Earth's Bow Shock Research Articles

Applicability of Bernoulli's theorem to upstream and downstream of the bow shock

In this research, the validity of Bernoulli's theorem for the solar wind crossing the Earth's bow shock has been investigated based on space explorations. Using solar wind plasma and interplanetary magnetic field data from the OMNI database, along with magnetosheath plasma and magnetic field data measured by the magnetospheric multiscale satellite in 2016 and 2018, 95 events were selected to study the Bernoulli invariant from the upstream to the downstream of Earth's bow shock (from the unshocked solar wind region to the shocked magnetosheath region). Statistical investigations show that the differences in the characteristic energy of plasmas between the solar wind region and the magnetosheath region are less than 5%, regardless of the upstream solar wind speed and the satellite's location in the magnetosheath. These results quantitatively support the generalization of Bernoulli's equation to the solar wind flowing through the bow shock, including the contribution of magnetic field energy.

Observation of chaotic fluctuations in turbulent plasma

Turbulence is a prevalent phenomenon in space and astrophysical plasmas, often characterized by stochastic fluctuations. While laboratory experiments and numerical simulations have revealed chaotic behavior, in situ observations of turbulent plasmas in natural environments have predominantly shown highly stochastic signatures. Here, we present unprecedented in situ evidence of chaotic fluctuations in the turbulent solar wind plasma downstream of the Earth's bow shock. By analyzing the relative location of magnetic-field fluctuations on the permutation entropy–complexity plane, we demonstrate that turbulence in the magnetosheath plasma exhibits characteristics of chaotic fluctuations rather than stochastic behavior, diverging from the expected traits of well-developed turbulence. This finding challenges established notions of plasma turbulence and reveals the need for caution when using the magnetosheath as a laboratory for studying plasma turbulence.

Origin of reconnecting current sheets in shocked turbulent plasma.

Magnetic reconnection, the rearrangement of magnetic field topologies, is a fundamental plasma process throughout the universe, which converts magnetic energy to plasma kinetic energy and results in particle energization. A current sheet is a prerequisite for the occurrence of magnetic reconnection. It has been well documented that reconnecting current sheets are prevalent in turbulent plasmas. However, how these current sheets are formed remains unclear. Among natural plasmas, the region downstream of the Earth's bow shock, the magnetosheath, is one of the most turbulent. Here, we show that the reconnecting current sheets in the turbulent magnetosheath originate from the waves in the region upstream of the shock. Once excited, the upstream waves are transmitted across the shock, compressed, and then transformed into current sheets in the downstream region. Magnetic reconnection subsequently occurs in these downstream current sheets. This process can be generalized to various shocked plasmas in astrophysical and laboratorial environments where turbulent magnetic reconnection should be common.

Comparing Information Theory Analysis With Cross‐Correlation and Minimum Variance Analysis of the Solar Wind Structures

AbstractThe space weather effects at the Earth's magnetosphere are mostly driven by the solar wind that carries the interplanetary magnetic field (IMF). In this paper, we use 2 years of data in the solar wind from lunar orbiting ARTEMIS and MMS spacecraft upstream of the Earth's bow shock to study the structure of the IMF. We determine the lag times of IMF structures and their dependence on spacecraft positions by conducting an information theory analysis and comparing it with two traditional analysis methods: cross‐correlation (CC) analysis and minimum variance of magnetic field analysis (MVAB). For the events with long time intervals (i.e., >4 hr) and with small‐spatial separation between the MMS and ARTEMIS along the yGSM‐direction (i.e., <40Re, where Re is the Earth's radius), the lag times based on the CC and the mutual information (MI) analyses statistically agree with each other, with p‐values of 1.675 × 10−7 and 4.833 × 10−9, with the confidence of 95%. Both the results based on MI and CC have a large deviation from the results from MVAB. For some of the events, such a deviation could be improved by taking the fast mode speed into account; however, p‐tests showed that they were not statistically significant to the 95% confidence level.

The Bow Shock and Magnetosheath Responses to Density Depletion Structures

AbstractHot flow anomalies (HFAs) are typical and important foreshock transients characterized by large flow deflection and plasma heating. HFAs can deform the Earth's bow shock by dynamic pressure perturbation resulting in disturbance in the magnetosphere and ionosphere. Traditionally, HFAs are believed to be associated with discontinuities. But recently, HFA‐like structures were simulated by a magnetohydrodynamics (MHD) model without the discontinuity prerequisite. In this study, three HFA‐like structures are shown to present the MHD formation mechanism. Multi‐points observation by the Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission tracks the formation of an HFA from a density depletion upstream. HFAs with high‐ or low‐ density cores are observed by the Magnetospheric Multiscale mission in the flank region. The comparisons of the observation and their simulation show general agreement, particularly in the presence of a core with strong heating and velocity deflection, and two compression regions (shocks) with clear maxima in the ram pressure with a strongly inclined normal boundary at the leading edge and moderately inclined at the trailing edge. Agreement is better when the MHD simulations used a transient change to quasi‐parallel solar wind magnetic field during the events.

Energy Transport and Conversion Within Earth's Supercritical Bow Shock: The Role of Intense Lower‐Hybrid Whistler Waves

AbstractDetailed analysis of a high Mach number quasiperpendicular Earth bow shock crossing by the Magnetospheric Multiscale (MMS) spacecraft fleet reveal that lower‐hybrid (LH) whistler waves generated in the shock foot region transport energy predominately along the shock surface and slightly toward the shock ramp in the shock normal incidence frame, where wave energy accumulates and is dissipated into the plasma. This suggests the LH whistlers play an integral role in energy reconfiguration at high Mach number collisionless shocks with ramifications to plasma heating. The multipoint observations are used to quantify the wave characteristic parameters (via interferometry), Poynting fluxes, and energy conversion rates D, and to assess their scale dependencies and spatial and temporal properties. The whistler associated energy transport and conversion are found to depend on scale and location within the layer. High‐frequency electrostatic waves yield largest values of D. However, the dominant net energy exchange contribution is from the LH whistlers. In the foot spatially temporally coherent net energy exchange from the plasma to whistlers is observed, whereas deeper in the ramp net wave energy dissipation to the plasma is observed exhibiting significant space‐time variability. These results are consistent with the modified two stream instability driven by the relative drift between reflected ions and electrons as the mechanism for wave growth in the foot. Owing to strong electron heating, whistler energy dissipation in the ramp is attributed to Landau damping, which out‐competes the destabilizing effect of the reflected ion and electron drift.

Statistical analysis of high-frequency whistler waves at Earth's bow shock: Further support for stochastic shock drift acceleration

We statistically investigate high-frequency whistler waves (with frequencies higher than ∼10% of the local electron cyclotron frequency) at Earth's bow shock using magnetospheric multi-scale (MMS) spacecraft observations. We focus specifically on the wave power within the shock transition layer, where we expect electron acceleration via stochastic shock drift acceleration (SSDA) to occur associated with efficient pitch-angle scattering by whistler waves. We find that the wave power is positively correlated with both the Alfvén Mach number in the normal incidence frame MA and in the de Hoffmann–Teller frame MA/cos θBn. The empirical relation with MA/cos θBn is compared with the theory of SSDA that predicts a threshold wave power proportional to (MA/cos θBn)−2. The result suggests that the wave power exceeds the theoretical threshold for MA/cos θBn≳30–60, beyond which efficient electron acceleration is expected. This aligns very well with previous statistical analysis of electron acceleration at Earth's bow shock [Oka et al., Geophys. Res. Lett. 33, 5–6 (2006)]. Therefore, we consider that this study provides further support for SSDA as the mechanism of electron acceleration at Earth's bow shock. At higher-Mach-number astrophysical shocks, SSDA will be able to inject electrons into the diffusive shock acceleration process for subsequent acceleration to cosmic-ray energies.

Nonlinear electron scattering by electrostatic waves in collisionless shocks

We present a theoretical analysis of electron pitch-angle scattering by ion-acoustic electrostatic fluctuations present in the Earth's bow shock and, presumably, collisionless shocks in general. We numerically simulate electron interaction with a single wave packet to demonstrate the scattering through phase bunching and phase trapping and quantify electron pitch-angle scattering in dependence on the wave amplitude and wave normal angle to the local magnetic field. The iterative mapping technique is used to model pitch-angle scattering of electrons by a large number of wave packets, which have been reported in the Earth's bow shock. Assuming that successive electron scatterings are not correlated, we revealed that the long-term dynamics of electrons is diffusive. The diffusion coefficient depends on the ratio $\varPhi _0/W$ between the wave packet amplitude and electron energy, $D\propto (\varPhi _0/W)^{\nu }$ . A quasi-linear scaling ( $\nu \approx 2$ ) is observed for sufficiently small wave amplitudes, $\varPhi _0\lesssim 10^{-3}W$ , while the diffusion is nonlinear ( $1<\nu <2$ ) above this threshold. We show that pitch-angle diffusion of ${\lesssim }1$ keV electrons in the Earth's bow shock can be nonlinear. The corresponding diffusion coefficient scales with the intensity $E_{w}^{2}$ of the electrostatic fluctuations in a nonlinear fashion, $D\propto E_{w}^{\nu }$ with $\nu <2$ , while its expected values in the Earth's bow shock are $D\sim 0.1\unicode{x2013}100$ $(T_{e}/W)^{\nu -1/2}\,{\rm rad}^{2}\,{\rm s}^{-1}$ . We speculate that in the Earth's quasi-perpendicular bow shock the stochastic shock drift acceleration mechanism with pitch-angle scattering provided by the electrostatic fluctuations can contribute to the acceleration of thermal electrons up to approximately 1 keV. The potential effects of a finite perpendicular coherence scale of the wave packets on the efficiency of electron scattering are discussed.

Whistler‐Mode Waves Inside Short Large‐Amplitude Magnetic Field Structures: Characteristics and Generation Mechanisms

AbstractShort large‐amplitude magnetic structures (SLAMS) frequently appear near the bow shock. Inside steepening SLAMS, whistler‐mode waves are coherently generated at their leading edges. These waves are crucial for electron dynamics, energy conversion and magnetic reconnection near the shock. Nevertheless, the characteristics and generation of the whistler‐mode waves inside SLAMS are still unclear. In this study, we conducted a statistical analysis of whistler‐mode waves within SLAMS near the Earth's bow shock to investigate their properties and generation mechanisms. We found that these waves are mainly excited by electron temperature anisotropy with 99% of them falling below the nonlinear resonant threshold. The combination of a low background magnetic field and intense steepening at SLAMS' leading edge makes it the main source region of whistler‐mode waves and may induce the nonlinear resonance of whistler‐mode waves.

Electron Acceleration at Earth's Bow Shock Due to Stochastic Shock Drift Acceleration

AbstractWe use the Magnetospheric Multiscale mission (MMS) to study electron acceleration at Earth's quasi‐perpendicular bow shock to address the long‐standing electron injection problem. The observations are compared to the predictions of the stochastic shock drift acceleration (SSDA) theory. Recent studies based on SSDA predict electron distribution being a power law with a cutoff energy that scales with upstream parameters. This scaling law has been successfully tested for a single Earth's bow shock crossing by MMS. Here we extend this study and test the prediction of the scaling law for seven MMS Earth's bow shock crossings with different upstream parameters. A goodness‐of‐fit test shows good agreement between observations and SSDA theoretical predictions, thus supporting SSDA as one of the most promising candidates for solving the electron injection problem.

Electron resonant interaction with whistler-mode waves around the Earth's bow shock. II: The mapping technique

Electron resonant scattering by high-frequency electromagnetic whistler-mode waves has been proposed as a mechanism for solar wind electron scattering and pre-acceleration to energies that enable them to participate in shock drift acceleration around the Earth's bow shock. However, observed whistler-mode waves are often sufficiently intense to resonate with electrons nonlinearly, which prohibits the application of quasi-linear diffusion theory. This is the second of two accompanying papers devoted to developing a new theoretical approach for quantifying the electron distribution evolution subject to multiple resonant interactions with intense whistler-mode wave-packets. In the first paper, we described a probabilistic approach, applicable to systems with short wave-packets. For such systems, nonlinear resonant effects can be treated by diffusion theory, but with diffusion rates different from those of quasi-linear diffusion. In this paper, we generalize this approach by merging it with a mapping technique. This technique can be used to model the electron distribution evolution in the presence of significantly non-diffusive resonant scattering by intense long wave-packets. We verify our technique by comparing its predictions with results from a numerical integration approach.

Electron resonant interaction with whistler-mode waves around the Earth's bow shock I: The probabilistic approach

Adiabatic heating of solar wind electrons at the Earth's bow shock and its foreshock region produces transversely anisotropic hot electrons that, in turn, generate intense high-frequency whistler-mode waves. These waves are often detected by spacecraft as narrow-band, electromagnetic emissions in the frequency range of [0.1, 0.5] of the local electron gyrofrequency. Resonant interactions between these waves and electrons may cause electron acceleration and pitch-angle scattering, which can be important for creating the electron population that seeds shock drift acceleration. The high intensity and coherence of the observed whistler-mode waves prohibit the use of quasi-linear theory to describe their interaction with electrons. In this paper, we aim to develop a new theoretical approach to describe this interaction, which incorporates nonlinear resonant interactions, gradients of the background density and magnetic field, and the fine structure of the waveforms that usually consist of short, intense wave-packet trains. This is the first of two accompanying papers. It outlines a probabilistic approach to describe the wave–particle interaction. We demonstrate how the wave-packet size affects electron nonlinear resonance at the bow shock and foreshock regions, and how to evaluate electron distribution dynamics in such a system that is frequented by short, intense whistler-mode wave-packets. In the Paper II, this probabilistic approach is merged with a mapping technique, which allows us to model systems containing short and long wave-packets.

Rankine–Hugoniot Relations and Magnetic Field Enhancement in Turbulent Shocks

In fast collisionless shocks, the density and magnetic field increase and the plasma is heated. The compression and heating are ultimately determined by the Rankine–Hugoniot relations connecting the upstream and downstream parameters. The standard Rankine–Hugoniot relations take into account only mean upstream and downstream parameters. Observations at the Earth's bow shock show that the downstream magnetic field does not always relax to a uniform state, but large amplitude magnetic oscillations persist. Here, these Rankine–Hugoniot relations are extended to such turbulent shocks where the mean downstream magnetic field is accompanied by magnetic fluctuations. It is shown that the turbulent magnetic field pressure may substantially exceed the pressure of the mean field, while the density compression and heating may be only weakly affected. Thus, strong amplification of the rms magnetic field can be achieved at the expense of a modest reduction of plasma heating.

Solar Wind Parameters Influencing Magnetosheath Jet Formation: Low and High IMF Cone Angle Regimes

AbstractMagnetosheath jets are localized flows of enhanced dynamic pressure that are frequently observed downstream of the Earth's bow shock. They are significantly more likely to occur downstream of the quasi‐parallel shock than the quasi‐perpendicular shock. However, as the quasi‐perpendicular geometry is a more common configuration at the Earth's subsolar bow shock, quasi‐perpendicular jets comprise a significant fraction of the observed jets. We study the influence of solar wind conditions on jet formation by looking separately at jets during low and high interplanetary magnetic field (IMF) cone angles. According to our results, jet formation commences when Alfvén Mach number MA ≳ 5. We find that during low IMF cone angles (downstream of the quasi‐parallel shock) other solar wind parameters do not influence jet occurrence. However, during high IMF cone angles (downstream of the quasi‐perpendicular shock) jet occurrence is higher during low IMF magnitude, low density, high plasma beta (β), and high MA conditions. The distribution of quasi‐parallel (quasi‐perpendicular) jet sizes parallel to flow peaks at ∼0.3 RE (∼0.1 RE). Some quasi‐perpendicular jets formed during high β and MA are particularly small. We show two examples of high β and MA quasi‐perpendicular shock crossings. Jets were observed in the transition region, but not deeper in the magnetosheath. A more detailed look into one jet revealed signatures of gyrating ions, indicating that gyrobunched ions near the shock may produce jet‐like enhancements. Our results suggest that jets form as part of the quasi‐perpendicular shock dynamics amplified by high solar wind MA and β.

Magnetohydrodynamic Shocks Revisited: Magnetically Constraining the Upstream Solar Wind Condition

Magnetohydrodynamic (MHD) shocks are commonly found in various space and astrophysical systems. One often encounters the problem that only the magnetic field data are available when studying interplanetary shocks and planetary bow shocks in the solar system, and not the plasma data due to instrumental and/or telemetry limitations. We aim to reveal the mathematical structure of the MHD Rankine–Hugoniot relation under the condition that magnetic field changes across the shock are known, which can be devised into a diagnosis tool to constrain the shock parameters. The perturbative solution is already known for the Rankine–Hugoniot equation. The solution is combined with the condition of the de Hoffmann–Teller frame for the vanishing convective electric field, and we obtain a two-dimensional matrix equation for the density jump and the Alfvén Mach number. The lesson is that the shock parameters (density jump and Alfvén Mach number) are analytically obtained by inverting the MHD Rankine–Hugoniot equation under the condition that the plasma beta is either a priori set or known. Further numerical studies indicate that the presented shock jump estimation works particularly well for quasi-perpendicular shocks. Moreover, a test case with the Earth bow shock crossing data supports our theoretical development. The presented method opens the door to determining or constraining the shock parameters even if the data are limited to the magnetic field only.

Electron Heating in Shocks: Statistics and Comparison

AbstractSupernova remnant (SNR) shocks are the highest Mach number non‐relativistic shocks in electron‐ion plasmas. These shocks are the most efficient particle accelerators in space. SNR shock parameters are inferred from measurements of electromagnetic radiation from heated and accelerated particles. Temperature of the shock heated electrons is one of the most important parameters in supernova remnant shocks. Knowledge of the downstream electron‐to‐ion temperature ratio or of the ratio of the downstream electron temperature to the incident ion energy is crucial for understanding physics of the very high‐Mach number SNR shocks. Heliospheric shocks have substantially lower Mach numbers than SNR shocks but can be extensively studied in in situ observations with further extrapolation of the findings to higher Mach numbers. Magnetospheric Multiscale mission observations of the Earth bow shock are used to analyze dependence of the electron heating on the shock Mach number. It is found that the ratio of the downstream electron temperature to the incident ion energy decreases with the increase of the Mach number. At high Mach numbers this ratio and stabilizes at about 2.5%. The electron‐to‐ion temperature ratio stabilizes at about 10%. The peak electron temperature occurs at the overshoot maximum, further downstream electrons cool down. The mean ratio of the 4.5 s averages of the downstream and maximum electron temperatures is 0.85. Electron heating does not follow the thermodynamic adiabatic law. The heating and cooling behavior implies that the energy is provided by the overall cross‐shock potential while small‐scale electric fields rapidly isotropize the electron distribution.

Intermittency at Earth's bow shock: Measures of turbulence in quasi-parallel and quasi-perpendicular shocks

Turbulent plasmas such as the solar wind and magnetosheath exhibit an energy cascade that is present across a broad range of scales, from the stirring scale at which energy is injected, down to the smallest scales where energy is dissipated through processes such as reconnection and wave–particle interactions. Recent observations of Earth's bow shock reveal a disordered or turbulent transition region exhibiting features of turbulent dissipation, like reconnecting current sheets. We used observations from magnetospheric multiscale (MMS) over four separate bow shock crossings of varying shock normal angle to characterize turbulence in the shock transition region and how it evolves toward the magnetosheath. These cases studies have been chosen to ensure validity of Taylor's hypothesis, which we discuss in depth. We observe the magnetic spectrum evolving by fitting power laws over many short intervals, finding that the power-law index in the shock transition region is separable from the upstream and downstream plasma, for both quasi-perpendicular and quasi-parallel shocks. Across the shock, we see a change in the breakpoint location between inertial and ion power-law slopes. We also observe the evolution of scale-independent kurtosis of magnetic fluctuations across the shock, finding a reduction of high kurtosis intervals downstream of the shock. Finally, we adapt a method for calculating correlation length to include a high-pass filter, allowing estimates for changes in correlation length across the shock. In a quasi-perpendicular shock, we find the correlation length to be significantly smaller in the magnetosheath than in solar wind; however, the opposite can occur for quasi-parallel shocks.

Jets and Mirror Mode Waves in Earth's Magnetosheath.

Magnetosheath jets are localized plasma structures with high dynamic pressure which are frequently observed downstream of the Earth's bow shock. In this work we analyze Magnetospheric MultiScale magnetic field and plasma data and show that jets can be found in the quasi-perpendicular magnetosheath in regions permeated by Mirror mode waves (MMWs). We show that structures identified as jets by their enhanced dynamic pressure can have very different internal structure, with variable signatures in magnetic field magnitude and components, velocity, and density and can be associated to ion distribution functions of various types. This suggests that jets observed in the quasi-perpendicular magnetosheath are generated by different mechanisms. We find that jets can be related to traveling foreshocks, flux transfer events, and some have MMWs inside them. Our results suggest that some jets have a local source and their formation does not depend on upstream structures. We find that different types of ion distributions can exist inside the jets, while in some cases anisotropic distributions are present, in others counterstreaming distributions exist. We also show that for jets with MMWs inside them, ion distributions can be modulated. This highlights the importance of using ion distributions to identify and classify different types of jets.

Efficient Energy Conversion through Vortex Arrays in the Turbulent Magnetosheath

Turbulence is often enhanced when transmitted through a collisionless plasma shock. We investigate how the enhanced turbulent energy in the Earth's magnetosheath effectively dissipates via vortex arrays. This research topic is of great importance as it relates to particle energization at astrophysical shocks across the universe. Wave modes and intermittent coherent structures are the key candidate mechanisms for energy conversion in turbulent plasmas. Here, by comparing in-situ measurements in the Earth's magnetosheath with a theoretical model, we find the existence of vortex arrays at the transition between the downstream regions of the Earth's bow shock. Vortex arrays consist of quasi-orthogonal kinetic waves and exhibit both high volumetric filling factors and strong local energy conversion, thereby showing a greater dissipative energization than traditional waves and coherent structures. Therefore, we propose that vortex arrays are a promising mechanism for efficient energy conversion in the sheath regions downstream of astrophysical shocks.

Theory of ion holes in plasmas with flat-topped electron distributions

Coherent bipolar electric field structures with negative unipolar potentials are widely observed in space plasmas. These bipolar structures are often found to be ion Bernstein Greene Kruskal (BGK) modes or ion holes. Most theoretical models of ion holes assume them to be stationary with respect to the background plasma that follows either Maxwellian or kappa-type distribution. In this paper, we present a new theoretical model of ion holes where the structures are non-stationary, and electrons follow flat-topped distribution. We use the classical BGK approach to derive the inequality separating allowed and forbidden simultaneous values of amplitude and spatial width of ion holes. The model reveals that the parametric space for the existence of ion holes decreases with their speed. We applied the developed model to the largest available dataset of ion holes obtained from the magnetospheric multiscale spacecraft observations in the Earth's bow shock region.