Abstract
Coulomb collisions provide plasma resistivity and diffusion but in many lowdensity astrophysical plasmas such collisions between particles are extremely rare. Scattering of particles by electromagnetic waves can lower the plasma conductivity. Such anomalous resistivity due to waveparticle interactions could be crucial to many processes, including magnetic reconnection. It has been suggested that waves provide both diffusion and resistivity, which can support the reconnection electric field, but this requires direct observation to confirm. Here, we directly quantify anomalous resistivity, viscosity, and crossfield electron diffusion associated with lower hybrid waves using measurements from the four Magnetospheric Multiscale (MMS) spacecraft. We show that anomalous resistivity is approximately balanced by anomalous viscosity, and thus the waves do not contribute to the reconnection electric field. However, the waves do produce an anomalous electron drift and diffusion across the current layer associated with magnetic reconnection. This leads to relaxation of density gradients at timescales of order the ion cyclotron period, and hence modifies the reconnection process.
Introduction
Most of the visible universe is composed of plasma, consisting of ions and electrons. The behavior of plasma is governed by electromagnetic forces. In lowdensity solar and astrophysical plasmas Coulomb collisions are typically extremely rare, meaning that collisions between particles do not play a role in the behavior of the plasma and cannot provide plasma resistivity and diffusion. However, the scattering of particles by electromagnetic waves can introduce effective collisions, lowering the plasma conductivity^{1,2}. Such anomalous resistivity due to waveparticle interactions is thought to be crucial to a wide variety of collisionless plasma processes^{3,4,5}. One process where anomalous effects are thought to be important is magnetic reconnection, which is a fundamental plasma process providing explosive energy releases by reconfiguring magnetic field topology^{6,7}. In particular, it has been suggested based on theoretical and numerical results that waves can provide both diffusion and resistivity, which can potentially support the reconnection electric field^{8,9}, the outofplane electric field responsible for sustaining reconnection.
One wave that has received significant attention as a source of anomalous effects is the lower hybrid wave^{10,11,12}. Lower hybrid waves are found at frequencies between the ion and electron cyclotron frequencies and are driven by plasma gradients and the associated crossfield currents^{11,13}. Previous attempts to calculate anomalous terms concluded that the anomalous resistivity was small^{14,15}, while crossfield particle diffusion associated could be significant^{16,17}. However, these estimates relied on density fluctuations inferred from the spacecraft potential, electron velocities inferred from the electric and magnetic fields assuming electrons remain frozen in, and often single spacecraft measurements. An external electric field can modify the spacecraft potential, making density fluctuations associated with waves inferred from the spacecraft potential unreliable^{18,19}. Similarly, it is unclear how well the frozenin approximation works without direct measurements. Recent observations have shown that electrons remain close to frozen in, although pressure fluctuations associated with the waves can cause some deviation from the ideal frozen in condition^{20}. Thus, calculations of anomalous resistivity, viscosity, and crossfield diffusion based on direct particle measurements are needed to determine the role of lower hybrid waves.
In this work, we directly measure and quantify anomalous resistivity, viscosity, and crossfield electron diffusion associated with lower hybrid waves using the highresolution fields and particle measurements from the four MMS spacecraft^{21}. We show that anomalous resistivity (drag) is balanced by viscosity (momentum transport), and thus the waves do not contribute to the reconnection electric field. However, the waves do produce an anomalous electron drift and diffusion across the current layer associated with magnetic reconnection. This can lead to the relaxation of density gradients at timescales of order the ioncyclotron period, which counteracts steepening of density gradients caused by magnetic reconnection and hence modifies the process.
Results
Magnetic reconnection and case study
A region where reconnecting current sheets and potential anomalous effects can be found is the terrestrial equatorial magnetopause, the boundary between the shocked solar wind in the magnetosheath and the magnetosphere (Fig. 1a). Magnetic reconnection occurs between the highdensity magnetosheath and the more tenuous magnetosphere. This results in reconnection being asymmetric with strong density gradients across the boundary. Figure 1b shows the result from a numerical simulation (see Methods, subsection Simulation description) designed to illustrate the magnetopause reconnection event presented in Fig. 2. The approximate orbit of MMS moving from the magnetosheath to the magnetosphere is indicated, with the turbulent magnetopause separating the two regions. The density fluctuations on the lowdensity side of the reconnection region are due to lower hybrid waves, which are driven by the strong density gradients in this region.
Figure 2 provides an overview of a magnetopause reconnection event observed by MMS. We use MMS electric^{22,23} and magnetic field data^{24,25}, and electron and ion data^{26}. In particular, to investigate fluctuations in the electron and ion distributions associated with waves, we use particle moments sampled at 7.5 and 37.5 ms, respectively^{27}. The electron sampling rate is high enough to resolve the local lower hybrid frequency and is unique and essential for comparisons with the lower hybrid waves.
Magnetic field data from one MMS spacecraft is shown in Fig. 2a in a local current sheet coordinate system: the current sheet normal points along N, L is along the antiparallel magnetic field direction, and M = N × L completes the righthand coordinate system. The local coordinates are determined using a minimum variance analysis of B. The magnetopause crossing is characterized by a reversal in B_{L} from negative in the highdensity magnetosheath to positive in the lowdensity magnetosphere (Fig. 2b). MMS crosses the magnetopause close to, but southward, of the electron diffusion region (EDR), as indicated by the electron and ion jets reported previously in ref. ^{28}. Based on fourspacecraft observations, we estimate the current sheet velocity to be ≈40 km s^{−1} sunward in the N direction.
The components of the electric field E perpendicular and parallel to the magnetic field B are shown in Fig. 2c. The most intense waves are observed on the lowdensity side of the current sheet, mainly perpendicular to B in the N and M directions, with some intermittent smalleramplitude higherfrequency fluctuations parallel to B (close to the L direction). We identify the waves as lower hybrid drift waves driven by the diamagnetic current at the density gradient (Fig. 2b). Lower hybrid waves occur between the ion and electron gyrofrequencies. The waves have a frequency of around ~10 Hz, a phase speed of v_{ph} ≈ 140 km s^{−1}, and a wavenumber of kρ_{e} ≈ 0.4^{28}, where ρ_{e} is the thermal electron gyroradius. The analysis techniques used to determine the wave properties are detailed in ref. ^{20}. These waves have been proposed as a source of anomalous resistivity and can be important for magnetic reconnection^{29}. Some recent studies concluded that the waves are relatively unimportant^{5,30,31}, while others conclude that the waves are important for ongoing reconnection^{9,12,32}.
Figure 2d shows the perpendicular and parallel components of V_{e} of the lower hybrid waves. The fluctuations are well resolved and the electron moments can be used to calculate the associated anomalous terms. Large V_{e} fluctuations are observed not only in the perpendicular but also in the direction parallel to B, indicating that the wave vector is not exactly perpendicular to B. This means the waves can potentially heat electrons. Figure 2e shows E and the electron and ion convection terms −V_{e} × B and −V_{i} × B in the M direction. Throughout the interval E ≈ −V_{e} × B meaning the electrons move together with the magnetic field (are approximately frozen in) as expected for lower hybrid waves. In contrast, −V_{i} × B remains close to zero. Although ion moments do not fully resolve the waves, this is consistent with the ions being unmagnetized, with only small perturbations in V_{i}. This results in largeamplitude fluctuating currents, which are in turn responsible for fluctuations in B (Fig. 2a). Figure 2f displays density fluctuations normalized to the background density, associated with the waves. Large normalized perturbations, δn_{e}/n_{e} > 0.1, and electric field fluctuations suggest that anomalous resistivity may be significant.
Anomalous terms associated with waves
To evaluate the effects of waves on the plasma we divide the quantities into fluctuating and quasistationary components, Q = 〈Q〉 + δQ where 〈Q〉 corresponds to spatial or temporal averaging over fast fluctuations and δQ corresponds to fluctuations. Anomalous resistivity is effectively a force on charged particles due to waves, so we analyse a momentum equation. The electron momentum equation for a collisionless plasma is
where e, n_{e}, m_{e}, V_{e}, and P_{e} are the unit charge, electron density, mass, bulk velocity, and pressure tensor, respectively, and E and B are the electric and magnetic fields. Introducing fluctuations, neglecting time derivatives, and averaging yields
Here D, T, and I are the anomalous drag (sometimes called resistivity), anomalous viscosity (momentum transport), and anomalous Reynold’s stress, respectively. These quantities are defined as
We define the total anomalous contribution to equation (2) as R = D + T + I. We find that the contributions of I are negligible compared with D and T, so they are neglected in the following analyses (see Methods, subsection Estimating the anomalous terms for an example and details).
We study the electron continuity equation to find anomalous flows due to fluctuations
A crossfield diffusion coefficient D_{⊥} relates the electron density and velocity fluctuations to the density gradient in the direction normal to the boundary
A gradient relaxation timescale can be estimated as
For lower hybrid waves, it has not been previously possible from observations to directly evaluate the terms involving electron density or velocity fluctuations, such as 〈δn_{e}δE〉.
Anomalous contributions from lower hybrid waves
Figure 3a shows the lower hybrid waves from one MMS spacecraft and Fig. 3b–e display the anomalous terms D, T, anomalous electron flow V_{N,anom} in the N direction, and the diffusion coefficient D_{⊥} in the N direction, obtained by combining data from all four spacecraft (see Methods, subsection Estimating the anomalous terms). The terms D and T have a maximum amplitude of 0.8 ± 0.2 mV m^{−1} (Fig. 3b, c), a small fraction (~2%) of the amplitude of the waves. For comparison, the reconnection electric field associated with magnetopause reconnection is expected to be ~1 mV m^{−1} for fast reconnection, comparable to the peak magnitudes of D and T. Both D and T are predominantly in the M direction and D ≈ −T. This is similar to the result found from the simulation in ref. ^{33}. The directions of D and T remain the same while the lower hybrid waves are observed, although some residual fluctuations remain. These fluctuations result from the fourspacecraft averaging used to approximate the spatial averaging needed to compute D and T and provide an indicator of the uncertainty in the averaging. Similarly, the magnitudes of D and T are larger than the estimated uncertainties based on the fields and particle measurements (indicated by the shaded regions associated with each anomalous term). The anomalous terms are significant only when largeamplitude waves are present and are localized to the density gradients on the lowdensity side of the boundary. Thus, the anomalous terms are negligible at the neutral point (B_{L} = 0). Overall, the contribution to the reconnection electric field is small because R ≈ 0, which results from E ≈ −V_{e} × B for lower hybrid waves, and the waves do not penetrate into the center of the current sheet.
Figure 3d shows a significant anomalous electron flow V_{N,anom} with a magnitude up to 20 km s^{−1} toward the lowerdensity side. Due to electrons being approximately frozen in V_{anom} ≈ −D × 〈B〉/〈∣B∣〉^{2}. Figure 3e shows a related large diffusion coefficient D_{⊥}, which peaks at about 10^{9} m^{2} s^{−1}, suggesting that significant broadening of the current layer can occur. Overall, the anomalous electric field [equation (2)] is not likely to affect the reconnection electric field and the reconnection rate. Rather, the lower hybrid waves can produce anomalous diffusion of electrons from higher to lower density regions, thus broadening the current layer. This can in turn affect the reconnection process by modifying the Hall electric and magnetic fields and contributing to the electron heating observed in the magnetospheric inflow region^{17,31}. From equation (8) the estimated relaxation time is ~1 s (comparable to the ioncyclotron period).
Examples of anomalous terms from lower hybrid waves
Figure 4 shows two different magnetopause crossings observed on 02 December 2015 (Fig. 4a–d) and 14 December 2015 (Fig. 4e–h). In Fig. 4a–d the spacecraft crossed from the magnetosphere to the magnetosheath. The spacecraft crossed the EDR at around 01:14:56 UT, close to the neutral point^{34}. The lower hybrid waves are observed for ≈10 s on the magnetospheric side of the boundary. In Fig. 4e–h the spacecraft crossed the EDR from the magnetosheath and magnetosphere^{35,36}, with the lower hybrid waves observed on the magnetospheric side for 1 s. The properties of the lower hybrid waves were investigated in detail in ref. ^{20}.
Overall, the properties of the magnetopause crossings are similar to the event in Figs. 2, 3. Namely, largeamplitude lower hybrid waves are observed on the magnetospheric side, D_{M} < 0 and T_{M} > 0, such that D + T is small, and a significant V_{N,anom} < 0 is observed. For the 14 December 2015 event D and T have a significant component in the L direction due to the significant B_{M} (guide field). In both events, the anomalous terms are negligible at the neutral point (indicated by the magenta lines in Fig. 4).
Although the amplitude of the lower hybrid waves are comparable in these two events, ∣V_{N,anom}∣ is significantly larger for the 14 December 2015 event. This is likely because B is smaller, corresponding to a larger amplitude δV_{e,⊥} ≈ δE × B/∣B∣^{2}. At the times when ∣V_{N,anom}∣ peaks we estimate D_{⊥} = 0.58 × 10^{9} m^{2} s^{−1} and D_{⊥} = 1.21 × 10^{9} m^{2} s^{−1}, respectively, for the 02 December 2015 and 14 December 2015 events. This corresponds to the diffusion of electrons across B from the magnetosheath to the magnetosphere in both cases. Thus the diffusion coefficients are significant and comparable to the values obtained in Fig. 2. In both cases, the uncertainties are smaller than the peak values of the anomalous terms. To summarize, the two events presented in Fig. 4 show the same qualitative behavior as the 06 December 2015 event: D and T both reach about 0.5 mV m^{−1} but have opposite signs so the contribution to E is small. Large anomalous flows and crossfield electron diffusion toward the magnetosphere are observed. In all three cases the peak values of ∣D∣ and ∣T∣ are ~2% of the maximum amplitude of δE.
Statistical results
Figure 5 shows statistics from magnetopause crossings where highresolution particle moments are available and lower hybrid waves are observed. We divide each event into (1) EDR crossings, where the waves are observed adjacent to EDR regions identified in ref. ^{37}. (2) Reconnection events, where the waves are observed at boundaries where reconnection signatures, such as ion outflows, are observed. (3) Nonreconnection events, where no clear reconnection signatures are observed.
In each case the largest D was in the −M direction and the largest T was in the M direction. Figure 5a shows the maximum T (\( {{{{{{{\bf{T}}}}}}}}{ }_{\max }\)) versus the maximum D (\( {{{{{{{\bf{D}}}}}}}}{ }_{\max }\)) and the associated uncertainties for each event. Here \( {{{{{{{\bf{D}}}}}}}}{ }_{\max }\) and \( {{{{{{{\bf{T}}}}}}}}{ }_{\max }\) can reach ≈1.5 mV m^{−1}, with \( {{{{{{{\bf{T}}}}}}}}{ }_{\max }\) increasing approximately linearly with \( {{{{{{{\bf{D}}}}}}}}{ }_{\max }\). Both peak at approximately the same time, so for all cases R ≈ 0. We find that \( {{{{{{{\bf{T}}}}}}}}{ }_{\max }\) tends to be slightly smaller than \( {{{{{{{\bf{D}}}}}}}}{ }_{\max }\), possibly due to small deviations from the frozenin condition for electrons due to fluctuations in the electron pressure due to density fluctuations. The largest D and T correspond to magnetic reconnection events and EDRs, although more nonreconnection events need to be analyzed.
Figure 5b shows the values of D_{⊥} where ∣V_{N,anom}∣ peaks versus the maximum −V_{N,anom}. Here D_{⊥} tends to increase as −V_{N,anom} increases. Each case corresponds to diffusion from higher to lower densities. We find that D_{⊥} ranges from 0.05 × 10^{9} to 2 × 10^{9} m^{2} s^{−1}, i.e., from small to very significant diffusion^{38,39}. The largest −V_{N,anom} and D_{⊥} tend to occur close to EDRs, although there are cases where −V_{N,anom} and D_{⊥} are also small near EDRs. Thus, crossfield diffusion and associated broadening are expected to be highly variable during magnetopause reconnection. The results suggest that D_{⊥} may be the largest close to the EDR, although further work and more events are required to confirm this.
Discussion
We find that for lower hybrid waves the anomalous terms D, T, and V_{anom} can be accurately determined from the data and that R = D + T ≈ 0, so the contribution to the reconnecting electric field is negligible because electrons are approximately frozen in. However, the diffusion coefficient D_{⊥} and V_{N,anom} can often be significant, corresponding to transport from the higherdensity magnetosheath to the lowerdensity magnetosphere, producing significant broadening of the magnetopause density gradient.
Overall, these direct observations of anomalous terms in collisionless plasma open a new window to investigate fundamental plasma physics. Directly evaluating all terms involved in waveparticle interactions will show which processes are important, and which are not, in many astrophysical plasmas. In many reconnection events, the lower hybrid waves are observed over several seconds at large amplitude, which suggests that the density gradient is driven by ongoing reconnection, while lower hybrid waves counter this.
Methods
Estimating the anomalous terms
For each event we rotate the vector quantities into LMN coordinates, where N is normal to the magnetopause pointing sunward, L is along the reconnecting magnetic field direction, and M completes the coordinate system and is close to the guidefield direction. We determine the coordinate system using a minimum variance analysis of B across the magnetopause. The reliability of the N direction is confirmed by determining the boundary normal velocity using fourspacecraft timing analysis, as well as minimum variance analysis of the current density J. In most cases, the uncertainty in the coordinate system directions are small and do not significantly affect the results.
Ideally, the quantities in equation (2) are computed from an ensemble average in the M direction. With MMS we must use a fourspacecraft average to estimate these quantities. For all events, the spacecraft were in tetrahedral configurations with spacecraft separations ranging from ~15 to ~5 km. These separations are well below ion spatial scales at the magnetopause, but larger than electron spatial scales, which is ideal for studying lower hybrid waves. To calculate the anomalous and background quantities we use the following procedure:

(1)
We resample all field data to the sampling frequency of the highresolution (7.5 ms) electron moments and perform a fourspacecraft timing analysis on B_{L} at the current sheet to determine the boundary normal velocity and the time delays between the spacecraft. Typical boundary normal speeds range from ~10 to ~100 km s^{−1}.

(2)
We use the time delays to offset the spacecraft times so all spacecraft cross the boundary layer at the same time as MMS1.

(3)
To obtain the nonfluctuating terms 〈Q〉 we average the timeshifted quantities over the four spacecraft and bandpass filter below 5 Hz. At the magnetopause the lower hybrid waves are typically found at frequencies 10 Hz < f < 30 Hz.

(4)
To obtain δQ associated with the lower hybrid wave fluctuations we bandpass filter Q above 5 Hz. The specific bandpass frequency does not significantly modify the results, as long as it is not too high to significantly remove lower hybrid wave power.

(5)
We obtain 〈δQ_{1}δQ_{2}〉 by averaging δQ_{1}δQ_{2} over the four spacecraft then lowpass filter the result below 5 Hz to remove any remaining higherfrequency fluctuating components.
To evaluate equation (4) we expand it to obtain
All terms are calculated to determine T, although we find that only the components involving 〈δn_{e}δV_{e}〉 are significant. The uncertainties in the anomalous terms are calculated from the uncertainties in the electron moments and assuming a 10% uncertainty in the gain of the electric field. The uncertainty in the electric field is based is on the fact that the gain is validated by comparing the electric field with the DC convection field caused by the spacecraft moving relative to a magnetized plasma, and there can be small changes in the gain due to changes in the plasma conditions. The uncertainties in the electron moments are based on the counting statistics of the particle distributions. This is only available at 30 ms sampling, so we assume that the uncertainties are four times larger for the 7.5 ms moments we use, due to the reduced azimuthal sampling. The magnitude of the uncertainties of the particle moments are compared with the magnitude of the envelope of the fluctuating quantities to estimate the relative uncertainties.
Estimates of the anomalous contributions from the electron inertial term and time derivative in equation (1) indicate that they are much smaller than D and T due to the m_{e}/e dependence. The M component of the anomalous inertial terms (anomalous Reynold’s stress^{5}) I can be well approximated by assuming that the anomalous terms in I vary primarily in the N direction, which is given by
Since the method of obtaining the anomalous terms equation (12) relies on fourspacecraft averaging, the four spacecraft cannot be used to calculate the gradient associated with these terms. Therefore, the gradient is approximated assuming these quantities move past the spacecraft at the boundary normal velocity, such that ∂N = −v_{N}∂t, where v_{N} is the boundary normal velocity estimated from the fourspacecraft timing of the current sheet. The values of I_{M} obtained from equation (12) are significantly smaller than D and T and do not significantly contribute to R.
As an example, Fig. 6 shows I_{M} estimated from equation (12) and a comparison with D_{M} and T_{M} for the 06 December 2015 event (Figs. 2, 3). Figure 6a shows the electric field associated with the lower hybrid waves from MMS1. In Fig. 6b we plot the anomalous terms 〈n_{e}〉〈δV_{M}δV_{N}〉, 〈V_{M}〉〈δn_{e}δV_{N}〉, and Γ = 〈n_{e}〉〈δV_{M}δV_{N}〉 + 〈V_{M}〉〈δn_{e}δV_{N}〉. We find that 〈V_{M}〉〈δn_{e}δV_{N}〉 < 0 due the term being proportional to V_{N,anom}. In contrast, 〈n_{e}〉〈δV_{M}δV_{N}〉 fluctuates with very little offset from zero. This is due to the lack of consistent correlation between the δV_{M} and δV_{N} associated with the lower hybrid waves. As a result, Γ fluctuates and is similar to 〈n_{e}〉〈δV_{M}δV_{N}〉. In Figure 6c we plot I_{M} for Γ and Γ bandpass filtered below 1 Hz to remove the fluctuations. We find that I_{M} fluctuates around zero when the 5 Hz lowpass filter is used. There is negligible largescale offset, as seen for the <1 Hz case. Thus, I_{M} for the 5 Hz lowpass filter is overestimated. In Fig. 6d we plot D_{M}, T_{M}, and I_{M} for the 5 Hz bandpass filter. We find that the I_{M} is much smaller than D_{M} and T_{M}. Both D_{M} and T_{M} have clear background components, in contrast to I_{M}. Similar results are found for the other events, and there is no clear evidence that I can significantly contribute to R for lower hybrid waves. We conclude that the contribution of I_{M} to the total anomalous electric field is negligible based on MMS observations.
This differs from the results of threedimensional simulations^{5,9,31,32}, which have found that I_{M} could be significant. Possible reasons for these differences are:

(1)
Artificial plasma conditions, such as reduced electron to ion mass ratio and reduced ratio of electron plasma to cyclotron frequency, are needed to run 3D simulations.

(2)
When spatially averaging over the M direction in simulations very lowfrequency fluctuations, such as current sheet kinking, are typically included, which can lead to large anomalous terms that are not due to lower hybrid waves^{5}. In observations, we used a highpass filter of 5 Hz, which removes such lowfrequency fluctuations, if they are present.
Simulation description
We model the 06 December 2015 event using the fully kinetic iPIC code^{40}. The code uses an implicit moment method, which allows the cell size to exceed the Debye length^{41}. The code x, y, and z coordinates point in the L, N, and −M directions used in this letter. The simulation is initialized with two thin current sheets of width 1d_{i} and 2d_{i} at y = L_{y}/4 and y = 3L_{y}/4, respectively, where d_{i} is the ion inertial length in the magnetosheath. The iontoelectron mass ratio is m_{i}/m_{e} = 256 and the speed of light to the reference Alfvén speed ratio is c/V_{A} = 103. The parameters used to set up the asymmetric force balance are [B_{L}, B_{M}, n_{e}, T_{e}, T_{i}] = [−37 nT, −16 nT, 14 cm^{−3}, 32 eV, 1200 eV] on the magnetosheath side and [73 nT, −16 nT, 1.85 cm^{−3}, 164 eV, 3900 eV] on the magnetospheric side.
The simulation is performed in two steps:

(1)
Asymmetric magnetic reconnection is first run in two dimensions in x − y coordinates in a double periodic domain^{42}. The size of the domain is L_{x} × L_{y} = 2822 × 1058 km^{2} and is resolved by 1728 × 648 cells. A weak localized perturbation at (L_{x}/2, L_{y}/4) is used to initiate reconnection^{43}.

(2)
The threedimensional (3D) simulation is initialized at time tΩ_{ci} = 35 once steadystate reconnection is reached, where Ω_{ci} is the angular ioncyclotron frequency. The initial conditions of the 3D simulation are the fields and particle information from the 2D run and replicated in the zdirection. The computational domain is L_{x} × L_{y} × Lz = 2822 × 1058 × 117.5 km^{3} and is resolved by 1728 × 648 × 72 cells. This replicated geometry is suitable for investigating instabilities, such as the lower hybrid drift instability, with wavelengths short compared with L_{z}.
Data availability
MMS data were available at https://lasp.colorado.edu/mms/sdc/public. The data can be found in the following directories: mms#/edp/brst/l2/dce/ for electric fields, mms#/fgm/brst/l2/ for the background magnetic field, mms#/scm/brst/l2/scb/ for the fluctuating magnetic field, mms#/fpi/brst/l2/desmoms/ for the background electron moments, mms#/fpi/brst/l2/desqmoms/ for the highest resolution electron moments, and mms#/fpi/brst/l2/disqmoms/ for the highest resolution ion moments. Source data required to generate the figures in this paper can be found at https://github.com/danbgraham/anomres^{44} and are available on request. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Code availability
All figures and data analyses were performed using the IRFUMatlab data analysis package, which is available at https://github.com/irfu/irfumatlab. The scripts and functions required to compute the anomalous terms and reproduce the figures in this paper can be found at https://github.com/danbgraham/anomres^{44}. The simulation code is available at https://github.com/CmPA/iPic3D.
References
Drummond, W. E. & Rosenbluth, M. N. Anomalous diffusion arising from microinstabilities in a plasma. Phys. Fluids 5, 1507–1513 (1962).
Papadopoulos, K. A review of anomalous resistivity for the ionosphere. Rev. Geophys. Space Phys. 15, 113–127 (1977).
Davidson, R. C. & Krall, N. A. Anomalous transport in hightemperature plasmas with applications to solenoidal fusion systems. Nucl. Fusion 17, 1313–1372 (1977).
Drake, J. F. et al. Formation of electron holes and particle energization during magnetic reconnection. Science 299, 873 (2003).
Le, A. et al. Drift turbulence, particle transport, and anomalous dissipation at the reconnecting magnetopause. Phys. Plasmas 25, 062103 (2018).
Priest, E. R. & Forbes, T. Magnetic Reconnection: MHD Theory and Applications (Cambridge Univ. Press, 2000)
Yamada, M., Kulsrud, R. & Ji, H. Magnetic reconnection. Rev. Mod. Phys. 82, 603–664 (2010).
Che, H., Drake, J. F. & Swisdak, M. A current filamentation mechanism for breaking magnetic field lines during reconnection. Nature 474, 184–187 (2011).
Price, L. et al. The effects of turbulence on threedimensional magnetic reconnection at the magnetopause. Geophys. Res. Lett. 43, 6020–6027 (2016).
Davidson, R. C. & Gladd, N. T. Anomalous transport properties associated with the lowerhybriddrift instability. Phys. Fluids 18, 1327 (1975).
Davidson, R. C., Gladd, N. T., Wu, C. S. & Huba, J. D. Effects of finite plasma beta on the lowerhybriddrift instability. Phys. Fluids 20, 301 (1977).
Silin, I., Buchner, J. & Vaivads, A. Anomalous resistivity due to nonlinear lowerhybrid drift waves. Phys. Plasmas 12, 062902 (2005).
Krall, N. A. & Liewer, P. C. Lowfrequency instabilities in magnetic pulses. Phys. Rev. A 4, 2094 (1971).
Shinohara, I. et al. Lowfrequency electromagnetic turbulence observed near the substorm onset site. J. Geophys. Res. 103, 20365–20388 (1998).
Mozer, F. S., Wilber, M. & Drake, J. F. Wave associated anomalous drag during magnetic field reconnection. Phys. Plasmas 18, 102902 (2011).
Vaivads, A. et al. Cluster observations of lower hybrid turbulence within thin layers at the magnetopause. Geophys. Res. Lett. 31, L03804 (2004).
Graham, D. B. et al. Lower hybrid waves in the ion diffusion and magnetospheric inflow regions. J. Geophys. Res. Space Phys. 122, 517–533 (2017).
Torkar, K. et al. Influence of the ambient electric field on measurements of the actively controlled spacecraft potential by MMS. J. Geophys. Res. Space Phys. 122, 12019–12030 (2017).
Graham, D. B. et al. Enhanced escape of spacecraft photoelectrons caused by Langmuir and upper hybrid waves. J. Geophys. Res. Space Phys. 123, 7534–7553 (2018).
Graham, D. B. et al. Universality of lower hybrid waves at earth’s magnetopause. J. Geophys. Res. Space Phys. 124, 8727–8760 (2019).
Burch, J. L., Moore, T. E., Torbert, R. B. & Giles, B. L. Magnetospheric multiscale overview and science objectives. Space Sci. Rev. 199, 5–21 (2016).
Lindqvist, P.A. et al. The spinplane double probe electric field instrument for MMS. Space Sci. Rev. 199, 137–165 (2016).
Ergun, R. E. et al. The axial double probe and fields signal processing for the MMS mission. Space Sci. Rev. 199, 167–188 (2016).
Russell, C. T. et al. The magnetospheric multiscale magnetometers. Space Sci. Rev. 199, 189–256 (2016).
Le Contel, O. et al. The searchcoil magnetometer for MMS. Space Sci. Rev. 199, 257–282 (2016).
Pollock, C. et al. Fast plasma investigation for magnetospheric multiscale. Space Sci. Rev. 199, 331–406 (2016).
Rager, A. C. et al. Electron crescent distributions as a manifestation of diamagnetic drift in an electron scale current sheet: magnetospheric multiscale observations using new 7.5 ms fast plasma investigation moments. Geophys. Res. Lett. https://doi.org/10.1002/2017GL076260 (2018)
Khotyaintsev, Y. V. et al. Electron jet of asymmetric reconnection. Geophys. Res. Lett. 43, 5571–5580 (2016).
Huba, J. D., Gladd., N. T. & Papadopoulos, K. The lowerhybriddrift instability as a source of anomalous resistivity for magnetic field line reconnection. Geophys. Res. Lett. 4, 125–128 (1977).
Pritchett, P. L. The influence of intense electric fields on threedimensional asymmetric magnetic reconnection. Phys. Plasmas 20, 061204 (2013).
Le, A., Daughton, W., Chen, L. J. & Egedal, J. Enhanced electron mixing and heating in 3d asymmetric reconnection at the earth’s magnetopause. Geophys. Res. Lett. 44, 2096–2104 (2017).
Price, L. et al. Turbulence in threedimensional simulations of magnetopause reconnection. J. Geophys. Res. Space Phys. 122, 11,086–11,099 (2017).
Price, L., Swisdak, M., Drake, J. F. & Graham, D. B. Turbulence and transport during guide field reconnection at the magnetopause. J. Geophys. Res. Space Phys. 125, e2019JA027498 (2020).
Khotyaintsev, Y. V. et al. Electron heating by debyescale turbulence in guidefield reconnection. Phys. Rev. Lett. 124, 045101 (2020).
Chen, L.J. et al. Electron diffusion region during magnetopause reconnection with an intermediate guide field: magnetospheric multiscale observations. J. Geophys. Res. Space Phys. 122, 5235–5246 (2017).
Graham, D. B. et al. Instability of agyrotropic electron beams near the electron diffusion region. Phys. Rev. Lett. 119, 025101 (2017).
Webster, J. M. et al. Magnetospheric multiscale dayside reconnection electron diffusion region events. J. Geophys. Res. Space Phys. 123, 4858–4878 (2018).
Sonnerup, B. U. O. Theory of the lowlatitude boundary layer. J. Geophys. Res. 85, 2017–2026 (1980).
Treumann, R. A., Labelle, J. & Pottelette, R. Plasma diffusion at the magnetopause  The case of lower hybrid drift waves. J. Geophys. Res. 96, 16009 (1991).
Markidis, S. et al. Multiscale simulations of plasma with ipic3d. Mathematics and Computers in Simulation 80, 1509–1519 (2010).
Brackbill, J. U. & Forslund, D. W. An implicit method for electromagnetic plasma simulation in two dimensions. J. Comput. Phys. 46, 271–308 (1982).
Shay, M. A. et al. Electron heating during magnetic reconnection: a simulation scaling study. Phys. Plasmas 21, 122902 (2014).
Lapenta, G., Markidis, S., Divin, A., Goldman, M. & Newman, D. Scales of guide field reconnection at the hydrogen mass ratio. Phys. Plasmas 17, 082106 (2010).
Graham, D. B. danbgraham/anomres: initial release (v1.0.0). Zenodo https://doi.org/10.5281/zenodo.6370048 (2022).
Acknowledgements
We thank the entire MMS team and instrument PIs for data access and support. This work was supported by the Swedish National Space Agency (SNSA), grants 128/17 and 176/15, and the Swedish Research Council grant 201605507. A.D. is supported by the Ministry of Science and Higher Education of the Russian Federation under agreement 075152021583. J. F. D. is supported by NASA Grant 80NSSC19K0396 and NSF Grant PHY2109083. The French involvement (SCM instruments) in MMS is supported by CNES and CNRS.
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D.B.G. carried out the data analysis, primary interpretation of the data, and manuscript writing. Y.V.K., M.A., and A.V. contributed to data interpretation and manuscript preparation. A.D. provided simulation support. J.F.D., C.N., J.L.B., K.J.H., and K.D. contributed data interpretation. O.L.C, P.A.L., A.C.R., D.J.G., and C.T.R. provided data support.
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Graham, D.B., Khotyaintsev, Y.V., André, M. et al. Direct observations of anomalous resistivity and diffusion in collisionless plasma. Nat Commun 13, 2954 (2022). https://doi.org/10.1038/s41467022305618
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DOI: https://doi.org/10.1038/s41467022305618
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