## 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 ∣*δ***B**(*f*)∣^{2} ∼ with *α*_{f} ≃ 5/3 and a strong characteristic wave vector anisotropy with greater fluctuation energy at propagation quasi-perpendicular to **B**_{o}, 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 ≲ *α*_{f} [*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 ≲ *α*_{f} ≲ 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 **B**_{o}, where **B**_{o} denotes an average, uniform background magnetic field, than at propagation parallel or antiparallel to **B**_{o}.

[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 **B**_{o} and at real frequencies *ω*_{r} < Ω_{p}, 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 *j*th species plasma frequency as *ω*_{j} ≡ , the *j*th species cyclotron frequency as Ω_{j} ≡ *e _{j}B_{o}*/

*m*, and

_{j}c*β*

_{∥j}≡ 8

*πn*

_{j}k_{B}T_{∥j}/

*B*

_{o}

^{2}. 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 **B**_{o} = *B*_{o} so that the subscripts *z* and ∥ represent the same direction. Thus **k** = *k*_{x} + *k*_{y} + *k*_{∥} and *k*_{⊥} = . We define two-dimensional reduced energy spectra by summation over the third Cartesian wave vector component, e.g.,

In contrast, the one-dimensional reduced spectra for *k*_{⊥} are obtained by summing the energy over both *k*_{∥} and concentric annular regions in *k _{x}-k_{y}* space. The total fluctuating magnetic energy density is obtained by summing over all simulation wave vectors, and the spectral anisotropy angle θ

_{B}is defined via

An isotropic spectrum corresponds to tan^{2}θ_{B} = 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*/*ω*_{e}∣ ≤ 3.0.