Aurora, Alfvén Waves and Substorms: A Tutorial Bob 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 Talk Overview 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 Magnetosphere

Field-Aligned Currents (FAC) and the 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 Light 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 widths (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 satellite 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 Auroral Zone Perpendicular and parallel field observations indicate “U-shaped” or “S-shaped potential structures (Mozer et al., 1980)

Adiabatic 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 (upward) parallel electric fields (Chiu and Schulz, 1978) ↑ v v|| → Key: M: magnetospheric; I: ionospheric; T: trapped; S: scattered Note: Ion and electron plots reversed for downward electric fields

Evidence for E|| in Auroral Particles Proton and Electron Velocity Distributions from S3-3 satellite (Mozer et al., 1980) “Monoenergetic Peak” in Electrons (Evans, 1974)

Knight (1973) Relation for Adiabatic Response 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.

Knight Relation (from Fridman and Lemaire, 1980) See Boström (JGR, April 2003) for a good description of this type of model

Self-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 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”!

So Why Does E|| form? (Song and Lysak, 2001, 2006) Magnetospheric processes twist magnetic field, Ampere’s Law gives: Note that if particles cannot carry required j||, parallel electric field must increase, leading to enhancement of current: Combining these equations, and assuming that oscillates at a frequency ω, we find So even though the displacement current is numerically small for low frequency, its presence is important for the development of parallel electric fields Use of displacement current formulation has numerical advantages: explicit treatment of E|| (Lysak and Song, 2001)

Steady-state E||: Plasma Double Layers Need to self-consistently maintain field with particle distributions: 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 distribution Ions are accelerated to left, electrons to the right.

Role of the Ionosphere: Electrostatic Scale Size (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

Some important details of ionospheric interaction 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.

MHD 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 (Note dispersion relations given are in low β limit)

Reflection of Alfvén Waves by the Ionosphere 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)

Alfvén Wave Simulation By Ex 4 RE r Ionosphere Fields from 100 km wide pulse, ramped up with 1 s rise time. Simulation shown in “real time”

Field-Aligned Currents vs. Alfvén Waves 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 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 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 currents Oversimplified picture! Wave reflections, parallel electric fields, kinetic effects all affect this ratio.

Effects of E|| on Alfven Wave Reflection: Alfvenic 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

Resonances of Alfvén Waves Alfvén 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 Observational Evidence for 0.1-1.0 Hz waves in the ionospheric Alfvén resonator 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 Alfvén Wave Pulse along auroral field line By Ex Peak of Alfven speed r Ionosphere

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

Ionospheric 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 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 Mixing Simulations of linear wave propagation including electron inertia effect were made in a overall perpendicular density gradient. Alfvén speed Density

Simulation results By Ex 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 Alfvén 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”

Kinetic Alfvén Wave: Local Theory

Results 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/VA2 (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 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 Observations (Arnoldy et al., 1999)

Electron acceleration in Alfvén 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 Polar Satellite at 4-6 RE (Wygant et al., 2000) 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. Left Panel: From Top to Bottom: Electric Field, Magnetic Field, Poynting Flux, Particle Energy Flux, Density

Alfvén Waves on Polar Map to Aurora and Accelerate Electrons 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) Left: Ultra-violet image of aurora taken from Polar satellite. Cross indicates footpoint of field line of Polar (Wygant et al., 2000)

Alfvénic 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? 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 space (Note: movie duration is 5 hours)

Phenomenology of Auroral Substorm Akasofu picture of the aurora during substorms: Quiet auroral arc before substorm Equatorward edge of aurora intensifies “Westward traveling surge” forms Poleward expansion of surge Aurora begins to fade; patchy “pulsating aurora” forms on dawn Auroral oval retreats to pre-substorm locations

Models for Substorm Initiation “Near Earth Neutral Line” model “Current Disruption” model THEMIS mission includes 5 spacecraft plus ground-based observatories to determine which model gives proper timing. Results inconclusive!!

Propagation 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 Note: fast mode speed in plasma sheet ~ 500 km/s, in lobe, 1500 km/s (Angelopoulos et al., 2008)

Comments 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 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 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)