Direct observations of anomalous resistivity and diffusion in collisionless plasma

Most of the visible universe is composed of plasma, consisting of ions and electrons, whose behavior is governed by electromagnetic forces. In low-density solar and astrophysical plasmas Coulomb collisions are extremely rare, so classical collisions do not provide the resistivity and diffusion that shape plasma dynamics. Wave–particle interactions can introduce effective collisions, reducing conductivity and generating anomalous resistivity, which has been proposed as crucial in a variety of collisionless plasma processes. Magnetic reconnection, a fundamental process that rapidly releases energy by reconfiguring magnetic topology, is one such process. Theory and simulations have suggested that waves could provide both diffusion and resistivity that might support the out-of-plane reconnection electric field. Lower hybrid waves, occurring between ion and electron gyrofrequencies and driven by plasma gradients and associated cross-field currents, have received particular attention as a source of anomalous effects. Prior observational estimates often inferred density fluctuations from spacecraft potential and electron velocities from E and B assuming electrons are frozen-in, frequently using single-spacecraft data, leading to uncertainties. External electric fields can bias spacecraft potential, making density fluctuation estimates unreliable, and the validity of the frozen-in approximation without direct particle measurements is uncertain. Recent observations indicate electrons remain close to frozen-in, with pressure fluctuations causing some deviations. Therefore, direct measurements of anomalous resistivity, viscosity, and cross-field diffusion based on particle data are required. In this work, using high-resolution fields and particle measurements from the four MMS spacecraft, we directly quantify anomalous resistivity, viscosity, and cross-field electron diffusion associated with lower hybrid waves during magnetopause reconnection.

Lower hybrid drift waves (LHDWs) have long been proposed to produce anomalous transport and resistivity in collisionless plasmas (e.g., Davidson & Gladd 1975; Huba et al. 1977), potentially affecting reconnection by providing anomalous dissipation. Prior observational estimates suggested anomalous resistivity is small while cross-field diffusion could be significant (e.g., Shinohara et al. 1998; Mozer et al. 2011; Vaivads et al. 2004; Graham et al. 2017). However, many studies inferred density fluctuations from spacecraft potential and electron flows from E × B drift, assuming electrons remain frozen-in, and often relied on single-spacecraft measurements. These approaches are prone to errors: ambient electric fields modify spacecraft potential, biasing density estimates, and deviations from ideal frozen-in conditions can occur due to electron pressure fluctuations. Simulations and theory have yielded mixed conclusions regarding the importance of LHDWs for reconnection, with some 3D studies suggesting significant anomalous terms and others downplaying their role. Recent observations established the ubiquity of lower hybrid waves at Earth’s magnetopause and that electrons are approximately frozen-in, motivating direct, multi-spacecraft, particle-based quantification of anomalous terms.

Observational setup: The study analyzes Magnetospheric Multiscale (MMS) mission data (electric and magnetic fields, electron and ion moments). High-resolution electron and ion moments sampled at 7.5 ms and 37.5 ms, respectively, are used to resolve wave-associated fluctuations at lower hybrid frequencies (~10–30 Hz). A local LMN current-sheet coordinate system is determined via minimum variance analysis of B; boundary normal velocity is obtained using four-spacecraft timing. Data are rotated into LMN. Decomposition and definitions: Physical quantities Q are decomposed into mean and fluctuating parts, Q = ⟨Q⟩ + δQ, where averaging removes fast fluctuations. Starting from the collisionless electron momentum equation, neglecting time derivatives and averaging yields an expression for the mean electric field involving ⟨V⟩ × ⟨B⟩, pressure-gradient terms, and anomalous terms: anomalous drag (resistivity) D = ⟨δn δE⟩/(⟨n⟩e), anomalous viscosity (momentum transport) T derived from correlations of δV and δB (expanded to include terms such as ⟨δV δV⟩), and anomalous Reynolds stress I from inertial terms. The total anomalous contribution is R = D + T + I. Cross-field anomalous flow is VN,anom = ⟨δn δv⟩/⟨n⟩, and the cross-field diffusion coefficient D⊥ is related to correlations of density and velocity fluctuations along the density gradient, D⊥ = ⟨δn δv·N⟩/|∇⟨n⟩|. A gradient relaxation timescale τ ≈ L^2/D⊥ is estimated to assess density-gradient evolution. Signal processing and averaging: To approximate spatial averaging along M, four-spacecraft averages are used. Steps: (1) resample field data to 7.5 ms; (2) use timing-derived delays so all spacecraft are aligned to a common boundary-crossing time; (3) obtain non-fluctuating components by four-spacecraft averaging and low-pass filtering below 5 Hz; (4) obtain δQ by high-pass filtering above 5 Hz to isolate lower hybrid wave power; (5) form correlations ⟨δQi δQj⟩ by averaging over spacecraft and low-pass filtering below 5 Hz. Uncertainties are assessed from particle counting statistics (scaled for 7.5 ms sampling) and a 10% electric-field gain uncertainty. Evaluation of I: The M component of I is approximated assuming primary variation in the N direction and using boundary-normal advection to estimate gradients (∂/∂N ≈ (1/vN)∂/∂t with vN from timing). I is found to be much smaller than D and T. Events analyzed: Detailed case study on 2015-12-06 magnetopause reconnection crossing with lower hybrid waves localized on the low-density side, plus two additional crossings (2015-12-02 and 2015-12-14) near EDRs, and a statistical survey of 22 current sheets with lower hybrid waves categorized as EDR, reconnection, and non-reconnection. Simulation: A complementary fully kinetic iPIC3D simulation models the 2015-12-06 event. A 2D asymmetric reconnection run is first performed then used to initialize a 3D run to capture short-wavelength instabilities like LHDI. Parameters include mi/me = 256, domain size Lx × Ly × Lz = 2822 × 1058 × 117.5 km^3, with fields and particle properties chosen to resemble the observed event.

- Lower hybrid waves are observed on the low-density (magnetospheric) side of magnetopause current sheets with frequencies ~10 Hz, phase speed ~140 km s−1, and kρe ≈ 0.428. Electrons remain approximately frozen-in (E ≈ −Ve × B), while ions are largely unmagnetized at wave scales. - Directly computed anomalous drag D and viscosity T peak at about 0.8 ± 0.2 mV m−1 (for the 2015-12-06 event), predominantly in the M direction. D and T have opposite signs and nearly cancel (D = −T), yielding R = D + T + I ≈ 0 and thus negligible contribution to the reconnection electric field at the neutral point. - Significant anomalous electron cross-field flow is observed: VN,anom up to ~20 km s−1 directed from higher to lower density across N. The associated diffusion coefficient D⊥ peaks near 1 × 10^9 m^2 s−1, implying strong cross-field electron diffusion and current-layer broadening. Estimated density-gradient relaxation timescale is ~1 s, comparable to the ion cyclotron period. - In two additional events: D and T each reach ~0.5 mV m−1 and cancel, while VN,anom is substantial. Peak diffusion coefficients estimated at D⊥ ≈ 0.58 × 10^9 m^2 s−1 (2015-12-02) and 1.21 × 10^9 m^2 s−1 (2015-12-14). - Statistical survey (22 events): Dmax and Tmax can reach ~1.5 mV m−1 and scale approximately linearly, yielding R ≈ 0 generally. D⊥ ranges from ~0.05 × 10^9 to 2 × 10^9 m^2 s−1, with larger values and |VN,anom| tending to occur near EDRs. Diffusion is consistently from higher to lower density across N. - The anomalous Reynolds stress I is negligible compared with D and T in observations, in contrast to some 3D simulation reports.

The observations demonstrate that lower hybrid waves generate correlated fluctuations that produce anomalous drag and viscosity of comparable magnitude but opposite sign, leading to R ≈ 0. Because electrons are approximately frozen-in to the magnetic field during the waves, these anomalous terms do not sustain the reconnection electric field and thus do not directly control the reconnection rate in the examined cases. However, the waves induce significant cross-field electron transport from the magnetosheath to the magnetosphere. The resulting large diffusion coefficients (up to ~10^9 m^2 s−1) act to relax density gradients on ion-gyro timescales (~1 s), broadening the magnetopause current layer. Such broadening can modify reconnection by altering Hall electric and magnetic fields and contributing to electron heating in the magnetospheric inflow region. The negligible contribution of the inertial Reynolds-stress term I in MMS data contrasts with some 3D simulations; the difference may arise from simulation constraints (reduced mass ratios and frequencies) and inclusion of low-frequency kinking in spatial averages, which are filtered out (>5 Hz high-pass) in the observations.

Direct, multi-spacecraft, particle-based measurements quantify anomalous resistivity, viscosity, and cross-field diffusion associated with lower hybrid waves at the magnetopause. Anomalous resistivity (drag) and viscosity nearly cancel, yielding no net anomalous contribution to the reconnecting electric field, while significant anomalous cross-field electron transport and diffusion broaden the current layer and relax density gradients on ion-cyclotron timescales. These results clarify the role of lower hybrid waves in collisionless reconnection: they modify plasma structure via transport rather than directly powering the reconnection electric field. Future work should expand the event database, assess variability under different upstream conditions and guide fields, investigate where and when diffusion is maximized (e.g., near EDRs), and reconcile observational results with simulations by addressing modeling constraints and separating true LHDI effects from low-frequency boundary dynamics.

- Spatial averaging: MMS approximates ensemble averaging along M using four-spacecraft averages with separations larger than electron scales but smaller than ion scales; residual fluctuations indicate imperfect averaging. - Filtering choices: A 5 Hz high-pass/low-pass separation is used to isolate lower hybrid waves; while results are not highly sensitive to cutoff choice, very low-frequency dynamics (e.g., boundary kinking) are removed and not included in anomalous term estimates. - Measurement uncertainties: Electric-field gain uncertainty (~10%) and particle counting statistics (extrapolated to 7.5 ms electron moments) contribute to error bars; ion moments do not fully resolve wave-scale fluctuations. - Estimation of inertial term I: Gradients for I are inferred using boundary-normal advection assumptions due to the averaging method; I is found negligible but its precise magnitude carries additional approximation uncertainty. - Event sampling: Although 22 events are analyzed, variability remains high and further events are needed, particularly to confirm where diffusion peaks (e.g., near EDRs) and to generalize to other reconnection regimes. - Simulation comparability: Differences with 3D simulations (reduced mi/me, ωpe/Ωce, inclusion of low-frequency modes in averages) limit direct quantitative comparison of observed and modeled anomalous terms.