Aurora, Alfvén Waves and Substorms: Aurora, Alfvén Waves and Substorms: A TutorialA Tutorial

Bob Lysak, University of MinnesotaBob Lysak, University of Minnesota

Auroral particle acceleration is the result of the transmission of electromagnetic energy along auroral field lines and its dissipation in the auroral acceleration region.

Electrostatic models have been widely used to understand parallel electric fields, but do not address dynamics.

Time-dependent transmission of electromagnetic energy is accomplished by shear Alfvén waves.

Strong Alfvénic Poynting flux observed at plasma sheet boundary: leads to field-aligned acceleration of electrons; implication for substorms.

Outline of the TalkOutline of the TalkOverview of the Auroral Zone

Single Particle Motions: the Knight relation

Parallel Electric Fields

The Ionosphere and Current Closure

Alfvén Waves

Particle Acceleration in Alfvén Waves

Sources of Alfvén Waves

Implications for Substorms

The Earth’s MagnetosphereThe Earth’s Magnetosphere

Field-Aligned Currents (FAC) and Field-Aligned Currents (FAC) and the Aurorathe Aurora

Currents can flow easily along magnetic field lines, but not perpendicular to the magnetic field

Pattern of FAC is similar to auroral oval

Field-aligned current pattern (Iijima and Potemra, 1976) UV Image from DE-1 satellite (Courtesy, L. Frank)

Production of Auroral LightProduction of Auroral Light

• Auroral Spectrum consists of various emission lines:

557.7 nm (“Green line”), 1S → 1D forbidden transition of atomic Oxygen ( = 0.8 s)

630.0 nm (“Red line”), 1D→ 3P forbidden transition of Oxygen ( = 110 s)

391.4 nm, 427.8 nm transitions in molecular Nitrogen ion N2

+ Hα (656.3 nm) and Hβ (486.1 nm) lines

due to proton precipitation

These lines are excited by electron and proton precipitation in 0.5-20 keV range. How do these particles get accelerated?

Bi-modal distribution of auroral arc Bi-modal distribution of auroral arc widthswidths

(Knudsen et al., Geophys. Res. Lett., 28, 705, 2001)

Auroral arcs show a bi-modal distribution, with a peak at very small scales of < 1 km and a second peak at about 10 km. Larger-scale structures are consistent with linear calculations; however, narrow-scale arcs are still not understood.

Recent Observations From FAST Recent Observations From FAST satellitesatellite

30 seconds of data from the Fast Auroral SnapshoT (FAST) satellite are shown.

Top 4 panels give energy and pitch angle of electrons and ions (red is most intense; 180 degrees is upward).

Next is perpendicular electric field. Strong perpendicular fields always are seen in auroral zone. Perpendicular fields separate different plasma regions.

(McFadden et al., 1998)

Electric Field Structures in the Electric Field Structures in the Auroral ZoneAuroral Zone

Perpendicular and parallel field observations indicate “U-shaped” or “S-shaped potential structures (Mozer et al., 1980)

Adiabatic Motion of Charged ParticlesAdiabatic Motion of Charged Particles

Motion of charged particles in a dipole magnetic field is governed by conservation of energy E = (1/2)mv2 + qΦ and magnetic moment μ = mv

2/2B where is pitch angle of particle.

Conservation of E and μ leads to magnetic mirror, creating “loss cone” in velocity space: particles with sin2 < B/BI, where BI is ionospheric field, are lost. Since on auroral field, LC = 1.8. Thus, very few particles lost.

For electrons, if > 0 (upward parallel electric field), loss cone becomes hyperboloid; therefore more particles lost. For ions, upward E|| leads to fewer particles in loss cone.

Velocity space in the presence of Velocity space in the presence of (upward) parallel electric fields (upward) parallel electric fields

(Chiu and Schulz, 1978)(Chiu and Schulz, 1978)

Key: M: magnetospheric; I: ionospheric; T: trapped; S: scattered

Note: Ion and electron plots reversed for downward electric fields

v|| →

↑

v

Evidence for EEvidence for E|| || in Auroral Particlesin Auroral Particles

“Monoenergetic Peak” in Electrons (Evans, 1974)

Proton and Electron Velocity Distributions from S3-3 satellite (Mozer et al., 1980)

Knight (1973) Relation for Adiabatic Knight (1973) Relation for Adiabatic Response to Parallel Potential DropResponse to Parallel Potential Drop

Consider bi-Maxwellian electron population at source region (density n0, temperatures T|| and T, magnetic field B0) in dipole field with upward parallel potential drop Φ. Total current corresponds to those particles that avoid mirroring before reaching the ionosphere. This gives:

Relation is linear for moderate Φ

For large potential drops, a saturation current is reached: j||,sat = nevthBI /B0

Important point: Knight relation only gives the field-aligned current resulting from an assumed potential drop. It does NOT explain the existence of parallel electric fields.

j n eB

B

e

xthI

xe T

||

/ ||

LNM

OQP

00

11

v

xT T

B BI

|| // 0 1

vth eT m || / 2

||, vlin th

ej ne K

T

Knight Relation Knight Relation (from Fridman and Lemaire, 1980)(from Fridman and Lemaire, 1980)

See Boström (JGR, April 2003) for a good description of this type of model

Self-consistent E parallelsSelf-consistent E parallels

To find E||, must combine adiabatic trajectories with Poisson’s equation to find self-consistent model.

For example, Ergun et al. (2000) used 7 populations to model FAST data.

Two “transition regions” found with large parallel electric fields.

Models for Parallel Electric FieldsModels for Parallel Electric Fields

High electron mobility would suggest electrons can short out parallel electric fields. Creating a significant E|| requires some inhibition of the electron motion, so consider electron momentum equation (“generalized Ohm’s Law”):

“Anomalous” resistivity: momentum transfer to ions due to wave-particle interactions.

Magnetic mirror effect: requires anisotropic pitch angle distributions

Electric “double layers”: self-consistent E|| on Debye length scales

Electron inertia: finite electron mass in time-dependent fields (linear) or spatially varying case (nonlinear): BUT this is “ma” not “F”!

||2|| || || ||* e e

e e e e e e e

p pnm v nm v neE nm v p B

t B

So Why Does ESo Why Does E|||| form? form?(Song and Lysak, 2001, 2006)(Song and Lysak, 2001, 2006)

Magnetospheric processes twist magnetic field, Ampere’s Law gives:

00

1Ej

t

B

Note that if particles cannot carry required j||, parallel electric field must increase, leading to enhancement of current:

2j neE

t m

Combining these equations, and assuming that oscillates at a frequency ω, we find

B

2

2 2 21 / p p

i cE

B

So even though the displacement current is numerically small for low frequency, its presence is important for the development of parallel electric fieldsUse of displacement current formulation has numerical advantages: explicit treatment of E|| (Lysak and Song, 2001)

Steady-state ESteady-state E||||: Plasma Double Layers: Plasma Double Layers

Need to self-consistently maintain field with particle distributions:

0/ E

A simple such structure is the plasma “double layer” Note when particles are reflected, their density increases. Thus, ion density is highest just to right of axis, and electron density to the left, making a “double layer” of charge.This is consistent with potential distributionIons are accelerated to left, electrons to the right.

Role of the Ionosphere: Electrostatic Role of the Ionosphere: Electrostatic Scale SizeScale Size(Lyons, 1980)(Lyons, 1980)

Ionosphere closes field-aligned currents:

For electrostatic conditions, uniform ionosphere, only Pedersen conductivity matters:

Assume the linear Knight relation is valid: j|| = K(ΦI – Φ0)

Combining these leads to equation for potential:

Here is electrostatic auroral scale length.

For ΣP = 10 mho and K = 10-9 mho/m2, L = 100 km

Parallel potential drops only exist on scales shorter than L

j E

�

2P Ij

2 201 IL

/PL K

Some important details of ionospheric Some important details of ionospheric interactioninteraction

Although Hall current doesn’t close current (in uniform ionosphere), it produces magnetic signature seen on ground

Fields in atmosphere attenuated as so structures small compared with ionospheric height (~ 100 km) are shielded from ground: so scales that produce potential drops are not seen at ground!

On very narrow scales (~ 1 km), collisional parallel conductivity becomes important (Forget et al., 1991)

At higher frequencies (~ 1 Hz), two effects:Hall currents lead to coupling to fast mode, signal can propagate

across field lines in “Pc1 waveguide”Effective height of ionosphere can be decreased by collisional skin

depth effect.

k ze

MHD Wave ModesMHD Wave Modes

Linearized MHD equations give 3 wave modes:Slow mode (ion acoustic wave):

Plasma and magnetic pressure balance along magnetic field

Electron pressure coupled to ion inertia by electric field

Intermediate mode (Alfvén wave):

Magnetic tension balanced by ion inertia

Carries field-aligned current

Fast mode (magnetosonic wave):

Magnetic and plasma pressure balanced by ion inertia

Transmits total pressure variations across magnetic field

/s sk c c p

0/A Ak V V B

2 2 2 2A sk V k c

(Note dispersion relations given are in low β limit)

The “Auroral Transmission Line”

The propagation of Alfvén waves along auroral field lines may be considered to be an electromagnetic transmission line. Energy is propagated in the “TEM” mode, the shear Alfvén wave at the Alfvén speed, 0/ AV B

Transmission line is filled with a dielectric medium, the plasma, with an inhomogeneous dielectric constant 2 21 / ( ) Ac V z

Can define a characteristic admittance for the transmission line

01/ A AV (= 0.8 mho for 1000 km/s)

Transmission line is “terminated” by the conducting ionosphere. In general, Alfvén waves will reflect from this ionosphere, or from strong gradients in the Alfvén speed.

Reflection of AlfvReflection of Alfvén Waves by the én Waves by the IonosphereIonosphere

Ionosphere acts as terminator for Alfvén transmission line.

But, impedances don’t match: wave is reflected

Usually P >> A, so electric

field of reflected wave is reversed (“short-circuit”)

Reflection coefficient:

(Mallinckrodt and Carlson, 1978)

up A P

down A P

ER

E

AlfvAlfvén Wave Simulationén Wave SimulationEx

By

Ionosphere

r

4 RE

Fields from 100 km wide pulse, ramped up with 1 s rise time. Simulation shown in “real time”

Field-Aligned Currents vs. AlfvField-Aligned Currents vs. Alfvén én WavesWaves

Field-aligned current is often quoted as energy source for aurora.

But, the kinetic energy of electrons is negligible: Poynting flux associated with FAC is responsible.

FAC closed by conductivity in ionosphere; electric and magnetic fields related by

0

1 800km/s

(mho)x

y P P

E

B

ΣP is usually > 1 mho, so ratio is less than 800 km/s

Alfvén waves have a similar electric and magnetic field signature, but for these waves

0

0

xA

y

E BV

B

VA is usually much greater than 1000 km/s, can be up to speed of light

Thus, large E/B ratios indicate Alfvén waves, smaller ratios static currentsOversimplified picture! Wave reflections, parallel electric fields, kinetic effects all affect this ratio.

Effects of EEffects of E|| || on Alfven Wave Reflection: on Alfven Wave Reflection:

Alfvenic Scale SizeAlfvenic Scale Size

If assume linear Knight relation j = KΦ, Alfven wave reflection is modified (Vogt and Haerendel, 1998)

Reflection coefficient same if replace Pedersen conductivity with effective conductivity

where

This leads to a new scale where the Alfvén wave is absorbed (providing energy to auroral particle acceleration) given by

2 21P

eff k L

/PL K

/ ~ 10 kmA AL K

( ) /( )A eff A effR

Resonances of AlfvResonances of Alfvén Wavesén WavesAlfvén can bounce from one ionosphere to the other: Field Line Resonance (periods 100-1000 s)

However, Alfvén speed has sharp gradient above ionosphere: wave can bounce between ionosphere and peak in speed: Ionospheric Alfvén Resonator (Periods 1-10 s)

Fluctuations in the aurora are seen in both period ranges. Feedback can structure ionosphere at these frequencies.

Profiles of Alfvén speed for high density case (solid line) and low-density case (dashed line). Ionosphere is at r/RE = 1. Sharp rise in speed can trap waves (like quantum mechanical well). Note speed can approach c in low-density case.

Observational Evidence for 0.1-1.0 Hz Observational Evidence for 0.1-1.0 Hz waves in the ionospheric Alfvwaves in the ionospheric Alfvén én

resonatorresonator

Above: Spectrogram from ground magnetic observations from Finland, showing waves at about 0.5 Hz (Koskinen et al., 1993)

Right: Electric field data and spectrum from Viking satellite, showing harmonics of resonator (Block and Fälthammar, 1990)

Simulations of AlfvSimulations of Alfvén Wave Pulse along én Wave Pulse along auroral field lineauroral field line

ExBy

r

Pe

ak o

f Alfv

en

spe

ed

Ionosphere

Spectral Structure of IARSpectral Structure of IAR

Spectrogram (left) and line plot (right) of the D-component of the magnetic field from Sodankylä, Finland, showing multiple harmonics of the ionospheric Alfvén resonator (Hebden et al., 2005) (Image courtesy of Darren Wright)

Models of IARModels of IAR

B/E ratio and phase shift for IAR model with ΣP/ΣAI = 1 (top) and 10 (bottom) (Lysak, 1991).

Model calculation of relative transmission to the ground for a model with ΣP = ΣH = 10 mho and a magnetic zenith angle of 14° to model Sodankylä data. (VAI=1000 km/s, h=400 km)

Ionospheric Alfvén Resonator from FASTIonospheric Alfvén Resonator from FAST

FAST evidence for the IAR (Chaston et al., 2002). Left panel shows oscillations in E and B at about 1 Hz with oscillating Poynting flux (after initial pulse). Right panel shows phase shifts consistent with standing waves in IAR (Lysak, 1991). Similar results have been obtained from Freja (Grzesiak, 2000) and Akebono (Hirano et al., 2005).

Phase mixing in Ionospheric Alfvén ResonatorPhase mixing in Ionospheric Alfvén Resonator

Gradients in the Alfvén speed lead to phase

mixing, producing smaller perpendicular scales

(basic mechanism behind field line resonance.)

Time scale for phase mixing given to a scale L

can be estimated by τ ~ (LA / L)T, where LA is perpendicular

scale length of Alfvén speed and T is wave period. For 1

second wave in IAR, 100 km scale reduced to <10 km in less

than a minute.

Suggests small-scale structure can be produced in presence of

large-scale density gradients.

VA

Simulations of Phase MixingSimulations of Phase Mixing

Simulations of linear wave propagation including electron inertia effect were made in a overall perpendicular density gradient.

Density Alfvén speed

Simulation resultsSimulation resultsEx

By

Simulation initiated with uniform pulse across system oscillating at 1 Hz. Interference between up and downgoing waves leads to structuring of fields. Series of harmonics seen due to change of IAR eigenfrequencies. Waves phase mix to ~ 1 km scale waves.

Kinetic AlfvKinetic Alfvén Wavesén Waves

Alfvén waves develop a parallel electric field on short perpendicular scales

Two-fluid theory gives modification to dispersion relation in two limits:

Cold plasma (vth << VA):

Warm plasma (vth >> VA):

The first is sometimes called “inertial Alfvén wave” and second “kinetic Alfvén wave,” but they are both limits of the full kinetic dispersion relation

Common misconception “ion gyroradius effect causes E||” but really it is electron inertia or pressure, through “acoustic gyroradius”

22 22 2 2

2 2 2 2

1

1 1ei

Ae e

E k kkk V

Ek k

2

2 2 2 2 2 22 2

1 ( )1

sA s i

i

E k kk V k

E k

/ /s s i e ic T m eB

Kinetic AlfvKinetic Alfvén Wave: Local Theoryén Wave: Local Theory

Kinetic Alfvén wave dispersion relation can be written as: 2|| ||

2|| ||

det 0

n n n

n n n

where

112

20c

VA

i

i

af

||||

1 102 2

e

DekZ

af afb g Dispersion relation is then solved to read:

22 2

2 2 2 2|| 0 0 ||

1

/ 1 / 1

s

A A i i e De

k

k V V c Z k

In cold electron limit ( / ||k ae ), dispersion relation becomes:

2 2

2 2 22 2

1

1(for )

A Aik

k Vk

cV

For warm electrons ( / ek a ), we find

2 2 2 2 2 2 21 1 / A i s ek V k k i k a

assuming 2 2, 1, and 1A e DeV c k .

Results from Local TheoryResults from Local Theory

Solutions for the local dispersion relation for equal ion and electron temperatures as a

function of perpendicular wavelength, kxc/pe (horizontal axis) and the ratio of

electron thermal speed to Alfvén speed, ve2/VA

2 (vertical axis). Left panel gives real

part of the phase velocity normalized to Alfvén speed; right panel gives damping rate

normalized to wave frequency (Lysak and Lotko, 1996).

Field-aligned acceleration on FASTField-aligned acceleration on FAST

Figure shows data from FAST satellite (Chaston et al., 1999). Note strong low energy electron fluxes (red regions at bottom of panel 4) which are field-aligned (0 degree pitch angle in panel 5).

These particle fluxes are associated with strong Alfvén waves (top 3 panels: electric field, magnetic field, and Poynting flux), suggesting wave acceleration.

Sounding Rocket ObservationsSounding Rocket Observations

(Arnoldy et al., 1999)

Electron acceleration in AlfvElectron acceleration in Alfvén Wavesén Waves

Parallel electric fields can develop in narrow-scale Alfvén waves due to finite electron inertia.

Test particle models have been used to determine distributions from this effect.

Results from a test-particle simulation of electron acceleration in Alfvén resonator, showing bursts at ~ 0.5 s (Thompson and Lysak, 1996)

Results from a similar simulation with more particles in pitch angle vs. energy format compared with FAST data (Chaston et al., 1999)

Observations of Poynting flux from Observations of Poynting flux from Polar Satellite at 4-6 RPolar Satellite at 4-6 REE (Wygant et (Wygant et

al., 2000)al., 2000)

Left Panel: From Top to Bottom: Electric Field, Magnetic Field, Poynting Flux, Particle Energy Flux, Density

Right Panel: Particle Data. Top 3 panels are electrons, bottom 3 are ions. Panels give particles going down the field line, perpendicular to the field, and up the field line.

AlfvAlfvén Waves on Polar Map to én Waves on Polar Map to Aurora and Accelerate ElectronsAurora and Accelerate Electrons

Left: Ultra-violet image of aurora taken from Polar satellite. Cross indicates footpoint of field line of Polar (Wygant et al., 2000)

Right: Electron distribution function measured on Polar. Horizontal direction is direction of magnetic field. Scale is ±40,000 km/s is both directions (Wygant et al., 2002)

Alfvénic Aurora as Transitional PhaseAlfvénic Aurora as Transitional Phase

Observations show that Alfvénic aurora occur at polar cap boundary, and during the onset of “auroral breakup” during magnetospheric substorms

Changes of field-aligned current require the passage of shear Alfvén waves along field line.

Thus, Alfvénic nature of onset arc should not be surprising

Similarly, at polar cap boundary, plasma is convecting from open to closed field lines, requiring transitional readjustment.

Alfvénic aurora can also occur within inverted-V’s: may indicate smaller changes in current structure.

Speculation: Alfvénic interaction prepares system to allow for quasi-static aurora, especially by excavating density cavity (e.g.., Chaston et al., 2006), creating low densities that are conducive to static parallel electric fields (Song and Lysak, 2006), and precipitating electrons into ionosphere to enhance conductivity and produce secondary and backscattered electrons.

How are these waves produced?How are these waves produced?

Linear mode conversion: Mode conversion can take place from a surface Alfvén wave (Hasegawa, 1976), from compressional plasma sheet waveguide modes (Allan and Wright, 1998), or from compressional waves in plasma sheet (Lee et al., 2001).

Reconnection at distant neutral line: Presence of finite By component in tail lobe gives rise to field-aligned currents on boundary layer (Song and Lysak, 1989). Bursty reconnection at this point will launch Alfvén waves along boundary layer.

Bursty Bulk Flows: Localized flow regions can generate Alfvén waves due to the twisting and compression of magnetic field lines (Song and Lysak, 2000), perhaps associated with localized reconnection. BBF association with Alfvénic Poynting flux observed by Geotail (Angelopoulos et al., 2001).

Auroral Substorm as seen from spaceAuroral Substorm as seen from space

(Note: movie duration is 5 hours)

Phenomenology of Phenomenology of Auroral SubstormAuroral Substorm

Akasofu picture of the aurora during substorms:

(a) Quiet auroral arc before substorm

(b) Equatorward edge of aurora intensifies

(c) “Westward traveling surge” forms

(d) Poleward expansion of surge

(e) Aurora begins to fade; patchy “pulsating aurora” forms on dawn

(f) Auroral oval retreats to pre-substorm locations

Models for Substorm InitiationModels for Substorm Initiation

THEMIS mission includes 5 spacecraft plus ground-based observatories to determine which model gives proper timing.

Results inconclusive!!

“Near Earth Neutral Line” model “Current Disruption” model

Propagation Speeds on 26 Feb 2008Propagation Speeds on 26 Feb 2008

Consider straight line distances to find minimum velocities required:

Reconnection (20.3 RE) to Auroral Intensification: 96 s, speed > 1284 km/s.

Reconnection to Expansion onset: speed > 893 km/s.

Reconnection to electrojet increase: > 520 km/s

Rec’n at P2 to flow at P3: > 375 km/s

Rec’n at P2 to dipolarization at P3: > 278 km/s

(Angelopoulos et al., 2008)

Note: fast mode speed in plasma sheet ~ 500 km/s, in lobe, 1500 km/s

Comments on 26 Feb eventComments on 26 Feb event

Reconnection site cannot communicate with auroral brightening by wave propagation through plasma sheet.

Propagation through lobe/boundary layer possible, but then, how could aurora expand more poleward?

Reconnection, flow at P3, and electrojet formation could be connected by flow or waves through plasma sheet: classic NENL signature (but not connected to aurora!).

Auroral activity before electrojet formation: consistent with Alfvénic nature of onset arc (e.g., Mende et al., 2003), followed by development of large-scale current system.

Space-Time Diagram for Substorm Space-Time Diagram for Substorm Events (N. Lin et al., 2009)Events (N. Lin et al., 2009)

Events from a number of THEMIS substorms were placed on a space-time diagram to get statistical picture of substorm timing relative to auroral expansion (t = 0).

Solid curve is model MHD wave travel time; dotted lines give variations of parameters within limits of data.

Three Regions of Auroral AccelerationThree Regions of Auroral Acceleration

Illustration of three regions of auroral acceleration: downward current regions, upward current regions, and the region near the polar cap boundary of Alfvénic acceleration (from Auroral Plasma Physics, International Space Science Institute, Kluwer, 2003, adapted from Carlson et al., 1998)