Compressive Structures in the Foreshock of Collisionless Shocks

Abstract

Collisionless shocks are fundamental accelerators of energetic particles; yet, the observations of nonlinear foreshock structures, which are essential in acceleration processes, differ significantly between Interplanetary (IP) shocks and planetary bow shocks. We present a direct comparison of two high-Mach-number, quasi-parallel shocks: an IP shock observed by Solar Orbiter and the Earth's bow shock measured by the Magnetospheric Multiscale (MMS) mission during the 2024-2025 ``string-of-pearls'' campaign. We show that Foreshock Compressive Structures (FCSs) initiate upstream of both shocks at similar normalized distances ($\lesssim$50 ion inertial lengths, $d_i$) when the suprathermal ($>10$ keV) ion density exceeds $\sim$1\% of the background. However, the IP shock lacks the fully evolved, high-amplitude Short Large Amplitude Magnetic Structures (SLAMS) characteristic of the terrestrial foreshock. We demonstrate that the ``growth zone'' capable of sustaining these structures is spatially limited ($\sim$135 $d_i$), which, due to the high speed of the propagating IP shock, corresponds to a brief observational window of $<10$ s. Beyond this observational constraint, we suggest an additional physical mechanism that can inhibit foreshock maturity at IP shocks. The lack of global curvature prevents the lateral supply (``cross-talk'') of energetic ions from different shock regions. These findings suggest that while the fundamental physics of FCS initiation is unified across collisionless shocks, the achievement of full nonlinearity can be regulated by the unique shock geometry and upstream properties, while ultimately remaining observationally challenging to identify.

Figures (5)

• Figure 1: Timeseries data for the collisionless shock observations and associated foreshock for Solar Orbiter and MMS, in RTN and GSE coordinates, respectively. Panels (a-c, f-h, k-m) show the magnetic field (nT), plasma density ($\text{cm}^{-3}$), and bulk ion velocity ($\text{km}\,\text{s}^{-1}$). Energetic and suprathermal ion fluxes (keV) are displayed in panels (d, i, n) from EPT+STEP and FEEPS, while the lower energy ion energy fluxes (eV) are shown in (e, j, o) from PAS and FPI. Shaded regions in the time-series panels denote different plasma environments denoted as “solar wind” (green), “far foreshock” region (orange), and “close foreshock” (blue) as described in the text and in Figure 2. The bottom row (p-u) details a foreshock compressive structure observed by Solar Orbiter and one of the compressive structures observed by MMS1. Panels (p, q, s, t) show zoomed-in time series of the magnetic field (nT) and density ($\text{cm}^{-3}$). The MVA interval is shaded in gray, while the wider interval of the zoomed-in panels is shaded in purple in the main timeseries panels. The corresponding hodograms in the L-M plane (nT) are shown in (r, u) with the green and red dots corresponding to start and end points, respectively. The shaded area of Solar Orbiter observations (p,q) and MMS (s,t) correspond to 1 and 10 seconds, respectively. Vertical black dashed lines denote the shock time for both the IP and the bow shock.
• Figure 2: A comparison of magnetic field power spectral densities (PSDs) in the foreshock of the IP shock and Earth's bow shock in the spacecraft frame. For both panels, power-law slopes for different turbulent regimes are shown for reference, with stronger dissipation observed at the IP shock. Panel (a) shows PSD of the magnetic field from the Solar Orbiter spacecraft on August 31, 2022, upstream of the shock. Three distinct plasma environments are compared: the “close foreshock” immediately upstream of the shock (blue), a “far foreshock” region (orange), and a “solar wind” upstream interval representing the background solar wind (green). Characteristic frequencies, including the local ion cyclotron frequency ($f_{\text{ci}}$) and the expected ULF wave frequency takahashi1984dependence, are indicated in the plot. (b) shows a similar analysis for Earth's bow shock on February 27, 2025, observed by the MMS mission. The plot compares the “close foreshock” (MMS1, blue) with the “far foreshock” (MMS4, orange) and a quiet upstream solar wind interval (MMS4, green). In addition to the $f_{\text{ci}}$ and expected frequencies, the approximate location of the secondary peak associated with the presence of whistler waves is shown ($f_w \approx 1$ Hz). The shaded area for the expected ULF range is obtained by assuming an error of $\pm$ 3 nT in upstream magnetic field and $\pm10^\circ$ degrees in the determination of $\theta_{\text{Bn}}$. The intervals used per line are also visualized as shaded areas in Figure 1.
• Figure 3: Comparative analysis of magnetic field and suprathermal particle properties across collisionless shocks. Panel (a) shows Solar Orbiter IP shock observations, while panel (b) presents MMS1 measurements of Earth's bow shock. For each shock event, the upper plots display the magnetic field magnitude $|B|$ (red line) and normalized suprathermal particle density ($n_{st}/n_{sw}$; blue line) as functions of distance along the shock normal. The lower plots show the magnetic field standard deviation $\sigma$(B) using a 15-data point moving window, which quantifies local variability levels. The x-axis represents distance in upstream ion inertial lengths ($d_i$) from the shock crossing ($S = 0$). The background magnetic field is defined as the average value from observations taken at $>$100 $d_i$. Red shaded regions highlight the presence of “compressive structures” occurring at the same relative distance of $\sim 25-50$$d_i$. The two horizontal axes on MMS plots indicate the spatial distance computed with the calculated shock speed along the normal and with the spacecraft speed, representing the upper and lower bounds of the distance from the shock, respectively, while the relative location of MMS4 bounds this in context to Figure \ref{['fig:1']}. More information and details about the suprathermal density and their profiles are detailed in Appendix \ref{['AppendixB']} and in the associated supplementary plots Figures \ref{['fig:appendix_1']} and Figures \ref{['fig:appendix_2']}.
• Figure 4: Derivation of suprathermal ion density ($n_{st}$) for Solar Orbiter and MMS. (a) Time-varying calibration factor for Solar Orbiter STEP data, derived to correct for dead-time saturation near the shock (red dashed line). (b) Solar Orbiter $n_{st}$ time series. The density (black) sums the STEP (cyan) and EPT (orange) components. The inset highlights the rapid density increase driven by STEP immediately upstream. (c) MMS $n_{st}$ time series, combining contributions from FPI (purple, $<$30 keV) and EIS (green, $>$30 keV). The vertical red dotted line marks the shock crossing. (d) Combined Solar Orbiter dynamic spectrum (STEP + EPT) used for integration. (e) Combined MMS dynamic spectrum (FPI + EIS). Cyan dashed lines on panels (d) and (e) show the minimum energy used for the suprathermal density integration
• Figure 5: Suprathermal density ratio ($n_{st,STEP}/n_{st,EPT}$) observed by Solar Orbiter in the last 50 seconds upstream of the IP shock. The plot specifically highlights the transition where lower-energy suprathermals (STEP) overtake the energetic tail (EPT) in terms of number density. Vertical dotted lines mark specific distances from the shock in ion inertial lengths ($d_i$). The dominance crossover (Ratio $>1$) starts at $\sim 315~d_i$ (23 sec), while the steepest gradient in the STEP-to-EPT suprathermal ratio occurs at about $\sim 135~d_i$ (10 sec). The gray dotted line indicates the equivalent spatial separation of MMS4 and MMS1 ($\sim 170~d_i$) for comparison and to associate the observations fo Figures \ref{['fig:1']} and \ref{['fig:3']}.