<figure><img src="image_1.png" /></figure>

# MMS Direct Observations of Kinetic-scale Shock Self-reformation

Zhongwei Yang<fnref n="1" />,<fnref n="2" />,<fnref n="3" /> ID, Ying D. Liu<fnref n="1" />,⁴ ID, Andreas Johlander<fnref n="5" /> ID, George K. Parks<fnref n="6" /> ID, Benoit Lavraud<fnref n="7" /> ID, Ensang Lee<fnref n="8" /> ID, Wolfgang Baumjohann<fnref n="9" /> ID, Rui Wang<fnref n="1" />, and James L. Burch<fnref n="10" /> ID

Received 2020 July 7; revised 2020 August 27; accepted 2020 August 28; published 2020 September 18

## Abstract

Studies of shocks have long suggested that a shock can undergo cyclical self-reformation on a timescale of ion cyclotron period. This process has been proposed as a primary mechanism for energy dissipation and energetic particle acceleration at shocks. Unambiguous observational evidence, however, has remained elusive. Here, we report direct observations for the self-reformation process of a collisionless, high Mach number, quasi-perpendicular shock using Magnetospheric Multiscale (MMS) measurements. We find that reflected ions by the old shock ramp form a clear phase-space vortex, which gives rise to a new ramp. The new ramp observed by MMS2 has not yet developed to a mature stage during the self-reformation, and is not strong enough to reflect incident ions. Consequently, these ions are only slightly slowed down and show a flat velocity profile from the new ramp all the way to the old one. The present results provide direct evidence of shock self-reformation, and also shed light on energy dissipation and energetic particle acceleration at collisionless shocks throughout the universe.

*Unified Astronomy Thesaurus concepts: Shocks (2086); Solar-planetary interactions (1472); Plasma astrophysics (1261)*

*Supporting material: animation*

## 1. Introduction

Collisionless shocks are a fundamental phenomenon in the inner heliospheric (Bale et al. 2005; Liu et al. 2019; Wilson et al. 2020), outer heliospheric (Richardson et al. 2008; Zank 2015; Zank et al. 2015), astrophysical (Guo et al. 2012; Plotnikov & Sironi 2019), and laboratory plasmas (Schaeffer et al. 2019). They are of great interest because they play key roles in energy dissipation and energetic particle acceleration throughout the universe (Blandford & Eichler 1987; Lembège et al. 2004; Burgess et al. 2005; Wilson et al. 2019). A fraction of the incoming ions can be reflected at high Mach number (supercritical) shocks (Lembège & Dawson 1987; Treumann 2009; Sundberg et al. 2017). Numerical simulations suggest that the reflected ions accumulate in front of the shock ramp, which results in a growing foot in the magnetic field. Under certain conditions, the reflected ions and decelerated incident ions form a vortex in the phase space, which enhances the foot into a new shock ramp (Hada et al. 2003; Scholer et al. 2003). As time goes on, the new ramp replaces the old one. This entire process is repeated periodically and named as shock front self-reformation (Lembege & Savoini 1992; Chapman et al. 2005; Umeda & Yamazaki 2006; Caprioli et al. 2015).

Observational studies of shock front nonstationarity have been performed based on only magnetic field measurements (Horbury et al. 2001; Mazelle et al. 2010). They point out that more detailed analysis, in particular those combining data from field and particle instruments, will greatly improve the

understanding of the collisionless shock phenomenon. Cluster satellites observed quasi-cyclically evolving events of reflected ions at the foot region of Earth's bow shock (Lobzin et al. 2007). However, features of ion velocity distribution functions at the reforming ramps cannot be discriminated because the cadence of their ion measurements is only 4 s. Hence the ion phase-space vortex in small scales responsible for the new ramp formation and corresponding shock microstructures cannot be identified unambiguously. Johlander et al. (2016) and Gingell et al. (2017) argue that shock front rippling could be another source of shock nonstationarity. Identification of the shock self-reformation is difficult because self-reformation and rippling are competing and mixed (Lembège et al. 2009; Hao et al. 2017; Umeda & Daicho 2018; Yang et al. 2018). The following conditions are required to identify shock self-reformation: (1) multi-spacecraft observations; (2) appropriate spacecraft separation and configuration; and (3) high-cadence measurements of particle velocity distributions and magnetic fields.

Here, we present direct in situ observations of full particle dynamics and electromagnetic microstructures associated with the self-reformation of Earth's bow shock using high-resolution measurements from the NASA Magnetospheric Multiscale (MMS) mission (Burch et al. 2016). In order to probe the evolving microstructures with multipoint measurements, the spacecraft separations are required to be as small as 10 km. Three-dimensional ion and electron velocity distributions are

<footnote n="1">¹ State Key Laboratory of Space Weather, National Space Science Center, Chinese Academy of Sciences, Beijing 100190, People's Republic of China;
zwyang1984@gmail.com, liuxying@swl.ac.cn</footnote>

<footnote n="2">² Key Laboratory of Polar Science, MNR, Polar Research Institute of China, Shanghai 200136, People's Republic of China</footnote>

<footnote n="3">³ Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, People's Republic of China
⁴ University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China</footnote>

<footnote n="5">⁵ University of Helsinki, Helsinki, Finland</footnote>

<footnote n="6">⁶ Space Sciences Laboratory, University of California, Berkeley, CA, USA</footnote>

<footnote n="7">⁷ Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, Toulouse, France</footnote>

<footnote n="8">⁸ Kyung Hee University, Yongin, Republic of Korea</footnote>

<footnote n="9">⁹ Space Research Institute, Austrian Academy of Sciences, Graz, Austria</footnote>

<footnote n="10">¹⁰ Southwest Research Institute, San Antonio, TX, USA</footnote>
<figure><img src="image_2.png" /></figure>

**Figure 1.** Geometry of MMS with respect to Earth's bow shock. (a) The locations of the four MMS spacecraft during an inbound bow shock crossing on 2016 January 11, 05:38:00 UT. The white solid curve is the spacecraft orbit on that day. An artistic illustration (modified from https://www.nasa.gov/) is taken as the background to represent the relative locations of Earth's bow shock (white dashed curve), magnetosphere, and the Earth. (b)–(d) The configuration of the four spacecraft in *n*–*t*₁–*t*₂ coordinates. Black, orange, green, and blue dots indicate the four spacecraft MMS1–4, respectively. The separation between MMS1 and MMS2 in the *n* direction (which can observe the shock transition one by one at a timescale of the ion cyclotron time) provides a good opportunity to investigate the shock front self-reformation process.

provided by the Fast Plasma Investigation (FPI) instrument at a time resolution of 150 ms and 30 ms, respectively (Pollock et al. 2016). These high-cadence, multi-spacecraft plasma measurements allow detailed kinetic-scale studies of shock self-reformation.

## 2. Observations and Results

An inbound crossing of Earth's bow shock is illustrated from MMS observations on 2016 January 11 (Figure 1(a)). Figures 1(b)–(d) show the relative positions of the satellites in a coordinate system *n*–*t*₁–*t*₂ as defined by Johlander et al. (2016, 2018), where *n* is the shock normal direction, *t*₂ = *n* × *B*<sub>u</sub>/*|*n* × *B*<sub>u</sub>|*, and *t*₁ = *t*₂ × *n*. Here *B*<sub>u</sub> is the upstream magnetic field vector averaged from 05:37:48 UT to 05:37:58 UT. The shock normal is calculated with a multi-spacecraft timing method (Russell et al. 1983; Schwartz 1998). Typical magnetic field peaks around the overshoot are used for the timing method. The obtained value of *n* is similar to those from single-spacecraft methods and bow shock models (Schwartz 1998). The shock is moving outward at a speed

of ~105 ± 20 km s⁻¹ at 05:38:08 UT in the Earth's frame (also given by the multi-spacecraft timing method; Russell et al. 1983; Schwartz 1998). In Figures 1(b)–(d), MMS1 is immediately followed by MMS3 and MMS4 in the *n* direction during the shock crossing. These three spacecraft (MMS1, 3, 4), which form a plane nearly parallel to the shock surface, observe roughly the same shock structure (see below). This seems to exclude the possibility of shock rippling. The fourth spacecraft (MMS2), which is separated along *n*, sees two shock ramps. For MMS1 and MMS2, their separation Δ*n*(1, 2) is the largest among the four spacecraft and is about 35 km (~0.5 *d*ᵢ, where *d*ᵢ is the ion inertial length and equals about 70 km in this case). Corresponding separations Δ*t*₁(1, 2) and Δ*t*₂(1, 2) are 0.18 *d*ᵢ and 0.04 *d*ᵢ, respectively, so MMS1 and MMS2 cross the shock front one after another at nearly the same location in the *t*₁ and *t*₂ directions. Such a unique configuration of spacecraft provides an opportunity to prove the self-reformation, which will be elaborated below.

Figure 2 shows an overview of the nonstationary shock profiles and corresponding ion velocity distributions observed by MMS1–4. This shock crossing at 05:38:05 UT is associated
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Overview of the Earth's bow shock crossing event on 2016 January 11 by MMS. (a), (c), (e), and (g) Four-spacecraft observations of magnetic field. (b), (d), (f), and (h) Ion velocity distributions with the same colorbar range. The magnetic field profiles observed by MMS1, 3, and 4 are quite similar. MMS2 observes multiple crossings of shock ramps (the old and new ramps are marked by vertical dashed lines). The magnetic field peaks observed at MMS2 are lower than those at MMS1, 3, and 4. The locations of the upstream (US), shock foot, ramp (marked by black dashed lines), overshoot, and downstream (DS) are labeled at the top of the figure.</figcaption></figure>

with an angle between the shock normal and upstream magnetic field $\theta_{Bn} \sim 78^\circ \pm 5^\circ$, Alfvén Mach number $M_A \sim 10.8 \pm 0.6$, and upstream ion $\beta_i \sim 0.3 \pm 0.02$.<fnref n="11" /> The distance between Wind and MMS is about 1.5 million km, and the solar wind speed upstream of the shock is about 400 km s⁻¹. We trace the solar wind parameters from MMS back to Wind, which is about an hour's time difference. Because the shock is observed by MMS at about 05:38 UT, we consider the Wind data in the time range from 04:38 UT to 04:39 UT in the above parameter calculation. This is a supercritical, quasi-

perpendicular shock with a relatively high ion thermal Mach number, which is in favor of shock front self-reformation (Hada et al. 2003; Scholer et al. 2003). In general, the spacecraft except MMS2 observe similar magnetic field profiles and particle distributions. A steep shock ramp with width <0.3dᵢ is observed by MMS1, 3, and 4 at about t = 05:38:05.750 UT (vertical dashed line in Figures 2(a), (e), (g)). If the shock front were stationary, MMS2 would see the same ramp about 0.3 s after MMS1. However, this is not the case. MMS2 observes a new ramp at the same time (marked by the left dashed line in Figure 2(c)), which is not yet as large as those seen at other spacecraft. This new ramp could contain two or more peaks due to the nonlinear evolution of the magnetic field during the self-reformation (Scholer et al. 2003; Yang et al. 2009), as observed here. A degraded old ramp is also seen at t = 05:38:07.400 UT (marked by the right dashed line in Figure 2(c)), about 1.7 s after the new ramp, with a lower

<footnote n="11">¹¹ The FPI instrument on MMS is primarily designed to make accurate measurements of the magnetospheric plasma. Measurements of the cold solar wind beam may be less accurate. However, once the plasma starts being heated from the shock foot the plasma measurements by FPI become reliable. To obtain more accurate plasma parameters far upstream of the shock we have used data from Wind (Ogilvie et al. 1995) that is situated upstream of MMS at L1.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Expanded view of the shock crossings at MMS2. (a) Magnetic field *B*. The old and new ramps are marked by dashed ellipses. (b) Electron density. (c) Electron (solid) and ion (dashed) perpendicular temperature. (d) Phase-space (*time*−*V*<sub>n</sub>) distribution of ions in log scale (log<sub>10</sub>*F*<sub>MMS2</sub>) observed by MMS2. The ion vortex associated with shock self-reformation is marked by the dashed ellipse. Reflected and incident ions are indicated by “R” and “I,” respectively. The reflection point of ions at the shock ramp is marked by “RP,” and the “Flat *V*<sub>n</sub>” in panel (d) indicates that the solar wind ions directly transmitted the new ramp. (e) Similar to (d) but for MMS1 with the same colorbar range. An animation of panel (d) (a 3D view of ion velocity distributions from MMS2) is available, which begins at 05:37:44 UT and ends at 05:38:30 UT.</figcaption></figure>

(An animation of this figure is available.)

magnetic field (<43 nT). The upstream proton cyclotron time
Ω⁻¹ci is about 1.6 s. The self-reformation period is about 1.7 Ω⁻¹ci
from previous simulations (Hada et al. 2003; Lee et al. 2005;
Yang et al. 2009), which is about 2.7 s in our case. The 1.7 s
interval enables MMS2 to observe the evolution of ion kinetic
features around the old and new ramps within one self-
reformation cycle. We will show that such multiple crossings
of shock ramps at MMS2 are caused by the shock self-
reformation, i.e., the growing foot ahead of the old ramp, rather
than shock surface ripples or the back and forth swings of the
shock.

In order to verify that the observed shock front is undergoing
self-reformation, we focus on the multiple crossings of shock
ramps observed by MMS2 (marked in Figure 3 by dashed lines
at 05:38:05.750 UT and 05:38:07.400 UT, respectively). Even
though the magnetic field values observed by MMS2 at the two
ramps are similar, the electron density N<sub>e</sub> (Figure 3(b)) at the
growing foot (i.e., the new ramp) is relatively low compared
with that at the old ramp. It is worth noting that the two large
amplitude peaks in the magnetic field are not correlated with
similar enhancements in the electron density. Simple MHD
theory suggests that for a perpendicular shock the magnetic
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Main features of self-reformation observations. (a) Schematic profiles of the magnetic field and ion phase-space distributions $f(V_n, time)$ for shock self-reformation. (b)–(d) Ion velocity distributions $(V_n-V_{r2})$ observed in the shock reformation case corresponding to the new and old ramps and the region in between.</figcaption></figure>

field and density should be compressed by the same ratio. This indicates that the new ramp may not be mature yet. The electron temperature peak (Figure 3(c)) at the growing foot is lower than that at the old ramp. The new ramp is associated with gyro-reflected ions (Figure 3(d)) and a decelerated solar wind beam, which leads to an increase in the ion temperature perpendicular to the magnetic field direction (Figure 3(c)) due to phase mixing (Scholer & Matsukiyo 2004). The density and temperature profiles should be similar if the multiple crossings are caused by the rippling effect or the back and forth swings of the shock. Obviously, this is not the case, which indicates the possibility of shock self-reformation. Note that the whole time range of Figure 3 is about 6.5 s. Cluster measurements can provide at most two moments of data for this range (4 s resolution) and could not distinguish between shock ripples and self-reformation.

For further confirmation of the shock front self-reformation, we examine the ion phase-space distributions (also see the associated animation of Figure 3, which shows a 3D view of ion velocity distributions observed by MMS2 during the shock crossing from 05:37:44 UT to 05:38:30 UT). Figure 3(d) shows that the incident and reflected ions form a phase-space vortex between the old and new ramps. Such a vortex is different from that observed at a quasi-perpendicular rippling shock; at rippling shocks, the ions are reflected at all the ramps (Johlander et al. 2018; also see below). The bulk velocity $V_n$ of the incident solar wind ions seen at MMS2 is slowed down (from about −350 to −225 km s⁻¹) and forms a flat profile from the leading edge of the new ramp all the way to the old

ramp (Figure 3(d)). In particular, ion reflection does not occur at the new ramp, and all incident ions are directly transmitted. The ion distributions are distinct from those at MMS1 (Figure 3(e)) and the other two spacecraft as well (Figure 2). The ion distribution features at MMS2 agree well with previous simulations of shock self-reformation (Hada et al. 2003; Umeda & Yamazaki 2006), which provides clear evidence for the shock front self-reformation.

Figure 4(a) shows a schematic profile of the magnetic field and ion phase-space distributions $f(V_n, time)$ for a typical self-reforming shock. At the old ramp (Figure 4(d), a high-$B$ region), a fraction of incident ions are being reflected. However, at the new ramp (another high-$B$ region) all the incident solar wind ions are directly transmitted, and the reflected ions by the old ramp and the incident solar wind ions are clearly separated in the velocity space (Figure 4(b)). It indicates that the ion reflection is not occurring in this high-$B$ region. This velocity distribution is similar to that observed in the low-$B$ region between the two ramps (Figure 4(c)). In contrast, in typical rippling shock observations, ion reflections are observed at all the high-$B$ regions (Johlander et al. 2018). These differences reinforce our interpretation of self-reformation for the present case.

### 3. Conclusions and Discussions

Based on the above analysis, we conclude that the multiple ramps observed at MMS2 around 05:38 UT on 2016 January 11 during an inbound crossing of Earth’s bow shock are caused by the shock front self-reformation along the shock normal,
rather than ripples in the shock surface or the back and forth
swings of the shock. The multipoint, high-resolution measure-
ments from MMS provide evidence for shock self-reformation,
i.e., the phase-space vortex in ion velocity distributions that
results in the new shock ramp. In particular, the incident ions
observed by MMS2 show a flat velocity profile from the new
ramp all the way to the old one. This is because the new ramp
has not yet developed to a mature stage during the
self-reformation, and is not strong enough to reflect these ions.
At the growing foot, the reflected ions by the old ramp and the
incident ions are clearly separated. The observed details of
the plasma and magnetic field profiles across the shock show
that it is the ion reflection that drives the self-reformation of
collisionless shocks.

In this Letter, the magnetic peaks observed by MMS2 at the
new ramp are not correlated with similar enhancements in the
electron density. This implies that there may be mechanisms
other than the ion accumulation that drive the growth of the
new ramp. For example, the decoupling of density and
magnetic field compression could imply sub-ion scale physics
driven by Weibel instability at high Mach number shocks
(Sundberg et al. 2017). Alternatively, since the reformation
process is continually dynamic, perhaps this signature can be
understood as a transition on, e.g., a whistler timescale
(Krasnoselskikh et al. 2002). Numerical studies also suggest
that, modified two-stream instability (Scholer et al. 2003;
Scholer & Matsukiyo 2004) may cause multiple magnetic field
peaks. Furthermore, observations of high Mach number shocks
indicate that shock parameters can affect the self-reformation
period (Sulaiman et al. 2015).

Insights gained from this study will help understand the
particle acceleration mechanism and microstructures (Yang
et al. 2009; Matsukiyo et al. 2011) associated with nonsta-
tionary collisionless shocks as well as radio emissions (Liu
et al. 2009; Morosan et al. 2019) and energy dissipation (Parks
et al. 2012; Yang et al. 2014) throughout the universe.

We are grateful to the valuable suggestions from A. Vaivads
from Swedish Institute of Space Physics, and S. J. Schwartz
from University of Colorado, Boulder. Z.W.Y. appreciates
helpful discussions with R. S. Wang from USTC, B. B. Tang
and W. Y. Li from NSSC. We acknowledge the use of data
from MMS (https://lasp.colorado.edu/mms/sdc/), the data-
base on NASA CDAWeb, and the clweb maintained by E.
Penou. The computations and visualizations are performed by
Numerical Forecast Modeling R&D and VR System of State
Key Laboratory of Space Weather, and HPC of Chinese
Meridian Project. This work is jointly supported by the NSFC
under grants 41574140, 41674168, and 41774179, the
Specialized Research Fund for State Key Laboratories of
China, Youth Innovation Promotion Association of the CAS
(2017188), the Open Research Program Key laboratory of
Polar Science, MNR (KP202005), NSF of Beijing Municipality
(1192018), Beijing Outstanding Talent Training Foundation
(2017000097607G049), and Beijing Municipal Science and
Technology Commission (grant Nos. Z191100004319001 and
Z191100004319003). A.J. is supported by Swedish National
Space Board Contracts 139/12 and 97/13.

ORCID iDs

Zhongwei Yang https://orcid.org/0000-0002-1509-1529
Ying D. Liu https://orcid.org/0000-0002-3483-5909
Andreas Johlander https://orcid.org/0000-0001-7714-1870
George K. Parks https://orcid.org/0000-0001-5580-5621
Benoit Lavraud https://orcid.org/0000-0001-6807-8494
Ensing Lee https://orcid.org/0000-0002-7737-0339
Wolfgang Baumjohann https://orcid.org/0000-0001-6271-0110
James L. Burch https://orcid.org/0000-0003-0452-8403

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