We present the analysis of 1,831 current sheets (CS) observed aboard four Cluster spacecraft in a pristine solar wind. Four-spacecraft estimates of the CS normal and propagation velocity are compared with different single-spacecraft estimates. The Minimum Variance Analysis (MVA) of the magnetic field is shown to be highly inaccurate in estimating the normal. The MVA normal often differs by more than 60 degrees from the normal obtained by multi-spacecraft timing method, likely due to ambient turbulent fluctuations. In contrast, the cross-product of magnetic fields at the CS boundaries delivers the normal with an uncertainty of less than 15 degrees at the confidence level of 90%. The CSs are essentially frozen into plasma flow, since their propagation velocity is consistent with local ion flow velocity within 20% at the confidence level of 90%. The single-spacecraft methodology based on the cross-product method and frozen-in assumption delivers the CS thickness and current density amplitude within 20% of their actual values at the confidence level of 90%. The CSs are kinetic-scale structures with half-thickness lambda from a few tenths to tens of local proton inertial length lambda p and scale-dependent shear angle and current density amplitude, Delta theta proportional to lambda/lambda p0.5 ${\Delta }\theta \propto {\left(\lambda /{\lambda }_{p}\right)}<^>{0.5}$ and J0 proportional to lambda/lambda p-0.5 ${J}_{0}\propto {\left(\lambda /{\lambda }_{p}\right)}<^>{-0.5}$. The classification of the CSs in terms of tangential and rotational discontinuities remains a challenge, because even the four-spacecraft normal has too large uncertainties to reveal the actual normal magnetic field component. The presented results will be valuable for the analysis of solar wind CSs, when only single-spacecraft measurements are available. Current sheets (CS) are sharp magnetic field rotations that could play a role in solar wind heating. These locally planar one-dimensional structures are highly likely produced by turbulence cascade, though some may originate in solar corona. Most of the studies of solar wind CSs were carried out using single-spacecraft measurements, the CS normal was typically computed using Minimum Variance Analysis (MVA), and the other CS properties were estimated using that normal. In this study we present multi-spacecraft analysis of 1,831 CSs observed aboard four Cluster spacecraft and determine the CS normal using four-spacecraft observations. We demonstrate that MVA is highly inaccurate in estimating the CS normal, while the cross-product of magnetic fields at the CS boundaries delivers accurate estimates of the normal. We show that the single-spacecraft methodology based on the cross-product normal and the assumption that solar wind CSs are frozen into local plasma flow provides accurate estimates of the CS thickness and current density. This single-spacecraft methodology assumes zero normal component of the magnetic field and does not allow classifying CSs in terms of tangential and rotational discontinuities, but we show that because of methodology uncertainties four-spacecraft observations do not allow carrying out such a classification either. Four-spacecraft observations are used to estimate the normal and propagation velocity for more than 1,800 current sheets at 1 AU MVA is highly inaccurate in estimating the current sheet normal, while the cross-product method delivers accurate estimates Single-spacecraft methodology based on the cross-product normal and frozen-in assumption is accurate in estimating current sheet properties

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.

We present the first observations of electrostatic solitary waves with electrostatic potential of negative polarity around a fast plasma flow in the Earth's plasma sheet. The solitary waves are observed aboard four Magnetospheric Multiscale spacecraft, which allowed accurately estimating solitary wave properties. Based on a data set of 153 solitary waves, we show that they are locally one‐dimensional Debye‐scale structures with amplitudes up to 20% of local electron temperature and they propagate at plasma frame speeds ranging from a tenth to a few ion‐acoustic speeds at arbitrary angles to the local magnetic field. The solitary waves are associated with multi‐component proton distributions and their velocities are around those of a beam‐like proton population. We argue that the solitary waves are ion holes, nonlinear structures produced by ion‐streaming instabilities, and conclude that once ions are not magnetized, ion holes can propagate oblique to local magnetic field.

Collisionless shocks are known to be natural sources of suprathermal particles, but the mechanism resulting in acceleration of thermal electrons to suprathermal energies still remains elusive. The problem is, that the Diffusive Shock Acceleration (DSA) becomes efficient only for suprathermal electrons, which fluxes in the far upstream region are relatively low. Recent studies have shown that the so-called Stochastic Shock Drift Acceleration (SSDA) mechanism can potentially provide the necessary pre-acceleration of incoming thermal electrons to suprathermal energies. In this mechanism, electrons are temporarily kept trapped in the shock transition region due to magnetic mirror reflection by the magnetic ramp and pitch-angle scattering of electrons trying to escape upstream by wave turbulence. Spacecraft measurements showed that broadband electrostatic turbulence is always present in the Earth’s bow shock, but its efficiency in scattering suprathermal electrons has not been estimated up to date. In this study we have quantified the electron scattering by the broadband electrostatic turbulence and, specifically, by electrostatic solitary waves (ion holes) substantially contributing to this turbulence in the Earth’s bow shock. Adopting the solitary wave and turbulence parameters typical of the Earth’s bow shock, we obtain quasi-linear scattering rates and compare these scattering rates to the results of test-particle simulations. This analysis showed that scattering of suprathermal electrons by the osberved electrostatic turbulence is relatively well estimated by the quasi-linear approach. We estimated the quasi-linear scattering rates at various energies and pitch-angles and demonstrated that the electrostatic turbulence in the Earth’s bow shock can provide pre-acceleration of thermal electrons from a few tens of eV to a few hundred eV via the SSDA mechanism. This work was supported by the Russian Scientific Foundation, Project No. 19–12-00313

We present analysis of about one hundred bipolar structures of positive polarity identified in ten quasi-perpendicular crossings of the Earth's bow shock by the Magnetospheric Multiscale spacecraft. The bipolar structures have amplitudes up to a few tenths of local electron temperature, spatial scales of a few local Debye lengths, and plasma frame speeds of the order of local ion-acoustic speed. We argue that the bipolar structures of positive polarity are slow electron holes, rather than ion-acoustic solitons. The electron holes are typically above the transverse instability threshold, which we argue is due to high values of the ratio ωpe/ωce between electron plasma and cyclotron frequencies. We speculate that the transverse instability can strongly limit the lifetime of the electron holes, whose amplitude is above a certain threshold, which is only a few mV/m in the Earth's bow shock. We suggest that electron surfing acceleration by large-amplitude electron holes reported in numerical simulations of high-Mach number shocks might not be as efficient in realistic shocks, because the transverse instability strongly limits the lifetime of large-amplitude electron holes at ωpe/ωce values typical of collisionless shocks in nature.

The electrostatic fluctuations are always present in the Earth's bow shock at frequencies above about 100 Hz, but the effects of this wave activity on electron dynamics have not been quantified yet. In this paper, we quantify electron pitch-angle scattering by electrostatic solitary waves, which make up a substantial part of the electrostatic fluctuations in the Earth's bow shock and were recently shown to be predominantly ion holes. We present analytical estimates and test-particle simulations of electron pitch-angle scattering by ion holes typical of the Earth's bow shock and conclude that this scattering can be rather well quantified within the quasi-linear theory. We use the observed distributions of ion hole parameters to estimate pitch-angle scattering rates by the ensemble of ion holes typical of the Earth's bow shock. We use the recently proposed theory of stochastic shock drift acceleration to show that pitch-angle scattering of electrons by the electrostatic fluctuations can keep electrons in the shock transition region long enough to support acceleration of thermal electrons by a factor of a few tens, that is up to a few hundred eV. Importantly, the electrostatic fluctuations can be more efficient in pitch-angle scattering of ≲1 keV electrons, than typically observed whistler waves.

We present Magnetospheric Multiscale observations of electrostatic double layers in quasi‐perpendicular Earth's bow shock. These double layers have predominantly parallel electric field with amplitudes up to 100 mV/m, spatial widths of 50–700 m, and plasma frame speeds within 100 km/s. The potential drop across a single double layer is 2%–7% of the cross‐shock potential in the de Hoffmann‐Teller frame and occurs over the spatial scale of 10 Debye lengths or one tenth of electron inertial length. Some double layers can have spatial width of 70 Debye lengths and potential drop up to 30% of the cross‐shock potential. The electron temperature variation observed across double layers is roughly consistent with their potential drop. While electron heating in the Earth's bow shock occurs predominantly due to the quasi‐static electric field in the de Hoffmann‐Teller frame, these observations show that electron temperature can also increase across Debye‐scale electrostatic structures.

Collisionless shocks are known to be natural sources of suprathermal particles, but the mechanism resulting in acceleration of thermal electrons to suprathermal energies still remains elusive. The problem is, that the Diffusive Shock Acceleration (DSA) becomes efficient only for suprathermal electrons, which fluxes in the far upstream region are relatively low. Recent studies have shown that the so-called Stochastic Shock Drift Acceleration (SSDA) mechanism can potentially provide the necessary pre-acceleration of incoming thermal electrons to suprathermal energies. In this mechanism, electrons are temporarily kept trapped in the shock transition region due to magnetic mirror reflection by the magnetic ramp and pitch-angle scattering of electrons trying to escape upstream by wave turbulence. Spacecraft measurements showed that broadband electrostatic turbulence is always present in the Earth’s bow shock, but its efficiency in scattering suprathermal electrons has not been estimated up to date. In this study we have quantified the electron scattering by the broadband electrostatic turbulence and, specifically, by electrostatic solitary waves (ion holes) substantially contributing to this turbulence in the Earth’s bow shock. Adopting the solitary wave and turbulence parameters typical of the Earth’s bow shock, we obtain quasi-linear scattering rates and compare these scattering rates to the results of test-particle simulations. This analysis showed that scattering of suprathermal electrons by the osberved electrostatic turbulence is relatively well estimated by the quasi-linear approach. We estimated the quasi-linear scattering rates at various energies and pitch-angles and demonstrated that the electrostatic turbulence in the Earth’s bow shock can provide pre-acceleration of thermal electrons from a few tens of eV to a few hundred eV via the SSDA mechanism. This work was supported by the Russian Scientific Foundation, Project No. 19–12-00313

We present a statistical analysis of >2,100 bipolar electrostatic solitary waves (ESWs) collected from 10 quasi-perpendicular Earth's bow shock crossings by Magnetospheric Multiscale spacecraft. We developed and implemented a correction procedure for reconstruction of actual electric fields, velocities, and other properties of ESW, whose spatial scales are typically comparable with or smaller than spatial distance between voltage-sensitive probes. We found that more than 95% of the ESW are of negative polarity with amplitudes typically below a few Volts and 0.1T(e) (5-30 V or 0.1-0.3T(e) for a few percent of ESW), spatial scales of 10-100 m or lambda(D)-10 lambda(D), and velocities from a few tens to a few hundred km/s that is on the order of local ion-acoustic speed. The spatial scales of ESW are correlated with local Debye length lambda(D). The ESW have electric fields generally oblique to magnetic field and they propagate highly oblique to shock normal N; more than 80% of ESW propagate within 30 degrees of the shock plane LM. In the shock plane, ESW typically propagates within a few tens of degrees of local magnetic field projection B-LM and preferentially opposite to N x B-LM. We argue that the ESW of negative polarity are ion holes produced by ion-ion streaming instabilities. We estimate ion hole lifetimes to be 10-100 ms, or 1-10 km in terms of traveling distance. The revealed statistical properties will be useful for quantitative studies of electron thermalization in the Earth's bow shock.

Slow electron holes, that are electrostatic solitary waves propagating with velocities comparable to the ion thermal velocity, can contribute to plasma heating and provide an anomalous resistivity in various space plasma systems. In addition, the analysis of electron holes allows revealing instabilities operating on time scales not resolved by plasma instruments. We present experimental analysis of more than 100 slow electron holes in the Earth’s bow shock and more than 1000 slow electron holes in the Earth’s nightside magnetosphere. We show that in both regions, the electron holes have similar parameters. The spatial scales are in the range from 1 to 10 Debye lengths, amplitudes of the electrostatic potential are typically below 0.1 of local electron temperature, velocities in the plasma rest frame are of the order of local ion-acoustic velocity. We show that in both regions the electron holes are most likely produced by Buneman-type instabilities. We develop theoretical models of the electron holes and compare them to MMS observations. The lifetime and the transverse instability of the electron holes are discussed. This work was supported by the Russian Scientific Foundation, Project No. 19–12-00313

We present analysis of more than 100 large-amplitude bipolar electrostatic structures in a quasi-perpendicular supercritical Earth's bow shock crossing, measured by the Magnetospheric Multiscale spacecraft. The occurrence of the bipolar structures is shown to be tightly correlated with magnetic field gradients in the shock transition region. The bipolar structures have negative electrostatic potentials and spatial scales of a few Debye lengths. The bipolar structures propagate highly oblique to the shock normal with velocities (in the plasma rest frame) of the order of the ion-acoustic velocity. We argue that the bipolar structures are ion phase space holes produced by the two-stream instability between incoming and reflected ions. This is the first identification of the ion two-stream instability in collisionless shocks.

We present a statistical analysis of large-amplitude bipolar electrostatic structures measured by Magnetospheric Multiscale spacecraft in the Earth's bow shock. The analysis is based on 371 large-amplitude bipolar structures collected in nine supercritical quasi-perpendicular Earth's bow shock crossings. We find that 361 of the bipolar structures have negative electrostatic potentials, and only 10 structures (< 3%) have positive potentials. The bipolar structures with negative potentials are interpreted in terms of ion phase space holes produced by ion streaming instabilities, particularly the two-stream instability between incoming and reflected ions. We obtain an upper estimate for the amplitudes of the ion phase space holes that is in agreement with the measurements. The bipolar structures with positive potentials could be electron phase space holes produced by electron two-stream instabilities. We argue that the negligible number of electron phase space holes among large-amplitude bipolar structures is due to the electron hole transverse instability, the criterion for which is highly restrictive at ωpe/ωce ≫ 1, a parameter range typical of collisionless shocks in the heliosphere and various astrophysical environments. Our analysis indicates that the original mechanism of electron surfing acceleration involving electron phase space holes is not likely to be efficient in realistic collisionless shocks.