work, by analyzing Magnetospheric Multiscale (MMS) spacecraft observations, we unambiguously identified shock reformation occurring in a quasi-perpendicular shock. A 2-D particle-in-cell simulation reproduces and explains the observed shock reformation. The simulation reveals that shock dynamics in the foot and ramp region generate two types of ion-scale whistler waves, respectively, each of which can drive shock reformation. Within one single wave period, the wave induces the magnetic field pile-up, ion accumulation and reflection, and upstream-pointing electric field, finally evolving into a new shock front. An interesting finding is that different shock dynamics compete and dominate the reformation at different stages. Our results not only provide evidence that the shock reformation in the present regime can be driven by ion-scale whistler waves, but also demonstrate the detailed kinetic processes how it happens, offering valuable insights into shock dynamics.\u003c/p\u003e","manuscriptTitle":"Shock Reformation Induced by Ion-scale Whistler Waves in Quasi-perpendicular Bow Shock","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-02-13 07:25:00","doi":"10.21203/rs.3.rs-6012766/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b0c2585b-e3a4-48c9-8990-72c03c3bab7d","owner":[],"postedDate":"February 13th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-02-13T07:25:01+00:00","versionOfRecord":[],"versionCreatedAt":"2025-02-13 07:25:00","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6012766","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6012766","identity":"rs-6012766","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}