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ESS 200C - Space Plasma PhysicsWinter Quarter 2006-2007
Raymond J. Walker
Date Topic1/8 Organization and Introduction
to Space Physics I1/10 Introduction to Space Physics II1/17
Introduction to Space Physics III1/19 The Sun I1/22 The Sun II1/24
The Solar Wind I1/29 The Solar Wind II1/31 First Exam 2/5 Bow Shock
and Magnetosheath2/7 The Magnetosphere I
Date Topic2/14 The Magnetosphere II2/16 The Magnetosphere
III2/21 Planetary Magnetospheres2/23 The Earth’s Ionosphere2/26
Substorms3/5 Aurorae3/7 Planetary Ionospheres3/12 Pulsations3./14
Storms and Review
Second Exam
Schedule of Classes
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ESS 200C – Space Plasma Physics
• There will be two examinations and homework assignments.• The
grade will be based on
– 35% Exam 1– 35% Exam 2– 30% Homework
• References– Kivelson M. G. and C. T. Russell, Introduction to
Space
Physics, Cambridge University Press, 1995.– Gombosi, T. I.,
Physics of the Space Environment, Cambridge
University Press, 1998– Kellenrode, M-B, Space Physics, An
Introduction to Plasmas and
Particles in the Heliosphere and Magnetospheres, Springer,
2000.– Walker, A. D. M., Magnetohydrodynamic Waves in Space,
Institute
of Physics Publishing, 2005.
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Space Plasma Physics
• Space physics is concerned with the interaction of charged
particles with electric and magnetic fields in space.
• Space physics involves the interaction between the Sun, the
solar wind, the magnetosphere and the ionosphere.
• Space physics started with observations of the aurorae.– Old
Testament references to auroras.– Greek literature speaks of
“moving accumulations
of burning clouds”– Chinese literature has references to auroras
prior
to 2000BC
-
• Aurora over Los Angeles (courtesy V. Peroomian)
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– Galileo theorized that aurora is caused by air rising out of
the Earth’s shadow to where it could be illuminated by sunlight.
(Note he also coined the name aurora borealis meaning “northern
dawn”.)
– Descartes thought they are reflections from ice crystals.–
Halley suggested that auroral phenomena are ordered by
the Earth’s magnetic field. – In 1731 the French philosopher de
Mairan suggested they
are connected to the solar atmosphere.
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• By the 11th century the Chinese had learned that a magnetic
needle points north-south.
• By the 12th century the European records mention the
compass.
• That there was a difference between magnetic north and the
direction of the compass needle (declination) was known by the 16th
century.
• William Gilbert (1600) realized that the field was
dipolar.
• In 1698 Edmund Halley organized the first scientific
expedition to map the field in the Atlantic Ocean.
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The Plasma State
• A plasma is an electrically neutral ionized gas.– The Sun is a
plasma– The space between the Sun and the Earth
is “filled” with a plasma.– The Earth is surrounded by a
plasma.– A stroke of lightning forms a plasma– Over 99% of the
Universe is a plasma.
• Although neutral a plasma is composed of charged particles-
electric and magnetic forces are critical for understanding
plasmas.
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The Motion of Charged Particles
• Equation of motion
• SI Units– mass (m) - kg– length (l) - m– time (t) - s–
electric field (E) - V/m– magnetic field (B) - T– velocity (v) -
m/s– Fg stands for non-electromagnetic forces (e.g. gravity) -
usually ignorable.
gFBvqEqdtvdm
rrrrr
+×+=
-
• B acts to change the motion of a charged particle only in
directions perpendicular to the motion.– Set E = 0, assume B along
z-direction.
– Equations of circular motion with angular frequency (cyclotron
frequency or gyro frequency
– If q is positive particle gyrates in left handed sense– If q
is negative particle gyrates in a right handed sense
2
22
2
22
mBvq
v
mBvq
mBvq
v
Bqvvm
Bqvvm
yy
xyx
xy
yx
−=
−==
−=
=
&&
&&&
&
&
mqB
c =Ω
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• Radius of circle ( rc ) - cyclotron radius or Larmorradius or
gyro radius.
– The gyro radius is a function of energy.– Energy of charged
particles is usually given in electron volts
(eV)– Energy that a particle with the charge of an electron gets
in
falling through a potential drop of 1 Volt- 1 eV =
1.6X10-19Joules (J).
• Energies in space plasmas go from electron Volts to
kiloelectron Volts (1 keV = 103 eV) to millions of electron Volts
(1 meV = 106 eV)
• Cosmic ray energies go to gigaelectron Volts ( 1 geV = 109
eV).
• The circular motion does no work on a particle
qBmv
v
c
cc
⊥
⊥
=
Ω=
ρ
ρ
0)()(2
21
=×⋅==⋅=⋅ Bvvqdtmvdv
dtvdmvF
rrrrr
rr
Only the electric field can energize particles!
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• The electric field can modify the particles motion.– Assume
but still uniform and Fg=0.– Frequently in space physics it is ok
to set
• Only can accelerate particles along• Positive particles go
along and negative particles go
along • Eventually charge separation wipes out
– has a major effect on motion. • As a particle gyrates it moves
along and gains energy • Later in the circle it losses energy.•
This causes different parts of the “circle” to have different radii
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it doesn’t close on itself.
• Drift velocity is perpendicular to and• No charge dependence,
therefore no currents
0≠Er
0=⋅ BErr
Er
Br
EE−
E
⊥EEr
2BBEuE
rrr ×
=
Er
Br
Br
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• Any force capable of accelerating and decelerating charged
particles can cause them to drift.
– If the force is charge independent the drift motion will
depend on the sign of the charge and can form perpendicular
currents.
• Changing magnetic fields cause a drift velocity.– If changes
over a gyro-orbit the radius of curvature will
change.– gets smaller when the particle goes into a region
of
stronger B. Thus the drift is opposite to that of motion.
– ug depends on the charge so it can yield perpendicular
currents.
2qBBFuF
rrr ×
=
Br
qBmv
c⊥=ρ
BErr
×
32
21
32
21
qBBBmv
qBBBmvug
∇×=
×∇= ⊥⊥−
rrr
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• The change in the direction of the magnetic field along a
field line can cause motion.– The curvature of the magnetic field
line introduces a drift
motion.• As particles move along the field they undergo
centrifugal
acceleration.
• Rc is the radius of curvature of a field line ( ) where
, is perpendicular to and points away from the center of
curvature, is the component of velocity along
• Curvature drift can cause currents.
cc
RR
mvF ˆ
2
=r
bbRn
c
ˆ)ˆ(ˆ
∇⋅−=
BBbr
=ˆ n̂ Br
v Br
2
2
2
2 ˆˆ)ˆ(
qBR
nBmv
qB
bbBmvu
cc
×−=
∇⋅×=
rrr
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• The Concept of the Guiding Center– Separates the motion ( ) of
a particle into motion
perpendicular ( ) and parallel ( ) to the magnetic field.– To a
good approximation the perpendicular motion can
consist of a drift ( ) and the gyro-motion ( )
– Over long times the gyro-motion is averaged out and the
particle motion can be described by the guiding center motion
consisting of the parallel motion and drift. This is veryuseful for
distances l such that and time scales τ such that
vr
⊥v v
Dv cvΩ
ccvvvvvvvv gcD ΩΩ⊥ +=++=+=rrrrrrrr
1
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• Maxwell’s equations– Poisson’s Equation
• is the electric field• is the charge density• is the electric
permittivity (8.85 X 10-12 Farad/m)
– Gauss’ Law (absence of magnetic monopoles)
• is the magnetic field
0ερ
=⋅∇ Er
Er
0=⋅∇ Br
Br
ρ
0ε
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– Faraday’s Law
– Ampere’s Law
• c is the speed of light.• Is the permeability of free space,
H/m• is the current density
tBE
∂∂
−=×∇r
r
JtE
cB
rr
r02
1 µ+∂∂
=×∇
0µ 70 104 −×= πµ
Jr
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• Maxwell’s equations in integral form
– A is the area, dA is the differential element of area– is a
unit normal vector to dA pointing outward.– V is the volume, dV is
the differential volume element
– is a unit normal vector to the surface element dF in the
direction given by the right hand rule for integration
around
C, and is magnetic flux through the surface. – is the
differential element around C.
∫∫ =⋅ dVdAnEA ρε 01ˆ
r
n̂
tdFn
tBsdE
AdnB
C
A
∂Φ∂
−=⋅∂∂
−=⋅
=⋅
∫∫
∫'ˆ
0ˆr
rv
r
'n̂
sdrΦ
dFnJdFntEsdB
cC ∫∫∫ ⋅+⋅∂∂
=⋅ '0'1 ˆˆ2 µ
rrr
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• The first adiabatic invariant– says that changing drives
(electromotive
force). This means that the particles change energy in
changing magnetic fields.
– Even if the energy changes there is a quantity that remains
constant provided the magnetic field changes slowly enough.
– is called the magnetic moment. In a wire loop the magnetic
moment is the current through the loop times the area.
– As a particle moves to a region of stronger (weaker) B it is
accelerated (decelerated).
EtB rr
×−∇=∂∂
Er
.2
21
constBmv
== ⊥µ
µ
Br
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• For a coordinate in which the motion is periodic the action
integral
is conserved. Here pi is the canonical momentum ( where A is the
vector potential).
• For a gyrating particle • The action integrals are conserved
when the
properties of the system change slowly compared to the period of
the coordinate.
constant == ∫ iii dqpJ
µπqmsdpJ 21 =⋅= ∫ ⊥
rr
Aqvmprrr
+=
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• The magnetic mirror– As a particle gyrates the current will
be
where
– The force on a dipole magnetic moment is
where
cTqI =
ccT Ω
=π2
µ
ππ
π
π ===
Ω==
⊥Ω
Ω
⊥
⊥
BmvIA
vrA
c
cqv
cc
2
2
2
2
22
2
2
dzdBBF µµ −=∇⋅−= r
r
b̂µµ =r
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• The force is along and away from the direction of increasing
B.
• Since and kinetic energy must be conserved
a decrease in must yield an increase in • Particles will turn
around when
Br
0=E
v ⊥v
µ221 mvB =
)( 22212
21
⊥+= vvmmv
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• The second adiabatic invariant– The integral of the parallel
momentum over one
complete bounce between mirrors is constant (as long as B
doesn’t change much in a bounce).
– Using conservation of energy and the first adiabatic
invariant
here Bm is the magnetic field at the mirror point.
.221
constdsmvJs
s== ∫
.)1(2 212
1
constdsBBmvJ
s
sm
=−= ∫
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– As particles bounce they will drift because of gradient and
curvature drift motion.
– If the field is a dipole their trajectories will take them
around the planet and close on themselves.
• The third adiabatic invariant– As long as the magnetic
field
doesn’t change much in the time required to drift around a
planet the magnetic flux inside the orbit must be constant.
– Note it is the total flux that is conserved including the flux
within the planet.
∫ ⋅=Φ dAnB ˆr
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• Limitations on the invariants– is constant when there is
little change in the field’s
strength over a cyclotron path.
– All invariants require that the magnetic field not change much
in the time required for one cycle of motion
where is the orbit period.
τ11
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The Properties of a Plasma• A plasma as a collection of
particles
– The properties of a collection of particles can be described
by specifying how many there are in a 6 dimensional volume called
phase space.
• There are 3 dimensions in “real” or configuration space and 3
dimensions in velocity space.
• The volume in phase space is• The number of particles in a
phase space volume iswhere f is called the distribution
function.
– The density of particles of species “s” (number per unit
volume)
– The average velocity (bulk flow velocity)
dxdydzdvdvdvdvdr zyx=dvdrtvrf ),,( rr
dvtvrftrn ss ),,(),(rrr
∫=
dvtvrfdvtvrfvtru ∫ sss ∫= ),,(/),,(),(rrrrrrr
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– Average random energy
– The partial pressure of s is given by
where N is the number of independent velocity components
(usually 3).
– In equilibrium the phase space distribution is a
Maxwelliandistribution
where
( ) ( ) ⎥⎦
⎤⎢⎣
⎡ −=
s
ssss kT
uvmAvrf2
21
exp,rr
rr
∫∫ −=− dvtvrfdvtvrfuvmuvm ssssss ),,(/),,()()( 221221rrrrrr
221 )(2( ss
s
s uvmNn
p rr−=
( )232 TkmnA ss π=
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• For monatomic particles in equilibrium
where k is the Boltzman constant (k=1.38x10-23 JK-1)• For
monatomic particles in equilibrium
• This is true even for magnetized particles.• The ideal gas law
becomes
( )( ) 2221 sss NkTuvm =− rr
2/)( 221 NkTuvm ss =−rr
sss kTnp =
sss kTnp =
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– Other frequently used distribution functions.• The
bi-Maxwellian distribution
– where – It is useful when there is a difference between the
distributions
perpendicular and parallel to the magnetic field • The kappa
distribution
Κ characterizes the departure from Maxwellian form.– ETs is an
energy.– At high energies E>>κETs it falls off more slowly
than a Maxwellian
(similar to a power law)– For it becomes a Maxwellian with
temperature kT=ETs
( )( ) ( )
⎥⎦
⎤⎢⎣
⎡ −−
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−=
⊥
⊥⊥
s
ss
s
ssss kT
uvmkT
uvmAvrf
221
221
' expexp,rr
rr
( ) ( )12
21
1,−−
⎥⎦
⎤⎢⎣
⎡ −+=
κ
κ κ Tsss
ss EuvmAvrfrr
rr
∞→κ
⎟⎟⎠
⎞⎜⎜⎝
⎛= ⊥ 2
123
'sssss TTTAA
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• What makes an ionized gas a plasma?– The electrostatic
potential of an isolated ion– The electrons in the gas will be
attracted to the ion and will reduce
the potential at large distances.
– If we assume neutrality Poisson’s equation around a test
chargeq0 is
– Expanding in a Taylor series for r>0 and for both electrons
and ions
–
( ) ( ) ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎟⎟
⎠
⎞⎜⎜⎝
⎛+−=
−=∇
ione kTe
kTeenrqr ϕϕ
εδ
εερϕ expexp
0
03
0
0
0
2 rr
rq
04πεϕ =
1
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•The Debye length ( ) is
where n is the electron number density and now e is the electron
charge.•The number of particles within a Debye sphere
needs to be large for shielding to occur
(ND>>1). Far from the central charge the electrostatic
force is shielded.
rqe D
r
04 επϕ
λ−
=
34 3D
DnN λπ=
Dλ21
20 ⎟
⎠⎞
⎜⎝⎛=
nekT
Dελ
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• The plasma frequency– Consider a slab of plasma of thickness
L.– At t=0 displace the electron part of the slab by
-
– The frequency of this oscillation is the plasma frequency
– Because mion>>me
ionpi
epe
pipep
mne
mne
0
02
2
0
02
2
222
εω
εω
ωωω
=
=
+=
pep ωω ≈
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• A note on conservation laws– Consider a quantity that can be
moved from place to place.
– Let be the flux of this quantity – i.e. if we have an element
of area then is the amount of the quantity passing the area element
per unit time.
– Consider a volume V of space, bounded by a surface S.
– If σ is the density of the substance then the total amount in
the volume is
– The rate at which material is lost through the surface is
– Use Gauss’ theorem
– An equation of the preceeding form means that the quantity
whose
density is σ is conserved.
fr
Ar
δAfrr
δ⋅
∫V
dVσ
∫ ⋅S
Adfrr
∫ ∫ ⋅−=V S
AdfdVdtd rrσ
0=⎭⎬⎫
⎩⎨⎧ ⋅∇+
∂∂
∫ dVftVrσ
ft
r⋅−∇=
∂∂σ
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• Magnetohydrodynamics (MHD)– The average properties are
governed by the basic
conservation laws for mass, momentum and energy in a fluid.
– Continuity equation
– Ss and Ls represent sources and losses. Ss-Ls is the net rate
at which particles are added or lost per unit volume.
– The number of particles changes only if there are sources and
losses.
– Ss,Ls,ns, and us can be functions of time and position.–
Assume Ss=0 and Ls =0, where Ms is
the total mass of s and dr is a volume element (e.g. dxdydz)
where is a surface element bounding the volume.
sssss LSuntn
−=⋅∇+∂
∂ r
sssss Mdrnm == ∫ ρρ ,
sdut
Mdrut
Mss
sss
s rrr ⋅+∂
∂=⋅∇+
∂∂
∫∫ ρρ )(sdr
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– Momentum equation
where is the charge density, is the current density, and the
last term is the density of non-electromagnetic forces
– The operator is called the convective derivative and gives the
total time derivative resulting from intrinsic time changes and
spatial motion.
– If the fluid is not moving (us=0) the left side gives the net
change in the momentum density of the fluid element.
– The right side is the density of forces• If there is a
pressure gradient then the fluid moves toward lower
pressure.
• The second and third terms are the electric and magnetic
forces.
sgssqsssssssss
s mFBJEpLSumuutu ρρρ +×++−∇=−+∇⋅+∂
∂ rrrrrrr
)()(
ssqs nq=ρ ssss unqJrr
=
)( ∇⋅+∂∂
sutr
-
– The term means that the fluid transports momentum with it.
• Combine the species for the continuity and momentum equations–
Drop the sources and losses, multiply the continuity
equations by ms, assume np=ne and add.
Continuity
– Add the momentum equations and use me
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• Energy equation
where is the heat flux, U is the internal energy density of the
monatomic plasma and N is the number of degrees of freedom– adds
three unknowns to our set of equations. It is usually
treated by making approximations so it can be handled by the
other variables.
– Make the adiabatic assumption (no change in the entropy of the
fluid element)
or
where cs is the speed of sound and cp and cv are the specific
heats at constant pressure and constant volume. It is called the
polytropic index. In thermodynamic equilibrium
mFuEJqupuUuUut g
rrrrrrr⋅+⋅=+++⋅∇++
∂∂ ρρρ ])[()( 221
221
)2/( nNkTU =qr
qr
.constp =−γρ )(2 ρρ
∇⋅+∂∂
=∇⋅+∂∂ u
tcpu
tp
srr
ργ pcs =2
vp cc=γ
3/5)2( =+= NNγ
-
• Maxwell’s equations
– doesn’t help because
– There are 14 unknowns in this set of equations -– We have 11
equations.
• Ohm’s law– Multiply the momentum equations for each individual
species
by qs/ms and subtract
where and σ is the electrical conductivity
JB
EtB
rr
rr
0µ=×∇
×−∇=∂∂
0=⋅∇ Br
0)( =×∇⋅−∇=∂
⋅∇∂ EtB rr
puJBE ,,,,, ρrrrr
)]}([11){( 2 uJtJ
nemBJ
nep
neBuEJ ee
rrr
rrrrrr⋅∇+
∂∂
−×−∇+×+= σ
sss unqJs
rr ∑=
-
– Often the last terms on the right in Ohm’s Law can be
dropped
– If the plasma is collisionless, may be very large so
• Frozen in flux– Combining Faraday’s law ( ), and
Ampere’ law ( ) with
where is the magnetic viscosity– If the fluid is at rest this
becomes a “diffusion” equation
– The magnetic field will exponentially decay (or diffuse) from
a
conducting medium in a time where LB is the system size.
)( BuEJrrrr
×+= σ
0=×+ BuErrr
tBE
∂∂
−=×∇r
r
JBrr
0µ=×∇
BButB
m
rrrr
2)( ∇+××∇=∂∂ η
01 µση =m
BtB
m
rr
2∇=∂∂ η
mBD L ητ2=
σ
)( BuEJrrrr
×+= σ
-
– On time scales much shorter than
– The electric field vanishes in the frame moving with the
fluid.– Consider the rate of change of magnetic flux
– The first term on the right is caused by the temporal changes
in B
– The second term is caused by motion of the boundary– The term
is the area swept out per unit time– Use Stoke’s theorem
– If the fluid is initially on surface s as it moves through the
system the flux through the surface will remain constant even
though the location and shape of the surface change.
)( ButB rrr
××∇=∂∂
∫∫ ∫ ×⋅+⋅∂∂
=⋅=Φ
CA AlduBdAn
tBdAnB
dtd
dtd )(ˆˆ
rrrr
r
ldurr
×
0ˆ)( =⋅⎟⎟⎠
⎞⎜⎜⎝
⎛××∇−
∂∂
=Φ
∫ dAnButB
dtd
A
rrr
Dτ
-
• Magnetic pressure and tension
– A magnetic pressure analogous to the plasma
pressure ( )
– A “cold” plasma has and a “warm”
plasma has
– In equilibrium
• Pressure gradients form currents
– The second term can be written as a sum of two terms
0021 )(2)(
0µµµ BBBBBBJFB
rrrrrrr∇⋅+∇−=××∇=×=
0
2
2µBpB =
p∇−
0
2
2µβ
Bp
≡ 1
-
– cancels the parallel component
of the term. Thus only the perpendicular component of the
magnetic pressure exerts a force on the plasma.
– is the magnetic tension
and is directed antiparallel to the radius of
curvature (RC) of the field line. Note that is
directed outward.
0
2
0 2ˆˆˆ
µµBbbBBb ∇=∇⋅
r
0
2
2µB∇−
)ˆ
(ˆˆ)(0
2
0
2
CRBnbbB
µµ−=∇⋅
n̂
-
•Some elementary wave concepts
–For a plane wave propagating in the x-direction with wavelength
and frequency f, the oscillating quantities can be taken to be
proportional to sines and cosines. For example the pressure in a
sound wave propagating along an organ pipe might vary like
–A sinusoidal wave can be described by its frequency
and wave vector . (In the organ pipe the frequency is f and .
The wave number is ).
ωkr
)}(exp{),( 0 trkiBtrB ω−⋅=rvrr
( ))sin()cos(),( 0 trkitrkBtrB ωω −⋅+−⋅= rrrrrr
λ
)sin(0 tkxpp ω−=
fπω 2= λπ2=k
-
• The exponent gives the phase of the wave. The phase velocity
specifies how fast a feature of a monotonic wave is moving
•Information propagates at the group velocity. A wave can carry
information provided it is formed from a finite range of
frequencies or wave numbers. The group velocity is given by
•The phase and group velocities are calculated and waves are
analyzed by determining the dispersion relation
kk
v phr
2
ω=
kv g r∂
∂=
ω
)(kωω =
-
• When the dispersion relation shows asymptotic behavior toward
a given frequency, , vg goes to zero, the wave no longer propagates
and all the wave energy goes into stationary oscillations. This is
called a resonance.
resω
-
• MHD waves - natural wave modes of a
magnetized fluid
– Sound waves in a fluid
• Longitudinal compressional oscillations which propagate
at
• and is comparable to the thermal speed.
21
21
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
=ρ
γρ
ppcs
21
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=
mkTcs γ
-
– Incompressible Alfvén waves• Assume , incompressible fluid
with background
field and homogeneous
• Incompressibility
• We want plane wave solutions b=b(z,t), u=u(z,t), bz=uz=0
• Ampere’s law gives the current
• Ignore convection ( )=0
∞→σ 0Br
z
y
x
B0
J b, u
0=⋅∇ ur
izbJ ˆ
0
1
∂∂
−= µr
∇⋅ur
BJptu rrr
×+−∇=∂∂ρ
-
• Since and the x-component of momentum becomes
• Faraday’s law gives
• The y-component of the momentum equation becomes
• Differentiating Faraday’s law and substituting the y-component
of momentum
0,0 ==∂∂
yJxp 0=zJ
iuBE
juu
JbBJxp
tu
zyyx
ˆ
ˆ
0)(
0
0
−=
=
=−+∂∂
−=∂
∂
r
r
ρ
zuB
zE
tb x
∂∂
=∂
∂−=
∂∂
0
zbBBJ
tuy
∂∂
=×−=∂
∂
ρµρ 00
01 rr
2
2
0
22
02
2
zbB
tzuB
tb
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛=
∂∂∂
=∂∂
ρµ
-
where is called the Alfvén velocity.
• The most general solution is . This is a
disturbance propagating along magnetic field lines at the
Alfvén velocity.
2
22
2
2
zuC
tb
A ∂∂
=∂∂
21
0
2
⎟⎟⎠
⎞⎜⎜⎝
⎛=
ρµBCA
)( tCzbb A±=
-
• Compressible solutions– In general incompressibility will not
always apply.– Usually this is approached by assuming that the
system
starts in equilibrium and that perturbations are small.• Assume
uniform B0, perfect conductivity with equilibrium
pressure p0 and mass density 0ρ
EE
JJ
uubBB
ppp
T
T
T
T
T
T
rr
rr
rr
rrr
=
=
=+=
+=+=
0
0
0 ρρρ
-
– Continuity
– Momentum
– Equation of state
– Differentiate the momentum equation in time, use Faraday’s law
and the ideal MHD condition
where
)(0 utr
⋅∇−=∂∂ ρρ
))((1 00
0 bBptu rrr
×∇×−−∇=∂∂
µρ
ρρρ
∇=∇∂∂
=∇ 20)( sCpp
0)))(((()(
)()(
22
2
0
=××∇×∇×∇×+⋅∇∇−∂∂
××∇=×∇−=∂∂
AAs CuCuCtu
BuEtb
rrrrr
rrrr
20 )( ρµ
1B
ACrr
=
0BuErrr
×−=
-
– For a plane wave solution
– The dispersion relationship between the frequency ( ) and the
propagation vector ( ) becomes
This came from replacing derivatives in time and space by
– Case 1
)](exp[~ trkiu ω−⋅ rrr
ωkr
0]2 =− )()())[(())(( 22 ⋅−⋅−⋅⋅+⋅++ AAAAAs
CukkuCukCkCkukCCurrrrrrrrrrrrrrrω
×→∇×
⋅→∇⋅
→∇
−→∂∂
ki
ki
ki
it
r
r
r
ω
0Bkrr
⊥
kukCCu Asrrrr ))(( 222 ⋅+=ω
-
• The fluid velocity must be along and perpendicular to
• These are magnetosonic waves– Case 2
• A longitudinal mode with with dispersion relationship(sound
waves)
• A transverse mode with and (Alfvénwaves)
kr
0Br
kr
ur0Br
21
)()( 22 Asph CCkkv +±==ωr
0Bkrr
0)()1)(()( 222222 =⋅−+− AAAsA CuCkCCuCkrrrrω
kurr
k ±=ω
0=⋅uk rr
ACk±=
ω
sC
-
• Alfven waves propagate parallel to the magnetic field.
•The tension force acts as the restoring force.
•The fluctuating quantities are the electromagnetic field and
the current density.
-
– Arbitrary angle between and kr
0Br
0Br
F
ISVA
VA=2CS
Phase Velocities
ESS 200C – Space Plasma PhysicsSpace Plasma PhysicsAurora over
Los Angeles (courtesy V. Peroomian)The Plasma StateThe Motion of
Charged Particles