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ESS 154/200C
Lecture 19
Waves in Plasmas 2
1
ESS 200C
Space Plasma Physics
M/W/F
10:00 – 11:15 AM
Instructors:
C.T. Russell (Tel. x-53188; Office: Slichter 6869)
R.J. Strangeway (Tel. x-66247; Office: Slichter 6869)
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1/4
1/6
1/8
1/11
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1/13
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1/25
1/27
1/29
2/1
2/3
2/5
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2/10
2/12
2/17
2/19
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ESS 154 Solar Terrestrial Physics
Geology 4677
Day
Topic
Instructor
M
A Brief History of Solar Terrestrial Physics
CTR
W
Upper Atmosphere / Ionosphere
CTR
F
The Sun: Core to Chromosphere
CTR
M
The Corona, Solar Cycle, Solar Activity
Coronal Mass Ejections, and Flares
CTR
PS1
W
The Solar Wind and Heliosphere, Part 1
CTR
F
The Solar Wind and Heliosphere, Part 2
CTR
W
Physics of Plasmas
RJS
F
MHD including Waves
RJS
M
Solar Wind Interactions: Magnetized Planets YM
W
Solar Wind Interactions: Unmagnetized Planets YM
F
Collisionless Shocks
CTR
M
Mid-Term
W
Solar Wind Magnetosphere Coupling I
CTR
F
Solar Wind Magnetosphere Coupling II;
The Inner Magnetosphere I
CTR
M
The Inner Magnetosphere II
CTR
W
Planetary Magnetospheres
CTR
F
The Auroral Ionosphere
RJS
W
Waves in Plasmas 1
RJS
F
Waves in Plasmas 2
RJS
F
Review
CTR/RJS
M
Final
Due
PS2
PS3
PS4
PS5
PS6
PS7 – 2/22
MHD – Harmonic Perturbation
wdr - k rux = 0
(continuity)
wru - k( x̂(b × B + d p) - Bx b) / m 0 = 0
d p - cs2dr = 0
(momentum)
(energy)
w b + k(Bx u - ux B) = 0
(Faraday)
• If we let B = (Bcosθ, 0, Bsinθ) and k = k x̂ where θ is angle
1
2
2
between B and k, define Alfvén speed as VA  ( B / 0  )
(w k ) u - ( x̂(bz sinq + ( k w ) cs2 rux / B) - b cosq )B / rm0 = 0
(Faraday)
(w k ) b + B(u cosq - ux ( x̂ cosq + ẑ sinq )) = 0
(cont., mom., energy)
3
MHD – Harmonic Perturbation
(w k ) by + Buy cosq = 0
(w k ) bz + B(uz cosq - ux sinq ) = 0
(w k ) u - ( x̂(bz sinq + ( k w ) cs2 rux / B) - b cosq )B / rm0 = 0
2
\ (w k ) u + x̂(VA2 (uz sin q cosq - ux sin 2 q ) - cs2 ux ) + (w k ) b cosq B / rm 0 = 0
[(w / k)2 -VA2 sin 2 q - cs2 ]ux +VA2 sinq cosq uz = 0
[(w / k)2 -VA2 cos2 q ]uy = 0
[(w / k)2 -VA2 cos2 q ]uz +VA2 sin q cosq ux = 0
• For cold plasma the dispersion relations are
( /k) 2  VA2 cos2 
( /k) 2  VA2
shear Alfven wave: ux  uz  0
compressional wave: uy  0
4
Wave Perturbations
•
In our mathematical development, we set k
along x and the magnetic field in the x-z plane.
If a wave is not compressional in this geometry,
the velocity and magnetic field perturbations (u
and b) must be along y (from y component of
Faraday’s law). E is along a direction
perpendicular to B in the ZY plane (as E=-vB).
•
If the wave is compressional then the magnetic
perturbation is along Z and j and E are along y.
•
If we draw the waves in a coordinate system
with B along Z with the wave vector in the x-z
plane, then a non-compressive wave has its
magnetic perturbation along Y. If we move the k
vector into the Y-Z plane, the wave becomes
compressional
•
Energy flow is along
•
Group velocity is
S  ( E  b) / 0
V A Bˆ for shear Alfven wave
V A kˆ for fast-mode wave
5
Waves in Warm Plasmas
• In a warm plasma, a third
mode appears called the
slow mode. It is
compressional but the field
and thermal pressure
fluctuations are in
antiphase.
• The shear Alfven wave
remains the same
 2 / k 2  VA2 cos 2 
• The fast and slow wave
dispersion relations are
 2 / k 2  0.5{cs2  vA2  [(cs2  vA2 )2  4cs2vA2 cos2  ] }
1
2
6
Topside Sounding
Z
O
X
F-layer maximum
• Data from the Alouette spacecraft
• Horizontal axis – frequency (MHz)
• Vertical axis – virtual range
• Local cut-offs have zero delay
• Waves at higher frequencies
propagate into the ionosphere
• Reflect at cut-off frequency
• Inversion techniques used to
determine density profile
7
Governing Equations
Harmonic Perturbation, first order quantities only:
¶/¶t º –iw
Ñ º ik
Faraday’s Law:
k ´ E = wb
Ampere’s Law:
k ´ b = –im 0 j– w2 E
c
i.e.
2
w
k – 2 E – k(k×E) = iwm 0 j
2
c
8
Cartesian Coordinates
Appleton-Hartree
Assume:
B 0 = B0z ,
Then:
2
w
k – 2 E x – k ^k ||E z = iwm 0 jx
2
||
k = (k ^, 0, k ||)
c
2
w
k – 2 E y = iwm 0 j y
2
c
2
w
k – 2 E z – k ||k ^E x = iwm 0 jz
2
^
c
Alternative formalism replaces j with the equivalent
dielectric tensor
9
Polarized Coordinates
Force Law:
–imwv = q(E + v´B0)
–imwv x = q(E x + v yB0)
–imwv y = q(E y – v xB0)
q
–iw(v x ± iv y) = m (E x ± iE y) – v yW ± iv xW
[W = – qB0 /m]
(± i)(± i)v y = – v y
Therefore
q
–iwv± = m E± ± iWv±
10
Electrons only, parallel
2
w
k – 2 E± = iwm0 j±
2
c
2
iw pe
e0 E±
j± =
(w ± We )
Therefore
2
w
pe
m 2 =1w (w ± We )
w 2 E = –iwm j
z
0 z
2
c
w 2pe
j z = i w e0E z
w 2 = w 2pe
For parallel propagation modes split into R, L, and P modes
11
Electrons only, perpendicular
O-mode:
w 2pe
j z = i w e0E z
2
w
k – 2 E z = iwm 0 jz
2
c
Therefore
X-mode:
w 2pe
m =1– 2
w
2
2
w
k – 2 E y = iwm 0 j y
2
c
w 2 E = –iwm j
x
0 x
2
c
2
iw pe
e0 E±
j± =
(w ± We )
12
A-H Dispersion Relation
w 2pe
w 2 pe
2 2 1– 2
w
w
m2 = 1 –
w 2pe We2 2
2 1 – 2 – 2 sin q ± G
w
w
G=
w 2pe
W
W
4
sin q +4
1 – 2 cos 2q
w
w
w
4
e
4
2
e
2
2
13
Appleton-Hartree
14
Quasi-longitudinal Approximation
w 2 pe
W
W
4
sin q << 4
1 – 2 cos 2q
w
w
w
4
e
4
m2 = 1 –
2
2
w 2 pe
w 2pe
2 2 1– 2
w
w
w 2pe We2 2
w 2pe
We
2 1 – 2 – 2 sin q ± 2 w 1 – 2 cosq
w
w
w
w 2 pe
sin q << 2 1 – 2
2
w
w
We2
2
e
2
m2 = 1 –
w 2pe
w2
W
1 ± we cosq
15
Quasi-transverse Approximation
w 2 pe
W
W
4
sin q >> 4
1 – 2 cos 2q
w
w
w
4
e
4
2
e
2
2
w 2 pe
1– 2
w
m2 = 1 –
w 2 pe 2
1 – 2 cos q
w
16
Appleton-Hartree Including Ions
17
Vlasov Equation
Vlasov Equation (Collisionless Boltzmann):
f
 ν  f  a   v f  0
t
Liouville’s theorem: phase space density is constant along a
particle trajectory, define Liouville operator L:
L
f
 ν    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 (ai):
n
Lf   ( La i )f / a i
i 1
18
Linearized Vlasov Equation
¶f1
q(v ´ B0 )
q
+ v × Ñf1 +
× Ñv f1 = - (E1 + v ´ B1 )× Ñv f0
¶t
m
m
The left hand side of this equation is the rate of change in f1
following an unperturbed particle trajectory. Formally
f1 (r, v, t) = - ò dt¢
t
-¥
q
(E1 + v ´ B 1 )× Ñv f0
m
Where the time integral is over the “past history” of the particle that
passes through r, v at time t.
19
Wave Solutions
Having obtained f1, the coupled Maxwell’s equations require either
charge density (ρ1) [electrostatic] or current density (j1)
[electromagnetic]
1 (r, t ) 
 q  f dv
1
species
j1 (r, t ) 
 q  νf dv
1
species
20
Harmonic perturbation
Past history integral:
i (  k  v) f1 
q
q
( v  B 0 )   v f1  [E1 (1  k  v /  )  kv  E1 /  ]   v f 0
m
m
Note that this form implicitly assumes that integrals
converge at t = -
21
Landau Damping
Parallel Propagation, electrostatic waves
i(w - k||v|| ) f1 -
q
q
(v ´ B0 )× Ñv f1 = [E|| (1- k||v|| / w ) + k||v||E|| / w ]× Ñv f0
m
m
q2
¶f / ¶v||
\ r = -i E|| ò dv|| 0
m
w - k||v||
22
Beam Generated Instabilities
23
Gyro-resonance
Parallel Propagation, electromagnetic waves
i(w - k||v|| ) f1 -
q
q
(v ´ B0 )× Ñv f1 = [E^ (1- k||v|| / w )+ k||v × E^ / w ]× Ñv f0
m
m
éæ k||v|| ö ¶f0 k||v^ ¶f0 ù
+
÷
êç1ú
2
è
ø
w
¶v
w
¶v
iq
ë
^
|| û
\ jr = Er ò dvv^
2m
(w - k||v|| - We )
éæ k||v|| ö ¶f0 k||v^ ¶f0 ù
+
÷
êç1ú
2
è
ø
w
¶v
w
¶v
iq
ë
^
|| û
jl = El ò dvv^
2m
(w - k||v|| + We )
24
Ion Pickup and Ion Gyro-Resonance
•
If neutrals at rest are ionized in a flowing
magnetized plasma, they are accelerated
by the electric field associated with the
flow so that they drift with the flowing
plasma perpendicular to the field and
form a ring (in velocity space) around the
magnetic field. A wave grows parallel to
the field resonating with the cyclotron
motion.
•
If the magnetic field is perpendicular to
the flow, it is easy to visualize that the
waves produced are not Doppler-shifted
because they are moving perpendicular to
the flow.
•
If the magnetic field has a component
parallel to the flow, the wave occurs at the
frequency Doppler shifted from the ion
gyro frequency by this component of the
flow but the observer sees the wave near
the gyro frequency because the observer
is moving along the field line in the plasma
flow.
25
Auroral Currents – Electron
Gyro-resonance
26
Consequences of Low Density
Because of low density electrons can be in direct gyroresonance with faster-than light R-X mode waves
27
Resonance Condition
Resonance condition must be modified to include
relativistic corrections [Wu and Lee, 1979]
Non-relativistic (R-mode gyro-resonance) :
éæ k||v|| ö ¶f0 k||v^ ¶f0 ù
+
÷
êç1ú
2
è
ø
w ¶v^ w ¶v|| û
iq
ë
jr = Er ò dvv^
2m
(w - k||v|| - We )
For non-relativistic gyro-resonance
waves damped because of tail of
distribution at large v
Relativistic Resonance condition is
an ellipse in velocity space:
  k||v||  e (1  v 2 / c 2 )
28
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
1. Acceleration by Electric Field
2. Mirroring by Magnetic Mirror
3. Diffusion through Auroral Kilometric Radiation
5x10 4
-16.3
Downgoing to
Ionosphere
1x105
-1x10 5
-5x104
0
Parl. Velocity (km/s)
5x10 4
1x105
-17.8
29