High-frequency short-wavelength whistler turbulences are often observed in space plasma (Beinroth and Neubauer, 1981; Lengyel-Frey et al., 1996; Narita et al., 2011, 2016). We define whistler turbulence as a broadband ensemble of incoherent field fluctuations in a magnetized plasma at frequencies between the lower hybrid and electron cyclotron frequencies and at wavelengths much shorter than the ion inertial length. There have been significant debates about the possible sources of whistler turbulence in recent years. One possible scenario is the cascade of fluctuations from the longer wavelength inertial range. Kinetic Alfvén waves and higher frequency magnetosonic-whistler fluctuations have been considered as the two candidates (Gary and Smith, 2009). Recent solar wind observations (Leamon et al., 1998; Sahraoui et al., 2009, 2010; Kiyani et al., 2012; Salem et al., 2012) and numerical simulations (Howes et al., 2008; TenBarge et al., 2013) have identified the existence of the kinetic Alfvén fluctuations with a wavelength around the ion inertial length or ion thermal gyro-radius. However, the mechanism on how such modes cascade fluctuation energy down to electron scale remains unclear. For instance, as the inertial range cascade preferentially transfers fluctuation energy to propagation directions relatively perpendicular to background magnetic field, where whistler fluctuations can be damped (Mithaiwala et al., 2012; Cerri et al., 2016), the cascade processes may not be able to provide a sufficiently large amplitude to feed to whistler turbulence.

Another possible scenario is kinetic whistler instabilities. A specific growing mode which can be a source for whistler turbulence at relatively long electron-scale wavelengths is the whistler anisotropy instability. We use subscripts “⊥″ and “‖″ to denote the directions perpendicular and parallel to the background magnetic field B ₀, respectively, subscripts e and i to denote electrons and ions, respectively, and k ⃗ to denote the wave-vector. This instability is driven by electron temperature anisotropy T _⊥e /T _‖e > 1 and propagates at k ⃗ × B 0 = 0 in a homogeneous plasma. Observations have indicated that this instability is operating in the terrestrial magnetosheath (Gary et al., 2005). In the solar wind, while adiabatic expansion of the solar wind would typically lead to T _‖e > T _⊥e which can excite the firehose instability, there is evidence that local compressions and turbulence in the solar wind may also create T _⊥e > T _‖e (Gary and Madland, 1985; Gary, 1993; Karimabadi et al., 2013), which can excite the whistler instability. Particle-in-Cell (PIC) simulations (Gary and Wang, 1996; Gary et al., 2000, 2014; Saito et al., 2008) have demonstrated that this mode can generate enhanced whistler fluctuations and spectral transfer (Gary and Wang, 1996; Gary et al., 2000, 2014; Saito et al., 2008; Chang et al., 2014, 2015).

To further investigate the aforementioned scenarios, one must first understand the competing effects from whistler turbulence and whistler anisotropy instability. For instance, one of the primary consequences of plasma turbulence is to produce sharp spatial gradients in the plasma which can produce enhanced anisotropies locally and forward cascade to dissipate the energy from long wavelength to short wavelength (Osman et al., 2010; Greco et al., 2012; Parashar and Matthaeus, 2016). On the other hand, simulations of whistler anisotropy instabilities driven by a bi-Maxwellian velocity distribution for electrons in a homogeneous plasma showed that the instability imposes an upper bound or constraint on that anisotropy uniformly across the plasma (Gary and Wang, 1996; Gary et al., 2000, 2014; Hughes et al., 2016). The electron anisotropy upper bound derived by Gary and Wang (1996) was verified by observations in the solar wind and magnetosphere (Gary et al., 2005; MacDonald et al., 2008; Štverák et al., 2008; An et al., 2017). However, past studies have mostly addressed the effects from whistler turbulence and whistler anisotropy instability separately.

Gary et al. (2008, 2010), Saito et al. (2008, 2010), and Saito and Gary (2012) presented the first 2-dimensional (2D) PIC simulations of whistler turbulence, and Gary et al. (2012, 2014) and Chang et al. (2011, 2013, 2014, 2015) presented the first 3-dimensional (3D) PIC simulations of whistler turbulence. These simulations considered a homogeneous, magnetized, collisionless plasma upon which an initial spectrum of relatively long wavelength whistler fluctuations is imposed. The results showed that the forward cascade leads to fluctuations which are consistent with the linear dispersion solution for whistler fluctuations. Electron temperature anisotropy were also found to form during forward cascade. Hughes et al. (2014, 2017) and Gary et al. (2016) further investigated electron/ion heating due to whistler turbulence as a function of the initial fluctuating magnetic field energy density, and found the maximum electron heat rate scales approximately linearly with the fluctuating field energy density. This suggests a quasi-linear type heating due to electron Landau damping (Gary et al., 2016).

In this paper, we consider a collisionless, magnetized plasma with co-existing whistler turbulence and the electron temperature anisotropy as the initial condition, and investigate the consequences of both microinstability growth and turbulent cascade using 3D fully kinetic PIC simulations. Qudsi et al. (2020) and Bandyopadhyay et al. (2020) carried out 2D PIC simulations of the Alfvénic turbulence, where ion temperature anisotropy is generated by the development of the turbulence. The results showed that microinstabilities can develop locally in response to ion temperature anisotropies generated by turbulence and may affect the plasma globally, and that there is an apparent correlation between linear instability theory and strongly intermittent turbulence. It was also speculated that a similar process might also occur on electron scale. In this paper, in order to evaluate the effect from whistler instability at a given temperature anisotropy, we prescribe electron temperature anisotropy as the initial condition to drive the instability.

We carry out four different ensembles of PIC simulations: 1) an initially quiet, anisotropic plasma with prescribed initial electron temperature anisotropy; 2) an isotropic plasma with prescribed initial whistler fluctuations; 3) an anisotropic plasma with initial whistler fluctuations (varying initial electron temperature anisotropy and fixed initial fluctuation field energy); and 4) an anisotropic plasma with initial whistler fluctuations (fixed initial electron temperature anisotropy and varying initial fluctuation field energy). Results from PIC simulation are linked with a statistical analysis to understand whether there is any interplay between whistler turbulence and whistler instability, and what are the competing effects from these two processes.

We consider a collisionless electron-ion plasma with a uniform background magnetic field B 0 = B 0 z ̂ . For the jth (j = e, i) species, we denote the plasma frequency as ω p j = 4 π n j e 2 / m j , the cyclotron frequency as Ω _j = eB ₀/m _j c, the thermal speed as v t j = T ‖ j / m j , and β j = 8 π n j T ‖ j / B 0 2 . We denote the angle of mode propagation θ by k ⋅ B ₀ = kB ₀ cos(θ).

The physical and numerical parameters are chosen to assure that the consequences of both microinstability growth and turbulent cascade can be accurately resolved in the simulation. In this paper, the initial electron plasma beta is taken to be of a typical value for the solar wind plasma, β _e = 0.1. To study the effects of whistler anisotropy instability, we consider a range of initial T _⊥e /T _‖e values that are below and above the instability threshold in the simulation. Through a sequence of test runs with varying initial T _⊥e /T _‖e values, we find the threshold to excite whistler anisotropy instability for the parameters considered is T _⊥e /T _‖e ≃ 2.3, close to the calculation using the linear theory from (Gary, 1993). In this paper, we present simulations with initial temperature anisotropy of T _⊥e /T _‖e = 1, 2, 3, 5, 7, 9. To study the effects of whistler turbulence, an ensemble of whistler fluctuations are imposed at t = 0. The initially loaded whistler fluctuations are set to be relatively long-wavelength with approximately isotropic wavevectors. The spectrum is the same as that used in our previous simulation studies on whistler turbulence (Chang et al., 2011; Gary et al., 2012; Chang et al., 2013; Hughes et al., 2014; Chang et al., 2015). The initial whistler modes include n = 0, ±1, ±2, and ±3 of the fundamental wavenumber in the perpendicular direction, and n = ±1, ±2, and ±3 of the fundamental wavenumber in the parallel direction, where the fundamental wavenumber corresponds to the maximum wavelength that can be contained in the domain. This leads to a total of N = 150 normal modes with random phases (Chang et al., 2013). The simulations will consider initial total fluctuating magnetic field energy density. ϵ = ∑ n = 1 N | δ B n t = 0 | 2 / B 0 2 (1) at ϵ = 0, 0.05, 0.25, and 0.5.

We apply a three-dimensional (3D) full particle electromagnetic particle-in-cell code, 3D-EMPIC by Wang et al. (1995), to simulate the evolution of plasma under four different sets of initial conditions. In Simulation Group A, the ions are set to follow an isotropic velocity distribution while the electrons follow an anisotropic bi-Maxwellian velocity distribution function with different initial values of T _⊥e /T _‖e , at T _⊥e /T _‖e = 2, 3, 5, 7, and 9. The plasma has no initial field fluctuations, ϵ = 0. Simulation Group A is a typical setup for simulations of whistler anisotropy instability (Gary and Wang, 1996). In Simulation Group B, both the ions and electrons are set to have an isotropic velocity distribution. An ensemble of whistler fluctuations are imposed at t = 0, with the initial total fluctuating magnetic field energy density at ϵ = 0, 0.05, 0.25, and 0.5. The initial condition in Simulation Groups C and D is a combination of that of Groups A and B, where the electrons follow an anisotropic bi-Maxwellian velocity distribution function and the plasma is initially loaded with an ensemble of whistler fluctuations. In Group C, we take the initial field fluctuation at ϵ = 0.25 and change the initial temperature anisotropy at T _⊥e /T _‖e = 2, 3, 5, 7, 9. In Group D, we take the initial temperature anisotropy at T _⊥e /T _‖e = 3 and change the initial field fluctuation field density at ϵ = 0, 0.05, 0.25, 0.50. The simulation groups are summarized in Table 1.

All the simulations are run using an artificial ion to electron mass ratio of m _i /m _e = 400. The ion initial temperature is set to be T _i = T _‖e . The ratio of the electron gyro-frequency to plasma frequency is Ω _e /ω _pe ≃ 0.447. The simulation box is a cube with a size in each direction at 51.2d _e , where d _e = c/ω _e is the electron inertial length. The grid spacing is set to be Δ = 0.10d _e , and hence the mesh size is 512 × 512 × 512. The time step is set to be Δtω _pe = 0.05. All the simulations are run for tω _pe > 1000 (tΩ _pe > 447.20), i.e. more than 20,000 steps. The macro-particles used is 48 ions and 48 electrons per cell or about 3.1 × 10¹¹ total macro-particles.

We use the Probability Density Function (PDF) in statistical analysis of magnetic fluctuations. The PDF of a random field B(x) may be defined as (Matthaeus et al., 2015) PDF B d B = probability that the random value lies between  B and B + d B (2) Then the increments of the field components are δ B x = e r ⋅ B x + r − B x (3) where r is the spatial separation length vector along the direction of any unit vector e _r . By summing over all the cells of a PIC simulation, one may construct a PDF for each component of the fluctuating fields as a function of the spatial separation r. If the random variable r is subject to a central limit theorem, the distribution is expected to be a Gaussian, whereas any departure from a Gaussian corresponds to a more strongly intermittent ensemble of fluctuations. An important advantage of the PDF analyses is that, by statistically averaging over a large body of observational and/or computational data, one may draw general conclusions which are less readily available via other means of data analysis. For example, the statistical analysis of solar wind magnetic fluctuations measured from the Cluster and ACE spacecraft by Kiyani et al. (2012) shows that the PDFs of both δB _‖ and δB _⊥ exhibit the same functional form in the kinetic range but not in the inertial range. The PDF analysis of the solar wind data from the Helios spacecraft (He et al., 2013) shows that, as the heliospheric distance of the spacecraft increases, the distribution of the local mean magnetic field vectors gradually broadens in the radial direction and becomes more scattered. The PDF analysis of 3D PIC simulations of whistler turbulence (Chang et al., 2014) shows distinct non-Gaussian “tails” in both the δB _‖ and δB _⊥ distributions as well as distinctly different functional forms between the two magnetic polarizations.

3D PIC simulations are carried out to study the dynamic evolution of a collisionless, magnetized plasma with co-existing whistler turbulence and electron temperature anisotropy as the initial condition, and the competing consequences of whistler turbulence cascade and whistler anisotropy instability growth. The results show that the operation of the whistler instability within whistler turbulence has no obvious effects on the fluctuating magnetic field. We find the overall fluctuating magnetic field energy and intermittency generated by turbulence are not influenced by the inclusion of an initial electron temperature anisotropy, while wavevector anisotropy is reduced somewhat by increased electron temperature anisotropy. In contrast, the results show that whistler instability has a major effect on electron temperature anisotropy. While whistler turbulence produces sharp gradients and enhanced electron temperature anisotropies locally, we find that increasing the initial electron temperature anisotropy actually leads to a reduction of the intermittency in the electron temperature anisotropy generated by turbulence, and the reduction of the average electron temperature anisotropy further below the upper bound as predicted by the linear theory for a homogeneous anisotropic plasma. The results suggest the small reduction of wavevector anisotropy and major reduction of electron temperature anisotropy are apparently due to whistler instability growth. Comparing to an isotropic plasma with whistler turbulence, an increase in the initial electron temperature anisotropy leads to a larger growth rate of the whistler anisotropy instability, resulting in faster damping of the energy in this perpendicular direction, and thus a small reduction of the growth rate of the wavevector anisotropy tan² θ _B . While turbulence produce sharp gradients and local enhanced anisotropies, the whistler anisotropy instability acts to reduce electron temperature anisotropy through wave-particle scattering. The results suggest that the fluctuation from the growth of a single mode of whistler anisotropy instability is more efficient in wave-particle scattering of the electrons than that of a spectrum of whistler modes. Thus, a larger initial global electron temperature anisotropy, when combined with enhanced local electron temperature anistropy, would lead to even stronger wave-particle scattering effects locally, thus leading to local temperature anisotropy reduction and a more homogeneous electron temperature landscape.

In conclusion, we find that the overall effect of whistler anisotropy instability on a plasma with co-existing whistler turbulence and global electron temperature anisotropy is minor comparing to that of whistler turbulence. This is because the fluctuating energy associated with the narrowband whistler instability is dominated by that from the broadband whistler turbulence. However, as field fluctuations from the growth of a single instability mode may be more efficient in the wave-particle scattering process than that from a spectrum of whistler modes, the whistler instability can significantly reduces the intermittency of electron temperature anisotropy generated by turbulence. This suggests that microinstability may act as a regulation mechanism on turbulence development. In this study, the whistler instability is generated by imposing a bi-Maxwellian electron velocity distribution as the initial condition. The competing consequences of whistler turbulence cascade and whistler anisotropy instability growth will need to be further evaluated in a more realistic setup where the instability develops naturally from turbulence in future study.