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ESS200C
Pulsations and Waves
Lecture 17
1
Magnetic Pulsations
•
The field lines of the Earth vibrate at different frequencies. The energy for these
vibrations can come from external (exogenic) sources or internal (endogenic)
sources.
•
Pc-1 waves (0.2 – 5s; 0.2 – 5 Hz) are produced by cyclotron resonance with ions.
•
Pc-2 waves (5 – 10s; 0.1 – 0.2 Hz) probably also produced by plasma resonance.
•
Pc-3 waves (10 – 45s; 22 - 100 mHz) produced by solar wind forcing of field-aligned
resonance.
•
Pc-4 waves (45 - 150s; 7 - 22 mHz) produced by solar wind forcing and/or Kelvin
Helmholtz instability.
•
Pc-5 waves (150 – 600s; 2 – 7 mHz) produced by Kelvin-Helmholtz instability or
magnetopause oscillations.
•
Pi-1 waves (1 - 40s; 0.025 – 1 Hz) associated with downward field-aligned currents in
auroral zone.
•
Pi-2 waves (40 – 150s; 2 – 25 mHz) produced by substorm triggered dynamics.
2
Field-Line Resonances
•
Proton beams moving back from
the bow shock are unstable as
they move through the incoming
solar wind.
•
The waves produced are
compressional and they push on
the magnetopause, periodically
generating compressional waves
in the magnetosphere that can
cross field lines.
•
Some field lines will resonate
(standing wave) at these
frequencies. Energy builds in the
azimuthal direction of perturbation.
•
These resonances can be seen in
ground magnetometers. They can
be used to determine the mass
content of the magnetic field line.
3
Further Examples of Resonances
in Magnetic Field data
4
Sources of Various ULF Waves
HUDSON ET AL. ANN. GEOPHYSICAE, 2004
TOROIDAL
POLOIDAL
5
Distribution of Pc 5 Waves
Measured in Magnetic Field Data
HUDSON ET AL. ANN. GEOPHYSICAE, 2004
6
Distribution of Pc 5 Waves
Measured in Electric Field Data
WENLON LIU ET AL. IN PREPARATION, 2009
Toroidal
Pc5
Pc4
Poloidal
7
Maxwell’s Equations and
Conservation Laws

   u  0
t
(
(continuity equation)
u
 u  u )  p  j  B (momentum equation)
t
B
   E
t
(Faraday’s law)
  B  0 j
(Ampere’s law)
B  0
(B is divergenceless)
E uB  0
(Ohm’s law)
(

p
 u  )(  )  0
t

(conservation of specific entropy)
8
Linear Waves
• Background quantities that can be large: B, ρ, p
• Perturbed quantities that are small: b, δρ, δp, u
E(=-u x B), j(= x b/μ0)
• Linearized equations become

   u  0
t
(continuity )
u

 p  (  b)  B /  0
t
b
   (u  B)
t
( Faraday )
(momentum)
9
One-Dimensional Cold Plasma
Waves (Dropped Thermal Pressure)
u

 x 0
t
x
(continuity )
 


 (b  B /  0 ) Bx b
u

  xˆ
( )
t
x
 0 x


b
u u x 
 Bx
(
)B
t
x
x
•
•
(momentum)
( Faraday )
For plane wave propagating in the x-direction oscillating quantities vary as expi(kx – ωt)
Then 
and
and we can rewrite

 i
 ik
t
x
i[  ku x ]  0
(continuity )
 


i[u  k ( xˆ (b  B)  Bx b ) /  0 ]  0



i[b  k ( Bx u  u x B)]  0
(momentum)
( Faraday )
10
One-Dimensional Cold Plasma
Waves (Dropped Thermal Pressure)
• If we let B = (Bcosθ, 0, Bsinθ) and k = k x̂ where θ is angle
between B and k
[( / k ) 2  VA2 sin 2  ]u x  VA2 sin  cos  u z  0
[( / k ) 2  VA2 cos 2  ]u y  0
[( / k ) 2  VA2 cos 2  ]u z  VA2 sin  cos  u x  0
1
where VA  ( B / 0  )
• Then the dispersion relations are
2
2
( / k ) 2  VA2 cos 2 
( / k ) 2  VA2
shear Alfven wave : v x  v z  0
compressio nal wave : v y  0
11
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
12
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  cA2  [(cs2  vA2 )2  4cs2vA2 cos2  ] }
1
2
13
Oscillations on Dipole Field Lines
•
•
•
•
Field lines are rooted in the conducting
ionosphere and the conducting Earth
and have natural resonating
frequencies depending on the strength
of the magnetic field, the plasma mass
density and the length of the field line.
If the field line were straight and the
density and field constant, the
frequencies of resonance would be
nB/2l(μ0ρ)1/2 where n is the harmonic
number, l is the length of the field line,
B the number density and ρ the mass
density.
Energy sources for these waves can
be solar wind pressure variations or
plasma anisotropies.
Mirror-mode grows when
1+β┴(1-β┴/βǁ)<0
where β is the ratio of plasma to
magnetic pressure and ┴ (ǁ) are the
perpendicular (parallel) directions.
14
Ion Pickup and Ion Cyclotron
Waves
•
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 Dopplershifted 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.
15
Waves in a Two-Fluid Plasma
•
Maxwell’s Laws
  E ( x, t )   2 ( x, t ) /  0
( Poisson )
  B ( x, t )  0
( Divergence of B)
  E ( x, t ) 
 B( x, t )
t
  B ( x , t )   0 j ( x , t )   0 0
•
( Faraday )

t
E ( x, t )
( Ampere)
Conservation Laws

ns    ( ns u s )  0
(continuity )
t
q
ps

F
u s  u s  u s  s ( E  u s  B ) 

t
ms
ns ms ns ms
(momentum)
where j   ns qs u s ,  q   qs ns , and F  is the force per unit volume excluding pressure and magnetic forces
s
s
Ps = constant x (ns)γs = nsTs
(polytropic law) where γs is the ratio of specific heats and Ts=kBT
Adiabatic approximation results in γs =(N+2)/N where N=number of degrees of freedom
γs = 5/3
3D adiabatic
=2
2D adiabatic
=3
1D adiabatic
=1
isothermal
=0
isobaric
16
Waves in an Unmagnetized Plasma
•
Assume ions are infinitely massive and geometry is one dimensional


ne  (neue )  0
(continuity )
t
x
u
 pe

me ne ( ue  ue e ) 
 ene E x
(momentum)
t
x
x
E x
 e(no  ne ) /  0
( Poisson )
x
•
•
Assume small perturbations, keeping only terms up to first order (linearization)
E1
 en1 ( x, t ) /  0
x
n1
u
 no 1  0
t
x
u
p
me no 1  eno E1  1
t
x
( Poisson )
(continuity )
(momentum)
Taking time derivative of continuity equation and spatial derivative of others and
substituting we get  2 n
n e2
t
1
2
(
where
o
 o me
)n1  0
no e 2 12
 pe  (
)  electron plasma frequency
 o me
17
Electrostatic Waves in an Unmagnetized
Plasma: Alternate Approach
•
Assume perturbations are plane parallel waves in 1D
~
E1 ( x, t )  E1 exp( it  ikx)
n ( x, t )  n~ exp( it  ikx)
1
1
u1 ( x, t )  u~1 exp( it  ikx)
•
Substitute in continuity, momentum and Poisson equations
 in~1  ikn0u~1  0
 iu~  e Eˆ  0
1
m
1
~
en~1  ikE1  0
•
For a solution the determinant must equal zero
n0 e 2
and    
me 0
2
2
pe
Group velocity
Here vg=0
vg 

k
 i
0
e / 0
ikn0
0
 i e / me  0
0
ik
18
Electrostatic Waves in a Warm
Magnetized Plasma
•
•
•
In a 1-D adiabatic situation, pressure
gradient is p1  3T0n1
Our three equations become:
 in~1  ikn0u~1  0
~
3ikT0 n~1  ime n0u~1  eno E1  0
~
en~1 /  0  ikE1  0
Using the fact that the determinant
must be zero, we obtain
2
2
 2   pe
 3k 2T0 / me   pe
 3 2 k 2ve2
•
where ve=(2T0/me)1/2 (thermal velocity of
electrons)
Rewriting we obtain dispersion relation
of Langmuir waves
  e (1  3k 22D )
•
1
2
The group velocity becomes
approximately
vg 

 3(kD )ve / 2
k
19
Electromagnetic Waves in an
Unmagnetized Plasma
•
Assume there is no unperturbed
magnetic field and that k·E1 = k·B1
=0
  E1 
 B1
t
( 0 ) 1   B1  j1   0 E1 / t
me n0
•
•
•
•
ue1
 pe1  en0 E1
t
( Faraday)
( Ampere)
(momentum)
After some algebra
ω2 = ωpe2+k2c2
Index of refraction n becomes
n = c/vph = ck/ω
= (1- ωpe2/ω2)1/2
Group velocity is
 kc2
vg 

 nc
k

Where index of refraction goes to
zero, is ω=ωpe, group velocity
goes to zero and wave is
reflected.
20