Slow electron holes in the Earth's bow shock

The Earth's bow shock is a natural laboratory for in situ analysis of plasma processes in supercritical collisionless shock waves,^2,31,72,73 in which experimental studies are continuously stimulated by remote observations^5,30,63 and simulations^1,11,23 of astrophysical shocks. Numerical simulations suggested that one of the viable mechanisms of thermal electron acceleration in high-Mach number astrophysical shocks can be electron surfing acceleration by electrostatic waves produced in a shock transition region.^{1,12,23,36,55} The electrostatic waves originally proposed to be efficient in electron surfing acceleration are bipolar electrostatic structures with positive polarity of electrostatic potential, known as electron holes, produced in a nonlinear stage of Buneman instability.^23,55,56 This instability is excited in the foot region due to the drift between incoming ions and electrons, which appears to compensate the current of ions reflected by the shock. The surfing acceleration mechanism consists of electron trapping into the bipolar structure and electron acceleration by the motional electric field.^23,56 The relevance of a specific process revealed by numerical simulations to realistic shocks is not always obvious, because numerical simulations typically cannot be performed at realistic geometry and background plasma parameters, such as ion-to-electron mass ratio and the ratio $ω p e / ω c e$ between electron plasma and cyclotron frequencies, critically affecting nonlinear development of plasma instabilities.^{9,22,31,40,57} In this study, we present observations of bipolar electrostatic structures of positive polarity in the Earth's bow shock and discuss the efficiency of these structures in electron surfing acceleration at background plasma parameters typical of realistic shocks.

The presence of large-amplitude (up to a few hundreds mV/m) bipolar electrostatic structures in the Earth's bow shock was originally revealed by electric field measurements aboard Wind spacecraft.^3,4 The bipolar structures were interpreted in terms of electron holes, hypothetically produced by electron-streaming instabilities, although the polarity of their electrostatic potential could not be determined using Wind spacecraft measurements. Surprisingly, the analysis of a few tens of bipolar structures observed in the Earth's bow shock aboard Cluster and Magnetospheric Multiscale (MMS) spacecraft revealed only bipolar structures of negative polarity,^6,21,65,69 which questioned the presence of electron holes in the Earth's bow shock. Quite recently, the analysis of more than 2100 bipolar structures observed aboard Magnetospheric Multiscale spacecraft in ten quasi-perpendicular Earth's bow shock crossings showed that about 95% of the bipolar structures have negative polarity of the electrostatic potential.⁷⁰ These structures were interpreted in terms of ion holes,⁷⁰ that is, electrostatic structures produced in a nonlinear stage of ion-streaming instabilities.^{8,13,20,39,67,68} About 5% of the bipolar structures have positive polarity of the electrostatic potential, but no detailed analysis and interpretation of these structures were performed. The low occurrence rate explains the absence of bipolar structures of positive polarity in the previous studies^6,21,65,69 limited to just a few tens of bipolar structures. Of critical importance are the nature and origin of the bipolar structures of positive polarity, as well as the reasons for their low occurrence rate in the Earth's bow shock.

In this paper, we present detailed analysis of about one hundred bipolar structures of positive polarity identified, but not considered, in Ref. 70. We demonstrate that these structures have speeds on the order of ion-acoustic speed, but they are not ion-acoustic solitons and can be only slow electron holes; such electrostatic structures are often observed in the Earth's plasma sheet^28,35,47 and magnetopause.^19,59 Importantly, the electron holes in the Earth's bow shock violate the criterion of the transverse instability,^25,45 which we argue is due to high values of $ω p e / ω c e$⁠, and can restrict electron hole lifetimes to less than 10 ms. Although the origin of these structures is still elusive, the presented observations demonstrate that at high $ω p e / ω c e$ values, typical of collisionless shocks in nature, the transverse instability substantially restricts electron hole lifetimes and, therefore, is a crucial factor potentially affecting the efficiency of electron surfing acceleration observed in one-dimensional simulations of high-Mach number shocks.^23,36,55

We consider 101 bipolar structures of positive polarity identified in a dataset of more than 2100 solitary waves selected among $≳ 10$ mV/m electric field fluctuations in ten quasi-perpendicular Earth's bow shock crossings.⁷⁰ We use MMS measurements in burst mode, which include electron moments at 0.03s cadence and ion moments at 0.15s cadence provided by the Fast Plasma Investigation instrument,⁴⁹ DC-coupled magnetic field at 128 S/s (samples per second) resolution provided by Digital and Analogue Fluxgate Magnetometers,⁵⁰ AC-coupled electric field at 8192 S/s resolution provided by Axial Double Probe,¹⁷ and Spin-Plane Double Probe.³⁴ Note that electric field measurements are also available at 65 kS/s resolution, but not continuously within a burst mode interval.

Figure 1 presents a schematic of axial and spin-plane double probes. The AC-coupled electric field at 8192 S/s resolution is computed using voltage signals V₁, V₂, V_3, and V₄ of four voltage-sensitive spherical probes mounted on tips of 60-meter antennas in the spacecraft spin plane and signals V₅ and V₆ of two probes mounted on tips of roughly 14.6-meter axial antennas along the spin axis. The voltages of the probes are measured with respect to spacecraft and available only at 8192 S/s resolution. The spin plane electric field components in the coordinate system related to the antennas are computed as $E 12 = 1.8 · ( V 2 − V 1 ) / 2 l 12$ and $E 34 = 1.8 · ( V 4 − V 3 ) / 2 l 34$⁠, while the axial electric field component is $E 56 = 1.65 · ( V 6 − V 5 ) / 2 l 56$⁠, where $l 12 = l 34 = 60$ m and $l 56 = 14.6$ m, and the frequency response factors of spin plane and axial antennas are 1.8 and 1.65. The optimal ratio between frequency response factors of axial and spin plane antennas of about 1.65/1.8 $≈$ 0.9 was determined previously by minimizing the angle between the electric field and propagation direction of ion holes observed in the Earth's bow shock.⁷⁰ While the ratio of the frequency response factors is around 0.9, the absolute value of the spin plane response factor may be actually around 1.35. The use of that factor instead of 1.8 would not affect any conclusions of this study though.

As a bipolar structure propagates across spacecraft, its electric field induces voltage signals on the voltage-sensitive probes. Time delays between voltage signals of the opposing probes can be used to estimate velocity and, hence, spatial scale of the bipolar structure.^58,65,66,70 If the time delay between voltage signals of each pair of opposing probes can be determined reliably (the correlation coefficient between voltage signals exceeds 0.75 and the time delay exceeds 0.06 ms, that is half of the electric field temporal resolution), then the speed V_s of the bipolar structure in the spacecraft rest frame and its direction of propagation $k = ( k 12 , k 34 , k 56 )$ can be determined as follows:^65,66

A solid indicator of a reliable velocity estimate is the smallness of angle θ_kE between propagation direction and electric field polarization direction, since for locally planar electrostatic structures, these directions must coincide. If the time delay between voltage signals of at least one pair of opposing probes cannot be determined reliably, we assume that k is parallel to unit vector $E ̂$ along the electric field polarization direction and use reliable time delays to estimate the speed as follows:^64,66,70

Importantly, to accurately compute $E ̂$⁠, we have to compensate amplitude reduction and widening of electric field signals E_ij, caused by the fact that the spatial scale of bipolar structures is typically comparable with spatial separation between opposing spin plane probes. The details of the corresponding correction procedure are described in Ref. 70.

Figure 2 presents an overview of all ten Earth's bow shock crossings, where 101 bipolar structures of positive polarity were collected. The left panels present magnetic field magnitude profiles across the Earth's bow shock crossings along with the moments of occurrence of the bipolar structures of positive polarity. The fast-magnetosonic Mach number and shock obliqueness are indicated in the panels. The bipolar structures of positive polarity are predominantly observed in the downstream region. More than 70% of the positive polarity structures collected in the ten shocks were observed in three of them crossed on November 2, 2017, and characterized by the lowest fast magnetosonic Mach numbers, M_F = 2.5, 2.7, and 2.8. In two of these shocks, the fraction of positive polarity structures among all the selected bipolar structures exceeds 25%, which is significantly higher than the overall average value of 5%. The right panels present a close view at about 10-ms intervals of parallel electric field waveforms measured at 8192 S/s or, if available, at 65 kS/s resolution. These panels show that the bipolar structures have typical peak-to-peak temporal widths of about 1 ms and peak-to-peak parallel electric field amplitudes up to about 100 mV/m. To address the nature of the bipolar structures of positive polarity, we estimated their velocity and other related properties using the interferometry analysis. The interferometry analysis for bipolar structures 1–3 highlighted in Fig. 2 is demonstrated below, and the revealed properties are given in Table I.

Figure 3 presents the interferometry analysis for bipolar structure 1. Panels (a)–(c) demonstrate voltage signals V_i of individual voltage-sensitive probes, measured with respect to the spacecraft, while panel (d) presents three electric field components E_ij. The fact that this bipolar structure has positive polarity can be revealed by a simple analysis of E₁₂, V₁, and V₂ signals. The bipolar structure propagates from probe 2 to probe 1, because signal $− V 2$ leads signal V₁, while E₁₂ is first directed from probe 2 to probe 1 and then in the opposite direction. This means that in physical space, E₁₂ has a divergent configuration and, hence, a positive polarity of the electrostatic potential. By analyzing voltage signals of two other pairs of opposing probes in a similar fashion, we reaffirm that the bipolar structure is indeed of positive polarity. The three electric field components in panel (d) have similar bipolar profiles, thereby indicating a locally 1D planar configuration of the bipolar structure. It is worth pointing out that signals E₁₂ and E₃₄ appear wider than signal E₅₆, which is an instrumental effect arising for bipolar structures with spatial scales comparable to the length of spin plane antennas. The associated effect is reduction of the actual amplitude of these electric field components.^65,70 After correcting the electric fields (Ref. 70), we use peak-to-peak values of electric field components E_ij to determine unit vector $E ̂$ along the electric field polarization direction. For the bipolar structure in Fig. 3, we found $E ̂ = ( 0.36 , − 0.37 , 0.86 )$ and the corrected electric field E_l along this direction is shown in panel (e).

The obvious correlations between signals V_i and $− V j$ of each pair of the opposing probes enable us to determine corresponding time delays $Δ t i j$⁠, which are indicated in panels (a)–(c). These time delays are used to estimate the speed of the structure V_s in the spacecraft rest frame, and the direction of propagation $k = ( k 12 , k 34 , k 56 )$ using Eq. (1). We found propagation direction $k = ( 0.32 , − 0.34 , 0.88 )$ and speed $V s ≈ 49$ km/s. The angle between vectors k and $E ̂$ is rather small, $θ k E ≈ 3.2 °$⁠, proving that the bipolar structure indeed has a locally 1D planar configuration. Interestingly, the bipolar structure propagates oblique to the local magnetic field at angle $θ k B ≈ 24 °$⁠. We use the estimated spacecraft frame velocity V_s to translate the temporal electric field profiles into spatial profiles and to compute the electrostatic potential of the bipolar structure. The spatial coordinate shown in Fig. 3 was computed as $x = ∫ V s d t$ with x = 0 corresponding to E_l = 0. The electrostatic potential shown in panel (d) is computed as $Φ = ∫ E l V s d t$⁠. We determine the typical spatial scale L as half of the distance between peaks of electric field E_l (the actual spatial width is about four times larger). Panel (d) shows that the spatial scale of the bipolar structure is $L ≈ 24$ m or about $4 λ D$ in units of local Debye length. The amplitude of the electrostatic potential is $Φ 0 ≈ 3.5$ V or about $0.1 T e$ in units of local electron temperature.

The velocity of the bipolar structure in the plasma rest frame is determined as $V s * = V s − k · U p$⁠, where $U p$ is the proton bulk velocity in the spacecraft rest frame measured at the moment closest to the occurrence of the bipolar structure. We found the plasma frame speed $V s * ≈ 20$ km/s, which is of the order of local ion-acoustic speed, $c IA ≈ 57$ km/s. The ion-acoustic speed was estimated as $c IA = ( ( T e + 3 T p ) / m p ) 1 / 2$⁠, where m_p is the proton mass, T_e is the local electron temperature, T_p is the temperature of incoming protons in the upstream region (Wind spacecraft data⁷¹), which is considered to be a proxy of the local temperature of incoming protons and cannot be measured aboard MMS spacecraft. Note that this is only an order of magnitude estimate of the actual ion-acoustic speed, because the ion distribution functions are typically not Maxwellian (Sec. III). Thus, the bipolar structure 1 is a Debye-scale positive polarity structure propagating relatively slowly (much slower than thermal electrons) quasi-parallel to local magnetic field.

We identified only 12 bipolar structures with all three time delays $Δ t i j$ determined reliably. For all these bipolar structures, the angle between propagation direction k and electric field $E ̂$ was within 15°. For 39 bipolar structures in our dataset, we could reliably determine two time delays, while only one time delay could be determined for the remaining 50 bipolar structures. For bipolar structures with one and two reliable time delays, we assumed k parallel to $E ̂$ and used reliable time delays $Δ t i j$ to estimate V_s using Eq. (2). The difference of $≲ 15 °$ between k and $E ̂$ will be used later to estimate uncertainties of our estimates of V_s and other related parameters (spatial scale and amplitude), resulting from our assumption that k is parallel to $E ̂$⁠.

Figure 4 presents the interferometry analysis of bipolar structure 2. The voltage signals of the opposing spin plane probes are well correlated and the corresponding time delays, $Δ t 12$ and $Δ t 34$⁠, are indicated in panels (a) and (b). The bipolar structure propagates from probe 2 to probe 1 and from probe 4 to probe 3, while both E₁₂ and E₃₄ are first positive and then negative. This implies that E₁₂ and E₃₄ have divergent configuration and, hence, the bipolar structure has a positive electrostatic potential. The voltage signals V₅ and $− V 6$ of the axial probes are not very well correlated, but the electric field E₅₆ has a bipolar profile similar to those of E₁₂ and E₃₄. The similarity of the electric field profiles indicates that the bipolar structure has locally 1D planar configuration. After correcting the electric fields, we found the unit vector along the polarization direction, $E ̂ = ( 0.51 , 0.77 , 0.4 )$⁠. Assuming k parallel to $E ̂$⁠, we obtain two independent velocity estimates in the spacecraft rest frame, $V s ( 1 ) = E ̂ 12 l 12 / Δ t 12 ≈ 144$ km/s and $V s ( 2 ) = E ̂ 34 l 34 / Δ t 34 ≈ 155$ km/s, and the averaged value $V s ≈ 150$ km/s. The consistency of independent velocity estimates confirms that the bipolar structure indeed has locally 1D planar configuration. The bipolar structure propagates quasi-parallel to local magnetic field, $θ k B ≈ 10 °$⁠. Panel (e) shows that the bipolar structure has a positive potential with amplitude $Φ 0 ≈ 8 V ≈ 0.07 T e$ and spatial scale $L ≈ 48 m ≈ 4 λ D$⁠. The plasma frame speed $V s * = V s − k · U p ≈ 40$ km/s is comparable with local ion-acoustic speed, $c IA ≈ 100$ km/s. Thus, bipolar structure 2 is a slow Debye-scale structure of positive polarity propagating quasi-parallel to local magnetic field.

Figure 5 presents interferometry analysis of bipolar structure 3. The voltage signals of all opposing probes are rather well correlated, and all the electric field components E_ij have similar bipolar profiles, but only the time delay between V₁ and $− V 2$ could be determined reliably. The bipolar structure propagates from probe 2 to probe 1, while E₁₂ is first positive and then negative, indicating positive electrostatic potential. After correcting the electric fields, we found the polarization direction, $E ̂ = ( 0.61 , − 0.38 , 0.68 )$⁠. Assuming k parallel to $E ̂$ and using the reliable time delay $Δ t 12$⁠, we estimate the spacecraft frame speed, $V s = E ̂ 12 l 12 / Δ t 12 ≈ 503$ km/s, and reveal rather oblique propagation to local magnetic field, $θ k B ≈ 54 °$⁠. The bipolar structure has a positive potential with amplitude $Φ 0 ≈ 6 V ≈ 0.18 T e$ and spatial scale $L ≈ 80 m ≈ 5 λ D$⁠. The speed in the plasma rest frame is $V s * = V s − k · U p ≈ 280$ km/s that is a few times larger than local ion-acoustic speed, $c IA ≈ 60$ km/s. Thus, bipolar structure 3 is also a slow Debye-scale structure of positive polarity, but propagating oblique to local magnetic field.

Figure 6 presents statistical distributions of amplitudes $Φ 0$⁠, spatial scales L and obliqueness θ_kB of all the 101 positive polarity structures. Panels (a) and (b) show that the bipolar structures have typical amplitudes within a few volts and spatial scales of 10–100 meters. Note that the electrostatic potential of the bipolar structures in their rest frame can be rather well fitted to the Gaussian model:

and, therefore, the actual spatial width of the bipolar structures is about four times larger, 40–400 m. Panel (c) shows that about 85% of the bipolar structures propagate quasi-parallel, $θ k B ≲ 30 °$⁠, to local magnetic field, while about 15% of the bipolar structures propagate obliquely, $θ k B ≳ 30 °$⁠.

Before presenting the results of statistical analysis, let us discuss uncertainties of the estimated parameters of the bipolar structures. The uncertainties of both amplitude $Φ 0$ and spatial scale L result from the uncertainty of speed V_s in the spacecraft frame. The experimental method to estimate the latter uncertainty was suggested in Ref. 70 and is based on comparison of independent speed estimates obtained using Eq. (2) for bipolar structures with at least two reliable time delays (Figs. 3 and 4). The analysis of bipolar structures of positive polarity with at least two reliable time delays (more than 60% of our dataset) showed that the two independent speed estimates are basically consistent within 30%. In what follows, we use 30% as typical uncertainty of both amplitude $Φ 0$ and spatial scale L of the bipolar structures.

Figure 7 presents analysis of possible correlations between various parameters of the bipolar structures. In panel (a), we present a scatterplot of normalized amplitudes $e Φ 0 / T e$ vs normalized spatial scales $L / λ D$⁠; error bars correspond to the uncertainty of 30%. First, $e Φ 0 / T e$ is in the range from $10 − 3$ to 0.2, while the spatial scales are typically between λ_D and $10 λ D$⁠. There is no distinct correlation between these parameters, but larger spatial scales clearly correspond to larger amplitudes. In panel (b), we compare speeds of the bipolar structures in the plasma rest frame, $V s * = V s − k · U p$⁠, to local ion-acoustic speed estimated as $c IA = ( ( T e + 3 T p ) / m p ) 1 / 2$⁠. The plasma frame speeds are in the range from a few to a few hundred km/s. Although the speeds of the bipolar structures are clustered around local ion-acoustic speed, there is no discernible correlation between these quantities. Moreover, the spread of the speeds around local ion-acoustic speed is rather wide, such that a substantial fraction of the bipolar structures has speeds several times smaller or larger than local ion-acoustic speed. The error bars in panel (b) represent uncertainties of $V s *$ (due to uncertainties of $k · U p$⁠), which were computed by varying k within $15 °$ of the electric field polarization direction. The relative uncertainties are rather large for bipolar structures with velocities below a few tens of km/s, which is a consequence of plasma flow velocities $U p$ being of a few hundred km/s.

Thus, the bipolar structures of positive polarity observed in the Earth's bow shock are Debye-scale structures propagating typically quasi-parallel to local magnetic field, though highly oblique structures are present too, at plasma frame speeds of the order of local ion-acoustic speed. These structures cannot be ion-acoustic solitons, because at small amplitudes theory predicts that ion-acoustic solitons must have their spatial scale strictly related to amplitude, $e Φ 0 / T e ∝ ( λ D / L ) 2$ (see Refs. 37, 52, and 61). It would be interesting to test the applicability of this scaling relation at the observed amplitudes using the full Sagdeev potential approach.^32,33 The only other interpretation is that these bipolar structures are electron holes, purely kinetic nonlinear structures (Bernstein–Green–Kruskal modes⁷) whose existence is due to a depressed phase space density of electrons trapped within the positive electrostatic potential.^16,24,44,54 Note though that in contrast to widely reported electron holes with speeds of a fraction of electron thermal speed (e.g., Refs. 18, 43, 48, and 60), electron holes in the Earth's bow shock are slow structures. Similar slow electron holes have been recently reported in the Earth's plasma sheet^28,29,35,47 and at the Earth's magnetopause.^19,59 Slow electron holes in these space plasma regions are hypothetically produced by Buneman or two-stream electron instabilities,^35,46 but operation of these instabilities has not been revealed by in situ particle measurements yet. In Sec. III, we present ion and electron velocity distribution functions (VDF) associated with the slow electron holes observed in the Earth's bow shock.

Previous studies suggested that slow electron holes can be produced by Buneman or two-stream electron instabilities (see, e.g., simulations in Refs. 10 and 14 and observations in Refs. 35 and 46). In addition, Kamaletdinov et al.²⁸ have recently shown that whatever is the origin of slow electron holes, they can avoid self-acceleration reported in simulations^15,74 only if their speed is around the local minimum of a double-humped velocity distribution function of background ions (see also detailed theory in Ref. 26). Otherwise, the interaction with ions accelerates an initially slow electron–hole to velocities much larger than ion thermal speed.²⁷ To address the origin and stability of speed of the slow electron holes in the Earth's bow shock, we inspected electron and ion velocity distribution functions (VDF) observed around them. Since the electron VDFs have 30-ms resolution and the typical electron gyroperiod is shorter than a few milliseconds, we present only gyrophase-averaged electron VDFs. The ion VDFs have 150 ms resolution (much smaller than typical ion gyroperiod), and therefore, a more detailed analysis of ion VDFs is presented.

Figure 8 demonstrates electron VDFs in the spacecraft rest frame observed around bipolar structures 1–3. We present phase space densities of electrons averaged over pitch angles in the ranges $( 0 ° , 15 ° ) , ( 75 ° , 105 ° )$⁠, and $( 165 ° , 180 ° )$⁠. For bipolar structures 1 and 2, we can see electron beams at energies of a few tens of eV propagating parallel or anti-parallel to local magnetic field. We inspected similar phase space densities measured around all other slow electron holes in our dataset and found electron beams for about 25% of them. It turned out however that instabilities potentially excited by these beams are unlikely to result in formation of the observed electron holes, because the electron holes typically, though not always, propagated opposite to these electron beams. For about 75% of the electron holes, we did not observe any electron beams, as in the case of bipolar structure 3.

Figure 9 presents ion VDFs in the spacecraft rest frame associated with bipolar structures 1–3. All the distribution functions are in the spacecraft rest frame and in the Geocentric Solar Ecliptic (GSE) coordinate system, whose $z$ axis is perpendicular to the ecliptic plane, while the $x$ axis is directed toward the Sun. Note that the coordinate system related to electric field antennas (Fig. 1) differs from the GSE system by a rotation around the $z$ axis. The propagation direction of bipolar structure 1 in the GSE coordinate system is almost within the yz plane, $k gse = ( 0.19 , 0.44 , 0.88 )$⁠. Panel (a) presents a reduced 2D distribution function of background ions, $∫ VDF d v x$⁠, and indicates the projection of the velocity of bipolar structure 1 onto the $v y v z$ plane. Also, panel (d) presents a 1D ion distribution function reduced over two velocities perpendicular to propagation direction $k gse$⁠. We can see that bipolar structure 1 is accompanied by a single-humped 1D ion distribution and its speed is at a positive slope of this distribution. Panels (b) and (e) present similarly computed reduced ion distribution functions for bipolar structure 2, whose propagation direction was predominantly in the xy plane, $k gse = ( 0.51 , 0.77 , 0.40 )$⁠. The 1D reduced distribution is clearly double-humped with about 300 km/s separation between the two beam-like populations. The speed of bipolar structure 2 is at the local minimum of the double-humped distribution that is exactly what is required for slow electron holes to avoid acceleration due to interaction with ions.^26,28 Panels (c) and (f) present ion distributions associated with bipolar structure 3, whose propagation direction was predominantly in the xz plane, $k gse = ( 0.64 , − 0.38 , − 0.68 )$⁠. The 1D ion distribution is double-humped with two beam-like populations separated by about 1000 km/s; the speed of bipolar structure 3 is between the beam-like populations, but much closer to the bulk velocity of one of them.

We inspected ion VDFs measured around all the slow electron holes in our dataset and found that about 15% of them are double-humped, which look similar to those presented in panels (b) and (e) and (c) and (f). The speed of the corresponding electron holes is either at the local minimum or in between the beam-like populations. However, 85% of the electron holes are associated with single-humped ion distributions similar to that in panels (a) and (d).

We have shown that bipolar structures of positive polarity in the Earth's bow shock are Debye-scale structures with plasma frame speeds of the order of local ion-acoustic speed. The electrostatic structures are most likely slow electron holes, because the scaling relation between spatial scale and amplitude, expected for small-amplitude ion-acoustic solitons, is not observed [Fig. 7(a)]. Note though that the applicability of this scaling relation at the observed amplitudes needs to be tested in the future using the full Sagdeev potential approach.^32,33 Electron holes have no constraints or correlations expected for ion-acoustic solitons, because of a free parameter controlling the phase space density depletion of trapped electrons.^16,24,44 Slow electron holes reduce to classical ion-acoustic solitons when there is no phase space density depletion of the trapped electron population.^51,61 We expect that slow electron holes are produced by Buneman^10,46 or two-stream^41,46 instabilities, but we could not reveal any of them in the Earth's bow shock (Sec. III). The most likely reason is that the saturation time of these instabilities is much shorter than 30-ms resolution of electron VDF measurements. Once produced, the electron holes have a finite lifetime, either because of their internal instabilities or dissipation processes. Clearly, the lifetime τ_L should be larger than typical bounce period of trapped electrons, $τ L ≳ 2 π / ω b$⁠, where

The statistical distribution of bounce periods $τ b = 2 π / ω b$ in Fig. 10(b) shows that the lifetime of the slow electron holes in the Earth's bow shock should be at least about 1 ms. The experimental measurements also indicate that the lifetime should be at least 1 ms, because it cannot be smaller than typically observed $≈ 1$ ms temporal width of the electron holes (Fig. 2). We show below that the slow electron holes in the Earth's bow shock are highly likely unstable to the transverse instability,^25,45 which can determine the upper threshold on their lifetime.

The multi-dimensional simulations of electron-streaming instabilities showed that electron holes disappear rather quickly in unmagnetized plasma,⁴² while a sufficiently strong background magnetic field can provide electron–hole stability for thousands of $ω p e − 1$ (Refs. 38 and 62). The detailed analysis in Refs. 45 and 25 showed that in weakly magnetized plasma electron holes disappear due to the transverse instability, and demonstrated that for a stable propagation, the electron cyclotron frequency ω_ce should roughly exceed typical bounce frequency ω_b of trapped electrons. Figure 10(a) shows that for almost all slow electron holes in the Earth's bow shock, we have the opposite relation, $ω b ≳ ω c e$⁠, which means the electron holes are susceptible to the transverse instability. In this regime, the lifetime of the electron holes is only several bounce periods of trapped electrons, that is, $τ L ≲ 10 τ b$ (Refs. 25, 27, and 45). This implies the slow electron holes in the Earth's bow shock cannot live longer than about 10 ms. Note that uncertainties of the bounce frequency estimates in Fig. 10(a) are within 15%, because $ω b ∝ ( Φ 0 / L 2 ) ∝ L − 1 / 2$ and the uncertainties of spatial scales L are within 30%.

A finite transverse width of the electron holes can certainly reduce the crucial effect of the transverse instability. The transverse width exceeds the spatial scale $L ∼ 10 – 100$ m by a factor of ten at least, since electric fields transverse to the polarization direction are by a factor of ten smaller than those parallel to the polarization direction (not shown here), but does not exceed 10 km, since different MMS spacecraft being spatially separated by about 10 km do not observe the same electron holes. Even though a finite transverse width may affect the transverse instability, there are still several additional strong indications that the lifetime of the slow electron holes in the Earth's bow shock is very limited. The first evidence for a relatively short lifetime is that more than 85% of the slow electron holes are typically associated with single-humped ion distribution functions (Sec. III). The fact that the electron holes were observed as slow implies they have not had enough time to be accelerated due to interaction with ions. Since the expected acceleration timescale is comparable with the bounce period τ_b (Ref. 27), the electron–hole lifetime should be less than a few τ_b. The second evidence is that about 20% of the electron holes propagate oblique to local magnetic field, $θ k B ≳ 30 °$ (Fig. 6). The previous observations^{28,35,48,59,60} and numerical simulations^38,62 reported only electron holes propagating roughly parallel to local magnetic field. In the case of a long lifetime, $τ L ≫ 2 π / ω c e$⁠, we expect quasi-parallel propagation, since only in this case a depletion of the phase space density of trapped electrons, required for the electron hole existence,^16,24,44 can be sustained. The oblique propagation implies the lifetime of such electron holes is not very different from the electron cyclotron period, $τ L ∼ 2 π / ω c e$⁠, that is of the order of τ_b again [Fig. 8(a)].

In contrast to the slow electron holes in the Earth's bow shock, similar electron holes in the Earth's plasma sheet and magnetopause were found to be stable to the transverse instability.^35,59 We believe the reason for this difference is due to different values of $ω p e / ω c e$⁠, which is below a few tens in the previous studies,^35,59 but on the order of one hundred in the Earth's bow shock. The electron holes are most likely produced by some electron-streaming instability, whose fastest growth rate γ is proportional to local electron plasma frequency, $γ ∝ ω p e$⁠. The saturation of the instability occurs when the amplitude of electric field fluctuations becomes sufficiently large so that the bounce frequency of trapped electrons becomes comparable to the growth rate, that is, $ω b ∼ γ$ (Ref. 53). Therefore, we expect that $ω b / ω c e ∝ ω p e / ω c e$ and electron holes observed at larger values of $ω p e / ω c e$ are more likely to be susceptible to the transverse instability. This argument is supported by Lotekar et al.,³⁵ who showed that in the Earth's plasma sheet electron holes observed at $ω p e / ω c e ≳ 10$ are more likely to be around or even above the transverse instability threshold than electron holes observed at $ω p e / ω c e ≲ 10$⁠. We conclude that the slow electron holes observed in the Earth's bow shock are typically above the transverse instability threshold because $ω p e / ω c e ∼ 100$ and this instability may limit their lifetime to below a few electron bounce periods, $τ L ≲ 10 τ b$⁠.

We should stress that, in principle, electron holes in the Earth's bow shock can have longer lifetimes if their amplitude is sufficiently small. The transverse instability criterion, $ω b ≲ ω c e$⁠, requires that the electric field amplitude E₀ of stable electron holes should be below a threshold given as follows:

Assuming typical background parameters, $T e ∼ 10 − 100$ eV, $λ D ∼ 10$ m, and $ω p e / ω c e ∼ 100$⁠, we find that Debye-scale electron holes can be stable only if they have amplitudes below a few mV/m. The bipolar structures considered in Ref. 70 and in this paper were selected among electric field fluctuations with amplitudes $≳$ 10 mV/m. Thus, we cannot rule out that there are plenty of small-amplitude electron holes, because their lifetime is not strongly restricted by the transverse instability.

What are the implications of all these findings for the electron surfing acceleration in collisionless shock waves? Although the Earth's bow shock has Mach numbers much smaller than those of typically simulated astrophysical shocks, the in situ observations can still contribute to understanding of various processes revealed in simulations. The early 1D simulations showed that large-amplitude electron holes can be produced in the shock transition region of a high Mach number shock by Buneman instability and can result in efficient electron acceleration via the surfing mechanism.^23,36,55 The larger amplitudes of these electron holes facilitate the efficiency of electron acceleration. The transverse instability cannot develop in 1D simulations and, thus, does not restrict the efficiency of the acceleration process. By contrast, the observations in the Earth's bow shock demonstrate that the lifetime of large-amplitude electron holes can be limited by the transverse instability. Under realistic values of parameter $ω p e / ω c e$⁠, only rather small-amplitude electron holes may have lifetimes longer than a few bounce periods. Since the transverse instability is a multi-dimensional effect, 1D simulations overestimate the contribution of large-amplitude electron holes to the surfing acceleration. Multi-dimensional simulations may overestimate their contributions too, because the threshold on electron hole amplitude depends on $ω p e / ω c e$⁠, whose values in realistic shocks are typically much larger than in simulations. Based on the observations in the Earth's bow shock, we conclude that electron holes might be not that efficient in electron acceleration in realistic shocks, because the transverse instability strongly limits the lifetime of large-amplitude electron holes at realistic $ω p e / ω c e$ values.

The work of S.R.K. (observation and data processing, visualization) and E.V.Y. (theoretical interpretation) was supported by the Russian Science Foundation through Grant No. 19-12-00313. The work of F.S.M., I.Y.V., and R.W. was supported by NASA MMS Guest Investigator Grant No. 80NSSC21K0730. The work of A.V.A. was supported by NASA Heliophysics Supporting Research Grant No. 80NSSC20K1325. I.Y.V. also thanks the International Space Science Institute (ISSI), Bern, Switzerland for support. We would like to thank Ian Hutchinson, Konrad Steinvall, and Yuri Khotyaintsev for their valuable contributions to this study.

The authors have no conflicts to disclose.

Sergey Railievich Kamaletdinov: Formal analysis (lead); Investigation (equal); Software (lead); Visualization (equal). Ivan Yurievich Vasko: Conceptualization (lead); Writing – original draft (equal); Writing – review & editing (lead). Rachel Wang: Data curation (supporting); Writing – review & editing (supporting). Anton Artemyev: Data curation (supporting); Writing – review & editing (supporting). Egor Yushkov: Writing – review & editing (supporting). Forrest Mozer: Writing – review & editing (supporting).

The data that support the findings of this study are openly available in MMS data repository, Ref. 75.