Whistler turbulence forward cascade: Three-dimensional particle-in-cell simulations



[1] The first fully three-dimensional particle-in-cell (PIC) simulation of whistler turbulence in a magnetized, homogeneous, collisionless plasma has been carried out. An initial relatively isotropic spectrum of long-wavelength whistlers is imposed upon the system, with an initial electron β = 0.10. As in previous two-dimensional simulations of whistler turbulence, the three-dimensional system exhibits a forward cascade to shorter wavelengths and broadband, turbulent spectra with a wave vector anisotropy in the sense of stronger fluctuation energy at k than at comparable k where the respective subscripts represent directions perpendicular and parallel to the background magnetic field Bo. However, the three-dimensional (3D) simulations display quantitative differences with comparable two-dimensional (2D) computations. In the 3D runs, turbulence develops a stronger anisotropic cascade more rapidly than in 2D runs. Furthermore, reduced magnetic fluctuation spectra in 3D runs are less steep functions of perpendicular wave numbers than those from 2D simulations. The much larger volume of perpendicular wave vector space in 3D appears to facilitate the transfer of fluctuation energy toward perpendicular directions.

1. Introduction

[2] Historically, the primary focus for the study of plasma turbulence in the solar wind has been on the inertial range at observed frequencies f ≲ 0.2 Hz [e.g., Horbury et al., 2005; Matthaeus and Velli, 2011]. In this regime, magnetic fluctuation energy spectra exhibit dependences on frequency of the form ∣δB(f)∣2image with αf ≃ 5/3 and a strong characteristic wave vector anisotropy with greater fluctuation energy at propagation quasi-perpendicular to Bo, the average background magnetic field, than at quasi-parallel propagation. At a frequency near 0.2 Hz ≲ f ≲ 0.5 Hz, measurements at 1 AU show a spectral break, a distinct change to spectra that are steeper than those of the inertial range [Leamon et al., 1998; Smith et al., 2006; Alexandrova et al., 2008]. At observed frequencies above this break, magnetic spectra exhibit steeper power laws, i.e., with 2.0 ≲ αf [Behannon, 1978; Denskat et al., 1983; Goldstein et al., 1994; Lengyel-Frey et al., 1996; Bale et al., 2005]. Recent analyses of data from the Cluster mission spacecraft have shown that magnetic spectra in the range 0.5 Hz ≲ f ≲ 20 Hz scale with 2.6 ≲ αf ≲ 2.8 [Sahraoui et al., 2009, 2010; Kiyani et al., 2009; Alexandrova et al., 2009], and suggest that there is a second break with still more steeply decreasing spectra at 20 Hz < f. Some of these observations [Chen et al., 2010; Sahraoui et al., 2010] also demonstrated that this turbulence is anisotropic in the sense of having more fluctuation energy at propagation perpendicular to Bo, where Bo denotes an average, uniform background magnetic field, than at propagation parallel or antiparallel to Bo.

[3] High frequency turbulence above the first spectral break is often described as being an ensemble of normal modes of the plasma with properties derived from linear dispersion theory. In this framework, there are two competing hypotheses as to the character of these fluctuations. One scenario is that this short-wavelength turbulence consists of kinetic Alfvén waves which propagate in directions quasi-perpendicular to Bo and at real frequencies ωr < Ωp, the proton cyclotron frequency. Both solar wind observations [Leamon et al., 1998; Bale et al., 2005; Sahraoui et al., 2009, 2010] and gyrokinetic simulations [Howes et al., 2008a, 2011; see also Matthaeus et al., 2008; Howes et al., 2008c] of turbulence above the first break have been interpreted as consisting of kinetic Alfvén waves, although others have indicated that such fluctuations do not necessarily provide a complete description of short wavelength turbulence [Podesta et al., 2010; Chen et al., 2010]. A fluid model of kinetic Alfvén turbulence, which does not include kinetic plasma effects, yields magnetic fluctuation energy spectra which scale as k−7/3, while inclusion of kinetic effects leads to still steeper spectra [Howes et al., 2008b]. A recent three-dimensional gyrokinetic simulation of kinetic Alfvén turbulence has exhibited a k−2.8 magnetic fluctuation spectrum [Howes et al., 2011].

[4] A second hypothesis is that whistler fluctuations at frequencies below the electron cyclotron frequency contribute to short-wavelength turbulence. Whistlers are often observed in the solar wind [Beinroth and Neubauer, 1981; Lengyel-Frey et al., 1996]. Simulations of whistler turbulence using three-dimensional electron magnetohydrodynamic (EMHD) fluid models [Biskamp et al., 1999; Cho and Lazarian, 2004, 2009; Shaikh, 2009] typically show a forward cascade to steep magnetic spectra with k−7/3, and anisotropies similar to those of the inertial range with greater fluctuation energy at quasi-perpendicular propagation. Two-dimensional particle-in-cell (PIC) simulations of whistler turbulence [Gary et al., 2008, 2010; Saito et al., 2008, 2010; Svidzinski et al., 2009], which include full kinetic effects such as Landau and cyclotron wave-particle interactions, also demonstrate forward cascades to anisotropic magnetic spectra with very steep wave number dependences (e.g., k−4). Camporeale and Burgess [2011] also have used two-dimensional PIC simulations to study the forward cascade of magnetic turbulence at electron scales. Here we describe the first PIC simulations to examine the forward cascade of whistler turbulence in a fully three-dimensional plasma model.

[5] Our simulation results are derived from a three-dimensional electromagentic particle-in-cell code 3D EMPIC described by Wang et al. [1995]. In this code, plasma particles are pushed using a standard relativistic particle algorithm; currents are deposited using a rigorous charge conservation scheme [Villasenor and Buneman, 1992]; and the self-consistent electromagnetic field is solved using a local finite difference time domain solution to the full Maxwell's equations.

[6] We denote the jth species plasma frequency as ωjequation image, the jth species cyclotron frequency as ΩjejBo/mjc, and βj ≡ 8πnjkBTj/Bo2. We consider an electron-proton plasma where subscript e denotes electrons and p stands for protons.

[7] Here “three-dimensional” means that the simulation includes variations in three spatial dimensions, as well as calculating the full three-dimensional velocity space response of each ion and electron superparticle. The plasma is homogeneous with periodic boundary conditions. The uniform background magnetic field is Bo = equation imageBo so that the subscripts z and ∥ represent the same direction. Thus k = equation imagekx + equation imageky + equation imagek and k = equation image. We define two-dimensional reduced energy spectra by summation over the third Cartesian wave vector component, e.g.,

equation image

In contrast, the one-dimensional reduced spectra for k are obtained by summing the energy over both k and concentric annular regions in kx-ky space. The total fluctuating magnetic energy density is obtained by summing over all simulation wave vectors, and the spectral anisotropy angle θB is defined via

equation image

An isotropic spectrum corresponds to tan2θB = 1.0. In the evaluation of each of these quantities, the wave number range of the summations is over the cascaded fluctuations; i.e., 0.55 ≤ ∣kc/ωe∣ ≤ 3.0.

2. Simulations

[8] This section describes results from three PIC simulations of the forward cascade of freely decaying whistler turbulence. For all three runs, as by Gary et al. [2008] and Saito et al. [2008], the grid spacing is Δ = 0.10c/ωe, where c/ωe is the electron inertial length, the time step is δt ωe = 0.05 and the number of superparticles per cell is 64. The system has a spatial length of 51.2c/ωe in each direction. For these parameters, the fundamental mode of all three simulations has wave number kc/ωe = 0.1227, and short wavelength fluctuations should be resolved up to the component wave number of kc/ω = kc/ωe = 4. As in the two-dimensional simulations of Saito et al. [2008] and Gary et al. [2008, 2010], the initial physical dimensionless parameters are mp/me = 1836, Te/Tp = 1.0, βe = 0.10, and ωe2e2 = 5.0.

[9] We impose a three-dimensional spectrum of right-hand polarized whistler waves at t = 0. Initial wave numbers parallel to Bo are kc/ωe = ±0.1227, ±0.2454, and ±0.3682, whereas initial perpendicular wave numbers are the six same values and kc/ωe = 0. Different runs have different number (N) of modes imposed in the system, however, the initial total fluctuating magnetic field energy density is the same for all three runs: Σn= 1NδBn(t = 0)∣2/Bo2 = 0.10. The method of loading such an initial spectrum is described by Saito et al. [2008]. The frequencies and relationships among the field components are derived from the linear dispersion equation for magnetosonic-whistler fluctuations in a collisionless plasma, but the subsequent evolution of the fields and particles are computed using the fully nonlinear particle-in-cell simulation code.

[10] Run 1 is a two-dimensional simulation with an initial spectrum similar to the corresponding runs of Gary et al. [2008] and Saito et al. [2008]. The domain size is 512 × 512 cells, which is equivalent to 51.2c/ωe × 51.2c/ωe. In this case, 42 modes are present in the y-z plane of the initial whistler spectrum. Runs 2 and 3 are three-dimensional simulations in which the system has a domain size of 512 × 512 × 512 cells and spatial dimensions Lx = Ly = Lz = 51.2c/ωe. The only difference between these two runs is in the initial spectra. Run 2 has 42 modes in the y-z plane and 36 modes in the x-z plane (omitting the 6 modes at k = 0), whereas Run 3 has the same 78 modes as Run 2, but an additional 72 modes corresponding to a rotation of the Run 2 modes through an angle of 45° about the z-axis. Figure 1 illustrates the distribution of the initial modes for the two three-dimensional cases. All three runs are computed to a final time of ∣Ωet = 447. The total particle plus fluctuating field energy of the system is conserved to within 0.6% between t = 0 and the final time.

Figure 1.

Two-dimensional reduced magnetic fluctuation energy spectra at t = 0: (left) Run 2 in the plane perpendicular to Bo, (middle) Run 3 in the plane perpendicular to Bo, and (right) Run 3 in the ky-kz plane containing Bo.

[11] Figure 2 shows two-dimensional reduced magnetic energy spectra from the three simulations at the final time. As in the two dimensional Run 1, the imposition of an initial spectrum of relatively long-wavelength whistlers in the three-dimensional runs leads to a forward cascade to shorter wavelengths and the development of a broadband, turbulent spectrum. The late-time spectrum in the kx-ky plane perpendicular to Bo is approximately gyrotropic, which is a consistency check on our homogeneous plasma simulation with initial conditions which are symmetric between x and y. The late-time reduced spectra in the two planes (kx-kz, ky-kz) containing Bo in all three runs show, just as in the two-dimensional PIC simulations of whistler turbulence [Gary et al., 2008, 2010; Saito et al., 2008, 2010], that the forward cascade is anisotropic, preferentially transfering energy to fluctuations with wave vectors quasi-perpendicular, rather than quasi-parallel, to Bo.

Figure 2.

Two-dimensional reduced magnetic fluctuation energy spectra at ∣Ωet = 447 from (left) Run 1, (middle) Run 2 and (right) Run 3. (top) Spectra in the ky-kz plane and (bottom) spectra in the plane perpendicular to Bo.

[12] The reduced magnetic ky spectrum from the two-dimensional Run 1 at ∣Ωet = 447 is proportional to ky−4.0 over 0.25 ≲ kyc/ωe ≲ 1.60 (not shown), similar to the ky−4.5 late-time results for the two-dimensional simulations with ε = 0.10 and initial βe = 0.10 of Gary et al. [2008] and Saito et al. [2008]. Figure 3 illustrates the reduced k spectra from the three-dimensional Runs 2 and 3 at ∣Ωet = 447. The results for the two runs are very similar, indicating that the late-time spectra are relatively independent of the detailed choice of initial fluctuations. The reduced spectra at relatively long wavelengths (0.36 ≲ kc/ωe ≲ 1.0) suggest a k−3.1 dependence, whereas at shorter wavelengths (1.0 ≲ kc/ωe ≲ 2.5) the spectra become steeper with a k−4.3 dependence. The upturn in the spectra at kc/ωe ≃ 2.5 corresponds to the noise level of the simulation. The suggestion of two distinct power law regimes of the turbulence is similar to the prediction of Meyrand and Galtier [2010] who used an EMHD model with isotropic fluctuations to derive a k−11/3 magnetic fluctuation spectrum at 1 < kc/ωe. Furthermore, solar wind turbulent spectra with breaks near the inverse electron inertial length were reported by Sahraoui et al. [2009, 2010]. However, Alexandrova et al. [2009] reported observations of solar wind turbulence with an exponential decrease at scales shorter than the electron gyroradius. These latter measurements are not necessarily in contradiction with our short-wavelength power law result because our simulations have been run at βe = 0.10 where Landau damping of whistler fluctuations is relatively weak, whereas Alexandrova et al.'s [2009] measurements were almost all at larger βe values where Landau damping is stronger and exponential decreases of spectra are more likely.

Figure 3.

Reduced k magnetic fluctuation energy spectra at ∣Ωet = 447 from Run 2 and Run 3 (as labeled). The dashed lines represent power law functions as labeled for comparison against the simulation results.

[13] Figure 4 compares the spectral anisotropies tan2θB of the three runs as functions of time. The results of the two-dimensional Run 1 are very similar to that of Run I from Saito et al. [2008], with tan2θB ∼ 4 at ∣Ωet ≃ 400. Both three-dimensional runs exhibit stronger anisotropies, with Run 2 reaching tan2θB≃ 6 and Run 3 attaining tan2θB > 10 at late times. Our interpretation of these results is that the much larger number of quasi-perpendicular modes available in the three-dimensional cases allows the nonlinear wave-wave interactions and perpendicular cascades to proceed much more efficiently than in two dimensions. The successively increasing number of modes with k components among these three runs implies more channels for perpendicular cascading and therefore a successively faster rate of such energy transfer for the fluctuations. Limits on our computational resources prevent us from continuing to move toward more dense modes in wave vector space, but the trend of these three runs indicates that such a condition should correspond to a highly anisotropic late-time condition with tan2θB ≫ 1.

Figure 4.

Time histories of the spectral anisotropy factor tan2θB from the two-dimensional Run 1 (red line) and the three-dimensional Run 2 (green line) and Run 3 (blue line) simulations. Evaluations here are taken on spectra over 0.65 ≤ kc/ωe ≤ 3.0.

3. Conclusions

[14] We have carried out the first fully three-dimensional particle-in-cell simulations of whistler turbulence in a homogeneous collisionless plasma with a uniform background magnetic field Bo. We imposed an initial spectrum of relatively long-wavelength whistler fluctuations and computed the free decay of these modes to a broadband shorter wavelength regime of turbulence. This cascade yielded transfer of fluctuation energy to wave vectors preferentially quasi-perpendicular to Bo. Both the forward cascade and its consequent wave vector anisotropy are qualitatively similar to previous results obtained from two-dimensional PIC simulations of whistler turbulence; quantitatively, however, the three-dimensional anisotropy develops faster and to a larger value than that in two-dimensional simulations. Furthermore, the more modes that are used as initial conditions, the more rapid the cascade and the more strongly anisotropic the turbulence becomes.

[15] The reduced magnetic fluctuation energy spectrum of our two-dimensional simulation shows a ky−4.0 dependence at kyc/ωe ≲ 1, similar to the steep power law behavior exhibited in earlier such simulations [Saito et al., 2008, 2010]. In contrast, our three-dimensional simulations of Runs 2 and 3 show a less steep reduced spectrum of k−3.1 at kc/ωe ≲ 1, a spectral break near kc/ωe ≃ 1, and a steeper spectrum with k−4.3 at shorter wavelengths, similar to recent observations of short-wavelength turbulence in the solar wind. Further three-dimensional simulations of whistler turbulence at different values of βe and initial fluctuation amplitude must be carried out before more conclusions concerning whistler turbulence are reached.


[16] The authors acknowledge useful exchanges with Homa Karimabadi, Kaijun Liu, and Shinji Saito, and in particular extended discussions with John Podesta. The Los Alamos portion of this work was performed under the auspices of the U.S. Department of Energy (DOE). It was supported by the Solar and Heliospheric Physics SR&T and Heliophysics Guest Investigators Programs of the National Aeronautics and Space Administration, and by the joint DOE/National Science Foundation program in fundamental plasma research. The USC portion of this work was performed under an ONR MURI project led by the University of Maryland (subaward Z882806). OC's research was also conducted as part of the Los Alamos Space Weather Summer School supported by the Institute of Geophysics and Planetary Physics, the Science, Technology and Engineering Directorate and the Global Security Directorate at Los Alamos National Laboratory. Computational resources supporting this work were provided by the USC High-Performance Computing and Communications (HPCC) as well as by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.

[17] The Editor thanks Gregory Howes and Fouad Sahraoui for their assistance in evaluating this paper.