Whistler waves are fundamental plasma waves frequently observed in space in association with transient phenomena, such as collisionless shocks e.g., and magnetic reconnection e.g.,. The waves have a right-hand polarization with respect to the ambient magnetic field, so they can couple with the electrons through the cyclotron resonance and give rise to pitch-angle scattering and parallel acceleration e.g.,. Such microscopic wave–particle interactions can cause anomalous transport in momentum and energy, resulting in anomalous magnetic dissipation in collisionless plasma where the classical Coulomb collision is negligibly weak. Regardless of the potential importance of whistler waves in collisionless plasma, the generation mechanism in transient configurations has been poorly understood.

Satellite observations have shown that whistler emissions usually have bursty properties continuing for a very short time compared to the dynamical timescales that are usually much longer than the ion timescales, such as the Alfvén transit time across the boundary layers. There are two explanations for these observations: the first is that the emissions are very localized in space, and the second is that they are intermittent within a short timescale. So far, a number of kinetic simulations have been carried out to investigate how whistler waves can be generated and what their roles are in the dynamical processes that control macroscopic configurations e.g.,. In particular, collisionless magnetic reconnection enables an explosive release of magnetic field energy into plasma kinetic energy. The associated plasma jets and pressure anisotropy can boost the activities of a variety of plasma waves, including whistler waves, which can in turn have an impact on the reconnection process through magnetic dissipation and plasma heating .

Previous kinetic simulations of collisionless reconnection have suggested that whistler waves are generated in the magnetic field pileup region downstream of the x-line and in the separatrix region separating the inflow and outflow regions . In the pileup region, the electrons are heated preferentially in the perpendicular direction due to betatron acceleration and have a temperature anisotropy leading to the whistler emission. Meanwhile, in the separatrix region, the electrons are strongly accelerated in the parallel direction due to a double layer formed locally in the separatrix region. The intense electron beam can be coupled with the stationary background electrons, which directly or indirectly triggers obliquely propagating whistlers. In both regions, the whistler emissions occur within a very narrow extent across the field lines, which explains the bursty nature in the observations. However, these models were based on a steady-state x-line configuration formed around the dissipation region relatively at the beginning of the reconnection process.

Recent large-scale and long-term particle-in-cell (PIC) simulations have shown the dynamical evolution of the reconnection current layer, repeating the elongation in the downstream direction and the formation of plasmoids . In fact, a chain of plasmoids and flux ropes has often been observed in association with reconnection and accompanied by energetic particles and intense wave activities, including whistlers . In this paper, a new generation mechanism for whistler waves is proposed in association with plasmoid collision through the use of large-scale PIC simulation. Contrary to previous scenarios in the magnetic field pileup region and separatrix region, the present model provides a short-term emission only after the plasmoid collision has been completed. The resultant whistler emission is bursty and has characteristics different from other emissions, so it can work as an indicator of plasmoid and flux rope collisions.

The simulations are carried out through the use of a two-dimensional (2-D) electromagnetic PIC model with the adaptive mesh refinement (AMR) and particle splitting–coalescence method . The refinement criteria for the AMR are provided by the local electron Debye length λDe and the out-of-plane electron flow velocity Vey defined at the center of each computational cell. Each cell is subdivided when ΔL≥2.0λDe or Vey≥2.0VA is satisfied; otherwise the child cells are removed when the computational cells are squares with a size of ΔL, and VA is the Alfvén velocity based on the asymptotic upstream magnetic field B0 and the plasma density n0 at the center of the initial current sheet. The system boundaries are “open” in both the inflow and outflow directions .

The initial current sheet profile employs a Harris-type equilibrium with the magnetic field Bx(z)=-B0tanh⁡(z/δ) and the number density n(z)=n0sech2(z/δ)+nbtanh⁡2(z/δ), where δ is the half width of the current sheet and the subscript b indicates the background parameter. We choose δ=λi and nb=0.044n0 with λi as the ion inertia length based on n0. Note that the background plasma is removed in the center of the current sheet in order to avoid artificial streaming instabilities (e.g., a two-stream instability between the plasma sheet and background components). Although such a density profile causes a weak pressure imbalance, the equilibrium is quickly established without any significant modification to the current sheet structure. The ion-to-electron mass ratio and velocity of light are mi/me=100 and c/VA=33, respectively, corresponding to ωpe/ωce=3.3, where ωpe and ωce are the electron plasma frequency and the cyclotron frequency based on n0 and B0, respectively. The temperature ratios are T0i/T0e=3.0, Tbi/Tbe=1.0, and Tbe/T0e=1.0 so that the background ions are colder than the plasma sheet ions. The system size is Lx×Lz=655λi×328λi, which is entirely covered by base-level cells (coarsest cells) with ΔLb=0.08λi and can be subdivided locally into finer cells up to the dynamic range level with ΔLD=0.02λi. Therefore, the highest spatial resolution is 32768×16384 (the effective number of the finest cells when they cover the whole domain) and the maximum number of particles used is ∼1010 for each species. The normalization parameters in the current study are mi for mass, e for charge, λi for length, and VA for velocity, unless otherwise noted.

The present study uses a small magnetic perturbation in order to initiate magnetic reconnection so that the x-line is generated at the center of the system . Once the simulation has started, a fast reconnection is soon achieved with a rate of ER≈0.12, where ER is evaluated by Ey at the most dominant x-line normalized to the upstream values. In the quasi-steady phase, the electron current layer formed around the x-line is elongated in the outflow direction and is subject to plasmoid formations. By repeating the electron layer elongation and plasmoid ejection, the reconnection exhaust expands to a distance far downstream of the x-line. Figure a–c show the evolution of a plasmoid indicated by the black arrow in the exhaust. This plasmoid has been ejected in the -x direction and eventually merged with the initial current sheet at the downstream edge of the exhaust. In association with the collision with the downstream current sheet, a secondary reconnection takes place between the field lines directing downward (-z) in the plasmoid and upward (+z) in the downstream current sheet. During the secondary reconnection, the electrons are strongly accelerated in the vicinity of the x-line in the out-of-plane (-y) direction (see Fig. a and b). Note that this direction is opposite to that in the main reconnection layer.

The localized structure of the secondary reconnection region is shown in Fig. d–f. The reconnection outflow jets associated with the plasmoid collision are ejected mainly in the z direction (Fig. d). Because the plasmoid scale is comparable to the local ion gyroradius, the electrons are more effectively accelerated than the ions. The ejected electrons are partially magnetized in the upper (+z) and lower (-z) regions of the secondary x-line, so the electrons are selectively heated in the perpendicular direction, resulting in Te⟂/Te∥>1 (Fig. e). The perpendicular heating is also significant in the pileup region further downstream of the exhaust, as described in . In Fig. g–i, the localized structure is presented after the secondary reconnection has ended. Since reconnection is over, the outflow jets do not exist (Fig. g). However, the intense temperature anisotropy of the electrons with Te⟂/Te∥>1 still remains off the Equator around the downstream edge of the exhaust (Fig. h). This is because the perpendicularly heated electrons, due to the reconnection process, move further downstream almost together with the field lines. At this time, an intense emission of microscopic waves with the wavelength λ≈7λe* occurs off the Equator plane, where λe* is the local electron inertia length. The waves typically appear in Ey, the out-of-plane electric field (Fig. i), which indicates that they are dominated by the electromagnetic properties. The emission is also evident in Movie S1 in the Supplement. It is interesting to note that the waves are not clearly excited during the secondary reconnection (Fig. f), regardless of the strong electron acceleration in the vicinity of the x-line.

Figure  shows the wave properties arising in Ey. One can see from Fig. a that the microscopic waves are excited at l/λi≈±10 and propagate along the field line away from the Equator point of l=0. Here, l represents the coordinate along a field line traced from (x,z)=(0,-18.5), which passes through the wave active region. Since the starting point of the field line is far away from the downstream edge of the exhaust (x≈150 at t=114), the field line at each time step is considered to be almost identical. The examples of the field line at tωci=106 and 114 are indicated with dashed curves in Fig. f and i, respectively. It is found that the wave emission is bursty: the emission of microscopic waves with k∥λe*>0.5 is only significant around 112<tωci<116 (Fig. b), which is a much shorter time compared to the dynamical scale, for instance, of the transit time of the plasmoid across the exhaust (∼30ωci-1 for the exhaust with a size of ∼100λi in the x direction).

To identify the generation mechanism of the waves, we compare the wave spectrum in the simulation with the dispersion relation derived from linear theory. The wave spectrum presented in Fig. c (color contour) shows a clear peak at ω≈0.5ωce* and |k∥λe*|≈0.9 for the electromagnetic waves, which is consistent with the typical characteristics of whistler waves, where ωce* is the local electron cyclotron frequency. The electric field data for calculating the wave spectrum are taken from the base-level cells of the AMR hierarchy. It is known that the wave dispersion depends on the grid size of the numerical simulation , i.e., on the cell level of the AMR. However, this dependency is only remarkable for the grid-scale waves with λ≈ΔLb. The dominant waves in the current analysis have a scale of λ≈7λe*≈23ΔLb based on the local density ne≈0.15n0. This indicates that the current waves of interest have a much larger scale than the base-level cell size, so the gridding effect on the wave dispersion is negligible.

The dispersion curve for the whistler waves excited by the temperature anisotropy is superposed on the simulation result in Fig. c. The wave dispersion is obtained from the linearized Vlasov equation for the electrons with temperature anisotropy so that X2-1Ω2Y2-YXβe∥Z(ξ)+12Te⟂Te∥-1Z′(ξ)=0, where X=k∥λe*, Y=ω/ωce* with ω=ωr+iγ, Ω=ωpe*/ωce*, Z(ξ)=1/π∫-∞∞exp⁡(-x2)/(x-ξ)dx is the plasma dispersion function with ξ=(Y-1)/Xβe∥, and βe∥ is the plasma beta based on the electron parallel pressure. The parameters used for calculating the dispersion curve are Te⟂/Te∥=2.5, βe∥=0.22, and Ω=1.9, which are taken at the wave active region in the simulation (i.e., inside the black dashed box in Fig. a). Figure c demonstrates that the simulation result is consistent with the linear dispersion, suggesting that the microscopic wave burst observed in the simulation is due to the whistler instability driven by the electron temperature anisotropy.

The temperature anisotropy is generated through the secondary reconnection in association with plasmoid collision. As shown in Fig. a, the key process leading to the temperature anisotropy is the intense perpendicular heating in the downstream region of the electron jets (indicated by sky blue arrows). This process is similar to that in the previous study , in which whistler waves are excited in the pileup region formed downstream of the main reconnection x-line. However, in the current simulation, we find that the whistler emission is not clear at this time. This is partly because the pileup process of the field lines is weak in the region downstream of the secondary x-line, so the magnetic field intensity remains low and insufficient for the high-energy electrons to be completely magnetized. Furthermore, the field line structure, including the out-of-plane Hall field (not shown), is highly nonuniform in this region in the scale below the ion inertia length. Therefore, the dispersion relation Eq. () derived for magnetized electrons under a uniform magnetic field is no longer valid in this region.

After the plasmoid merging and the associated reconnection process have been completed, the electrons with temperature anisotropy move further downstream of the main reconnection jet. The temperature anisotropy is almost preserved during convection because the whistler emission that can relax the anisotropy is absent. The electrons finally reach the edge of the downstream current sheet and are compressed at the off-Equator regions (Fig. b) due to the pileup of the magnetic field. The whistler waves are triggered at this moment and propagate along the field line. Once the whistler waves are excited, the temperature anisotropy is quickly decreased due to the pitch-angle scattering, and the wave emissions are suppressed. The timescale of the isotropization is estimated as tωpe*∼40 , which corresponds to tωci∼0.3 for the local parameters (n=0.15n0 and B=0.67B0) in the current simulation. Therefore, the whistler emissions associated with the plasmoid collision are very bursty with a timescale smaller than the typical ion timescales.

During the whistler emissions, the electrons can be accelerated in the parallel direction. Figure  shows the electron distribution functions averaged over the field line (-50≤l/λi≤0) traced from the point ((x,z)=(0,-18.5)) far away from the wave active region. At each time, the thermal spread is much larger than in the upstream electrons (dashed curves) in both the parallel and perpendicular directions. This is due to the energization process in the reconnection exhaust. In particular, a flat-top type of distribution is produced in the parallel direction due to the counter-streaming components along the field line, which is in good agreement with the observations typical in the boundary region between the current sheet and magnetospheric lobe . At tωci=106 (green curves), before the whistler emissions and during the secondary reconnection, the electrons typically have a temperature anisotropy of Te⟂/Te∥<1 in this region. The high-energy electrons seen at v∥/VA≲-20 are generated in the course of the secondary reconnection. On the other hand, at tωci=114 (red curves) after the secondary reconnection and during the whistler emissions, the electron temperature is increased in the perpendicular direction because of the convection of the perpendicularly heated electrons with Te⟂/Te∥>1. The whistler waves triggered by the temperature anisotropy scatter the electrons in a pitch angle. As a result, the high-energy electrons with v∥/VA≲-20 are drastically increased, even though the secondary reconnection is over. This high-energy component quickly disappears at tωci=120 (blue curves) after the whistler activities are suppressed.

The present study proposes a new mechanism for the generation of whistler waves in the course of collisionless magnetic reconnection. The present model can explain the bursty nature of whistler emissions as observed in the Earth's magnetosphere e.g.,. The key processes to generate the whistlers are intense perpendicular heating of the electrons in association with plasmoid collision and the subsequent compression of the ambient magnetic field. The perpendicular electron heating occurs during a secondary reconnection forced by the plasmoid collision with the current sheet formed downstream of the main reconnection x-line. During the secondary reconnection, the electrons are strongly accelerated in the vicinity of the secondary x-line and ejected downstream, leading to perpendicular heating. However, the whistler instability due to the temperature anisotropy is not excited at this moment because of the weak intensity and complicated structure of the magnetic field. Instead, the whistler emissions occur when the further compression of the ambient magnetic field is yielded after the secondary reconnection has ended. Once the whistler waves are excited, the temperature anisotropy is quickly decreased due to the pitch-angle scattering, so the wave emissions are suppressed.

The present study has focused on the collision between a plasmoid ejected from the reconnection current layer and the current sheet located at the downstream edge of the reconnection exhaust. However, a similar process for the whistler emissions is also expected in a collision between two plasmoids if the plasmoid size is comparable with or larger than the typical ion scales. During such a collision, the electrons are efficiently accelerated through the secondary reconnection, and a large temperature anisotropy is formed downstream of the electron jets. Subsequent magnetic compression occurs due to the pressure enhancement at the core region of the merged plasmoid, leading to the whistler emissions. In fact, a recent satellite observation shows whistler waves associated with the collision of two plasmoids . The whistler waves triggered by plasmoid collision have several characteristics different from the other whistlers in reconnection . The plasmoid-induced whistlers are produced in a short-term burst, propagate away from the Equator plane, and have a peak intensity off the Equator plane. In other words, the detection of bursty whistler emissions with these characteristics could be an indicator of plasmoid collisions and associated particle energization during collisionless magnetic reconnection.