Shock ripples observed by the MMS spacecraft: ion reflection and dispersive properties

We study one slow partial crossing by MMS of Earth's quasi-perpendicular bow shock on the 6 January 2016, 00:33 UT. Magnetic field data are from the fluxgate magnetometer (Russell et al 2016). Ion and electron data are from the fast plasma investigation (FPI) instruments dual ion/electron spectrometer, FPI-DIS and FPI-DES respectively (Pollock et al 2016). FPI can sample the full 3D distribution function every 150 ms for ions and 30 ms for electrons. The FPI instruments on MMS are primarily designed to make accurate measurements at the magnetopause. Plasma measurements of the cold solar wind beam may therefore be less accurate. As soon as the plasma starts being heated and slowed by the shock the plasma measurements by FPI become reliable. However, to obtain more accurate plasma parameters upstream of the shock we use ion temperature data from the solar wind experiment (Ogilvie et al 1995) on the Wind spacecraft that is situated upstream of MMS at Lagrange point 1. The data from Wind are time-shifted to the bow shock and provided by the OMNI database.

At the time of the event, all four MMS spacecraft experience a partial crossing of the bow shock. The spacecraft are flying in a tetrahedron formation with inter-spacecraft separations of ∼35 km. MMS1 is positioned furthest earthward and therefore reaches furthest downstream in the shock. In this paper, MMS1 will be used as the reference spacecraft. Figure 1 shows magnetic field and ion data for the event. In the figure, MMS starts out in the downstream (shocked) magnetosheath and then crosses the bow shock to the upstream (unshocked) solar wind. The bow shock then slowly moves sunward, leading to MMS encountering first the shock foot then ramp and overshoot before the shock moves back and MMS returns upstream. We adopt the naming inbound phase for the first half of the crossing and outbound phase for the later half. In the partial shock crossing, MMS never reaches the asymptotic downstream plasma as can clearly be seen when comparing to the previous shock crossing on the same day, particularly in ion density. During this almost 1 min long encounter with the shock, MMS observes large fluctuations in magnetic field and plasma parameters with a frequency of close to 0.5 Hz. As we will show below, these fluctuations are due to ripples or surface waves traversing the shock surface. Since the fluctuations are more periodic and less noisy in the outbound phase, we will focus on this period in our analysis. The goal of this paper is to study the physical properties of the shock ripples. Since this partial shock crossing is particularly slow, MMS observes many ripple periods, which allows for detailed analysis of fluctuations and their impact on ion dynamics and reflection.

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Our analysis of the shock requires an accurate determination of the shock normal vector. The inter-spacecraft separation is smaller than the scale of the ripples and the shock crossing is relatively slow. This means that it is impossible to perform four-spacecraft timing on the overall shock structure for a determination of the shock normal vector $\hat{{\bf{n}}}$. Since the shock crossing is partial, MMS never reaches the asymptotic downstream plasma, and therefore mixed mode methods using up- and downstream plasma values (Abraham-Shrauner 1972) are also inapplicable. Instead, we employ a bow shock model (Slavin and Holzer 1981, Schwartz 1998), fitted to the spacecraft position to calculate $\hat{{\bf{n}}}$. The shock speed on the outbound side of the shock crossing in the spacecraft frame was determined using the shock foot thickness method by Gosling and Thomsen (1985) using a foot crossing time of 30s. The resulting normal vector and shock speed are presented in table 1. The acute angle between the shock normal and upstream magnetic field θ_Bn is ∼64° measured immediately after the crossing, and ∼60° measured right before the crossing. The measurement right before the crossing has larger uncertainty in θ_Bn due to larger B fluctuations. From now on we assume that the shock angle is ∼64° and remains constant throughout the shock crossing.

Table 1.  Shock and plasma parameters. The subscript u denotes upstream or unshocked values. Upstream parameters are measured after the shock crossing unless otherwise stated. The Mach numbers are calculated in the normal incidence frame.

Parameter	Value
Magnetic field magnitude Bu	12 nT
Solar wind density Nu	12 cm−3
Solar wind speed Vu	500 km s−1
Alfvén Mach number MA	7
Magnetosonic Mach number Mms	5
Solar wind ion βi,u	0.6
Shock normal $\hat{{\bf{n}}}$ in GSE	(0.95, −0.30, −0.05)
${\hat{{\bf{t}}}}_{1}$ in GSE	(0.03, 0.24, −0.97)
${\hat{{\bf{t}}}}_{2}$ in GSE	(0.33, 0.92, 0.24)
Shock speed, outbound phase Vsh	4 km s−1
Shock normal angle θBn	64 ± 6°
θBn before crossing	60 ± 25°
Ion inertial length di,u	61 km
Ion gyrofrequency fci,u	0.2 Hz
Alfvén speed in overshoot, vA,o	140 km s−1

We use the normal incidence (NI) frame to facilitate comparison to simulations. In the NI frame, the incoming upstream plasma travels along the shock normal, and the shock itself is at rest. The shock aligned coordinate system we use in our analysis is based on $\hat{{\bf{n}}}$ and the average upstream magnetic field B_u. We define ${\hat{{\bf{t}}}}_{2}=\hat{{\bf{n}}}\times {{\bf{B}}}_{u}/| \hat{{\bf{n}}}\times {{\bf{B}}}_{u}|$ and ${\hat{{\bf{t}}}}_{1}={\hat{{\bf{t}}}}_{2}\times \hat{{\bf{n}}}$ completes the right-handed (n, t₁, t₂) system. Now, $n-{t}_{1}$ is the coplanarity plane and ${\hat{{\bf{t}}}}_{2}$ is the out-of-plane direction and is close to the direction of the upstream convection electric field, see figure 1.

The strong fluctuations observed throughout the shock crossing suggests that the shock is non-stationary. Here we investigate the nature of the non-stationarity. The major competing non-stationarity processes in this case are the shock self-reformation (e.g. Lembège and Savoini 1992, Lobzin et al 2007) and shock rippling (e.g. Winske and Quest 1988, Johlander et al 2016). Simultaneous measurements at four different points in the shock can help us to determine whether the shock undergoes self-reformation or is rippled. Figure 2(b) shows ${\rm{\nabla }}B$ calculated from four-spacecraft measurements (Chanteur 1998). In the case of a continuous and laminar shock, one would expect to see the shock ramp as a single gradient increase with ${\rm{\nabla }}B\cdot \hat{{\bf{n}}}\,\lt$ 0. As can be seen in figure 2(b), we observe several peaks in ${\rm{\nabla }}B$. In the case of self-reformation, a new shock front is formed upstream of the shock ramp and then convects downstream, past the spacecraft. An example of 2D simulations with clear reformation signatures can be seen in figures 2(a) and 3 in Lembège and Savoini (2002). In this case, one would expect that ${\rm{\nabla }}B\cdot \hat{{\bf{n}}}$ should show significant positive values in the shock foot when the new shock ramp convects past MMS. These positive values are expected to be comparable in magnitude to the negative values in the ramp. However, as can be seen in figure 2(b) ${\rm{\nabla }}B\cdot \hat{{\bf{n}}}$ is dominantly negative throughout the shock foot. If instead the shock is rippled, the shock is similar to a continuous and laminar shock, but with a wave-like structure of the surface. In this case, ${\rm{\nabla }}B\cdot \hat{{\bf{n}}}$ is also expected to be mainly negative in the shock foot and ramp, in agreement with observations. The fluctuations in ${\rm{\nabla }}B\cdot \hat{{\bf{n}}}$ can be explained by the shock ramp repeatedly moving up- and downstream relative to MMS due to the rippling. Since the ripples exist in the gradient of the shock ramp, ${\rm{\nabla }}B\cdot \hat{{\bf{n}}}$ would not necessarily be positive even if the ripple phase speed has a component downstream. The slightly positive values close to the overshoot are likely because MMS1 reaches the downstream edge of the overshoot and therefore observes a lower B. The behavior of mainly negative gradient in the shock foot is validated by gradients of electron density and temperature (not shown). In addition, as we shall see below, four-spacecraft observations also show that the fluctuations propagate with a large component in the tangential direction i.e. along the shock surface. Thus, we conclude that the multipoint observations from within the shock clearly favor ripples over self-reformation.

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We also study the ion dynamics to investigate the non-stationarity mechanism. Figures 2(c)–(g) shows reduced ion phase-space density along the three principal velocity directions $\hat{{\bf{n}}}$, ${\hat{{\bf{t}}}}_{1}$,and ${\hat{{\bf{t}}}}_{2}$. The ion phase-space density f_i is integrated along the other two velocity directions so that ${F}_{i}({V}_{1})=\int {f}_{i}({\bf{V}}){\rm{d}}{V}_{2}{\rm{d}}{V}_{3}$. Panel (c) shows the full ion distribution as a function of V_n in the spacecraft frame. The speed of the NI frame ${{\bf{V}}}_{{\rm{NI}}}\cdot \hat{{\bf{n}}}={V}_{{\rm{sh}}}$ is marked with a dashed line. In the following panels,we only show partial distributions corresponding to V_n < V_sh for incident ions and V_n > V_sh for reflected ions. We can make several important observations in figure 2. First,there are phase-space hole-like structures in V_n and periodic variability in the amount of reflected ions. Similar behavior is expected from simulations both of self-reformation (Lembège and Savoini 1992) and of ripples (McKean et al 1995). In both cases,the phase-space holes are due to the spacecraft periodically moving between the thermalized shocked downstream plasma and the upstream plasma,which is characterized by counter-streaming solar wind and shock reflected ion populations. Secondly,as can be seen in figure 2(d),the incident solar wind exhibits large fluctuations in V_t1. From the Rankine–Hugoniot relations (Landau and Lifshitz 1960),we find that the change in V_t1 from up- to downstream should be ∼10 km s⁻¹. Any fluctuations arising from a change in shock strength (as in self-reformation) would be of this order. However,the observed fluctuations in V_t1 are >100 km s⁻¹, clearly greater than expected for self-reformation. The V_t1 fluctuations can however be explained by local changes in $\hat{{\bf{n}}}$ due to shock ripples. To estimate the magnitude of the fluctuations from this effect, we assume the Rankine–Hugoniot relations are locally fulfilled in a rippled shock. This assumption is approximative since shock ripples is smaller than the fluid scales required for the Rankine–Hugoniot relations. We find that a relatively small turning of ∼10° in $\hat{{\bf{n}}}$ can explain the observed velocity fluctuations. Also the reflected ion velocity fluctuates greatly in V_t1, consistent with near-specular ion reflection of a shock surface that is periodically turning due to ripples. In figures 2(f)–(g) in the outbound shock foot, we can see the solar wind and reflected ions with large positive V_t2. Further downstream, the solar wind fluctuates in V_t2 due to a variability in the amount of reflected ions that together with the solar wind gyrate around the common center of momentum. The variability in reflected ions can clearly be seen in figure 2(g). A large portion of these reflected ions turn around before they reach upstream to the foot, see panel (f). Overall, the ion distribution functions in figure 2 support the case of non-stationarity in the form of shock ripples.

The ripples appear as quasi-periodic waves in the shock overshoot. Here, we determine the dispersive properties of these waves. The phase speed and direction of propagation, $\hat{{\bf{k}}}$ is determined using timing analysis of low-pass filtered B_n from all four spacecraft. We use the timing method by Vogt et al (2011), which also gives error estimates of speed and propagation direction. Figure 3(a) shows B_n for all four MMS spacecraft as well as the time interval used in the timing. Figure 3(b) also shows B_n time-shifted so that the mean square deviation between the signals is minimized. The resulting phase speed in the spacecraft frame is 133 ± 12 km s⁻¹ and $\hat{{\bf{k}}}=(-0.63,-0.78,0.00)$ in shock aligned coordinates (n, t₁, t₂) with an angular uncertainty of ∼7°. The timing shows that the waves propagate almost entirely in the coplanarity plane. The phase speed in the NI frame is 144 ± 23 km s⁻¹ or $1.0\pm 0.2{V}_{A,o}$, where ${V}_{A,o}$ is the local Alfvén speed in the overshoot. Furthermore, the angle between the local background magnetic field and propagation θ_kB is 42 ± 7°. In the spacecraft frame the frequency f^SC is 0.46 ± 0.04 Hz, where the errors are the FWHM of the peak of the ripples in Fourier-space. In the NI frame, the frequency is ${f}^{{\rm{NI}}}=0.50\pm 0.04\,\mathrm{Hz}$ or $2.8\pm 0.2{f}_{{ci},u}$. This yields a wavelength of λ = 290 ± 40 km or $4.7\pm 0.6{d}_{i,u}$. Overall, the dispersive properties are in good agreement with those observed in 2D and 3D hybrid simulations (Lowe and Burgess 2003, Ofman and Gedalin 2013, Burgess et al 2016), and previous in situ observations, (Johlander et al 2016, Gingell et al 2017).

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The observations show that the shock ripples propagate only in one direction. In our case the projection of wave vector is anti-parallel to an average magnetic field direction. In simulations, it is commonly observed that the waves propagate at the same time parallel and antiparallel to B (Lowe and Burgess 2003, Burgess et al 2016, Lee 2017). We note however that the simulations were performed with θ_Bn close to 90° and that the smaller shock angle in the observations could be the cause for the symmetry breaking of wave propagation direction. In a recent simulation, Gingell et al (2017) observed shock ripples that propagated along B. However, in the simulations, the upstream magnetic field is pointing away from the shock, ${{\bf{B}}}_{u}\cdot \hat{{\bf{n}}}\gt 0$, and one might therefore expect a different direction of propagation. What determines the preferred propagation direction remains an open question and requires further investigation.

The polarization of the ripples provides important information on magnetic structure and wave mode of the ripples. We therefore perform a minimum variance analysis (MVA) (Sonnerup and Scheible 1998) using the magnetic field. For the MVA, we use B measured by MMS1 in the time interval shown in figures 3(a), (b) band-pass filtered between 0.27 and 0.79 Hz. The resulting minimum variance vector, $\hat{{\bf{N}}}$, is within 10° from $\hat{{\bf{k}}}$, in reasonable agreement with the timing analysis. It is also clear that the waves are nearly linearly polarized. Figures 3(c), (d) shows hodograms of B in the minimum N, intermediate M, and maximum L variance directions. The eigenvalue ratio between the maximum and intermediate variance vectors is ∼13. The direction of $\hat{{\bf{L}}}$ is nearly perpendicular to ${\hat{{\bf{t}}}}_{2}$. These observations show that in addition to propagating in the coplanarity plane, the magnetic field fluctuations are also in this plane in the form of linearly polarized ripples.

Here, we in detail analyze how ion reflection depends on the phase of the ripples and on distance to the shock. In order to do this, we divide the shock into phase bins based on B fluctuations. Due to the quasi-periodic behavior of the waves, the phase bins are picked by hand based on the low-pass filtered B for each spacecraft. The phase φ = 0° is defined as the local maximum of B and φ = 180° as the local minimum of B . The bin width is set to 60°, and the rest of the bins are equally spaced between the 0°- and 180°-bins. The time intervals defined for all four spacecraft for every bin are shown in figure 4. After being divided into bins given the phase of the ripples, the data can be further divided into bins defined by the distance along the shock normal. We do this by assuming a constant overall motion of the shock in the normal direction. The time of an observation can then be transformed to distance from the shock. The width of these bins is set to 20 km. By assuming a constant wavelength of the ripples, we can further transform phase into tangential distance along the shock surface. This allows us to create maps of the shock in the $n-{t}_{1}$ plane. An example of such a map is shown in figure 5(a) where the color denotes the average value of all measurements of B by any spacecraft in each bin. In the figure, the data is repeated two times side-by-side. Bins are slanted since the ripples propagate ∼40° downstream with respect to the shock surface. A constant wave phase φ then translates to a line ${t}_{1}=-{{nk}}_{n}/{k}_{t1}+\varphi \lambda /2\pi$. Also included in figure 5(a) are magnetic field lines constructed from the observed B_n and B_t1. For the field lines, we enforce ${\rm{\nabla }}\cdot {\bf{B}}=0$ while preserving the field as close as possible to the original. This does however mean that the B_n fluctuations are somewhat exaggerated in the field lines, likely due to the exclusion of B_t2. The magnetic field lines showing the structure of the ripples together with the maps of different physical parameters are helpful tools when analyzing the ripples.

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Next, we investigate how ions are reflected in the shock. We have previously seen that ion reflection is highly variable in time. To determine how this translates to spatial variations we construct a map showing the density of ions that are moving upstream away from the shock, ${N}_{i,r}$. Most of the reflected ions after a gyration will be moving downstream towards the shock, which can be seen in figure 2(f) as incident ions with high positive values of V_t2. Thus, ${N}_{i,r}$ can be considered as the density of newly reflected ions that are between the point of reflection and the turnaround point. The map is shown in figure 5(b). The figure shows that the density of reflected ions is phased by the ripples and the highest density of reflected ions is observed in the region with the lowest magnetic field. Further upstream ${N}_{i,r}$ is displaced in the positive t₁ direction. This is a clear indication that ion reflection is much more efficient in localized regions of the ripples.

The observed ion distribution functions indicate that ions are being reflected in regions of high B. Ion projections for two representative times are shown in figures 5(c)–(f). Here, F_i is the integrated phase-space density along the out-of-plane velocity component. The distribution in figures 5(c)–(d) is from a region with high B. Here, the solar wind is strongly heated and deflected and a part of the distribution has started to turn around to the upstream. The distribution in figures 5(e)–(f) is from a time with low B. Here instead the incoming solar wind is colder than in the previous case and less deflected. At the same time we see a relatively cold beam of reflected ions. The incoming solar wind and reflected populations are clearly separated in velocity space indicating that ion reflection is not occuring in this region. Instead, these ions were reflected in another part of the rippled shock because the reflected has positive V_t1 and that the ripples move in the negative t₁-direction. We have drawn two approximate ion trajectories in figure 5(b) that explains the observed ion distributions. In the high-B region, the solar wind is slowed down and reflected. In the low-B region, the solar wind is relatively undisturbed, while at the same time ions reflected from the high-B region are observed here due to the motion of the ripples.