Electrodynamic Coupling of the Ionosphere and
Magnetosphere
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.
R. L. Lysak GEM 2003 Tutorial
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
Focus on:
Auroral zone: But low and mid-latitude coupling important
Electrodynamics: But mass coupling also important
R. L. Lysak GEM 2003 Tutorial
The Earth’s Magnetosphere
R. L. Lysak GEM 2003 Tutorial
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)
R. L. Lysak GEM 2003 Tutorial
Production 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
d
u
e
t
o
p
r
o
t
o
n
p
r
e
c
i
p
i
t
a
t
i
o
n
These lines are excited by electron and proton precipitation in 0.5-20
keV range. How do these particles get accelerated?
R. L. Lysak GEM 2003 Tutorial
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.
R. L. Lysak GEM 2003 Tutorial
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)
R. L. Lysak GEM 2003 Tutorial
Electric Field Structures in the
Auroral Zone
Perpendicular and parallel field observations indicate “U-shaped” or “SR. L. Lysak GEM 2003 Tutorial
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.
R. L. Lysak GEM 2003 Tutorial
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
R. L. Lysak GEM 2003 Tutorial
Evidence for E|| in Auroral Particles
“Monoenergetic Peak” in
Electrons (Evans, 1974)
Proton and Electron Velocity
Distributions from S3-3 satellite (Mozer
et al., 1980)
R. L. Lysak GEM 2003 Tutorial
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:
L
M
N
BI
e xe/ T
j|| n0 evth
1
B0
1 x
Relation is linear for moderate Φ
j||,lin  nevth
||
O
P
Q
T /T
x  || 
BI / B0 
1
vth  T|| / 2me
e
 K
T
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.
R. L. Lysak GEM 2003 Tutorial
Knight Relation
(from Fridman and Lemaire, 1980)
See Boström (JGR, April 2003) for a good description of this type of model
R. L. Lysak GEM 2003 Tutorial
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.
R. L. Lysak GEM 2003 Tutorial
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”):
pe  pe||

2
nme ve    nme ve  neE||  nme  *ve||  || pe|| 
B

t
B


“Anomalous” resistivity: momentum transfer to ions due to waveparticle 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”!
R. L. Lysak GEM 2003 Tutorial
So Why Does E|| form?
(Song and Lysak, 2001)
Magnetospheric processes twist magnetic field, Ampere’s Law gives:
0
E
t

1
  B   j
0
Note that if particles cannot carry required j||, parallel electric field
must increase, leading to enhancement of current:
j
ne2

E
t
m
Combining these equations, and assuming that B oscillates at a
frequency ω, we find
i
c2
E 
  B
2
2
2 
1  / p p
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) R. L. Lysak GEM 2003 Tutorial
Steady-state E||: Plasma Double Layers
Need to self-consistently maintain
field with particle distributions:
 E   / 0
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.
R. L. Lysak GEM 2003 Tutorial
Role of the Ionosphere: Electrostatic
Scale Size
(Lyons, 1980)
Ionosphere closes field-aligned currents:

j       E

For electrostatic conditions, uniform ionosphere, only Pedersen
conductivity matters:
j   2 
P

I
Assume the linear Knight relation is valid: j|| = K(ΦI – Φ0)
Combining these leads to equation for potential:
1  L   
2
2

I
 0
Here L   P / K 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
R. L. Lysak GEM 2003 Tutorial
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 e  k z 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.
R. L. Lysak GEM 2003 Tutorial
MHD Wave Modes
Linearized MHD equations give 3 wave modes:

 Slow mode (ion acoustic wave):   k cs cs  p / 

Plasma and magnetic pressure balance along magnetic field
Electron pressure coupled to ion inertia by electric field

 Intermediate mode (Alfvén wave):   k VA VA  B / 0

Magnetic tension balanced by ion inertia
Carries field-aligned current
2 2
2 2
 Fast mode (magnetosonic wave):   k VA  k cs
Magnetic and plasma pressure balanced by ion inertia
Transmits total pressure variations across magnetic field
(Note dispersion relations given are in low β limit)
R. L. Lysak GEM 2003 Tutorial
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, VA  B / 0

Transmission line is filled with a dielectric medium, the plasma, with
an inhomogeneous dielectric constant   1  c 2 / VA2 ( z )

Can define a characteristic admittance for the transmission line
 A  1/ 0VA (= 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.
R. L. Lysak GEM 2003 Tutorial
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:
R
(Mallinckrodt and Carlson, 1978)
Eup
Edown
 A  P

 A  P
R. L. Lysak GEM 2003 Tutorial
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”
R. L. Lysak GEM 2003 Tutorial
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
Ex
1
800 km/s
than 800 km/s


By 0 P  P (mho)
Alfvén waves have a similar electric and magnetic field signature, but for
these waves
Ex
B0
VA is usually much greater than 1000 km/s,
 VA 
can be up to speed of light
By
 0
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.
R. L. Lysak GEM 2003 Tutorial
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 R  ( A   eff ) /( A   eff ) if replace
Pedersen conductivity with effective conductivity
P
 eff 
1  k2 L2
where L   P / K
This leads to a new scale where the Alfvén wave is absorbed
(providing energy to auroral particle acceleration) given by
LA   A / K ~ 10 km
R. L. Lysak GEM 2003 Tutorial
Resonances of Alfvén Waves
Alfvén can bounce from one
ionosphere to the other: Field
Line Resonance (periods 1001000 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 lowdensity case.
R. L. Lysak GEM 2003 Tutorial
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)
R. L. Lysak GEM 2003 Tutorial
Simulations of Alfvén Wave Pulse along
auroral field line
By
Peak of Alfven
speed
Ex
r
Ionosphere
R. L. Lysak GEM 2003 Tutorial
Ionospheric Feedback
Ionospheric feedback instability (Atkinson, 1970; Miura and Sato, 1980; Lysak,
1991) can produce structuring of auroral arcs through ionospheric modification.
Upward current carried by energetic downward electrons can lead to localized
enhanced conductivity.
Secondary field-aligned currents develop at conductivity gradients. The Alfvén
waves carrying these currents can be reflected at conjugate ionosphere or
ionospheric Alfvén resonator. If returning wave reinforces conductivity change,
instability develops.
Growth rate proportional to wave travel time: few minutes for conjugate
ionosphere (FLR), few seconds for ionospheric resonator.
Instability damped by recombination, so strong damping for large background
conductivity. Recombination time ~ 50 s for 1 mho, 5 s for 10 mho.
j
j
→
(Lysak, 1990)
R. L. Lysak GEM 2003 Tutorial
What sets lower limit on scale size?
Feedback instability favors short wavelength waves. What can limit
how small the waves are?
Some basic scale sizes:
 Electron inertial length le = 5 km/n1/2. For n = 104-106 cm–3, this gives 50-5
m.
 Electron/ion gyroradius: for e–, e = 5 cm  T(eV)1/2; for ions, H = 2 m 
T1/2 and O = 8 m  T1/2. All < 100 m for temperatures < 100 eV in
ionosphere (B = 0.5 G).
Electron parallel resistivity (not anomalous!) becomes important in
ionosphere. Gives diffusion in current on scale Lres  le e /  where
e is electron collision frequency (103-104 s–1 in ionosphere). This gives
150 m-5 km for ionospheric resonator ( ~ 1 s–1) and 1.5-50 km for
FLR’s ( ~ 0.01 s–1).
Shear in EB flow can give instabilities when dv/dx ~ 0.1 Wi (e.g.,
Ganguli et al., 1988). For E = 1 V/m (upper limit!), this gives 40 m for
H+ and 640 m for O+.
These suggest that parallel resistivity is most likely limiting mechanism.
R. L. Lysak GEM 2003 Tutorial
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):
2 2
1

k
 i
2  k 2VA2
1  k2 l e2

 Warm plasma (vth >> VA):   k V 1  k (   )  
2
2
2
A
2

2
s
2
i
E
E
E
E

k k l e2
1  k2 l e2

k k2s
1  k2 i2
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”
s  cs / Wi  Te mi / eB
R. L. Lysak GEM 2003 Tutorial
Kinetic Alfvén Wave: Local Theory
    n||2
 Kinetic Alfvén wave dispersion relation can be written as: det 
 n|| n
af
c 2 1 0 i

 1  2
VA
i
where
a fb afg
n||n 
0
||  n2 
0 e


1

1 
Z 
||
2 2
k|| De
 Dispersion relation is then solved to read:
2
  
k2 2s
1


  2 2
2 2
k
V
V
/
c

1



/



1


Z


k
l De










||
A
A
0
i
i
0
e
||


 In cold electron limit ( / k|| 
ae), dispersion relation becomes:
1  k2i2
 k V
1  k2 l 2
2
For warm electrons (  / k
2
2
A
(for VA
c)
ae ), we find

2  k 2VA2 1  k2i2  k22s 1  i    / k ae  
assuming VA
c,  e
1, and k 2l 2De
1.

R. L. Lysak GEM 2003 Tutorial
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).
R. L. Lysak GEM 2003 Tutorial
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.
R. L. Lysak GEM 2003 Tutorial
Sounding Rocket Observations
(Arnoldy et al., 1999)
R. L. Lysak GEM 2003 Tutorial
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)
R. L. Lysak GEM 2003 Tutorial
Non-local theory of Alfvén waves on auroral field lines
(e.g., Rankin et al., 1999; Tikhonchuk and Rankin, 2000)
 Idea is to integrate Vlasov equation over past history of a particle.
Trajectory is defined by considering constants of motion: magnetic
moment   mv2 / 2 B and total energy
1
W  mv 2  B  z   q  z 
2
 Linearized Vlasov equation can then be integrated to get
perturbation in the distribution function; calculation of first
velocity moment gives field-aligned current.
 Since distribution function is linear in the parallel electric field,
this integral can be given in terms of a non-local conductivity
relation:
j  z    dz   z , z   E  z  
R. L. Lysak GEM 2003 Tutorial
Phase Space Trajectories: Ionospheric
Particles
R. L. Lysak GEM 2003 Tutorial
Phase Space Trajectories: Magnetospheric
Particles
R. L. Lysak GEM 2003 Tutorial
Observations of Poynting flux from
Polar Satellite at 4-6 RE (Wygant et
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.
R. L. Lysak GEM 2003 Tutorial
Alfvén Waves on Polar Map to
Aurora 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) R. L. Lysak GEM 2003 Tutorial
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).
R. L. Lysak GEM 2003 Tutorial
Simulations of Linear Mode Conversion
Left: compression of magnetic field: Blue
area is plasma sheet; red is lobe. Yellow
region is compression pulse on boundary
layer.
Right: Field-aligned currents: Blue is parallel to
magnetic field; red is anti-parallel. Pre-existing
currents are on bottom; currents in upper part are
generated at the boundary layer.
Acknowledgements: T. W. Jones, D. Ryu for code; D. Porter for visualization software
R. L. Lysak GEM 2003 Tutorial
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)
R. L. Lysak GEM 2003 Tutorial
R. L. Lysak GEM 2003 Tutorial