A REVIEW OF WHISTLER TURBULENCE BY THREE- DIMENSIONAL PIC SIMULATIONS A REVIEW OF WHISTLER TURBULENCE BY THREE- DIMENSIONAL PIC SIMULATIONS S. Peter Gary, Space Science Institute Ouliang Chang, Oracle Corporation R. Scott Hughes and Joseph Wang University of Southern California Queenstown, New Zealand 9 February 2015

A Viewpoint for Short- Wavelength Turbulence n Short-wavelength turbulence is fundamentally nonlinear and must be treated with fully nonlinear techniques such as particle-in-cell simulations. n At short wavelengths, fluctuation amplitudes are relatively weak (|  B| << B o ). So linear kinetic wave theory can be useful in describing some aspects of such turbulence.

Short Wavelength Turbulence in the Solar Wind: Sahraoui et al. (2010) n Cascade of long wavelengths to dissipation at short wavelengths. n Inertial range:  Hz < f < 0.5 Hz n Kinetic range (aka “Dissipation range”):  0.5 Hz < f < 100’s Hz n Kinetic Alfven waves  0.5 Hz < f < 10 Hz n KAWs or whistlers?  10 Hz < f

Scenario for Short-Wavelength Turbulence Shaikh & Zank, MNRAS, 400,1881 (2009)

T hree-Dimensional Whistler Particle-in-cell (PIC) Simulations n Buneman particle-in-cell 3D EMPIC code. n Homogeneous magnetized electron-ion plasma. n Initial conditions:  Turbulence: Almost isotropic spectrum of whistler fluctuations at kc/ω pe < 1.  Instability: T/T ||e > 1 leads to whistler anisotropy instability.  Instability: T  e /T ||e > 1 leads to whistler anisotropy instability.

3D PIC Simulations of Whistler Turbulence n Chang et al. (2011), Geophys. Res. Lett., 38, L n Gary et al. (2012), Astrophys. J., 755, 142 (Variations with initial wave amplitude). n Chang et al. (2013), J. Geophys. Res., 118, 2824 (Variations with β e ). n Chang et al. (2014), Phys. Plasmas, 21, (Linear vs. nonlinear dissipation). n Gary et al. (2014), J. Geophys. Res., 119, 1429 (Whistler anisotropy instability). n Hughes et al. (2014), Geophys. Res. Lett., 41, 8681 (Electron and ion heating). n Chang et al. (2015), Astrophys. J., in press (Inverse vs. forward cascade)

3D PIC Simulations of Whistler Turbulence: Forward vs. Inverse Cascades n Run 2: Large-box simulation n Initial spectrum:  0.24 < kc/ω pe < 0.49 n Fluctuation energy in forward cascade ~ 80 times greater than energy in inverse cascade. n So from here on, we emphasize forward cascade results.

3D PIC Simulations of Whistler Turbulence: Forward Cascade n Magnetic fluctuations show:  Whistler-like dispersion  Decay of energy * Likely cause: wave-particle interactions (Electron Landau damping)  Forward cascade to larger wavenumbers and k  >> k || * Likely cause: Wave-wave interactions  Spectral break at k  c/ω pe ~ 1 * Likely causes: Dispersion + dissipation

3D Whistler Turbulence: Satisfies Linear Whistler Dispersion n Colors: Dispersion from PIC simulations. n Black lines: Dispersion from linear kinetic dispersion theory.

2D Whistler Turbulence: Magnetic Fluctuation Ratios n Saito et al. [2008] n Circles: 2D PIC simulation of whistler turbulence. n Dashed lines: Linear kinetic dispersion theory n Red: |B || | 2 /|B| 2 n Red: |  B || | 2 /|  B| 2 n Blue: |B| 2 /|B| 2 n Blue: |  B  | 2 /|  B| 2 n Green: |B| 2 /|B| 2 n Green: |  B  | 2 /|  B| 2

3D Whistler Turbulence: Dissipation Rate Increases with Increasing β e 3D Whistler Turbulence: Dissipation Rate Increases with Increasing β e

3D Whistler Turbulence: Wavevector Anisotropy Decreases with β e 3D Whistler Turbulence: Wavevector Anisotropy Decreases with β e

3D Whistler Turbulence: Spectral Break 3D Whistler Turbulence: Spectral Break n PIC s i mulations at β e =0.1 [Gary et al., 2012] have spectral break at k  c/ω pe ~1. n But no “universal” power-law scaling; rather, slopes become less steep as initial amplitude is increased.

Turbulent Dissipation n Forward cascade of turbulence carries fluctuating field energy to dissipation at short wavelengths. Possible mechanisms:  Linear wave-particle interactions: * Landau damping. * Cyclotron damping. Nonlinear Landau damping.Nonlinear Landau damping.  Nonlinear reconnection at small-scale current sheets.  Nonlinear nonresonant stochastic heating.

3D Whistler Turbulence: Electron Heating n Electron heating rate increases with increasing β e. n Forward cascade yields k  >> k ||, yielding δE ||, yielding electron heating with T ||e > T n Forward cascade yields k  >> k ||, yielding δE ||, yielding electron heating with T ||e > T  e.

3D Whistler Turbulence: Linear Damping vs. Total Dissipation 3D Whistler Turbulence: Linear Damping vs. Total Dissipation n Total damping rates: solid lines. n Linear theory damping rates: dashed lines. n Agreement at high β e and low initial fluctuation amplitudes (ε e ). n Chang et al. (2014)

3D Whistler Turbulence: Scaling with Simulation Box Size n Lω pe /c = 25.6 (black lines) n Lω pe /c = 51.2 (blue lines) n Lω pe /c = (red lines)

Whistler Anisotropy Instability: Particle-in-cell Simulation n 3D PIC simulation in homogeneous plasma [Gary, Hughes et al., 2014]. n Fluctuating fields driven by the instability grow, saturate, then gradually decay. n Wave-particle scattering reduces electron anisotropy, but does not yield full isotropy.

3D Whistler Turbulence: Satisfies Linear Whistler Dispersion n Turbulence from initial whistler fluctuations:  Dashed line: Linear dispersion theory. n Turbulence from whistler anisotropy instability:  Dashed line: Linear dispersion theory.

Whistler Anisotropy Instability: Spectral Evolution Early times: Short-wavelength whistler instability grows at kc/ω pe ~ 1 with k  << k || Later times: Inverse cascade to long wavelengths and k  >> k || Forward cascade to very short wavelengths and k  << k ||. But instability-driven spectra do not show power-law behavior characteristic of turbulent cascade.

Whistler Anisotropy Instability: Anisotropy Upper Bound n Instability constrains value of T/T ||e. n Instability constrains value of T  e /T ||e. n PIC simulation [Gary & Wang, 1996]: n Magnetosheath observations [Gary et al., 2005]:

3D PIC Simulations of Whistler Turbulence Cascades: Conclusions n Forward cascade 80 x faster than inverse cascade. n Forward cascade yields k  >> k || wavevector anisotropy. n Two distinct power-law spectra with break at k  c/ω pe ~1. n At weak amplitudes fluctuations  Satisfy linear theory dispersion.  Heat electrons by Landau damping with T ||e > T  Heat electrons by Landau damping with T ||e > T  e.  T ||i < T  Heat ions by Landau damping with T ||i < T  i.

Conclusions: Whistler Turbulence Scaling Relations n Increasing β e yields  Faster forward cascade rates.  Less anisotropic magnetic spectra.  Less anisotropic electron velocity distributions.  Hotter electron velocity distributions. n Increasing simulation box size yields  Weaker overall dissipation.  Stronger ion heating.  Weaker electron heating.