Observational evidence of accelerating electron holes and their effects on passing ions

Electron holes are Debye-scale, nonlinear electrostatic solitary structures (BGK modes) ubiquitous in space and laboratory plasmas. They have been implicated in particle acceleration, anomalous resistivity, and magnetic energy dissipation, especially near reconnection sites. Despite extensive theory, direct observational evidence that symmetric potential electron holes can impart a net acceleration to ambient particles has been lacking. Prior work largely assumed constant-velocity holes, for which theory predicts no net momentum change for passing particles; however, theory also predicts that non-zero hole acceleration can change the net momentum of passing electrons or ions. Slow electron holes, with drift speeds comparable to ion thermal speed, are frequently observed but their stability and kinematics remain debated. Recent observations suggest that slow holes can persist near minima of a double-humped ion velocity distribution function (VDF). This study asks: Do electron holes accelerate or decelerate in response to the local ion VDF gradient, and can accelerating/decelerating holes impart a measurable net velocity change to passing ions? Using MMS multispacecraft measurements, the paper investigates accelerating slow electron holes and their effects on local ion VDFs.

Electron holes have been reported across diverse space plasma environments (plasma sheet, bow shock, auroral region, reconnection sites) and in laboratory plasmas. Studies have proposed roles in particle energization and anomalous resistivity. While electron acceleration near holes has been suggested, direct evidence for acceleration by symmetric potential holes has been missing. Theory and simulations indicated that slow holes should be destabilized by ion interactions, yet statistical observations show slow holes can persist near VDF minima. Theoretical work predicts hole acceleration or deceleration depending on the ion VDF gradient at the hole speed, but observational confirmation of hole acceleration had not been shown. High-resolution MMS data previously found no electron acceleration around a single hole drifting at constant speed. This work builds on these findings by targeting cases with measurable hole acceleration to test the predicted coupling between ion VDF gradients and hole kinematics, and the consequent net momentum change of passing ions.

Observational setting: MMS observed several slow electron holes on 2017-05-28 in the plasma sheet at GSE coordinates approximately (−19, −11, 3) RE. Local magnetic field was about (4.5, 8.6, 2.2) nT and plasma density <0.1 cm−3. Electric fields were measured by the Electric Double Probe at 8192 samples/s; ion 3D distributions (integrated to 1D VDF along B) were measured by FPI at 150 ms cadence.

Hole identification and basic properties: Four bipolar parallel electric field structures (A–D) were identified as positive-potential slow electron holes drifting nearly anti-parallel to B, with propagation angles >174° and speeds 800–1000 km/s (slightly above local ion thermal speed ~490 km/s, far below electron thermal speed ~3.4×10^4 km/s). An anti-parallel ion beam near −1500 km/s was present.

Velocity and acceleration estimation:

• Four-spacecraft timing (multispacecraft timing): Solve D m = T with m = n/V to obtain normal direction n and speed V assuming constant velocity and direction across the tetrahedron. This provides drift velocities and propagation angles but cannot resolve acceleration.
• Two-spacecraft interferometry along B: Assuming strictly anti-parallel propagation to B, derive drift velocities from time delays across spacecraft pairs to assess changes in speed between pairs. For hole C, velocities inferred from MMS1–MMS3, MMS1–MMS4, MMS1–MMS2 were −1250, −1077, and −982 km/s, respectively, indicating deceleration.
• Uniform acceleration fit: Assuming anti-parallel propagation and constant acceleration over the detection interval, fit position–time (spacecraft separation along B vs observation time delay) data from the four spacecraft to x = U0 t + (U) t^2 / 2 to estimate acceleration/deceleration rate U and initial speed U0. Spacecraft motion (<1 km/s) is neglected relative to hole speeds.

Link to ion VDF gradient: Compare fitted acceleration signs/magnitudes with the local ion VDF gradient evaluated at the hole velocity VEH. Positive gradient corresponds to hole deceleration (for negative VEH), negative gradient to acceleration.

Ion transit/velocity-change modeling: Using a kinematic model (after Hutchinson) in the accelerating hole frame, ions experience an inertial force −mi U. For constant hole acceleration U and hole potential Φ(x) that vanishes at the hole boundaries x1 and x2, the velocity change in the hole frame is v2 − v1 = −2 U (x2 − x1). Transforming to the inertial frame adds the hole’s velocity change during transit and integrates dv/dx across the structure to obtain Δv (Eq. 9). The ion transit time δt = ∫ dx / v is tens of ms. Model inputs are the hole parallel scale L, acceleration U, and central potential Φ, estimated from observations: L ≈ vEH t_pp / 2 (t_pp is peak–valley time of E||), Φ ≈ ∫ E|| vEH dt (maximum). Using measured parameters for holes A–C (vEH ≈ 841, 817, 996 km/s; Φ ≈ 360, 260, 230 V; L ≈ 9.8, 8.4, 7.0 km), compute Δv for passing ions (−1500 to −2000 km/s) and compare mapped VDFs between two regions (F to E) across the holes.

Assumptions and checks: Holes propagate strictly anti-parallel to B with constant acceleration over the multi-spacecraft crossing; electron holes have much larger perpendicular than parallel scale so E⊥ is small; possible asymmetry in E|| due to ion reflection produces only a few volts net potential difference, negligible for the high-speed passing ions considered.

• Observational detection of accelerating/decelerating slow electron holes: Four holes (A–D) were observed with drift nearly anti-parallel to B and speeds 800–1000 km/s. Two-spacecraft interferometry for hole C showed decreasing speed across spacecraft pairs (−1250, −1077, −982 km/s), evidencing deceleration.
• Quantified acceleration rates: Uniform-acceleration fits yielded U (km s−2): A = 5.7×10−3 (with variable sign using different spacecraft subsets), B = −1.5×10−3 (accelerating), C = 1.3×10−3 (decelerating), D = 9.6×10−3 (decelerating). Sign convention: positive U indicates slowing down since holes drift anti-parallel to B.
• Coupling to ion VDF gradient: Hole acceleration sign correlates with the local ion VDF gradient at VEH. Holes at positive gradient decelerate; holes at negative gradient accelerate. Hole A lay near the VDF peak, leading to unstable acceleration (first deceleration then acceleration) and larger fit uncertainty.
• Ion beam context: A distinct anti-parallel ion beam near −1500 km/s coexisted with the slow holes; ion core speeds were <500 km/s.
• Hole parameters and scales: For holes A–C, vEH ≈ 841, 817, 996 km/s; central potentials Φ ≈ 360, 260, 230 V; parallel scales L ≈ 9.8, 8.4, 7.0 km. Local Debye length ≈ 2.5 km; electron and proton gyroradii ≈ 29 km and 733 km. Hole parallel scales are several Debye lengths and well below gyroradii.
• Ion transit times and net velocity changes: Modeled ion transit times δt across a hole are tens of ms. Predicted inertial-frame Δv for passing ions are several to tens of km/s (a few percent of v||), increasing with larger L, |U|, and Φ (Φ has the strongest effect). Using observed parameters, mapping of the ion VDF from region F to E reproduces the observed enhancement and shift for passing ions (−1500 to −2000 km/s). An excess of reflected ions in region F is consistent with reflection by positive potentials of nearby holes.
• Primary conclusion: Electron holes with non-zero acceleration impart a net velocity change to ions passing through them, increasing ion speed when ions traverse parallel to the hole’s acceleration direction (and decreasing for anti-parallel), in agreement with theoretical predictions.

The observations establish that slow electron hole acceleration is governed by the local ion VDF gradient at the hole velocity, and that non-zero acceleration enables net momentum exchange with passing ions. In the hole frame, ions moving parallel to the hole’s acceleration gain energy while those moving anti-parallel lose energy, providing an energy exchange channel in collisionless plasmas. This coupling suggests a feedback: the ion VDF gradient drives hole acceleration/deceleration, while accelerating holes in turn modify the ion VDF, tending toward stabilization at the VDF’s local minimum. The presence of an anti-parallel ion beam helps sustain slow electron holes by maintaining a positive VDF gradient that prevents self-acceleration to fast-hole speeds. These insights bear on the microphysics of plasma sheet dynamics, reconnection regions, and other collisionless environments, informing how small-scale electrostatic structures mediate particle energization and distribution evolution.

Using MMS multi-satellite, high-time-resolution measurements, the study provides direct observational evidence that slow electron holes can accelerate or decelerate, with the acceleration sign determined by the ion VDF gradient at the hole speed, and that such accelerating/decelerating holes cause a net velocity change of passing ions consistent with theoretical expectations. Quantitative fits yield acceleration rates for four holes and, combined with measured hole scales and potentials, predict ion transit times and velocity changes that match observed VDF mappings. The results bridge kinetic ion distribution features with electron hole kinematics, revealing a coupled pathway for energy exchange in collisionless plasma and explaining how slow holes stabilize near VDF minima.

• Four-spacecraft timing assumes constant propagation speed and direction; it cannot directly resolve acceleration and may yield inaccurate velocities if the hole accelerates.
• Two-spacecraft interferometry requires the assumption of strictly parallel or anti-parallel propagation to the local magnetic field to infer acceleration; deviations from this geometry would introduce errors.
• Uniform-acceleration fits assume constant acceleration over the multi-spacecraft crossing; hole A showed variable acceleration near an ion VDF peak, leading to larger fit errors and lower R^2.
• Parameter estimates (central potential Φ and parallel scale L) may be lower bounds because spacecraft likely did not traverse hole centers.
• Asymmetric ion reflection can make parallel E-field waveforms slightly asymmetric; the resulting net potential differences are a few volts and deemed negligible for high-speed passing ions, but still introduce small uncertainties.
• Spacecraft motion is neglected (<1 km/s) relative to hole speeds; while justified, this is an approximation.
• The analysis focuses on ions; analogous electron effects are inferred by charge sign but not directly quantified here.