Richard B. Horne British Antarctic Survey

Part 1

Wave-Particle Interactions in the Magnetosphere

Lecture, Advanced Summer School in Solar System Physics, U of Sheffield, 6th September 2016.

Outline

• Importance of wave-particle interactions• The Zoo of waves in a magnetised plasma• Dispersion relation for a cold plasma• Dispersion relation for a hot plasma

– Landau resonance– Doppler shifted cyclotron resonance

• Concepts of quasi-linear diffusion• Acceleration and loss• Application to the Earth’s radiation belts• Examples of applications• Research needed

Importance of Wave-particle Interactions

• Particle precipitation into the atmosphere– Loss of radiation belt particles (MeV)– Diffuse aurora (~ keV particles)

• Particle heating and acceleration– Electron acceleration in the radiation belts– Ion heating

• Magnetic reconnection– Anomalous resistivity– Acceleration and drag in the outflow region

• Applications– Plasma thrusters on spacecraft– Klystrons – transmitters– Tokamaks – plasma fusion

Nuclear Detonation at High Altitude

• Earth’s radiation belts

• Starfish nuclear detonation, July 1962• 1.4 Megaton at 400 km

Atmospheric Tests

• Injection of energetic electrons from nuclear detonation

• Electron flux observed to decay

• Loss process ?

Electron Lifetimes

• Lifetimes less than that due to collisions• Another loss mechanism required

Dispersion Relation

• Plasma is a dispersive medium• Phase velocity depends on k

• Relates frequency w to the wavenumber k

• Determines– Frequency range for propagating waves– Polarisation– Phase velocity– Group velocity– Resonant frequencies k = infinity (where

waves interact strongly with particles)– Cut-off frequencies k = 0 (where waves are

reflected)– Wave growth or damping

Dispersion Relation

• The dispersion relation is derived from Maxwell’s equations• Assume small amplitude, plane wave solutions• Assume cold, infinite magnetised plasma

• Elements of the dielectric tensor, to be determined• E is the wave electric field• k is the wavevector• n = ck/w is the refractive index• Can also be written in terms of R, L, P, S, D – the Stix parameters• We use the form above as it can be used for a hot plasma

Cold Plasma Dispersion Relation

• Equivalent to the Appleton Hartree equation• Quadratic in n2

• (Refractive index n = ck/w) • Propagating solutions for n2 > 0• Evanescent (purely damped) waves for n2 < 0• At most 2 solutions at any frequency (in fact 4 for +k and –k)

• Non-trivial solutions when the determinant is zero

Dielectric Tensor for a Cold Plasma

• Plasma frequency

For a cold plasma

• Assumed phase velocity is much greater than thermal velocity

• No Landau damping• No cyclotron damping

Obtain • Resonance at gyrofrequency of

each particle species

• Strong interaction between waves and particles at a resonance• Cyclotron frequency

Propagation Parallel to B

• R mode• Right hand circularly polarised

• L mode• Left hand circularly polarised

• Whistler mode– Below fpe or fce, whichever is

lowest

• By solving the dispersion relation we can find propagating solutions n2 > 0 and non-propagating solns n2 < 0

• Find how w is related to k

Wave Polarisation - 1

• Right hand circular polarisation• Ex/Ey = - i

• E field rotates clockwise as wave propagates passed a fixed point along B0

• Same sense as electron gyration

• Left hand polarisation• Ex/Ey = + i

• E field rotates anti-clockwise• Same sense as ion gyration

• In general Ex/Ey is complex and waves are elliptically polarised

Propagation Perpendicular to B

• Only solutions in certain frequency bands

• Depend on fpe and fce

• O mode• E field along B

• X mode• E field perpendicular to B

Wave Polarisation - 2

• O mode polarisation• E field along B

• X mode polarisation• E field in x-y plane• E field perpendicular to B• E field can still rotate

• In general Ex/Ey is complex and waves are elliptically polarised• If Ex/Ey is real then linear polarisation• If Ex/Ey = 0 then linear with E along Ey• If Ex/Ey = i infinity then linear with E along Ex

Electrostatic Waves

• When k and E lie in the same direction we have simple solutions

• Since k x E=0, the induced magnetic field B is very small and can be neglected – hence electrostatic waves

• We can write

• Electrostatic waves usually have low phase velocities and can interact strongly with thermal particles

• BUT, n2 >> 1 is not sufficient at ion cyclotron frequencies

• Propagation along B, and perpendicular to B at certain frequencies

Propagation at Any Angle

• Electron only plasma

Low Frequency Propagation Parallel to B

Alfven wavePlasma and B field move together

Becomes whistler and ion cyclotron waves at high frequencies

Fast and intermediate

Slow mode

Compressions in the plasma along B

Low Frequency Propagation Perpendicular to B

• Fast compressionalmagnetosonic wave

• B field and plasma compressions

Low Frequency Propagation at Any Angle • Multi-ion plasma H+ and He+

Whistler Mode Waves at Halley Antarctica

Power Line Harmonic Radiation

• Lines with a frequency separation of about 50 Hz

• Seen in space and on the ground

• 1970s controversy• Origin of line magnetospheric line

radiation?• Power lines

• Suggested that power line radiation can deplete the radiation belts

• Suggested it triggers other emissions in space and releases free energy

• But – signals are very weak

Triggered Emissions

• Navy transmitters seen to excite risers and fallers• Experiment

• Transmitter put into Siple, Antarctica – 21.2 km dipole!!!• Transmitted a series of narrow frequency pulses (usually a few kHz range)• Pulses triggered rising frequency elements (risers) and fallers• Cyclotron resonance and nonlinear effects• Importance of nonlinear wave-particle interactions

Antarctica - Space• Antarctica – observe very low

frequency radio waves• Most originate in space• We have shown they accelerate

electrons and form the radiation belts• Changed ideas going back 40 years

Antarctic observations

Satellite observations

kHz

10

0Time (s)

Whistler Mode Hiss and Chorus Waves

• Hiss• Band of noise – no structure

• Chorus• Nonlinear, discrete rising tones

Chorus as a Source of Hiss

• Bortnik et al., Nature, [2008]

Waves in a Hot PlasmaKinetic Theory

• Consider a distribution of particles• Temperature, pressure

• Vlasov equation

• Perturbation expansion – linear theory• Solve with Maxwell’s equations• Obtain dispersion relation

• Note• MHD equations are derived by taking moments (integrating over velocity

space) of the Vlasov equation• MHD does not include resonant wave-particle interactions

Wave Growth and Decay• For a hot plasma we have

• Wave growth/decay depends on the gradients in the distribution function at the resonant velocity

• Contour integral• Strong resonant interactions when the denominator is zero

• Doppler shifted cyclotron resonance condition• Note nΩ does not appear in cold plasma theory

• For weak growth expand D in a Taylor series to obtain the growth rate:

Growth of Whistler Mode Waves

• Temperature anisotropy

• Temperature anisotropy occurs due to inward plasma convection in the magnetosphere and conservation of adiabatic invariants

• Growth depends on• Number of particles in resonance• Temperature anisotropy A

Landau Damping

• Hot plasma effect

• Landau resonance, n=0• Wave electric field component along B• Particle velocity close to phase velocity

of the wave• If gradient of f(v) negative

• Wave damping• If gradient positive

• Wave growth• Can generate waves via a beam

• Electron plasma• ion acoustic

Doppler Shifted Cyclotron Resonance• For resonance with electrons, wave

frequency is Doppler shifted by motion along B.

• For propagation along B, whistler waves and electrons must propagate in opposite directions

• Electric field rotates in sense as electrons

• E field remains in phase with particle

• Efficient exchange of energy

Resonant Ellipse

• In the relativistic case, the resonance condition is an ellipse

• The minimum resonant energy (Eres) is where the ellipse crosses the vz axis

• To solve - require the phase velocity – obtained from the dispersion relation

• Dependence on • Plasma frequency fpe• Gyro-frequency fce• Propagation angle• Wave frequency

• For f < fce, Eres smaller for R mode

• For f < fci, Eres smaller for L mode

Resonant DiffusionSingle Wave Characteristics

• Gendrin [1981] showed that small amplitude waves diffuse particles along constant energy surfaces

• Force on an electron

• For transverse plane waves

• Transform to the wave frame

• The force is orthogonal to the electron displacement – no net transfer of energy

• In the wave frame energy is conserved

Resonant DiffusionSingle Wave Characteristics

• In the wave frame

• Electrons are scattered along circles in velocity space• Transform back to the lab frame

• Single wave characteristics are circles centred on the phase velocity along which the particles are scattered

• If the phase velocity is small – electrons scattered mainly in pitch angle• If the phase velocity is large – electrons scattered in energy and pitch angle• Pitch angle - is the angle between the particle velocity and Bo

Single Wave Characteristics – High Phase Velocity

• Particle distribution (blue) anisotropic Tp > Tz(red = constant energy)

• Particle diffusion along single wave characteristics (black)– To lower phase space density

• At Vres, direction must be anti-clockwise• Scattered in pitch angle and energy (energy loss)• Contribute to wave growth

Broad Band Waves• Single wave characteristics provide insight

• Real world– Broad band waves– Overlapping resonances

• Quasi-linear diffusion approach– Waves uncorrelated– Small scattering with each wave– Large enough bandwidth– Diffusion is proportional to wave power

• Quasi-linear diffusion can give us the effects of the waves on the particles and a way to simulate on a global scale for timescales of days

Richard B. Horne British Antarctic Survey

Part 2

Wave-Particle Interactions in the Magnetosphere

Lecture, Advanced Summer School in Solar System Physics, U of Sheffield, 6th September 2016.

Non Linear Wave-particle Interactions

• Linear Theory• Perturbation approach, small amplitude waves• Obtain wave growth and decay, frequencies, etc.

• Quasi-linear theory• Take account of wave interaction back on the particles

• Particle diffusion caused by the waves• Stochastic approach

• Non-linear (turbulence)• Includes coupling between waves• Saturation effects• Formation of structures

• Solitons, cavitons, electron holes

Quasi-linear Theory - Weak Turbulence

• Separate the distribution function into 2 components• Slowly evolving spatially averaged distribution, f0• Rapidly varying part, f1

• Expand Vlasov equation by perturbation approach and spatially average

• Wave power depends on f, and evolves in time• Diffusion rate is time dependent

• Evolution of f0 is due to second order terms• Calculate f1 from linear theory, in terms of E1• We have the form of a diffusion equation• Diffusion depends on wave power

Quasi-linear Theory

• Validity• Assumes small perturbation to f, so small growth rates• Assumes f1 is calculated from linear theory

• Breaks down for particle trapping• Valid for times less than bounce time

• Assumes wave-wave coupling is small• OK for broad band spectrum of waves

• Conserves energy• Conserves entropy

• Diffusion rates• In general - Pitch angle and Energy• Obtain timescale for diffusion• Requires gradients in the distribution function• Diffusion tries to remove the gradients

The Earth’s Radiation Belts

• We will apply Quasi-linear theory of wave-particle interactions to the radiation belts

• Key questions:

• How are electrons accelerated to ~ MeVenergies?

• What controls the variability of the radiation belts?

• Wave-particle interaction play a major role

Earth’s Radiation Belts

• Electrons and ions trapped inside the magnetic field

• Only one proton belt

• Two electron belts• Energies > 1 MeV• Peaks near 1.6 and 4.5 Re

• Outer electron belt highly variable

• Hazardous for spacecraft and humans• Extend to Geostationary orbit

• GPS + Galileo satellites fly through the heart of the radiation belts

Electron Radiation Belts –The Haloween 2003 Magnetic Storms

Baker et al. Nature [2004]

• Outer belt depleted

• Reformed in “slot region”

• Plasmapause L < 2.5

• 23rd Oct to 6th Nov 2003• 47 satellites reported malfunctions – 1 total loss• 10 satellites – loss of service for more than 1 day

Particle Motion in the Earth’s Magnetic Field

• Waves at frequencies comparable to the characteristic frequencies can cause resonance – break the invariants and scatter the particles

• Efficient exchange of energy - acceleration• Change direction – can cause loss into the atmosphere

Magnetopause

Plasmapause

Sun

c). Magnetosonic waves

a). EMIC waves

13:00 14:00 15:00UT

0.0

1.01.52.0

0.5

Freq

uenc

y (H

z)

d). Chorus

13:01:12 13:01:17 13:01:22UT

0.0

0.5

2.0

1.0

1.5

Freq

uenc

y (k

Hz)

10-1310-1210-1110-10

10-8

10-9

V2 m

-2H

z-1

SC1 Rumba

b). Plasmaspheric hiss

13:49:24 13:49:29 13:49:340.0

2.0

1.0

0.5

1.5

Freq

uenc

y (k

Hz)

UT

10-15

10-14

10-13

10-12SC1 Rumba

V2 m

-2H

z-1

16:00 17:0015:00UT

10-8

10-610-4

10-2

mV

2 m-2

Hz-1

CLUSTER 4 – STAFF SA

10

100

1000

Freq

uenc

y (H

z)

10-2

100

102

nT2 H

z-1Electron drift path

CRRES - FMI

CRRES Survey of fpe/fce

Concept

• Injection of ~1 - 100 keV electrons excites whistler mode chorus waves

• Whistler mode chorus accelerates fraction of population to ~ MeV energies

• Solve Fokker-Planck equation to get timescale

Summers et al. [1998]

• Fokker-Planck Equation

• Drift & bounce averaged diffusion coefficients DLL , Dαα , DEE

are activity, location and energy dependent

• Details in: Glauert et al. [2014]

BAS Radiation Belt Model

),()(

)(1)(

)(1

22

Ef

EfDEA

EEAfDg

gLf

LD

LL

tf

LEE

ELJ

LL

Energy diffusion LossesRadial transport Pitch angle diffusion

Pitch Angle and Energy Diffusion Ratescalculated from wave properties

Radiation Belt from Chorus Alone

• Initial soft electron spectrum (~ 10 keV) along the low energy boundary

• Chorus wave diffusion only

• Kp = 2

• Time delay for higher energies

• Glauert et al., JGR [2014]

Importance of Wave-Particle Interactions

Satellite data - Electrons

No waves - Just radial transport

Radial transport and hiss waves

Radial transport, hiss and chorus waves

New Wave Acceleration Concept

Horne, Nature Physics [2007]