Whistler waves generated by nongyrotropic and gyrotropic electron beams during asymmetric guide field reconnection

Magnetic reconnection is an important process of energy release and plasma transport that occurs in laboratory and space plasmas. It converts energy from magnetic fields to plasma particles. In the electron diffusion region (EDR), electrons are decoupled from ions and become demagnetized, and they meander across the current sheet when the guide field is small.^1–15 In contrast, electrons outside the EDR are magnetized and have a circular motion around the magnetic field. To identify the kinetics of electrons at various locations in magnetic reconnection, electron velocity distribution functions (VDFs) are excellent physical quantities to analyze. Intensive studies have been done by particle-in-cell (PIC) simulations to understand electron kinetics during reconnection, and well-organized electron VDFs^7–17 have been found inside and outside the EDR, including VDF shapes of rings, swirls, arcs, triangles, and crescents. By theory and two-dimensional (2D) PIC simulations, the origin of a crescent VDF,^8,11–13,18 the relation between the thickness of a crescent VDF and the reconnection electric field,¹⁴ and the effect of the guide field on a crescent VDF^15,17 have been revealed.

Perpendicular/parallel electron crescent VDFs have been observed in current sheets in the Earth's magnetopause reconnection by NASA's Magnetospheric Multiscale mission (MMS).^18–32 The perpendicular crescent VDF presents a crescent shape in the velocity plane perpendicular to the magnetic field, while the parallel crescent VDF exhibits a crescent shape toward the velocity parallel/anti-parallel to the magnetic field. Perpendicular crescent shapes in electron VDFs indicate meandering particles in the current sheet during reconnection when they are found in the EDR, and the gyrodiameters of those particles are larger than the distance between the measurement point and the magnetic neutral line. Perpendicular/parallel crescent VDFs are used to identify the EDR and indicate the kinetics of particles including the generation of nongyrotropy, particle acceleration, and finite gyroradius effects in reconnection.

Through space measurements in current sheets in the Earth's magnetopause and magnetotail, including MMS observations, various types of waves such as upper-hybrid waves,^26,33–38 whistler waves,^{32,35,39–53} electrostatic solitary waves,^{33,41,44,49,54–56} electron cyclotron waves,^41,48,55,57 and Langmuir waves^{35,44,51,55,57} are observed inside/outside the diffusion region. Among them, whistler waves can be excited by electron-scale kinetic physics during reconnection. For example, space observations show that whistler waves are associated with a temperature anisotropy instability due to electron perpendicular heating^{41,43,44,46–52} or a loss cone instability.^42,47 In contrast, PIC simulations and space observations demonstrate that whistler waves can also be generated due to electron beams.^{40,46,49,53,58,59} Further studies are necessary to understand the roles of these waves in reconnection.

In this paper, by means of a 2D PIC simulation of asymmetric guide field reconnection, we show enhancements of wave intensities during reconnection with frequencies less than the electron cyclotron frequency. At the same time of the wave power enhancements, gyrotropic ring VDFs and nongyrotropic perpendicular/parallel crescent VDFs are observed with electron beams. We investigate the relation between electron beams and the wave activity inside/outside the EDR using electron VDFs and the fast Fourier transform (FFT) analysis. We demonstrate that both gyrotropic and nongyrotropic electron beams exist inside the EDR and investigate the effect of the beam speed, density, and nongyrotropy on the wave intensity. In Sec. II, we explain the simulation setup and structures of asymmetric guide field reconnection. In Sec. III, we investigate wave properties and the relation between wave intensities, electron VDFs, and electron temperature anisotropy. In Sec. IV, we will present the conclusions of this study.

We employ a 2D PIC simulation to study asymmetric reconnection with a small guide field whose magnitude is less than the reconnecting magnetic field. Periodic boundary conditions are used in the x direction, which is parallel to the reconnecting magnetic field. Conducting walls are applied as the boundary condition in the z direction, which is normal to the current sheet. The y coordinate is parallel to the current in the current sheet. The system size is $Lx×Lz=51.2di×25.6di$ with $2048×1024$ grids, where 1 grid = 0.025d_i. The length $di=c/(4πn0e2/mi)1/2$ is the ion skin depth based on $n0$⁠, which is the density in the magnetosheath, and we use the speed of light c, the elementary charge e, and the ion mass $mi$⁠. The initial magnetic field and density at t = 0 are $Bx=B0(tanh(z/w)+α1)$ and $n=n0[1−α2−α2tanh(z/w)+(α2/2α1)sech2(z/w)]$ with $α1=1/6$ and $α2=7/16$⁠, respectively, where $B0$ is the average asymptotic field and $w=0.5di$ is the current sheet width. With these $α1$ and $α2$ values, the initial magnetic fields in the magnetosphere and the magnetosheath are (7/6)B₀ and (5/6)B₀, respectively. The initial electron densities are also n₀ in the magnetosheath and n₀/8 in the magnetosphere. These initial values are comparable with values in MMS observations²⁵ during magnetopause reconnection. At the initial time t = 0, $vdi/vde=−Ti/Te$ and $B02/8π=n0(Ti+Te)(α2/2α1)$ are satisfied, where $Ti,e$ and $vdi,de$ are temperatures and the y-directed drift speeds for each species, respectively. A perturbation to the magnetic flux function is added by $Ψ=0.1diB0sech2(x/w)sech2(z/w)$ to initiate reconnection. The mass ratio of ion to electron is $mi/me=200$⁠, the temperature ratio is $Ti/Te=2$⁠, and the ratio of the plasma frequency to the electron cyclotron frequency is $ωpe/Ωe=2$⁠, where $ωpe=(4πn0e2/me)1/2$ and $Ωe=eB0/mec$⁠. The simulation time step is Δt = 0.026 $ωpe−1$ = 0.000 638 2$/Ωi$⁠, where $Ωi$ is the ion cyclotron frequency based on $B0$⁠. The Alfvén speed based on B₀ and n₀ is $Va=B0/(4πn0mi)1/2=c/202$⁠. A guide field (⁠$BG$⁠) is applied in the y direction, and $BG=0.3B0$ at t = 0, which is also a typical value in magnetopause reconnection. We rotate the system clockwise around the z axis to maximize the reconnection rate,⁶⁰ and the rotation angle for the above guide field is 2.69°. In the following, all the simulation results are shown in the rotated system.

Figure 1 shows color contours of various quantities at $t=44.67 Ωi−1$⁠. Every plot except for (n) in Fig. 1 also includes the projection of magnetic field lines on the x–z coordinate plane at $t=44.67 Ωi−1$⁠.

The plots in Fig. 1 present magnetic fields $Bx,By,and Bz$ in (a), (b), and (c), electric fields $Ex,Ey,and Ez$ in (d), (e), and (f), current densities $Jx,Jy,and Jz$ in (g), (h), and (i), electron fluid velocities $Uex,Uey,and Uez$ in (j), (k), and (l), and the electron density n_e in (m). Magnetic fields $Bx$ and $Bz$ [Figs. 1(a) and 1(c)] present field patterns of reconnection where the X-line is located at (1.15d_i, 0.05d_i) in the x–z plane. The magnetic field $By$ [Fig. 1(b)] shows a bipolar magnetic field pattern in the x direction, which is a characteristic of the Hall effect in asymmetric reconnection.⁶¹ Figure 1(f) shows a pattern of the Hall electric field $Ez,$⁶¹ which is dominant on the separatrices in the magnetospheric side (z > 0). The electric field $Ex$ in Fig. 1(d) is also dominant on the separatrices in the magnetospheric side, and the direction is away from the X-line. The reconnection electric field $Ey$ near the X-line is shown in Fig. 1(e), and the value of the reconnection rate at the X-line is 0.11 $B0Va/c$ at $t=44.67Ωi−1$⁠. Figure 1(n) presents the reconnection electric field (reconnection rate) as a function of time. The red point in Fig. 1(n) indicates the time $t=44.67/Ωi$⁠, which is at the peak stage in the reconnection rate.

The current densities [Figs. 1(g)–1(i)] have strong magnitudes on the separatrices and near the X-line. $Jy$ is enhanced near the X-line and the separatrices. $Jx$ directs into the X-line in the positive z side (magnetosphere) and away from the X-line in the negative z side (magnetosheath) on the separatrices. $Jz$ points to the negative z region on all the separatrices. The in-plane current flows along the contours of B_y in the 2D plane, flowing into the X-line in the magnetospheric side and out of the X-line in the magnetosheath side. The electron fluid velocities in Figs. 1(j) and 1(l) show the electron outflows from the X-line along the separatrices in the magnetospheric side, and the inflows toward the X-line along the separatrices in the magnetosheath side. In the left side of the X-line (negative x region), the outflow speed $Uex$ [Fig. 1(j)] becomes larger than in the right side of the X-line (positive x region). This asymmetry in the x direction is because of the guide field. The velocity $Uey$ [Fig. 1(k)] indicates that electrons are directed out of the x–z plane near the X-line and the separatrices in the magnetospheric side. The electron density in Fig. 1(m) shows that more electrons are in the magnetosheath side. The density becomes small above the magnetospheric separatrices, but there is asymmetry in the x direction because of the effect of the guide field. We also investigated field quantities in a test simulation where the system size is doubled (51.2d_i $×$ 102.4d_i, data not shown), and they are similar to the ones in the original simulation.

We investigate waveforms of the electric fields E_x and E_z [Figs. 2(a) and 2(b)] and magnetic field B_y [Fig. 2(c)] in the reconnection region, using data from t = 44.674 to 46.269 $Ωi−1$⁠. Figure 2 shows spatial field variations, ΔE_x, ΔE_z, and ΔB_y, due to waves at $t=45.120 Ωi−1$⁠, obtained by subtracting the time averaged field quantities ⟨E_x⟩, ⟨E_z⟩, and ⟨B_y⟩ in 44.674 $Ωi−1$ < t < 46.269 $Ωi−1$ from the original fields E_x, E_z, and B_y, respectively. White lines represent magnetic field lines.

Wave activity is found in both the positive and negative x regions, but clearer waveforms in the electric and magnetic fields are seen on/near the separatrix in the magnetospheric side (z > 0) and in the negative x region, where we study the wave activity and the kinetics of electrons in this section. Both the electric and magnetic field fluctuations on/near the separatrix in the negative x region show similar wavelengths and frequencies: the wavelengths range from 0.5 to 0.8d_i, and the frequencies range from 0.1 to 0.2 $Ωel$⁠, which are less than the local electron cyclotron frequency $Ωel$⁠. The waves are propagating almost along the separatrices, and almost antiparallel to the magnetic field in both the positive and negative x regions: waves in the positive x region (right to the X-line) are propagating toward the X-line (see the movie files in the supplementary material), while waves in the negative x region (left to the X-line) are propagating away from the X-line. The measured angle of wave propagation to the magnetic field ranges from around 140° to around 170°. The wave amplitude in the negative x region is much larger than that in the positive x region. The negative x side from the reconnection X-line has the electron outflow speed larger than the positive x side [see U_ex in Fig. 1(j)], which suggests that the larger electron outflow enhances the wave activity in the negative x side.

The amplitudes of E_x and E_z fluctuations near the magenta position (x, z) = (−1.475d_i, 0.35d_i) in Fig. 2 are of the order of 0.01B₀, and the amplitudes in the region left to the magenta position, x < −1.475d_i, become larger and larger as we move away from the X-line. In Fig. 2(a), we can separate electrostatic fluctuations by $Ek=kxExkx2+kz2=0.96Ex$⁠, where $Ek$ means the electric field parallel to the wave vector k with components (k_x and k_z). However, since the wave vector almost points in the x direction, E_x fluctuations are virtually exclusively electrostatic fluctuations. As we will discuss later, there is a wave mode destabilized by an electron beam and that has an electrostatic fluctuation, and these strong E_x fluctuations in Fig. 2 can be partially explained by the beam mode. However, the amplitude of E_x fluctuation is large in this region, and the formation of such a large-amplitude electrostatic fluctuation (mostly in E_x), which is due to the nonlinear evolution of the waves, will be discussed elsewhere. In this paper, we focus mainly on the transverse (electromagnetic) fluctuations in the same region along the separatrix, analyzing the magnetic field fluctuations. In the supplementary material, we included a space-time plot of waves (fluctuations in B_y) along a line parallel to the wave vector k in this region as well as its Fourier transformed plot (in the k-ω space).

We investigate the excitation mechanisms of the whistler waves near the EDR, focusing on the waves on/near the magnetospheric separatrix left to the X-line. Let us consider two mechanisms of whistler wave excitation: instabilities due to electron temperature anisotropy and electron beams. The electron temperature anisotropy is observed inside/around the EDR, as shown in Fig. 3, where the EDR ranges roughly −1.5d_i < x < 3.0d_i in the x direction and −0.6d_i < z < 0.7d_i in the z direction, corresponding to the gray shaded region in Fig. 4(a) surrounded by the four boundary points on the separatrices within which nongyrotropic electron VDFs are observed. In Fig. 4(a), the shaded region covers the strong current region (blue, corresponding to the region where J_y > 0.46 times the maximum of J_y). It is possible that whistler waves are generated when the electron temperature perpendicular to the magnetic field (⁠$Te⊥$⁠) is larger than the temperature parallel to the magnetic field (⁠$Te∥$⁠).⁶² 

Figure 3(a) shows that in the negative x area of the EDR, $Te⊥$ is larger than $Te∥$ only in a region near z = 0. In most of the regions where we observe significant wave activity of the whistler waves in the magnetospheric side (z > 0), $Te∥$∼$Te⊥$ or $Te∥$ >$Te⊥$ holds. Figure 3(b) is a blowup of the region around the position (x, z) = (−1.475d_i, 0.35d_i) (red dot), where we analyze the wave properties using a linear dispersion solver. Around the magnetospheric separatrix near the red dot position, strong electron beams exist, so that $Te∥$ > $Te⊥$⁠; however, even in the region of $Te∥$ > $Te⊥$⁠, there exists a highly anisotropic electron beam component whose $Te⊥$ is much greater than $Te∥$⁠, as shown in the discussion below, and such an anisotropic beam can excite a whistler wave.

We performed a numerical analysis using a linear dispersion solver, WHAMP.⁶³ We analyzed the wave at the red dot in Figs. 3(a) and 3(b), (x, z) = (−1.475d_i, 0.35d_i), at which we observed a relatively small temperature anisotropy (⁠$Te⊥/Te∥=0.890$⁠) and an anti-parallel electron beam. The 2D reduced VDF at the red position is shown in Fig. 3(c) in the velocity plane (⁠$V∥$⁠, $V⊥2$⁠). The velocity $V∥$ is parallel to the local magnetic field, and the velocities $V⊥1$ and $V⊥2$ are defined to be parallel to the following vectors: $b×Ue×b$ and $−Ue×b$⁠, respectively, where $Ue$ is the electron fluid velocity and b is the unit vector of the local magnetic field. Each velocity is normalized by the Alfvén speed (V_a) based on B₀ and n₀. The values at the color bar are the number of electrons in the log scale. Figure 3(d) shows a model VDF at this position with two electron populations, and we used the parameters as follows. Electron component 1 has a parallel temperature $Te1∥=1.91meVal2$⁠, a perpendicular temperature $Te1⊥=14.6meVal2$ (the ratio $Te1⊥/Te1∥$ = 7.64), and a parallel drift speed $Ve1∥=−6.97Val$⁠. Here, we use $Val=0.0345c$ as the Alfvén speed based on the local electron density and the local magnetic field, and the mass ratio $mi/me$ = 200. Electron component 2 has a parallel temperature $Te2∥=30.9meVal2$⁠, a perpendicular temperature $Te2⊥=35.4meVal2$ (the ratio $Te2⊥/Te2∥$ = 1.15), and a parallel drift speed $Ve2∥=−1.33Val$⁠. The densities of components 1 and 2 are 0.3n_l and 0.7n_l, respectively, where n_l is the local density. Under these parameters, the parallel and perpendicular temperatures for the total electron population (combining component 1 and component 2) become $Te∥=28.8 meVal2$ and $Te⊥=29.2 meVal2$⁠, respectively, and the temperature anisotropy for the total electron is almost unity. In the linear dispersion analysis, we used a single component of ions: the ion parallel temperature is $Ti∥=40.5meVal2$⁠, the ion perpendicular temperature is $Ti⊥=79.2meVal2$⁠, and the ion parallel drift speed is $Vi∥=−0.530Val.$ Note that both the electron and ion distribution functions in the simulation are actually nongyrotropic, but to make the analysis simple, we used a shifted bi-Maxwellian model. The effect of the nongyrotropy is not taken into account in this linear analysis, which is beyond the scope of this paper.

Figures 3(e) and 3(f) show the results of the linear dispersion analysis. We found two modes: mode 1 (panel e) is excited by the cyclotron resonance due to the anisotropic fast electron beam (component 1), and mode 2 (panel f) is excited by the Landau resonance due to the slow electron beam (component 2). The blue and red curves show the real part and the imaginary part of the frequency, respectively, of the unstable wave that propagates in the direction with an angle of 140° (solid line), 160° (dashed line), and 180° (dotted lines), respectively, with respect to the magnetic field. At this position (x, z) = (−1.475d_i, 0.35d_i), the propagation angle in the simulation is 140° because of a relatively large B_y magnetic field B_y ∼ B_x, but near the separatrix, there is also a region with a small B_y, where the propagation angle is around 170°, and VDFs are similar to Fig. 3(c). These two wave modes in Figs. 3(e) and 3(f) show that the maximum growth rate occurs in a range of wavenumbers 20 < $kdil$ < 30 for Mode 1 and 15 < $kdil$< 30 for Mode 2, respectively, where $dil$ is a local ion skin depth, and a range of the real frequencies 0.15Ω_el < ω < 0.24Ω_el for Mode 1 and 0.1Ω_el < ω < 0.16Ω_el for Mode 2, respectively, which are consistent with the wavenumber $kdil$ = 25.5 (λ ∼0.25d_il) and the frequency ω = 0.17Ω_el in the simulation at the red point (the frequency will be shown in Fig. 4).

Mode 1 is excited by the cyclotron resonance due to the fast electron beam (⁠$Ve1∥=−6.97Val$⁠), which has a very large anisotropy $Te1⊥/Te1∥$ = 7.64. This mode is propagating close to the direction anti-parallel to the magnetic field. Let us discuss the nature of this mode, considering the simplest case, which is the wave whose propagation angle is 180° (similar discussions can be made for waves with other propagation angles). The phase speed of this wave at the maximum growth rate is $ω/k∥=−1.62Val$ (⁠$k∥=−k$⁠). The resonant velocity based on the cyclotron resonant condition, $ω−k∥Vres=−Ωel$⁠, is V_res = −8.4V_al, which is within the thermal spread (−8.9V_al < $V∥$ < −5.0V_al) of component 1 electron beam. Here, we used the negative frequency $−Ωel$ in the right-hand side, because the wave in the simulation frame is left-handed, as we explain below. Therefore, we conclude that this mode is generated by the cyclotron resonance of the resonant electrons whose V_ǁ is near the resonant velocity. In the simulation frame, the fast electron beam is moving much faster than the wave phase speed in the negative direction. To obtain the result in Fig. 3(e), we performed the WHAMP analysis for Mode 1 in the reference frame where the fast electron beam speed becomes −1 $Val$⁠, and let us call this frame as the analysis frame. In the analysis frame, the excited wave is moving opposite to the fast electron beam, with the same wavelength, but $k∥$ is positive. In the analysis frame, the wave is excited due to the cyclotron resonance because of the temperature anisotropy $Te1⊥/Te1∥$ = 7.64, and the wave has the right-handed polarization, propagating in the positive direction. When we change the reference frame to the original frame, where the fast electron beam has a speed $V∥=$−6.97 $Val$⁠, the wave polarity is changed to left-handed. In this original frame, although the observed wave polarity is left-handed, the resonant electrons in the fast beam will see the wave polarity as right-handed because the beam electrons are moving faster than the excited wave. In this way, if this mode is excited in the simulation, we will see this wave as a left-handed wave, but the electrons in the fast beam will resonate with this mode in the same way as the regular resonance with the whistler wave, whose polarity is right-handed. In the WHAMP analysis, we found that the frequency in Mode 1 in the fast electron beam frame saturates to the electron cyclotron frequency (data not shown), consistent with the whistler wave property.

Mode 2 is excited by the Landau resonance due to the slow beam, whose $V∥=−1.33Val$⁠. Let us consider again the simplest case, the wave whose propagation angle is 180°. The phase speed of this mode at the maximum growth rate is $ω/k∥=−1.17Val$⁠, which corresponds to the velocity $V∥$ in the electron VDF where the slope of the reduced 1D distribution function f (⁠$V∥$⁠) has a negative slope, df(⁠$V∥$⁠)/d $V∥$ < 0. Note that the wave is propagating in the negative $V∥$ direction, and df(⁠$V∥$⁠)/d $V∥$ < 0 in $V∥$ < 0 corresponds to the wave growth due to the Landau resonance. Therefore, Mode 2 at the 180° propagation angle is considered to be an electrostatic mode due to the slow electron beam. In the oblique propagation (not 180° propagation angle), Mode 2 is also excited by the slow beam in which there are both the Landau resonance and the cyclotron resonance. The ellipticity calculated by WHAMP shows 0.005, mostly a linear polarization but slightly right-handed. An elliptical right-handed polarization is due to the cyclotron resonance of electrons in component 2, for which the resonant velocity due to the condition $ω−k∥Vres=Ωel$ is $Vres=6.6Val$ for the propagation angle 160°. Here, we used the positive frequency $Ωel$ in the right-hand side because the wave in the simulation frame is right-handed. Near this positive resonant velocity, there are almost no electrons due to component 1, and the resonance occurs due to electrons in component 2. We also performed the linear analysis where component 1 (fast beam) was removed, and the result is almost the same as shown in Fig. 3(f). Therefore, we conclude that Mode 2 is due to component 2 (slow beam). Strong electrostatic fluctuations observed in the simulation [see Fig. 2(a)] can be due to Mode 2.

In the simulation, the whistler waves show both right-handed and left-handed polarizations, as we found in Mode 1 and Mode 2. Figures 3(g)–3(i) show hodograms of magnetic field fluctuations (including only a few wave oscillations, removing high frequency fluctuations using a 50Δt smoothing window) in the plane perpendicular to the mean magnetic field. Figure 3(g) shows the hodogram at the red point, where the WHAMP analysis was applied, (x, z) = (−1.475d_i, 0.35d_i), and the wave polarity is the mixture of left-handed and right-handed. The wave polarity is mostly right-handed in Fig. 3(h), which is for the wave at (x, z) = (0.65d_i, 0.25d_i) [white point in panel (a)], close to the X-line, where the beam speed and its intensity are still small (the VDF will be discussed later, at the top panel of column C in Fig. 7). In contrast, Fig. 3(i) shows that the wave polarity is mostly left-handed, which is for the wave at (x, z) = (−3.975d_i, 1.05d_i) [black point in panel (a)], where the fast beam intensity [see the VDF in Fig. 4(d)] is larger than the VDF shown in Fig. 3(c). Based on the consistency between the simulation result and the WHAMP analysis, we conclude that the whistler waves near the separatrix in the negative x side are the mixture of Mode 1 and Mode 2 identified by the linear analysis.

We interpret this result as follows. In the region around the separatrix, there are multiple electron beams generated by reconnection, both fast beams and slow beams, although the origin of these multiple beams remain to be investigated. In such a region, Mode 1 (electromagnetic wave) and Mode 2 (electrostatic wave with electromagnetic fluctuations) can be unstable. Mode 2 is the mode that has the properties of both an electrostatic wave (due to the Landau resonance) and a whistler wave (due to the cyclotron resonance), whose frequency at the maximum growth rate is around 0.1–0.2Ω_el, as we can see in Fig. 3(f). Since this frequency is close to the frequency of Mode 1 (whistler wave due to temperature anisotropy), Mode 1 and Mode 2 are coupled, and they propagate in the negative direction together. Therefore, we are seeing the whistler wave with large electrostatic fluctuations as in Fig. 2(a).

The result that the whistler wave is excited by the temperature anisotropy of the high energy beam (Mode 1) may be consistent with space observation studies^43,49 in which whistler waves were observed where there exist high energy electrons with large perpendicular energy, although the total temperature shows $Te∥$ > $Te⊥$ or $Te∥$ ∼ $Te⊥$⁠. Also, the generation of strong electrostatic field in the whistler wave (Mode 2) can be consistent with observations of large amplitude parallel electric fields together with whistler waves.⁴⁴ 

Note that Fig. 2(c) shows that there are B_y-fluctuations not only near the separatrices in the magnetospheric side, but also in the region near z = 0 in the negative x area. Figure 3(a) around z = 0 shows a temperature anisotropy $Te⊥/Te∥$ between 1 and 2, which can be the source of whistler waves generated by the temperature anisotropy instability in the region of the magnetic flux pileup and perpendicular heating.⁶⁴ Since there are no electrostatic fluctuations around z = 0 [Figs. 2(a) and 2(b)], these whistler waves are due to only the cyclotron resonance in the anisotropic electrons (Mode 1), not due to the Landau resonance as in Mode 2.

Let us investigate more details of the whistler waves in the region near the left-side separatrix at $t=44.67/Ωi$⁠, analyzing electron VDFs, which are plotted in a parallel velocity plane (⁠$V∥$ vs $V⊥2$⁠) and in the perpendicular velocity plane (⁠$V⊥1$ vs $V⊥2$⁠). VDFs on the separatrix show that there are both gyrotropic and nongyrotropic electron beams. Figure 4 shows the locations of VDFs we will discuss, VDFs in two velocity planes (reduced VDFs, integrated in the third velocity direction), and the wave activity at that time and those locations.

Figure 4(a) shows three locations in which we observe nongyrotropic beams, at (−1.475d_i, 0.35d_i), (−2.0d_i, 0.70d_i), and (−3.975d_i, 1.05d_i) in the (x, z) coordinates, and magnetic field lines are also shown. The red position and the blue position are near the edge of the EDR, and the green position is outside the EDR. These locations are close to the separatrix in which the wave activity is strong as in Fig. 2. The left panels in Figs. 4(b)–4(d) are the reduced VDFs in the velocity plane (⁠$V∥$⁠, $V⊥2$⁠), while the right panels are the reduced VDFs in the (⁠$V⊥1$⁠, $V⊥2$⁠) plane. The left panels of Figs. 4(b)–4(d) present intense electron beams in $V∥<0$ (away from the X-line), which produce Mode 1 (cyclotron resonance mode) of the whistler wave. In addition, in Figs. 4(b) and 4(c), there are less-intense slow-beam populations near $V∥∼0$ with small negative $V∥$⁠, which generate Mode 2 (beam mode) of the whistler wave. In the previous section, we have approximated the VDF in Fig. 4(b) with two components (component 1 and component 2). The VDF in Fig. 4(d) does not show a slow beam component with $V∥$ ∼ 0, and the hodogram in Fig. 3(i) for the same location shows the left-handed polarization in most of the time because of the lack of the slow beam component (see the wave excitation mechanisms in the previous section). The right panels in Figs. 4(b)–4(d) show asymmetric VDFs in the velocity plane perpendicular to the magnetic field, and the asymmetry in the $V⊥1$ direction means that the electron beam is nongyrotropic. Compared with Figs. 4(b) and 4(d), the VDF in Fig. 4(c) shows more nongyrotropy.

In Fig. 4(e), the temporal FFT of B_y-fluctuations at the red point (x, z) = (−1.475d_i, 0.35d_i) (left panel) and the B_y-fluctuations as a function of time (right panel) is shown. We use 2048 time steps of field data (2048Δt for the time interval from 44.674 $/Ωi$ to 45.981 $/Ωi$⁠), averaged over nine spatial points (3 $×$ 3 grids) around the measurement point for the FFT analysis. The horizontal axis in the FFT plot (left panel) is the frequency normalized by the local electron cyclotron frequency Ω_el, and the vertical axis is the square of the mode amplitude in the power spectrum. The left panel of Fig. 4(e) shows that the dominant mode has a peak at $ω=0.17Ωel$⁠. This mode has been increasing its amplitude with time during the reconnection evolution up to the investigation time in Fig. 4(e) (see the supplementary material). Note that a large amplitude in the frequency range <0.1 $Ωel$ in the FFT plot is not due to the waves we are investigating, but due to the extension of the EDR, and there is an almost linear trend of the field variation during the investigation time interval [see the right panel of Fig. 4(e)]. Therefore, in the following discussion, we will focus on the wave frequency larger than 0.1 $Ωel.$ The period $T=2π/ω$ of the dominant B_y-fluctuation in the right panel of Fig. 4(e) is about $0.38Ωi−1$⁠, which matches the peak frequency $ω=0.17Ωel$ in the FFT plot as $TΩi=2πΩi/0.17Ωel≃0.34$⁠. Figure 4(f) shows the FFT plots at the blue point (left panel) and the green point (right panel), and there are peaks in a range 0.15 $Ωel$ < $ω$ < 0.2 $Ωel$⁠.

The wave activity in the magnetospheric side depends on the existence of electron beams, and waves are ignorable if no electron beam is observed above the separatrix. In Fig. 5(a), we consider two locations, a blue point (−4.975d_i, 1.35d_i) and a red point (−4.975d_i, 3.225d_i), at $t=44.67/Ωi$⁠. The VDF in the (⁠$V∥$⁠, $V⊥2$⁠) plane at the blue point (−4.975d_i, 1.35d_i) [Fig. 5(d)] shows that a strong electron beam exists in the negative magnetic field direction (away from the X-line). Many electrons are concentrated in a region between $V∥=−10Va$ and $−4Va$⁠, moving away from the X-line. In contrast, there is no electron beam at the red point (−4.975d_i, 3.225d_i), where the VDF [Fig. 5(b)] shows a symmetric oval-shape whose center is at $V∥ ∼0$⁠. The symmetric circular shapes in the perpendicular velocity plane (⁠$V⊥1,V⊥2$⁠) at both locations indicate that they are almost gyrotropic [Figs. 5(c) and 5(e)]. The FFT plot for B_z-fluctuations in Fig. 5(f) shows that there is no wave activity where no electron beam exists. Figure 5(g) shows that the dominant mode in the region of the electron beam has a peak at a frequency of 0.14 $Ωel$⁠.

If a VDF has a component with $V∥$ ∼ 0 in the (⁠$V∥$⁠, $V⊥2$⁠) plane and if that component has a large density, it affects the wave activity near the separatrix. The anti-parallel beam enhances the wave power, but a large density of the component with $V∥$ ∼ 0 causes to lower the wave power. Figure 6(a) shows three locations: three red dots along x = −4.975d_i at (−4.975d_i, 1.35d_i), (−4.975d_i, 1.175d_i), and (−4.975d_i, 0.975d_i). The top red dot (the same position as the blue dot in Fig. 5) is located near the separatrix, and the others are slightly away from the separatrix. Figures 6(b)–6(d) show the VDFs in the (⁠$V∥$⁠, $V⊥2$⁠) plane and the (⁠$V⊥1$⁠, $V⊥2$⁠) plane, and FFT plots for B_y-fluctuations (red) and B_z-fluctuations (black) at each position, respectively. In Fig. 6(b), which is for the position located nearest to the separatrix, the VDF in the parallel velocity plane shows a strong anti-parallel electron beam (between $V∥=−10Va$ and $V∥∼−4Va$⁠), and there is a little component with $V∥$ ∼ 0. In the middle location [Fig. 6(c)], the VDF shows that the electron beam consists of electrons with a continuous parallel velocity range from $V∥=−10Va$ to $V∥∼+3Va$⁠, and there are many electrons with $V∥ ∼0$⁠, where they move only perpendicular to the magnetic field. In Fig. 6(d), which is farthest from the separatrix among the three positions, the VDF shows that the electron beam ranges from $V∥=−10Va$ to $V∥∼+5Va$⁠. The VDFs away from the separatrix have a larger population of the component with $V∥$ ∼0 than the VDF near the separatrix.

In the FFT plots at the three positions, the dominant modes in B_y (red) and B_z (black) show the largest amplitudes at the top position (−4.975d_i, 1.35d_i), which is the closest position to the separatrix and where the VDF has a component with $V∥$ ∼ 0 with the least density among the three VDFs. In contrast, at the other positions, which are slightly away from the separatrix and where the VDFs show components with $V∥$ ∼ 0 with larger densities than the top VDF, the wave powers become smaller as the population of the component with $V∥$ ∼ 0 becomes larger. This decrease in the wave power is due to the fact that the density of the component with $V∥$ ∼ 0 becomes larger, and the anisotropy (⁠$Te⊥/Te∥$⁠) of the electron beam (the fast beam plus the $V∥$ ∼ 0 component) becomes smaller, which results in a smaller growth of the anisotropy-induced wave instability (Mode 1).

Next, we investigate the dependence of VDFs and the wave intensities on positions inside the EDR near the X line located at (1.15d_i, 0.05d_i). Figure 7 shows a map of VDF locations and VDFs in the (⁠$V∥$⁠, $V⊥2$⁠) plane at those locations at $t=44.67/Ωi$⁠. The VDFs in the (⁠$V∥$⁠, $V⊥2$⁠) plane present various densities, shapes, and speeds of electron beams. In Fig. 7, the upper twelve VDFs (columns A–D) are located in the left side of the X-line including the X-line itself (the bottom panel in column D), while the lower nine VDFs (columns E to G) are located in the right side of the X-line. The three rows in each column of VDFs correspond to the three rows of the locations (red boxes) in the 2D map. Note that VDFs at the two blue boxes in columns A and B will be presented in Fig. 10. In Fig. 7, the electron beams are moving away from the X-line at each (left/right) side of the X-line, and not symmetric about the X-line. This asymmetry in the left and right electron motion from the X-line is due to the guide field and the reconnection electric field E_y. On the left side of the X-line, there are many electrons with negative $V∥$ as we see in Fig. 7, and they are accelerated anti-parallel to the magnetic field by the parallel electric field, which is due to the reconnection electric field and the guide field. In contrast, on the right side of the X-line, the electrons with positive $V∥$ are decelerated by the parallel electric field. Therefore, the electron velocities are asymmetric about the X-line due to the acceleration and the deceleration at each side of the X-line.⁶⁵ 

There are stronger anisotropic electron beams in the negative x side from the X-line (columns A–C) than the positive x side (columns E–G). At the right end (column G) in the lower nine VDFs, which are for the right side of the X-line, there is no electron beam. This is because column G is located near the EDR boundary in the positive x side, where the EDR boundary is around x = 3.0d_i. In contrast, there are electron beams at the left end (column A) in the upper twelve VDFs, which are for the left side of the X-line. In the left side of the X-line (negative x region), the outflow edge of the EDR is around x = −1.5d_i, and column A is inside the EDR. The middle panels in columns B and C show that the electron beam has a significant anisotropy in the negative x side from the X-line, which results in strong wave activity in the negative x region as in Fig. 9 (middle panels in columns B and C). Note that in column C, we observe a parallel crescent VDF (the middle row), which is composed of a negative parallel beam at $V∥ ∼−10Va$ and $V⊥ ∼0$⁠, and there are enhancements of the VDF density along the circular edge of the VDF in the negative $V∥$ region. This VDF resembles the parallel crescent electrons observed by MMS in the Earth's magnetopause.^19,27–30

Figure 8 presents VDFs in the velocity plane perpendicular to the magnetic field (⁠$V⊥1$⁠, $V⊥2$⁠) at the locations in the 2D map (top panel in Fig. 7). There are various shapes such as a ring, a crescent, and a circle in the VDFs. The symmetric ring shape in the VDFs in column A (middle and bottom) and column B (middle and bottom) is formed due to the vertical component with $V∥$ ∼ 0 in the (⁠$V∥$⁠, $V⊥2$⁠) plane in the corresponding rows of columns A and B in Fig. 7. The velocity range from −10 $Va$ to +10 $Va$ in the (⁠$V⊥1$⁠, $V⊥2$⁠) plane in the outer ring matches the perpendicular velocities in those corresponding VDFs in Fig. 7. Crescent shapes are seen in some panels in columns A–E. The VDFs in column G (the right end in the map) show almost gyrotropic electrons because they are near the edge of the EDR.

In Fig. 9, we present the FFT plots for B_y (red line) and B_z (black line) at the locations of columns A, B, and C, which are in the left side of the X-line, and we discuss the relation between the wave intensities and the VDFs. All VDFs in column A in Fig. 7, which are below the separatrix, have a vertical bar component, i.e., the component with $V∥$ ∼ 0 broadly spread in the perpendicular velocity (⁠$V⊥2=−10Va$ to $+10Va$⁠), which corresponds to the gyrotropic ring in the VDFs in the perpendicular velocity plane in Fig. 8, although the top panel in column A shows a little nongyrotropy. In contrast, in column C, there is no vertical bar component near $V∥$ ∼ 0 in the parallel velocity plane in Fig. 7, and Fig. 8 for the VDFs in the perpendicular velocity plane shows that all the VDFs in column C have a nongyrotropic crescent component. The top panel of column C in Fig. 7 shows a dense and narrow-$V⊥2$-range component (a red region covers −$6Va$ < $V⊥2$ < $6Va$⁠) with a negative $V∥$ (−7 $Va$ < $V∥$< 0), which corresponds to the crescent in the top panel of column C in Fig. 8. Figure 9 shows that the mode amplitudes (red and black) over 0.2 $Ωel$ in the top panel for column A are comparable with those in the middle panel in column A. Additionally, the top and middle FFT plots in column B and C show that the middle panel has a larger mode amplitude over 0.2 $Ωel$ and 0.1 $Ωel$⁠, respectively, in B_y (red line) than the top panel, even though the top panel is closer to the separatrix. The wave activity is not concentrated near the separatrix close to the X line because there is a large density of a component with $V∥$ ∼ 0 in column B and $V∥∼$ −4V_a in column C in the (⁠$V∥$⁠, $V⊥2$⁠) plane in the top panels in Fig. 7, and the anisotropy (⁠$Te⊥/Te∥$⁠) becomes small, which makes the instability weaker at the top of columns B and C than the middle positions in columns B and C. In contrast, the middle panels of columns B and C in Fig. 7 show a significant anisotropy in the electron beam, which makes the instability stronger than the VDFs in the top panels.

The nongyrotropy in the crescent component also affects the wave activity. Figure 10 shows the VDFs and the FFT plots at locations on the separatrix, indicated by the two blue boxes in Fig. 7, columns A and B, and the top red box in column C. At all the three locations, the VDFs in the (⁠$V∥$⁠, $V⊥2$⁠) plane (top panels in Fig. 10) have an elongated component in the range of −15V_a <$V∥$< 15V_a, and a beam component around $V∥$ ∼ −5V_a. They also show a crescent (nongyrotropic meandering electrons) in the perpendicular velocity plane (⁠$V⊥1$⁠, $V⊥2$⁠) (middle panels). The elongated component is due to electrons coming from the magnetospheric side, while the crescent component with $V∥$ ∼ −5 $Va$ is due to electrons coming from the magnetosheath. The density of the crescent in the VDFs is decreasing as we move away further from the X-line along the separatrix, and the VDF in column C (closest to the X-line) has the largest crescent. Although the VDFs in the (⁠$V∥$⁠, $V⊥2$⁠) plane (top panels) show similar shapes, the mode amplitudes in the frequency range 0.15 $Ωel<ω<0.3 Ωel$ in the FFT plots are decreasing (bottom panels) as we move away further from the X-line, in both B_y and B_z fluctuations. The FFT plot in column C shows the largest wave powers among the three positions, while the mode amplitudes in column A are the smallest. This is because the VDF in (⁠$V⊥1$⁠, $V⊥2$⁠) in column C has the largest crescent density among the three positions, indicating that the larger the nongyrotropic population, the larger the wave power in the EDR close to the X-line. The larger the density of the nongyrotropic population (column C in Fig. 10), the larger the perpendicular electron temperature becomes, which enhances the temperature anisotropy (⁠$Te⊥/Te∥)$⁠. The enhanced temperature anisotropy generates a stronger instability, which eventually can increase the wave power as in Fig. 10.

Figures 9 and 10 show that the wave activity is not concentrated on the separatrix near the X-line, while it is concentrated near the separatrix if the location is far from the X-line, as shown in Fig. 6. If we compare the wave amplitudes on/near the separatrix in Figs. 9 and 10 with those in more negative x locations, such as the locations in Figs. 4 and 5 at x = −1.475d_i and −4.975d_i, the wave power is further enhanced in the area with x <−1.0d_i, outside the region we discussed in Figs. 7–10. This is consistent with Fig. 2, which presents whistler waves propagating along the separatrix in the magnetospheric side, enhancing the amplitude as the wave propagates in the negative x direction from the X-line. This increase in the wave power along the separatrix may be due to the nonlinear effects as the whistler waves are excited near the X-line and propagating along the separatrix to the downstream region. Nonlinear interactions between the anti-parallel beam and waves are not investigated in this study, but they may be important in the region near the outflow area of the EDR and the downstream region. Further studies are necessary to understand such a nonlinear stage of the wave growth and the saturation.

Note that in the test simulation where the system size is doubled (51.2d_i $×$ 102.4d_i, data not shown), we observed similar results of whistler waves (frequency, intensity, regions where the waves are generated, shapes of electron VDFs, etc.), and the results do not significantly depend on the system size.

In this paper, we only focus on the left side region of the X-line, where waves are propagating away from the X-line, in the same direction as the anti-parallel electron beams. However, as we see in Fig. 2 (and the movie files in the supplementary material), in the right-side region of the X-line, waves are propagating toward the X-line, opposite to the electron beam direction seen in the VDFs in columns E and F in Fig. 7. Further investigations about the wave excitation and propagation in the right-side region remain to be conducted.

We have studied electron distribution functions and wave activities in the diffusion region in asymmetric guide field reconnection using a 2D PIC simulation, focusing on the EDR, and found waves with frequencies less than the electron cyclotron frequency (ω ∼0.1–0.2Ω_e). The waves are propagating near the separatrices in the magnetospheric side almost antiparallel to the magnetic field, and the wavelengths are less than the ion skin depth. The wave intensity is larger in the side of the stronger electron outflow than the other outflow side.

The observed waves have been identified as whistler waves. We have investigated the wave excitation mechanisms for the whistler wave: the instabilities due to electron beams and the electron temperature anisotropy. The numerical analysis using a dispersion solver has confirmed that the whistler waves near the separatrices away from the X-line are generated by the temperature anisotropy in the fast beam, and we also have found a wave mode due to the slow beam. The anisotropic fast beam excites a whistler wave through the cyclotron resonance, and the whistler wave is propagating slower than the beam speed away from the X-line. As a result, the whistler wave shows a left-handed wave polarization in the simulation frame. In contrast, the slow beam mode shows a right-handed but almost linear polarization, and an almost electrostatic property. These two modes are coupled, propagating in the left side of magnetospheric separatrix away from the X-line, with properties of both left- and right-handed polarizations, with strong electrostatic fluctuations.

We have investigated electron VDFs in the regions where the whistler waves are observed. The wave activity depends on the properties of electron beams: the beam density, the anisotropy of the beam, and the density of the component with $V∥$ ∼ 0. Some electron beams around the EDR show nongyrotropy, which is seen as an asymmetric shape of the VDF in the velocity plane perpendicular to the magnetic field. The wave activity is primarily determined by the intensity of the anisotropic electron beam, and the component with $V∥$ ∼ 0 is another key to determine the wave intensity. In the separatrix outside the EDR or near the EDR boundary, the whistler wave intensity becomes strong along the separatrix, while the wave intensity is decreased as we move away from the separatrix in the z direction because of the increase in the population of the component with $V∥$ ∼ 0 in VDFs.

Next, we have investigated the whistler wave activity near the X-line and the relation between the wave intensities and the shapes of electron VDFs. Below the separatrix near the X-line, the component with $V∥$ ∼ 0 shows a ring structure at some places. Most of VDFs near the X-line have nongyrotropic meandering electrons, which correspond to a crescent-shaped component. We have found that the whistler wave activity is enhanced in the EDR due to two key components: one is the anisotropic fast electron beam, which enhances the wave intensity away from the separatrix. The other is the nongyrotropic crescent component, which enhances the wave activity along the separatrix. In both cases, the instability is due to the increase in the temperature anisotropy.

The results of this study are useful to interpret space measurements of magnetic reconnection in the Earth's magnetopause, such as observations by MMS. MMS observed whistler waves in/near the EDR, and both electron beams and electron temperature anisotropy. Our study indicates that whistler waves on/near the magnetospheric separatrices away from the X-line are due to an anisotropic electron beam, which excites a wave with a left-handed polarization as well as due to the Landau resonance in a slow electron beam, which excites an almost-electrostatic wave with a right-handed polarization. However, this study is based only on one strength of the guide field, 0.3B₀, and further studies are necessary to fully understand the excitation mechanisms and the dependence on the guide field strength and other parameters such as the density ratio and the temperature ratio between the magnetosheath and magnetosphere. Also, investigations about wave-particle interactions remain to be conducted, to understand the heating and acceleration of electrons inside and outside the EDR. High-energy electrons up to a few hundred keV associated with whistler waves were observed by MMS.⁴⁵ Also, PIC simulation studies of symmetric reconnection^58,66 show that electrons are heated by waves excited in the separatrix region due to an electron–electron stream instability with a frequency similar to whistler waves. Our simulation also shows strong electrostatic fluctuations, due to Mode 2 generated by the Landau resonance, which may heat electrons. Energization by those whistler waves will be addressed in the future.

See the supplementary material for a time evolution of B_y-fluctuations, a space-time plot of B_y-fluctuations and its Fourier transformed plot in the k–ω space, and movie files for wave propagations.

The work was supported by the DOE under Grant No. DESC0016278, the NASA under Grant No. 80NSSC21K1795, and the NASA MMS project. PIC simulations were performed on Cori at the National Energy Research Scientific Computing Center.

No conflict of interest to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.