Space Plasmas as a Natural Laboratory
R. J. Strangeway (IGPP/UCLA)
PSTI Research Lectures – RJS–9
June 4, 2003 – 1
Space Plasmas as a Natural Laboratory
Recap
1.
Single particle dynamics –drifts, magnetization currents,
adiabatic invariants, particle drift paths in the Earth’s
magnetosphere.
2.
Magnetohydrodynamics – moments of the Vlasov equation,
truncating the hierarchy, “ideal MHD,” frozen in theorem, does
MHD work? Introduction to reconnection.
3.
The mechanical view of space plasmas, collisional MHD and
magnetosphere-ionosphere coupling.
PSTI Research Lectures – RJS–9
June 4, 2003 – 2
Space Plasmas as a Natural Laboratory
Recap Continued
4.
Waves in cold plasma –Alfven waves, “inertial effects,”
Appleton-Hartree dispersion, the zoo of plasma waves.
5.
Vlasov (kinetic theory) – inverse Landau damping and gyroresonance.
6.
Auroral kilometric radiation – a fundamental plasma instability.
PSTI Research Lectures – RJS–9
June 4, 2003 – 3
Particle Drifts
Electric field drift:
Gradient drift
vE = E ×2 B
B
vG = W⊥ B × ∇B
qB 3
⋅
B× Β ∇ B
Curvature Drift:
vC = 2W ||
Polarization Drift:
vP = m 2 dE
qB dt
qB 4
(Alfvén wave)
Note: particle drifts do not necessarily imply current –
Magnetization current must be included
PSTI Research Lectures – RJS–9
June 4, 2003 – 4
PSTI Research Lectures – RJS–9
June 4, 2003 – 5
What Is Magnetohydrodynamics?
Magnetohydrodynamics (MHD) is a formalism that describes a plasma in
terms of fluid properties, such as density, flow velocity, pressure, etc.
The basic assumption of MHD is that the plasma is “local” in its behavior.
This means that the rates of change of fluid properties are dependent
only on the local values of those properties.
In a classical gas “localization” is provided by collisions.
In a magnetized collisionless plasma, the gyration of particles around the
field gives localization in the perpendicular direction.
Along the field other processes (e.g., wave-particle interactions) have to
be invoked to provide parallel localization.
What happens when the mean free path is much larger than any gradient
scales within the system?
PSTI Research Lectures – RJS–9
June 4, 2003 – 6
Boltzmann Equation
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June 4, 2003 – 7
Moment Equations - Closure?
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June 4, 2003 – 8
Momentum Equations
Electron Momentum Equation leads to generalized Ohm’s Law
∇ Pe
dU e
me
= − e E + Ue × B − n
dt
e
Sum of Electron and Ion Momentum Equations leads to total
momentum equation – No electric field term!
ρ DU = j × B − ∇P
Dt
PSTI Research Lectures – RJS–9
June 4, 2003 – 9
Frozen-in Field
PSTI Research Lectures – RJS–9
June 4, 2003 – 10
Why Does MHD work so well?
From Faraday’s Law, it is ∇×E that matters, not E
Under assumption of scalar pressure, and a simple
adiabatic relationship between pressure and density, the
curl of the momentum equation reduces to
mi
B′ = B + e ω,
∂B′
− ∇ × U × B′ = 0
∂t
where ω is the vorticity.
For “ideal MHD” ω = 0
PSTI Research Lectures – RJS–9
June 4, 2003 – 11
Magnetic Field Reconnection
Mechanism by which energy and momentum are transferred to
the magnetosphere. Requires local violation of MHD.
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June 4, 2003 – 12
Force Balance - MI Coupling
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June 4, 2003 – 13
Maxwell Stress and Poynting Flux
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June 4, 2003 – 14
Currents and Ionospheric Drag
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June 4, 2003 – 15
MI Coupling - FAST Observations
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June 4, 2003 – 16
Auroral Currents
PSTI Research Lectures – RJS–9
June 4, 2003 – 17
Alfvén Wave Governing Equations
The electron momentum equation provides the
“inertial” correction to the MHD modes. In particular
electron inertia can allow the wave to carry a parallel
electric field.
PSTI Research Lectures – RJS–9
June 4, 2003 – 18
“Dungey Triads”
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June 4, 2003 – 19
Shear (Alfvén) Mode
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June 4, 2003 – 20
Fast Mode
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June 4, 2003 – 21
Governing Equations – Recap
Harmonic Perturbation, first order quantities only:
∂/∂t ≡ –iω
∇ ≡ ik
Faraday’s Law:
k × E = ωb
Ampere’s Law:
k × b = –iµ 0 j– ω2 E
c
i.e.
PSTI Research Lectures – RJS–9
2
ω
k – 2 E – k(k⋅E) = iωµ 0 j
c
2
June 4, 2003 – 22
Cartesian Coordinates
Assume:
B 0 = B0z ,
Then:
2
ω
k – 2 E x – k ⊥k || E z = iωµ 0 jx
c
k = (k ⊥, 0, k || )
2
||
2
ω
k – 2 E y = iωµ 0 j y
c
2
2
ω
k – 2 E z – k | |k ⊥E x = iωµ 0 jz
c
2
⊥
Alternative formalism replaces j with the equivalent
dielectric tensor
PSTI Research Lectures – RJS–9
June 4, 2003 – 23
Alfvén Waves (Again!)
Frozen-in ions:
E + U×B = 0
Momentum:
– iωnmU = j×B – ikδ P
δ P = C s2δρ
Energy:
Mass Conservation:
Therefore
i.e.
[E x = –U yB0, E y = U xB0]
[C s2 = γ P/nm]
ωδρ = nmk⋅U
C s2
j×B = –iωnm[U – 2 k(k⋅U)]
ω
k 2⊥C s2
2
j y = –iωnm(1 –
)E
/B
y
0
ω2
j x = –iωnm E x /B02
[V a2 = B02 /nmµ 0]
ω 2pe
Electron momentum: j z = i ω ε 0 E z
PSTI Research Lectures – RJS–9
June 4, 2003 – 24
Fast mode dispersion
2
ω
k – 2 E y = iωµ 0 j y
c
2
k 2⊥C s2
2
µ 0 j y = –iω(1 –
)E
/V
y
a
ω2
Therefore:
k 2V a2 + k 2⊥C 2s = ω 2(c 2 + V a2)/c 2
Aside: Slow mode comes from
parallel component of momentum
PSTI Research Lectures – RJS–9
C s2 2
U z – 2 k || U z = 0
ω
June 4, 2003 – 25
Shear-mode dispersion
2
ω
k – 2 E x – k ⊥k || E z = iωµ 0 jx
c
2
||
µ 0 j x = –iω E x /Va2
Therefore
2
2
ω
ω
k – 2 – 2 E x = k ⊥k | |E z
c
Va
2
||
2
ω
k – 2 E z – k | |k ⊥E x = iωµ 0 jz
c
2
⊥
ω 2pe
µ 0 jz = i 2 E z
ωc
Therefore
k
2
⊥
PSTI Research Lectures – RJS–9
(ω 2 pe– ω 2)
+
E z = k ||k ⊥ E x
c2
June 4, 2003 – 26
Appleton-Hartree
Maxwell’s equations again
2
ω
k – 2 E x – k ⊥k || E z = iωµ 0 jx
c
2
||
2
ω
k – 2 E y = iωµ 0 j y
c
2
2
ω
k – 2 E z – k | |k ⊥E x = iωµ 0 jz
c
2
⊥
PSTI Research Lectures – RJS–9
June 4, 2003 – 27
Polarized Coordinates
Force Law:
–imωv = q(E + v×B 0)
–imωv x = q(E x + v yB0)
–imωv y = q(E y – v xB0)
q
–iω(v x ± iv y) = m (E x ± iE y) – v yΩ ± iv xΩ
[Ω = – q B0 /m]
(± i)(± i)v y = – v y
Therefore
PSTI Research Lectures – RJS–9
q
–iωv± = m E± ± iΩv±
June 4, 2003 – 28
Electrons only, parallel
2
ω
k – 2 E± = iωµ0 j±
c
2
iω 2 peε 0 E±
j± =
(ω – ±Ωe)
Therefore
2
ω
pe
µ2 = 1 –
ω (ω – ±Ωe)
ω 2 E = –iωµ j
0 z
c2 z
ω 2pe
j z = i ω ε0E z
ω 2 = ω 2pe
For parallel propagation modes split into R, L, and P modes
PSTI Research Lectures – RJS–9
June 4, 2003 – 29
Electrons only, perpendicular
O-mode:
Therefore
X-mode:
ω 2pe
j z = i ω ε0E z
2
ω
k – 2 E z = iωµ 0 jz
c
2
ω 2pe
µ =1– 2
ω
2
2
ω
k – 2 E y = iωµ 0 j y
c
2
ω 2 E = –iωµ j
0 x
c2 x
iω 2 peε 0 E±
j± =
(ω – ±Ωe)
PSTI Research Lectures – RJS–9
June 4, 2003 – 30
A-H Dispersion Relation
ω 2pe
ω 2 pe
2 2 1– 2
ω
ω
µ2 = 1 –
ω 2pe Ωe2 2
2 1 – 2 – 2 sin θ ± Γ
ω
ω
Γ=
PSTI Research Lectures – RJS–9
2
ω 2pe
Ω
Ω
4
sin θ +4
1 – 2 cos 2θ
ω
ω
ω
4
e
4
2
e
2
June 4, 2003 – 31
Appleton-Hartree
PSTI Research Lectures – RJS–9
June 4, 2003 – 32
Quasi-longitudinal Approximation
ω 2 pe
Ω
Ω
4
sin θ << 4
1 – 2 cos 2θ
ω
ω
ω
4
e
4
µ2 = 1 –
2
e
2
2
ω 2 pe
ω 2pe
2 2 1– 2
ω
ω
ω 2pe Ωe2 2
ω 2pe
Ωe
2 1 – 2 – 2 sin θ ± 2 ω 1 – 2 cosθ
ω
ω
ω
ω 2 pe
Ωe2 2
sin θ << 2 1 – 2
ω2
ω
PSTI Research Lectures – RJS–9
ω 2pe
2
ω
2
µ =1–
Ω
1 ± ωe cosθ
June 4, 2003 – 33
Quasi-transverse Approximation
ω 2 pe
Ω
Ω
4
sin θ >> 4
1 – 2 cos 2θ
ω
ω
ω
4
e
4
2
2
e
2
µ2 = 1 –
PSTI Research Lectures – RJS–9
ω 2 pe
1– 2
ω
ω 2 pe 2
1 – 2 cos θ
ω
June 4, 2003 – 34
Appleton-Hartree Including Ions
PSTI Research Lectures – RJS–9
June 4, 2003 – 35
Vlasov Equation
Vlasov Equation (Collisionless Boltzmann):
∂f
+ v⋅∇
∇ f + a⋅∇
∇v f = 0
∂t
Liouville’s theorem: phase space density is constant along a
particle trajectory, define Liouville operator L:
L = ∂ + v⋅∇
∇ + a⋅∇
∇ v,
∂t
Lf = 0
Jeans’s theorem: Any phase distribution that satisfies the Vlasov
equation is a function of the constants of the motion (αi):
n
Lf = Σ Lα i ∂f ∂α i
i=1
PSTI Research Lectures – RJS–9
June 4, 2003 – 36
Linearized Vlasov Equation
∂ f1
q
q
+ v⋅∇
∇ f1 +
v × B0 ⋅∇
∇ v f1 = −
E + v × B1 ⋅∇
∇v f0
m
m 1
∂t
The left hand side of this equation is the rate of change in f1
following an unperturbed particle trajectory. Formally
f1 r,v,t = −
t
−∞
q
dt′ m E 1 + v × B1 ⋅∇
∇ v f0
Where the time integral is over the “past history” of the particle that
passes through r, v at time t.
PSTI Research Lectures – RJS–9
June 4, 2003 – 37
Wave Solutions
Having obtained f1, the coupled Maxwell’s equations require either
charge density (ρ1) [electrostatic] or current density (j1) [electromagnetic]
ρ 1 r,t
j1 r,t
PSTI Research Lectures – RJS–9
=
Σ
species
=
Σ
species
q
f1dv
q v f 1dv
June 4, 2003 – 38
Harmonic perturbation
Past history integral:
⋅
×
∇
q
q
v ω + k v⋅E1 ω ⋅∇
∇v f0
∇v f1 = m E1 1 − k⋅v
i ω − k⋅v f1 − m v × B0 ⋅∇
⋅
⋅
∇
Note that this form implicitly assumes that integrals
converge at t = -∞
PSTI Research Lectures – RJS–9
June 4, 2003 – 39
Landau Damping
Parallel Propagation, electrostatic waves
q2
∂ f0 ∂v||
ρ = − i m E || dv||
ω − k ||v||
PSTI Research Lectures – RJS–9
June 4, 2003 – 40
Beam Generated Instabilities
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June 4, 2003 – 41
Gyro-resonance
Parallel Propagation, electromagnetic waves
iq 2
jr = −
E
2m r
iq 2
jl = −
E
2m l
PSTI Research Lectures – RJS–9
dvv⊥
dvv⊥
k ||v|| ∂ f0 k ||v⊥ ∂ f0
1− ω
+ ω
∂v⊥
∂v||
ω − k ||v|| − Ωe
k ||v|| ∂ f0 k ||v⊥ ∂ f0
1− ω
+ ω
∂v⊥
∂v||
ω − k ||v|| + Ωe
June 4, 2003 – 42
Auroral Currents
PSTI Research Lectures – RJS–9
June 4, 2003 – 43
Liouville’s Theorem and Aurora
Liouville’s theorem states that phase space density is constant
along a particle trajectory.
For an auroral flux tube with a parallel potential drop constants
of the motion are (subscript “0” indicates top of the
acceleration region, where Φ = 0):
W + qΦ = W 0, W ⊥ = µB = W ⊥0B B0
i.e., for electrons
W || = W ||0 + W ⊥0 1 − B B0 + eΦ
PSTI Research Lectures – RJS–9
June 4, 2003 – 44
Flux into atmosphere
Electrons in the loss-cone are the current carriers
eΦ
Flux at atmosphere = flux in local loss cone x BI/B (flux-tube area)
PSTI Research Lectures – RJS–9
June 4, 2003 – 45
Phase Space Mapping
Theoretical and Observed Distributions
(Ergun et al., GRL, 27, 4053-4056, 2000)
Acceleration Ellipse and Loss-cone Hyperbola
PSTI Research Lectures – RJS–9
June 4, 2003 – 46
Current Density – Flux in the Loss-Cone
The auroral current is carried by the particles in the loss-cone.
Without any additional acceleration the current carried by the
electrons is the precipitating flux at the atmosphere:
j0 = nevT/2π1/2 ≈ 1 µA/m2 for n = 1 cm-3, Te = 1 keV.
A parallel electric field can increase this flux by increasing the
flux in the loss-cone. Maximum flux is given by the flux at the
top of the acceleration region (j0) times the magnetic field ratio
(flux conservation - with no particles reflected).
jm = nevT/2π1/2 × (BI/Bm).
PSTI Research Lectures – RJS–9
June 4, 2003 – 47
Knight Relation
j/j0
Asymptotic Value
= BI /Bm
1+eΦ/T
eΦ/T
[Knight, PSS, 21, 741-750, 1973; Lyons, 1980]
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June 4, 2003 – 48
Consequences of Low Density: 1
Because of low density electrons can be in direct gyroresonance with faster-than light R-X mode waves
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June 4, 2003 – 49
Consequences of Low Density: 2
Resonance condition must be modified to include
relativistic corrections [Wu and Lee, 1979]
Non-relativistic (R-mode gyro-resonance) :
iq 2
jr = −
E
2m r
dvv⊥
k v ∂ f0 k ||v⊥ ∂ f0
1 − ω|| ||
+ ω
∂v⊥
∂v||
ω − k ||v|| − Ωe
For non-relativistic gyro-resonance
waves damped because of tail of
distribution at large v⊥
Relativistic Resonance condition is
ω − k ||v|| = Ωe
an ellipse in velocity space:
PSTI Research Lectures – RJS–9
1 − v2 c2
June 4, 2003 – 50
Consequences of Low Density: 3
Dispersion relation must be modified to include relativistic
corrections [Pritchett, 1984; Pritchett and Strangeway, 1985]
Resonance condition:
ω − k ||v|| = Ωe
In a cold plasma R-X mode cut-off =
Weakly relativistic:
ωr = Ωe
1 − v2 c2
ωr = Ωe + ω pe2 Ωe
1 − v 2 c 2 + ω pe2 Ωe
Modified dispersion relation allows k|| = 0
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June 4, 2003 – 51
Free Energy Source
"Horseshoe" Distribution
1997-01-31/06:43:48.899 - 06:44:25.617
1ï10 5
-12.0
Resonance
ellipse, k = 0
Resonance
circle, k = 0
5ï10 4
-13.4
0
-14.9
-5ï10 4
-16.3
Loss-cone
Hyperbola
-1ï10 5
-1ï10 5
PSTI Research Lectures – RJS–9
-5ï10 4
Acceleration
Ellipse
0
Para. Velocity (km/s)
5ï10 4
1ï10 5
-17.8
June 4, 2003 – 52
Energy flow in the AKR Source Region
Electron Distribution in
Density Cavity
-1x105
-12.0
Upgoing to
Magnetosphere
-5x104
-13.4
2
3
Energy Flow
Loss Cone
-14.9
0
1
5x104
1. Acceleration by Electric Field
2. Mirroring by Magnetic Mirror
3. Diffusion through Auroral Kilometric Radiation
-16.3
Downgoing to
Ionosphere
1x105
-1x105
-5x104
0
Parl. Velocity (km/s)
PSTI Research Lectures – RJS–9
5x104
1x105
-17.8
June 4, 2003 – 53