General aspects of whistler wave generation in space plasmas

K. Sauer and R. Sydora

Institute of Geophysics, University of Alberta, Canada

ISSS-10, Banff, Canada, July 24-30, 2011

• Introduction: whistler wave observations

in space and laboratory

• Two kinds of whistler waves:- unstable waves, - stationary nonlinear waves: oscillitons (Sauer et al., 2002)

• Results of PIC simulations showing the transition

from unstable waves to oscillitons

• Summary

General aspects of whistler wave generation in space plasmas

HF electromagnetic waves in an electron-proton plasma

G=ΩΩΩΩe/ωωωωe < 1 (solar wind)

and

G=ΩΩΩΩe/ωωωωe > 1 (auroral regions)

Whistler waves in an electron-proton plasma

with

G=ΩΩΩΩe/ωωωωe< 1

Power spectral density of the magnetic field

versus the generalized McIllwain coordinate L*

Polar spacecraft measurements, adapted from Santolik et al. (2010).

The black line

marks fce/2

Large-amplitude whistlers observed in Earth’s radiation belt by STEREO

Cattel et al., 2008

ωωωω ~ 0.2 ΩΩΩΩe

Vph ~ 0.2VAe

θθθθ ~ 500

Large-amplitude whistlers observed in Earth’s radiation belt by Wind

Wilson III et al., 2011

B0~400nT

dBy/B0~0.02

f/fce≤0.5

mostly oblique

propagation

Auroral hiss near Saturn’s moon Enceladus: Cassini observations

Gurnett et al., 2011

The frequency is terminated at f/fce~0.75.

Parameters:

fce=9 kHz, B~300 nT

fpe= 65 kHz, n~45 cm-

G=fce/fpe~0.15

VAe=4.5⋅109 cm/s,

VTe= 2.5⋅108 cm/s (Te~40eV)

ββββe~0.007

Vb ~ 4 108 cm/s (~100eV)

Vb/VAe ~ 0.1 (0.2)

Electron beam experiment aboard Spacelab 2 Farrell et al., 1998

Parameters:

fce=1 MHz, B~3 104 nT

fpe= 3 MHz, n~105 cm-3

G=fce/fpe~0.33

VAe~ 1010 cm/s,

VTe ~ 108 cm/s (Te~10eV),

VAe~100 VTe :

ββββe~2⋅⋅⋅⋅10-4

Vb~109 cm/s (~1 keV)

Vb/VAe~0.1 (0.2)

Laboratory experiment, Stenzel 1977Parameters:

fce~220 MHz, B~30 G

fpe~ 1 GHz, n~108-109 cm-3 G=fce/fpe~0.2-0.6

VAe~ 2⋅109 cm/s, VTe ~ 108 cm/s (Te~2eV), VAe~100 VTe : ββββe~10-4

Vb~3⋅108 cm/s (≤50 eV) Vb/VAe~0.1 (0.2)

nb/n0=0.005

Two kinds of whistler waves

• Unstable waves driven by beams or

electron temperature anisotropy

• Nonlinear stationary waves:

whistler oscillitons

(nonlinear Gendrin mode waves)

G=ΩΩΩΩe/ωωωωe <<<<1

θ=00

ω = f(θ)

Unstable whistler waves driven by temperature anisotropy

a) Warm plasma (βe=3⋅10-2)

with temperature

anisotropy (T⊥/T//=2),

parallel propagation (θ=00).

b) Cold (βe= 4⋅10-5) and hot

anisotropic population

(nh/nc=0.15, Th/Tc=7,

T⊥/T//=10, θ=400)

Unstable whistler waves driven by electron beams

c) Cold plasma (βe=2⋅10-4)

with sub-Alfvenic beam

(nb/nc=0.01, Vb/VAe=0.2,

Tb/Tc=1), θ=400):

Cherenkov-type instability

ω=k⋅Vb

d) Cold plasma (βe= 2⋅10-4)

with super-Alfvenic beam

(nb/nc=0.01, Vb/VAe=2.5,

Tb/Tc=1), θ=600:

Doppler-shifted cyclotron

mode ω = -Ωe + k⋅Vb

Unstable whistler waves: maximum growth rate at oblique propagation

a) Cold plasma and hot

anisotropic populationb) Cold plasma and

super-Alfvenic beam

Gendrin mode waves propagating obliquely to the magnetic field

θθθθ

magnetic field

direction

B0

wave propagation

direction

z

x

k

Gendrin mode

waves:

kc/ωe=1,

ω/Ωe=(1/2)cosθ

Vph ׀׀=Vph/cosθθθθ=Vgr

=VAe/2

The component of the

phase velocity parallel to

B0 and the group velocity

have the same value!

Gendrin, 1961; Helliwell, 1995

Whistler wave dispersion, stationary waves and oscillitons

Whistler wave dispersion

at parallel and oblique

propagation, θ=00,700

ωωωω = ωωωω(k)

Dispersion of stationary

waves:

ωωωω k⋅⋅⋅⋅U

k = k(U)

Whistler oscillitons

(nonlinear Gendrin

mode waves):

By , Ez= f(x)

θ=700θ=00

⋅ ⋅

⋅ ⋅

Gendrin point

Sauer et al., 2002

Basic equations are:

(1)

Equation of

motion for

electrons and

protons

(2)

Ampere‘s law

and

(3)

Faraday‘s law

xU

t ∂

∂−→

∂

∂:

Nonlinear stationary waves (whistler oscillitons), nonlinear Gendrin mode waves;

Sauer et al., 2002

Governing equations of whistler oscillitons

Equations of

motion for

electrons and

protons: i=e,p

M=U/VAe

)v/(MBuBuEx

uexxeyyexz

ez −−+−=∂

∂)(

Conservation of longitudinal momentum:

zy BME +=

Ampere‘s law:

yz BME −=Faraday‘s law:

)v/(MBuBuEx

uexxeyyexz

ez −−+−=∂

∂)(

em

1

)ixxizzixyiy v-/(M)BuBu(Ex

u+−−=

∂

∂

im

1

)v/(MBuBuEx

uixxiyyixz

iz −−+−=∂

∂)(

im

1

pzpeze vnvn +−=∂

∂

x

By

pypeye vnvn ++=∂

∂

x

Bz

Bo

x

,

,

1)(B2M

1ex

upx

u 2

p

−=≅µµµµ

Waveform of nonlinear Gendrin mode waves(whistler oscillitons)

θ=700, U=0.172VAe

Sauer, Sydora; 2010

Particle in-cell (PIC) simulations of the evolution of unstable whistler waves

Temporal evolution of unstable whistler waves, transition to oscillitons

magnetic energy

temperature ratio T⊥/T׀׀

(transition from T⊥/T9=׀׀ to ~3)

wave number(transition from

kc/ωe~2 to 1)

0

4

8

T⊥

/T׀׀

Wave number shift from unstable waves to oscillitons

unstable waves

oscillitons

Transition to oscillitons: nonlinear Gendrin waves

Spatial profiles

from oscilliton

theory

(Sauer et al., 2002, 2010)

Spatial profiles

from PIC

simulations

(Sydora et al., 2007)

kc/ωe~1

ωet

Time: 3276.800/3276.800

Magneti

c E

nerg

y

0 500 1000 1500 2000 2500 3000

10-6

10-4

10-2

two electron populations: cold+hot anisotropic

kc/ωe

ωet

log

By

0 5 10 15 200

200

400

600

800

x 10-3

-6

-5

-4

-3

Wave number shift of unstable waves

Wave number shift seen in other studies

Schriver et al., 2010 - Whistler wave generation

PIC simulation

Silin et al., 2010 - EMIC waves

Vlasov simulation

Usanova, 2010 - EMIC waves

(thesis, Univ. Alberta) - Hybrid code simulation

Wave number shift due to nonlinear wave-wave interaction

Schriver et al., 2010

Superposition of two anisotropic populations: cool and hot one

βc=0.003

nc/ne =0.9

Ac=5

θmax~ 450

βh=0.01

nh/ne =0.1

Ah=7

θmax~100

Gary et al., AGU Radiation belt physics

conference, St. John‘s, July 2011

Gendrin mode

Explanation of banded structure of whistler wave emission

The spectrum of whistler waves is essentially determined by the electron plasma

beta (βe):

Only one warm population (βe>0.01) – whistler emission only in the lower

frequency band: ω=(1/2)Ωecos(θ)

Only one cool population (βe<<0.01) – whistler emission only in the upper

frequency band: ω=Ωecos(θ)

Explanation of frequency bands and gaps at whistler wave emission

Schriver et al., 2011

Nunn et al., 2009

Summary

There are two kinds of whistler waves:

a) unstable waves with kc/ωωωωe>1,

b) whistler oscillitons (nonlinear Gendrin modes)

PIC simulations have shown that oscillitons can

be excited by unstable waves owing to nonlinear

wave number shift, obviously caused by wave-

wave interaction.

Frequency bands and gaps in observed whistler

wave spectra and the particular role of one-half

the cyclotron frequency (ΩΩΩΩe/2) can be explained by

the transition from unstable waves to oscillitons.

END