Coherent whistler waves and oscilliton formation: Kinetic simulations



[1] Coherent wave emission in the whistler frequency range (0.1–0.5Ωe, where Ωe is the electron cyclotron frequency), consisting of nearly monochromatic wave packets, has recently been observed by several spacecraft in the Earth's plasma environment. In this paper whistler waves generated by an electron temperature anisotropy is studied using particle-in-cell (PIC) simulations. It is shown that during the initial phase of the evolution the exponential wave growth is consistent with linear kinetic dispersion theory. Simultaneously, a reduction of the temperature anisotropy takes place, mainly caused by electron heating parallel to the magnetic field. In the final stage the system reaches conditions of marginal stability which is characterized by wave packets of coherent emissions with a frequency about one half the initial value at maximum instability and such that the phase speed and group velocity coincide. These signatures are shown to relate to whistler oscillitons which are a class of nonlinear stationary waves with superimposed spatial oscillations. The calculated structures for a plasma with βe = 1.0 quantitatively resemble magnetospheric whistler events.

1. Introduction

[2] Wave emissions in the whistler frequency range (0.1–0.5Ωe, where Ωe is the electron cyclotron frequency), called ‘lion roars,’ are a characteristic feature of the wave activity in the magnetosheath [Smith and Tsurutani, 1976; Zhang et al., 1998; Baumjohann et al., 1999] and magnetosphere [Baumjohann et al., 2000; Dubinin et al., 2007]. It has been shown previously that these emissions could be generated by a cyclotron resonant instability in the magnetosheath plasma with anisotropic electron temperature, Te > Te, where Te and Te are temperatures perpendicular and parallel to the magnetic field, respectively [Thorne and Tsurutani, 1981]. The main observed features of the waveforms are coherency and wave packet structure which are difficult to explain using linear theory which predicts a broader frequency range of excited waves with random phases. A scenario for generating nearly monochromatic whistler waves involves wave trapping in a mirror wave cavity which selects a narrow range of wavenumbers [Treumann et al., 2000]. Another mechanism is based on the nonlinear properties of whistler waves where a class of solitary structures with embedded smaller-scale oscillations resembling wave packets has recently been found [Sauer et al., 2002]. These nonlinear stationary waves, termed ‘whistler oscillitons,’ have been investigated using a two-fluid description which is valid for the low electron plasma beta (βe ≡ 2μonkTe/B2 ≪ 1) regime.

[3] As a basic concept for explaining coherent whistler emission it has been suggested that the final stage of temperature anisotropy-driven whistlers has essential signatures of whistler oscillitons. In order to apply this to magnetosheath and magnetospheric plasma observations we must consider the effects of wave-particle interaction and this motivates us to employ electromagnetic particle-in-cell (PIC) simulations. A detailed discussion about this topic on the background of Cluster spacecraft observations is contained in a recent paper by Dubinin et al. [2007]. Generally, the appearance of stationary nonlinear waves is related to the existence of a ‘resonance point’ where phase and group velocity coincide. In the cold plasma approximation this point is at (ω = Ωe/2, kc/ωe = 1), where ωe is the electron plasma frequency, and according to the kinetic dispersion theory, it shifts to smaller values of ω and k as βe increases. Accordingly, results of PIC simulations are presented for βe = 0.01 and βe = 1.0.

[4] The paper is organized as follows; section 2 contains the results of PIC simulations starting from an unstable plasma which reaches a quasi-stationary state. In section 3 the dispersion of stationary whistler waves is discussed and using the kinetic approach it is shown how frequency and wavelength vary with parameters, such as βe, which allows predictions of the most dominant solitary waves. In addition, the structure of oscillitons is calculated and compared with that of corresponding PIC simulations of a cold plasma. The time evolution of whistlers obtained from PIC simulations for βe = 1.0 is found to be in good agreement with Cluster observations in the magnetosphere. Finally, summary and conclusions are given.

2. Kinetic Simulations

[5] The kinetic simulations of the whistler electron thermal anisotropy instability were performed using a 1D-3V electromagnetic PIC model [Sydora, 1999], initialized with a bi-Maxwellian electron distribution function. A magnetic field is imposed in the x-direction and periodic boundary conditions are employed. The normalized physical parameters used in the simulation are: system length, Lx = 400c/ωe, ωee = 5, mi/me = 1837, Ti/Te = 1, normalized electron skin depth, c/ωeΔ = 5 (βe = 1 case), and 50 (βe = 0.01 case). The initial thermal anisotropy Te/Te(t = 0) = 9, for βe = 0.01 and Te/Te(t = 0) = 3, for βe = 1. The numerical parameters are time step, ωeΔ t = 0.1, and average number of particles per cell is 100. The wavenumbers up to the maximum of kmaxc/ωeΔ = 6 are retained in the simulation.

[6] The time evolution of the total magnetic energy and electron temperature anisotropy are shown in Figures 1a and 1b, respectively, for two different electron beta values. In Figure 1a the linear growth of the magnetic field energy is consistent with the theoretically predicted values determined from numerical solution of the full electromagnetic kinetic dispersion relation for parallel propagating whistler modes [Gary, 1993]

equation image

where equation image = kc/ωe, equation image = ωe, z = (equation image − 1)/equation image , and W is the plasma dispersion function. The lower initial anisotropy results in a decreased fluctuation level at later time evolution. From Figure 1b, the initial anisotropy reduces as the electric and magnetic fluctuation energy increase until the approximate linear marginal stability value is reached. For βe = 1.0 the saturated level is Te/Te ≈ 1.2 and for βe = 0.01 the level reaches Te/Te ≈ 3.0. Only the time interval Ωet = 0–200 is displayed to make the linear and saturation phase visible. Figure 1c displays the magnetic fluctuation amplitude versus time for various wavenumbers using the βe = 1.0 case. Now the full time interval of the simulation is shown (Ωet = 0–2000) and one clearly observes the shift of the most unstable wavenumbers to lower k, with asymptotic evolution to kc/ωe ≃ 0.2. For this same case, the spatial profiles of the transverse magnetic field are given at two different time steps in Figures 1d and 1e. The shift to longer wavelength is also evident along with the wave packet structure. The complete spatial and temporal evolution of the transverse magnetic field is given in Figure 1f and reveals the forward and backward propagating whistler wave packets. The dashed line in Figure 1f indicates the group velocity of a wavepacket which closely matches the phase speed, Vph ∼ 0.25 VAe where VAe is the electron Alfven speed.

Figure 1.

Time evolution of the (a) total magnetic field energy and (b) electron temperature anisotropy for βe = 0.01 and βe = 1.0. (c) Transverse magnetic field energy versus wavenumber and time. Spatial profiles of the magnetic field fluctuations, for the βe = 1.0 case, taken at (d) Ωet = 20, (e) Ωet = 2000, and (f) for the entire spatial and temporal range. Dashed line indicates the group velocity of a wave packet coinciding with the phase velocity.

[7] Figure 2 shows the real frequency and growth rate for the βe = 1.0 case obtained from the numerical solution of the kinetic dispersion relation given by equation (1). The dashed line indicates the solution for the initial electron thermal anisotropy Te/Te = 3.0 and the solid line for Te/Te = 1.2 corresponding to the marginally stable value. In Figure 2a the intensity of the transverse magnetic field fluctuation(Bz) versus frequency and wavenumber is superimposed on the linear dispersion curves. The initially unstable wavenumber band (kc/ωe ≃ 0.25–1.4) shifts to a peak value of kc/ωe ≤ 0.3 and the related (real) frequency is ωe ≈ 0.15.

Figure 2.

Kinetic dispersion relation illustrating the (a) real and (b) imaginary frequency versus wavenumber, for βe = 1.0 and temperature ratios Te/Te = 3.0 (dashed line) and Te/Te = 1.2 (solid line). In Figure 2a the intensity of the transverse magnetic field fluctuation (gray scale) at saturation from the PIC simulation is superimposed. During the linear phase (not shown) the peak intensity occurs at wavenumbers corresponding to the maximum linear growth along the dashed curve.

3. Relation to Oscillitons

[8] The PIC simulations have shown that a transition from the initially unstable regime to quasi-stationary whistler emission takes place when the most dominant wavenumber shifts to smaller values. The final stage of marginal stability is characterized by packets of nearly coherent waves which originate from the vicinity of that (ω, k) ‘resonant point’ where phase and group velocity coincide. As discussed in earlier papers [Sauer et al., 2002; Dubinin et al., 2003] the existence of such a ‘resonant point’ is a necessary condition for the occurrence of whistler oscillitons and has implications on the dispersion of stationary waves.

[9] Starting with a cold plasma and replacing ω by kU, where U is the velocity of the moving structure, in the corresponding dispersion relation D(ω, k) = 0, one gets a quadratic form in k which has the solution

equation image

where M = U/VAe is the oscilliton velocity normalized to the electron Alfven speed. It is easily seen that the wavenumber equation image becomes complex for M > 0.5 and when M = 0.5 the velocity U = VAe/2. This corresponds to the point (kc/ωe = 1, ωe = 0.5) where the phase velocity is maximum and coincides with the group velocity. The appearance of complex wave numbers for M > 0.5 indicates the existence of spatially localized solutions which are superimposed by spatial oscillations with a wavelength of λ = 2π/kr, where kr is the real part of k. It should be noted that for velocities M slightly larger than 0.5, the wavelength of the oscillating growing wave structure is λ ≈ 6c/ωe with krc/ωe ∼ 1. Furthermore, the imaginary part, ki, approaches unity if M increases above 0.5.

[10] For a warm plasma one must use equation (1) and by replacing ω with kU, one obtains the (kinetic) dispersion relation of stationary whistler waves which determines the wavenumber as a function of the oscilliton velocity U with βe and Te/Te as parameters. For βe = 0.01 and βe = 1.0 the results are shown in Figure 3 where the temperature ratios Te/Te are taken at marginal stability (Te/Te = 3.0 for βe = 0.01 and Te/Te = 1.2 for βe = 1.0). There are remarkable differences compared to the cold plasma case (βe = 0) described by equation (2). As shown in Figure 3a, ki now has a maximum near M = 0.5 and continuously decreases with increasing βe. Also, kr in Figure 3b is strongly modified, especially for βe near unity. A significant feature is that kr at maximum ki decreases if βe increases. Whereas this value is krc/ωe ≈ 1 for βe ≤ 0.1, it goes down to krc/ωe < 0.5 for βe ≈ 1. This has consequences for the frequency ωr = krU displayed in Figure 3c which decreases from ωre ≈ 0.5 for βe ≤ 0.01 to ωre ≤ 0.15 for βe ≥ 1.0. Assuming that both the wavenumber kr and frequency ωr, at maximum increment ki, retain the main characteristics as the linear wave develops to its fully nonlinear stationary configuration, the described tendency with respect to βe dependence can be used to predict the essential signatures of an initially unstable plasma if it reaches the quasi-stationary state.

Figure 3.

Dispersion of stationary whistlers. (a) Imaginary part of k, (b) real part of k, and (c) frequency ωr = krU versus the oscilliton velocity U (all in normalized units) for βe = 0.01 and βe = 1.0.

[11] To test this hypothesis, the PIC results are compared with predictions derived from Figure 3. For βe = 0.01, representing the cold plasma case, a wavenumber of krc/ωe ≈ 1, phase velocity of U/VAe ≈ 0.5 and real frequency ωre ≈ 0.5 have been obtained for both approaches. In the case βe = 1.0, Figure 3 predicts smaller values; krc/ωe ≈ 0.25, U/VAe ≈ 0.3, and ωre ≤ 0.1 which are in good agreement with the PIC results.

[12] Proceeding further into the nonlinear theory, the two-fluid Hall approach is used to calculate the spatial profile of whistler oscillitons which allows a comparison with PIC results in the cold plasma approximation. For this, the full set of nonlinear stationary equations is derived from the fluid equations of motion for electrons and protons together with Faraday's and Ampere's law which are then solved numerically [Sauer et al., 2002; Dubinin et al., 2003]. Contrary to the linear theory, the proton dynamics cannot be excluded because it makes a non-negligible contribution to the total momentum balance. As predicted for cold plasmas, whistler oscillitons are found for M ≥ 0.5 and an example is shown in Figure 4a for M = 0.5025. The spatial profile obtained from the PIC simulations (βe = 0.005, Ωet = 300) is given in Figure 4b which consists of several wave packets. For a best fit to the simulations, a finite amplitude of the magnetic field component at infinity, Bz,∞/Bo = 0.002, has been chosen. The wavelength of the spatial structure in both cases match very well, λ ≈ 6c/ωe, and this value corresponds to the linear theory prediction of kc/ωe ≈ 1. These results support our conclusion that the final state of temperature anisotropy-driven whistler waves ends in structures which exhibit characteristics of oscillitons as a particular class of nonlinear stationary waves.

Figure 4.

(a) Whistler oscillitons obtained from stationary two-fluid Hall approach for M = 0.5025 and (b) spatial profile of whistlers from PIC simulations at Ωet = 300 for βe = 0.005 and an initial temperature anisotropy Te/Te = 9.0.

[13] Lastly, the waveform of transverse magnetic field variation in a single whistler wave packet measured by the Cluster spacecraft in the magnetosheath [Dubinin et al., 2007] is directly compared with the outcome of PIC simulations taking βe = 1.0. As seen in Figure 5, there is good agreement between observation and theory. The oscillation period is Ωet ≈ 70 which corresponds to a frequency of emission of ωe ≤ 0.1. The (maximum) wave amplitude is Bz/Bo ≈ 0.015 for both cases. The time variation from the PIC simulations appears to contain a higher-frequency contribution, the origin of which is not clear at present.

Figure 5.

Single whistler wave packet. (a) Cluster spacecraft measurement from Dubinin et al. [2007] and (b) PIC simulation for βe = 1.0 taken at spatial position e/c = 100. The simulation time range is from Ωet = 1000–2000 and the origin has been shifted to zero.

4. Summary and Conclusions

[14] The aim of this paper is to illuminate the relation between coherent whistler wave emission in magnetospheric plasmas and whistler oscillitons. Two facts have led to the original idea that both phenomena are closely interrelated [Sauer et al., 2002; Dubinin et al., 2003]. One is the existence criteria of oscillitons which only occur for particular forms of wave dispersion having a ‘resonance point’ where phase and group velocity coincide. This leads to a sharp selection of frequency and wave number and suggests an explanation of the observed coherence. The other important property is the wave packet structure of emission which indicates the crucial role of nonlinear phenomena. These two facets have guided our kinetic studies of whistler wave generation using PIC simulations.

[15] Their results can be summarized as follows; the initial phase of the whistler instability caused by an electron temperature anisotropy develops in agreement with the predictions of linear kinetic dispersion theory with respect to growth rate and frequency range of unstable waves. During this period, electron heating takes place preferentially parallel to the magnetic field. The subsequent saturation process leading up to marginal stability is characterized by a shift of frequency and wave number to smaller values. Finally, in the saturated state the whistler oscillitons are observed. The corresponding frequency and wave number agree with the predictions obtained from the kinetic dispersion relation of stationary waves. A comparison between Cluster spacecraft measurements and PIC simulations for a single wave packet are quantitatively similar in amplitude and frequency.

[16] In conclusion, we remark on the general property of coherent waves at marginal stability. After initial excitation involving resonant electrons they survive due to the onset of momentum exchange between the whole particle ensemble and the waves manifested in form of oscillitons. Their existence requires another particular type of ‘resonance,’ namely the coincidence of phase and group velocity.


[17] This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Computing resources were made available through the WestGrid system, Canada. One of the authors (KS) acknowledges the support and hospitality of the University of Alberta.