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Unified view of nonlinear wave structures associated with whistler-mode chorus

A range of nonlinear wave structures, including Langmuir waves, unipolar electric fields and bipolar electric fields, are often observed in association with whistler-mode chorus waves in the near-Earth space. We demonstrate that the three seemingly different nonlinear wave structures originate from the same nonlinear electron trapping process by whistler-mode chorus waves. The ratio of the Landau resonant velocity to the electron thermal velocity controls the type of nonlinear wave structures that will be generated.

Whistler-mode chorus [1][2][3] is a coherent electromagnetic emission found widely in the near-space region of the Earth and other magnetized planets. Chrous waves are the Earth's own "cyclotron accelerator" that accelerates the radiation belt electrons [4,5]. They can also scatter the energetic electrons out of their trapped orbit and light up the pulsating aurora in the upper atmosphere [6]. Nonlinear wave structures, for example, Langmuir waves [7][8][9][10], unipolar electric fields [11][12][13][14][15][16] and bipolar electric fields [16,17], are often observed in association with chorus waves. These nonlinear wave structures are considered to be important since they have the potential for significant particle scattering and acceleration [12,[18][19][20]. Despite several past attempts [14,15,[21][22][23][24] to explain the generation of such nonlinear wave structures and their relation to chorus, their linkage is not yet understood, and direct measurements of electron phase space structures responsible for these nonlinear wave structures have been difficult to obtain.
In this Letter we demonstrate the link between several different nonlinear wave structures and whistler-mode chorus, by observing the associated electron phase space structures using computer simulations. When the tail of the electron distribution is trapped by chorus, trapped electrons form a spatially modulated bump-on-tail distribution and excite Langmuir waves. When the thermal electrons are trapped by chorus, they form phase space holes and hence produce bipolar electric fields. Between these two regimes, trapped electrons generate nonlinear electron acoustic waves, which in turn disrupt the trapped electrons and accumulates them in a limited spatial region, leading to the unipolar electric fields.
This study connects a variety of seemingly unrelated nonlinear field structures and provides a simple, integrated picture of the microscopic interactions between whistler waves and electrons.
The three basic types of nonlinear wave structures are illustrated using data from the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) [25] on board NASA's Van Allen Probes. High-frequency Langmuir waves are seen to occur primarily near the negative phase of the whistler parallel electric field (i.e., parallel with respect to background magnetic field) and shown in Fig. 1(a). Langmuir waves are a class of electrostatic plasma waves, naturally found in the Earth's near-space environment at frequencies near the electron plasma frequency [7,9] (ω pe , corresponding to the electrostatic oscillation frequency of electrons in response to a small charge separation). In the other examples, the parallel electric field of the chorus is highly distorted and appears as either a unipolar electric field [ Fig. 1(c)] or a bipolar electric field structure [ Fig. 1(e)]. The unipolar electric field is also called a 'double layer' because it resembles a net potential drop from a layer of net positive charges to an adjacent layer of net negative charges, whereas the bipolar electric field is also referred to as an 'electron hole' since it resembles the field created by a collection of positive charges. As the propagation direction of whistler is reversed (See Supplemental Material [26] for the propagation direction), the excitation location of Langmuir waves changes to occur primarily near the positive phase of the whistler parallel electric field [ Fig. 1b]. The polarity of unipolar electric fields and bipolar electric fields also change to the opposite sense [ Fig. 1(d), Fig. 1  To gain insight into the generation process of the nonlinear wave structures, we performed a series of 1D spatial, 3D velocity Particle-In-Cell (PIC) simulations [28][29][30][31] which are able to capture complex nonlinear interactions between the whistler waves and any electrons that are potentially trapped by the wave's electromagnetic fields (See Supplemental Material [32] for details of simulation setup). We set up the whistler wave field by driving the plasma with an external pump field for a prescribed time interval. After the pump field is turned off, the electromagnetic field of whistler wave continues to propagate, and is self-consistently To test our hypothesis further, we confirmed that Langmuir waves are excited near the positive phase of δE by reversing the propagation direction of whistler waves (not shown), consistent with observations.
As the whistler waves begin to resonate with electrons closer to the bulk of the distribution, the unipolar electric field structure starts to become more prevalent. An example with v r /v th = 2.1 is shown in Fig. 3 and Fig These beams also generate beam-mode waves, propagating in small phase velocities in both forward and backward directions [ Fig. 3(c)]. It is also worthwhile to note that the net potential drop across the simulation domain is zero due to the periodic boundary condition for fields. In the space environment, the unipolar electric fields are, however, not subject to the periodic boundary condition and hence can be viable for particle acceleration.
When the Landau resonant velocity is lowered further and becomes comparable to electron thermal velocity, the bipolar electric field is seen to be generated. An example with A large number of simulations [37] were performed in order to study the development and transition of the different nonlinear electric field structures, in the range 1 v r /v th 4. We found that beam-mode Langmuir waves were modulated by whistler waves at approximately 2.2 v r /v th 4 [38]. In this range, the whistler wave field was only slightly distorted, corresponding to weak harmonic structure in the whistler wave field. In the intermediate range 2, beam-mode electron acoustic waves were generated, and the whistler wave field became highly distorted into the observed unipolar structure, simultaneously resulting in strong harmonics of whistler. In the range 1 v r /v th 1.8, the bipolar electric field structure was generated. A direct comparison of v r /v th between simulations and observations is difficult to perform at this stage. The cold, dense electron component of the particle-in-cell simulations is not practical at this stage, since the electric field noise (due to the limited number of particles per cell) disrupts the trapping dynamics before the effect of wave amplitude threshold comes into play.
In summary, we have demonstrated that the ratio v r /v th in the simulation is the controlling parameter that determines the type of electrostatic nonlinear feature that will be

SIMULATION SETUP
We used a spectral Darwin particle-in-cell code in this study, which was developed as part of the University of California, Los Angeles Particle-In-Cell (UPIC) framework [28]. The Darwin approximation neglects the transverse displacement current and hence eliminates light waves in the system, but does not affect the physics of whistler and electrostatic waves [29][30][31]. The simulation has one dimension (x) in configuration space and three dimensions (v x , v y , v z ) in velocity space. The boundary conditions for both particles and fields are periodic. The time step ∆ t is 0.2 ω −1 pe . The cell length ∆ x is 2 λ D , where λ D = v th /ω pe is the initial electron Debye length. The background magnetic field B 0 is oriented at 30 • with respect to x-direction in the x-z plane. The electron cyclotron frequency ω ce is equal to 0.1 ω pe . This uniform background magnetic field is used since the excitation of the nonlinear wave structures is typically observed in the region around the magnetic equator [9,16]. and three dimensional simulations. The detailed parameters specific for each simulation are listed in Table I. To set up the whistler wave field, we need a particle distribution that supports the wave field. To achieve this goal, we use an external pump electric field during a prescribed time interval. Therefore each electron experiences an external acceleration from the pump electric field given by Here v α and E α are the particle velocity and the pump electric field, respectively. e is the elementary charge, m e is the electron mass and t is time. We add the pump electric field to the self-generated electric field as the total electric field, and add the background magnetic field to the self-generated magnetic field as the total magnetic field. The total electric and magnetic fields are used in the particle push. The wave number and frequency of the pump field is k 0 and ω 0 . k 0 is connected to the mode number M through k 0 = 2πM/(N x ∆ x ), meaning the pump field has M wave lengths in the system. Here N x is the number of cells in the system and ∆ x is the cell length. The associated wave magnetic field is set up naturally by the particle response. The time profile of the pump electric field is The pump field starts with a linear up-ramp until t = t rmp , then keeps a constant amplitude until t = t off − t rmp and finally ends with a linear down-ramp until t = t off . The simulation stops at t = t end . The relative amplitude of the pump field, i.e., E y0 /E x0 and E z0 /E x0 , is determined by the dispersion relation of whistler mode. The magnitude and duration of the pump field is chosen so that the normalized magnetic field δB/B 0 of the whistler wave reaches 0.1 when the pump field is turned off. Such a large amplitude wave is needed to overcome the incoherent field fluctuations in the simulation. The parameters relevant to the pump field are listed in Table I.
Finally, it is worthy to note that the setup of the problem in the simulation is a temporal problem whereas it is a spatial problem in the space environment. In other words, the nonlinear trapping occurs across the simulation domain but for a limited time span in sim- ulations, while in the space environment it occurs in a limited spatial range but for a longer time span when the whistler waves propagate from lower to higher latitudes. Nevertheless, the underlying physics of nonlinear trapping is the same in both scenarios.