A REVIEW OF WHISTLER
TURBULENCE BY THREEDIMENSIONAL 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 ShortWavelength Turbulence
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Short-wavelength turbulence is
fundamentally nonlinear and must be
treated with fully nonlinear techniques
such as particle-in-cell simulations.
At short wavelengths, fluctuation
amplitudes are relatively weak (| B| <<
Bo). 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)
Cascade of long
wavelengths to
dissipation at short
wavelengths.
Inertial range:
 10-4 Hz < f < 0.5 Hz
Kinetic range (aka
“Dissipation range”):
 0.5 Hz < f < 100’s Hz
Kinetic Alfven waves
 0.5 Hz < f < 10 Hz
KAWs or whistlers?
 10 Hz < f
Scenario for Short-Wavelength
Turbulence
Shaikh & Zank, MNRAS, 400,1881 (2009)
Three-Dimensional Whistler
Particle-in-cell (PIC) Simulations
Buneman particle-in-cell 3D EMPIC
code.
Homogeneous magnetized electron-ion
plasma.
Initial conditions:
 Turbulence: Almost isotropic spectrum of whistler
fluctuations at kc/ωpe < 1.
 Instability: Te/T||e > 1 leads to whistler anisotropy
instability.
3D PIC Simulations of Whistler
Turbulence
Chang et al. (2011), Geophys. Res. Lett., 38, L22102.
Gary et al. (2012), Astrophys. J., 755, 142 (Variations with initial
wave amplitude).
Chang et al. (2013), J. Geophys. Res., 118, 2824 (Variations
with βe).
Chang et al. (2014), Phys. Plasmas, 21, 052305 (Linear vs.
nonlinear dissipation).
Gary et al. (2014), J. Geophys. Res., 119, 1429 (Whistler
anisotropy instability).
Hughes et al. (2014), Geophys. Res. Lett., 41, 8681 (Electron
and ion heating).
Chang et al. (2015), Astrophys. J., in press (Inverse vs. forward
cascade)
3D PIC Simulations of Whistler Turbulence:
Forward vs. Inverse Cascades
Run 2: Large-box simulation
Initial spectrum:
 0.24 < kc/ωpe < 0.49
Fluctuation energy in forward
cascade ~ 80 times greater
than energy in inverse
cascade.
So from here on, we
emphasize forward cascade
results.
3D PIC Simulations of Whistler
Turbulence:
Forward Cascade
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/ωpe ~ 1
* Likely causes: Dispersion + dissipation
3D Whistler Turbulence:
Satisfies Linear Whistler Dispersion
Colors: Dispersion from PIC simulations.
Black lines: Dispersion from linear kinetic
dispersion theory.
2D Whistler Turbulence:
Magnetic Fluctuation Ratios
Saito et al. [2008]
Circles: 2D PIC
simulation of whistler
turbulence.
Dashed lines: Linear
kinetic dispersion
theory
Red: |B|||2/|B|2
Blue: |B|2/|B|2
Green: |B|2/|B|2
3D Whistler Turbulence:
Dissipation Rate Increases with Increasing βe
3D Whistler Turbulence:
Wavevector Anisotropy Decreases with βe
3D Whistler Turbulence:
Spectral Break
PIC simulations at
βe=0.1 [Gary et al.,
2012] have spectral
break at kc/ωpe~1.
But no “universal”
power-law scaling;
rather, slopes
become less steep
as initial amplitude is
increased.
Turbulent Dissipation
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 reconnection at small-scale current
sheets.
 Nonlinear nonresonant stochastic heating.
3D Whistler Turbulence:
Electron Heating
Electron heating
rate increases with
increasing βe.
Forward cascade
yields k >> k||,
yielding δE||,
yielding electron
heating with T||e >
Te.
3D Whistler Turbulence:
Linear Damping vs. Total Dissipation
Total damping
rates: solid lines.
Linear theory
damping rates:
dashed lines.
Agreement at
high βe and low
initial fluctuation
amplitudes (εe).
Chang et al.
(2014)
3D Whistler Turbulence:
Scaling with Simulation Box Size
Lωpe/c = 25.6 (black lines)
Lωpe/c = 51.2 (blue lines)
Lωpe/c = 102.4 (red lines)
Whistler Anisotropy Instability:
Particle-in-cell Simulation
3D PIC simulation in
homogeneous plasma
[Gary, Hughes et al.,
2014].
Fluctuating fields driven
by the instability grow,
saturate, then gradually
decay.
Wave-particle scattering
reduces electron
anisotropy, but does not
yield full isotropy.
3D Whistler Turbulence:
Satisfies Linear Whistler Dispersion
Turbulence from
initial whistler
fluctuations:
 Dashed line: Linear
dispersion theory.
Turbulence from
whistler anisotropy
instability:
 Dashed line: Linear
dispersion theory.
Whistler Anisotropy Instability:
Spectral Evolution
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•
•
•
•
Early times: Short-wavelength whistler instability grows at
kc/ωpe ~ 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
Instability constrains
value of Te/T||e.
PIC simulation
[Gary & Wang,
1996]:
Magnetosheath
observations [Gary
et al., 2005]:
3D PIC Simulations of Whistler
Turbulence Cascades: Conclusions
Forward cascade 80 x faster than inverse
cascade.
Forward cascade yields k >> k|| wavevector
anisotropy.
Two distinct power-law spectra with break at
kc/ωpe~1.
At weak amplitudes fluctuations
 Satisfy linear theory dispersion.
 Heat electrons by Landau damping with T||e > Te.
 Heat ions by Landau damping with T||i < Ti.
Conclusions: Whistler Turbulence
Scaling Relations
Increasing βe yields
 Faster forward cascade rates.
 Less anisotropic magnetic spectra.
 Less anisotropic electron velocity distributions.
 Hotter electron velocity distributions.
Increasing simulation box size yields
 Weaker overall dissipation.
 Stronger ion heating.
 Weaker electron heating.