High-Frequency Waves Driven by Agyrotropic Electrons Near the Electron Diffusion Region.

National Aeronautics and Space Administration's Magnetosphere Multiscale mission reveals that agyrotropic electrons and intense waves are prevalently present in the electron diffusion region. Prompted by two distinct Magnetosphere Multiscale observations, this letter investigates by theoretical means and the properties of agyrotropic electron beam-plasma instability and explains the origin of different structures in the wave spectra. The difference is owing to the fact that in one instance, a continuous beam mode is excited, while in the other, discrete Bernstein modes are excited, and the excitation of one mode versus the other depends on physical input parameters, which are consistent with observations. Analyses of dispersion relations show that the growing mode becomes discrete when the maximum growth rate is lower than the electron cyclotron frequency. Making use of particle-in-cell simulations, we found that the broadening angle Δ in the gyroangle space is also an important factor controlling the growth rate. Ramifications of the present finding are also discussed. ©2020. The Authors.

Introduction

Magnetic reconnection is a fundamental physical process in plasmas. It converts magnetic energy into kinetic energy of charged particles by reconfiguring topologies of magnetic field lines. The recently launched National Aeronautics and Space Administration's Magnetosphere Multiscale (MMS) mission enables investigations of electron‐scale phenomena in reconnection sites as the four MMS spacecraft provide high‐resolution data (Burch et al., 2016). Specifically, they reveal details of the electron diffusion region (EDR) of magnetic reconnection. In this region, electrons are demagnetized. Consequently, the kinetic physics of electrons becomes dominant (Birn et al., 2001; Vasyliunas, 1975).

One of most important findings is the existence of crescent‐shaped agyrotropic electrons in EDRs (Burch et al., 2016). They are generated by meandering motions of electrons (Hesse et al., 2014; Scudder & Daughton, 2008), and their structures have been intensely studied (Bessho et al., 2014; Egedal et al., 2016; Shuster et al., 2015). Agyrotropic electrons play crucial roles in generating the reconnection electric field via off‐diagonal electron pressure tensor terms (Hesse & Winske, 1994; Hesse et al., 2014; Lyons & Pridmore‐Brown, 1990).

Moreover, the MMS spacecraft observes that agyrotropic electrons generate various waves, for example, Langmuir waves, upper‐hybrid waves (Burch et al., 2018, 2019; Dokgo et al., 2019; Graham et al., 2017; Tang et al., 2019), and electron Bernstein waves (Graham et al., 2018; Li et al., 2020). As such waves are generated near EDRs, they could significantly affect the environment of EDRs via wave‐particle interactions. They can energize and diffuse electrons and are also a possible source of anomalous resistivity or viscosity that supports the reconnection electric field. Previous studies of agyrotropic‐beam instability are incomplete in that either agyrotropy is ignored or that effects of magnetic field are not included. This letter rectifies such shortcomings and test the theory against recent MMS observations.

We study two MMS events when the spacecraft observed agyrotropic electrons near EDRs. These events show two different types of high‐frequency wave (HFW) activities. By employing an electrostatic linear dispersion relation, which is appropriate for analyzing instabilities driven by agyrotropic electrons, we accurately interpret the observed wave phenomena in terms of the agyrotropic beam‐plasma instability. Particle‐in‐cell (PIC) simulations show that the broadening angle is also an important factor, which can change the growth rate by more than an order of magnitude and eventually affect the wave structure generated. In this study, we use burst mode data from the fluxgate magnetometer (Russell et al., 2016), the electric field double probes (Ergun et al., 2016; Lindqvist et al., 2016), and the fast plasma investigation (Pollock et al., 2016).

Two Different Types of Waves in MMS Observations

Figure 1 presents MMS observations of two events: (left) the Event 1 on 3 July 2017 near 05:26:50 UT (Burch et al., 2019; Dokgo et al., 2019) in the inflow region near the EDR of the symmetric reconnection at the magnetotail and (right) the Event 2 on 24 December 2016 near 15:30:32 UT (Li et al., 2020) in the electron exhaust region of the asymmetric reconnection at the dayside magnetopause. The yellow‐shaded regions in the top panels are in or near the EDR. The field data are presented in LMN coordinates calculated by minimum variance analysis (Paschmann et al., 1998) of the magnetic field, which are L = [0.97, 0.25, 0.04], M = [0.23, 0.93, 0.29], N = [ 0.11, 0.27, 0.96] for Event 1 and L = [0.09,0.08,0.99], M = [0.23, 0.97,0.06], N = [0.97, 0.22, 0.10] for Event 2 in geocentric solar ecliptic coordinates. (L: reconnecting direction, M: out‐of‐plane guide field direction, and N: normal direction)

Figure 1

MMS Observations of (left) Event 1 and (right) Event 2. (a, f) Magnetic field, (b, g) electric field, (c, h) burst of HFWs, (d, i) field spectrum, and (e, j) electron distribution in the perpendicular plane. Black dashed lines in (d) and (i) are ( 4.8 kHz) and ( 0.86 kHz), respectively.

For Event 1, MMS spacecraft crossed the EDR when reverses from negative to positive (Figure 1a) (Dokgo et al., 2019). As shown in Figure 1b, HFWs are observed in the field before the EDR crossing. We enlarge the interval exhibiting the most intense wave burst (Figure 1c) and present the power spectrum in Figure 1d, where horizontal lines corresponding to integer multiples of electron plasma frequency, , , ( 4.8 kHz) are indicated. Figure 1e shows agyrotropic electrons in the perpendicular plane. Ignoring effects of the magnetic field on the agyrotropic beam, Burch et al. (2019) found that the linear beam‐plasma interaction between core and agyrotropic electrons generate the fundamental mode near .

We present the observation for Event 2 in the same format (Figures 1f–1j). HFWs observed in the electron exhaust region are shown in Figure 1g. Note that in this case multiple electron cyclotron harmonics, , , ( 0.86 kHz) are indicated with horizontal dashed lines in Figure 1i, together with the wave power spectrum. Highly discrete wave signals at multiples of characterize this event. As shown in Figure 1j, the agyrotropic electrons are also observed in this event. By assuming a gyrotropic ring electron distribution, Li et al. (2020) interpreted such a feature in terms of the customary electron Bernstein mode excitation.

While dispersion relations calculated by Burch et al. (2019) and Li et al. (2020) for each event show a superficial agreement with observations, strictly speaking both of their assumptions are not supported by MMS observations, since observations show agyrotropy of electron distributions with the ambient magnetic field. Moreover, as the free energy source for both events resides in the common agyrotropic nature of the electron distribution, the only difference being plasma parameters, it is desirable to formulate a unified theory that may account for both events. In the present letter we calculate the dispersion relation including effects of agyrotropic electrons and the magnetic field, so as to offer unified explanations for the two events showcased in Figure 1.

A Unified Kinetic Theory for Both Types of Waves

Derivations of electrostatic dispersion relations are based upon the approach taken by Romeiras and Brinca (1999) with modifications. Cylindrical coordinates are used, where the ambient magnetic field is directed along axis and is the gyrophase angle in perpendicular ‐ plane with the axis defines . The agyrotropic electrons propagate to the direction.

The zeroth‐order distribution is given by , where is a thermal ring Maxwellian and the subscript indicates species: eM (electron Maxwellian), pM (proton Maxwellian), and b (agyrotropic beam). stands for Maxwellian. Thermal speed is defined by , where and are mass and temperature, respectively. is the anisotropy. is a perpendicular drift speed; thus, . Note that Romeiras and Brinca (1999) considered only . is the normalization factor, which is given for each species by , and , where . We assume gyrotropic cores and monoenergetic beam, so it is given by and , following Romeiras and Brinca (1999). Note that is a periodic delta function, known as the Dirac comb, because is a ‐periodic variable. We consider that all perturbations propagate along axis, so the first‐order perturbation in ( being the density) can be expressed in the form .

Calculating the perturbed charge density and inserting to Poisson equation, we may derive the dispersion relation: where is the gyrofrequency, is the charge, is the plasma frequency for species , is the vacuum permittivity, is the modified Bessel function of the first kind of order , is the Bessel function of the first kind of order , , , , and . Solving equation (1) leads to the dispersion relations describing wave frequency and growth for instability excited by agyrotropic beams.

Making use of equation (1) and plasma parameters obtained from two MMS events (Burch et al., 2019; Dokgo et al., 2019; Li et al., 2020), we calculate dispersion relations. We focus on electron interactions, so components of electron core and beam are considered while assuming the ions are immobile. Parameters are given by , , , , , , and for Event 1 and , , , , , , and for Event 2, where is the speed of light, is the total density, and . We have summed the second and third terms in equation (1) from to +100 in order to fully include effects of the magnetic field above the electron plasma frequency for both cases.

Figure 2 shows dispersion relations and comparisons between observations. Figure 2a is the dispersion relation corresponding to Event 1 in space. The horizontal axis represents the wave number , while the vertical axis is the frequency . Black lines are the real parts, while the blue line is the imaginary part (growth rate). The red line emphasizes the growing part of the dispersion curves where growth rates are positive. The horizontal black curves (near integer multiples of ) arise from including the effects of ambient magnetic field, which in the unmagnetized plasma theory are absent (Burch et al., 2019; Dokgo et al., 2019). For Event 1, the cyclotron harmonics are closely packed so that the growth rate envelop forms a rather smooth and continuous curve. As a result, the dispersion relation is almost the same as the unmagnetized analysis carried out by Burch et al. (2019). Figure 2b shows the averaged power of spectrum as observed by MMS. The frequency range of the fundamental mode agrees with the growth rate converted to the real frequency domain shown in Figure 2c. As already explained in Dokgo et al. (2019), higher harmonics are results of nonlinear mode excitations, which is beyond scope of the present linear theory.

Figure 2

Comparisons between MMS observations and theory for (left) Event 1 and (right) Event 2. (a, d) Dispersion relations and growth rates in space by equation (1) using plasma parameters obtained from MMS data, (b, e) averaged powers of observed electron field data, and (c, f) growth rates computed from theory.

Effects of the ambient magnetic field are very important for Event 2. In this case, gyroharmonic separations in frequency space are sufficiently wide so that the growth rate envelop features distinct local maxima. This is the result of beam mode intersecting horizontal cyclotron modes, which leads to the growth rate curve becoming discretized. This is shown in Figure 2d, where horizontal and vertical axes are normalized by and , respectively. Indeed, the growth rate has local maxima near integer multiples of , which resembles the electron Bernstein waves, but it is important to emphasize that the free energy source is primarily provided by the agyrotropic electrons. The averaged power spectrum of (Figure 2e) and the theoretical growth rate (Figure 2f) agree with each other; the most intense modes are in the range from 4 to 10 , and each mode is located between n and (n+1) . We note that the frequency range of this event is from 0.1 to 0.3 in the unit of and this is very low compared to Event 1. The frequency range is downshifted from because of high beam density (Cairns, 1989).

We carry out analyses of general properties of agyrotropic beam‐plasma interactions. As a reference case, we use input parameters given by , , , , , and . Figure 3 shows real frequencies and growth rates for unstable modes in space. Four sets of upper and lower panels show the influence of varying the magnetic field, beam speed, beam density, and the temperature. The reference case is plotted by black lines in all the panels.

Figure 3

Dispersion relations (upper panel) and growth rates (lower panel) derived by changing (a, b) ratio, (c, d) beam drift speed, (e, f) beam density, and (g, h) temperatures of beam and core.

It is interesting to note that the instability property undergoes a gradual yet distinct change near the threshold corresponding to . That is, for higher than in magnitude, it is seen that the growing modes turn into a continuous beam mode. As the maximum growth rate falls below , however, the modes discretizes and take on the characteristics of Bernstein modes. Such a description is born out by all the cases considered in Figure 3 This behavior is not too difficult to understand, however, since the effects of the ambient magnetic field increasingly becomes important as the wave growth time scales exceed the inverse cyclotron period, .

Figure 3a shows the influence of varying on the real frequency. It is seen that the unstable range monotonically decreases until it shrinks to a narrow range near Figure 3b shows that increasing from the reference value of 0.1 leads to a reduction in the growth rate. As field increases, the individual cyclotron harmonic structure associated with the growth rate becomes more distinct. In contrast, when is reduced to 0.05, the growth rate envelop becomes a smooth curve, which resembles the unmagnetized beam‐plasma instability growth rate.

Figure 3c shows that increasing beam speed steepens the beam mode curve in space, which is expected. Figure 3d, which plots the growth rate, indicates that decreasing/increasing the beam speed has a similar effect as increasing/decreasing field intensity. That is, for higher beam speed the growth rate curve resembles the unmagnetized case in that individual harmonic peaks merge, while for lower beam speed, gyroharmonic peaks become more prominent.

Effects of varying beam density on the real frequency are generally quite small until the beam density is increase to 0.3 , which is shown in Figure 3e. Figure 3f shows that decreasing beam density leads to the reduction in growth rate as well as more prominent harmonic structure. For beam density corresponding to 0.3 , the growth rate increases significantly in magnitude, while the envelop becomes continuous. Note that the real frequency where the growth rate is maximum decreases to when the beam density is 0.3 . Consequently, frequency ranges of growing modes can be an order of as in Event 2.

The effects of varying thermal speed are noteworthy. In general, the lower the thermal speed, the more continuous the growth rate curve becomes while increasing in magnitude. However, when thermal speed becomes very slow, and , separate branch(es) of unstable mode begin(s) to appear, as shown in the higher‐ region of Figure 3g. This secondary (and tertiary) modes is (are) related to upper‐hybrid waves from the core having slow thermal speeds. The envelop of growth rate curve in Figure 3h shows that the secondary unstable branch can be identified by small bumps at the high domain. The tertiary branch, which is barely visible in the growth rate plot, can nonetheless be identified as well.

We further investigate the dependence of growth rates on the broadening angle in the gyrophase space using 2‐D PIC simulation, where means that the crescent‐shape electrons exist within the range of gyroangle from to forming a Maxwellian in the gyrospace centered at . Therefore, the ring distribution corresponds to °. Our theory assumes a delta function shape in the gyrophase space, and this assumption cannot be implemented by simulations. On the other hand, it is difficult to formulate and solve the theoretical dispersion relation with finite distribution. Thus, theory and simulation are complementary. The particle loading and exchanging methods in Dokgo et al. (2019) are used to make crescent‐shaped distributions.

We plot maximum growth rates derived from theory and simulation in Figure 4, where axis represents the broadening angle and axis is the maximum growth rate . The red square denotes the theoretical maximum growth rate of the reference case ( ). Each black cross presents the maximum growth rate calculated from the slope of energy history of the field in each simulation run using different . The black dotted line is a multiplication of and . The growth rate from simulations converges to as approaches 0, which means that theory and simulation are in agreement. The maximum growth rate decreases as increases. Their trend follows (dotted line) when °; however, it has a finite value when . In the range of 90° °, the growth rate is not changed much.

Figure 4

Dependence of the maximum growth rate on the broadening angle . A Red square is the theoretical value of the reference ( ), and black crosses come from PIC simulations using different values of ( ). The black dotted line is .

The simulation results show that the effects of on the growth rate, especially in the range of 60°  °, can be quite complex. In this range the simple formula predicts that the growth rate should rapidly decrease, as indicated by dotted curve. Instead, the growth rate deduced from simulations shows that the reduction in the growth is arrested and it gradually settles down to some finite level. We also found that gradual transition from the beam mode to Bernstein mode takes place as increases, but we also found that the transition is rather smooth and not sudden (not shown). As can be seen from Figure 4, the growth rate crosses the threshold condition of as is varied such that our previously stated conclusion on the property of dispersion relations (see Figure 3) applies herewith. Recall that this threshold defines the transition from beam like to Bernstein mode.

Discussion and Conclusion

In this letter, we have investigated the detailed property associated with HFWs driven by agyrotropic electrons near the EDRs by employing MMS observations, kinetic theory, and PIC simulations. It was shown that these waves are basically discrete Bernstein modes when the effect of the ambient magnetic field is significant as in Event 2. As their growth rates increase depending on plasma parameters, each mode may overlap with adjacent modes. As a result, they may become a continuous beam mode as in Event 1.

We note that the types of reconnection, symmetric and asymmetric, are not the main reason for generations of different types of waves because upper‐hybrid waves have been reported for both types of reconnection. However, different types of magnetic reconnection make different profiles of plasma parameters. Moreover, considering EDR environments, plasma parameters can vary depending on the locations of observations. Accordingly, various properties of HFWs could be detected near EDRs, such as broad frequency ranges from to and continuous or discrete structures, as shown above. Our theoretical approach thus shows that generations of those waves can be explained by a unified theory, and the underlying physics is the same. We have identified that the maximum growth rate could be a threshold condition that delineates one type of wave versus the other, namely, discrete versus continuous.

The amplitude of HFWs near EDRs can be extremely high, reaching up to 400 mV/m (Graham et al., 2017). Thus, HFWs can affect reconnection processes by energizing electrons effectively, which in turn may contribute to the origin of anomalous resistivity and viscosity. In addition, nonlinear processes of HFWs can generate electromagnetic radio emissions (Dokgo et al., 2019), which could, in principle, facilitate remote sensing of occurrences of magnetic reconnection. Consequently, investigating the properties of HFWs may be crucial for understanding the physics of EDRs and reconnection processes. Our study provides a more precise and unified treatment of HFWs driven by agyrotropic electrons near EDRs. With the help of quasi‐linear theory and kinetic simulation, future studies can quantitatively estimate anomalous resistivity and viscosity by HFWs in EDRs, which could eventually lead to the identification of energy conversion process in EDRs.

We have focused on electron phenomena near EDRs ignoring ions in this study, however, equation (1) is equally applicable for waves driven by agyrotropic ions. Since agyrotropic ions are ubiquitous in space, such as in ion diffusion regions of magnetic reconnection (Graham et al., 2017; Wang et al., 2016), the Earth's foreshock region (Eastman et al., 1981; Gosling et al., 1982), thin current sheets (Zhou et al., 2009), and near the space shuttle (Cairns, 1990), interactions between ion cores and agyrotropic ions are an important topic of fundamental plasma physics contributing energizations of both ions and electrons. Our work can provide detailed physics of generations of low‐frequency waves, such as lower‐hybrid waves or ion‐acoustic waves, driven by gyrobunched ions.