ESS 154 - Solar Terrestrial Physics A Brief Introduction
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Transcript ESS 154 - Solar Terrestrial Physics A Brief Introduction
ESS 200C - Space Plasma Physics
Winter Quarter 2009
Vassilis Angelopoulos
Robert Strangeway
Schedule of Classes
Date
1/5
Topic
[Instructor]
Organization and Introduction
to Space Physics I [A&S]
1/7
Introduction to Space Physics II [A]
1/12
Introduction to Space Physics III [S]
1/14
The Sun I [A]
1/21
The Sun II [S]
1/23 (Fri) The Solar Wind I [A]
1/26
The Solar Wind II [S]
1/28
--- First Exam [A&S]
2/2
Bow Shock and Magnetosheath [A]
2/4
The Magnetosphere I [A]
Date
Topic
2/9
The Magnetosphere II [S]
2/11
The Magnetosphere III [A]
2/18
Planetary Magnetospheres [S]
2/20 (Fri) The Earth’s Ionosphere [S]
2/23
Substorms [A]
2/25
Aurorae [S]
3/2
Planetary Ionospheres [S]
3/4
Pulsations and waves [A]
3/9
Storms and Review [A&S]
3/11
--- Second Exam [A&S]
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.
– Chen, F. F., Introduction of Plasma Physics and Controlled Fusion, Plenum
Press, 1984
– 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.
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
Cro-Magnon “macaronis” may be earliest
depiction of aurora (30,000 B.C.)
• Aurora over Los Angeles (courtesy V. Peroomian)
– Galileo theorized that aurora is caused by air rising out of
the Earth’s shadow to be illuminated by sunlight. (He also
coined the name aurora borealis meaning “northern dawn”.)
– Descartes thought aurorae 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.
• 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.
By the beginning
of the space age
auroral eruptions
had been placed
in the context of
the Sun-Earth
Connection
• 1716 Sir Edmund Halley
Aurora is aligned with Earth’s field…
• 1741
Anders Celsius
and has magnetic disturbances.
• 1790
Henry Cavendish
Its light is produced at 100-130km
• 1806
Alexander Humboldt but is related to geomagnetic storms
• 1859
Richard Carrington
and to Solar eruptions.
• 1866 Anders Angström
Auroral eruptions are self-luminous and…
• 1896
Kristian Birkeland
… due to currents from space:
• 1907
Carl Störmer
in fact to field-aligned electrons…
• 1932
Chapman & Ferraro accelerated in the magnetosphere by…
• 1950
Hannes Alfvén
the Solar-Wind–Magnetosphere dynamo.
• 1968
Single Satellites
Polar storms related to magnetospheric activity
• 1976
Iijima and Potemra
… communicated via Birkeland currents
• 1977
Akasofu
Magnetospheric substorm cycle is defined
• 1997
Geotail
Resolves magnetic reconnection ion dynamics
• 1995-2002 ISTP era
Solar wind energy tracked from cradle to grave
• 2002- Cluster era
Space currents measured, space-time resolved
• 2008
Substorm onset is due to reconnection
THEMIS
It All Starts from the Sun…
Solar Wind Properties:
• Comprised of protons (96%),
He2+ ions (4%), and electrons.
• Flows out in an Archimedean
spiral.
• Average Values:
– Speed (nearly Radial): 400 - 450 km/s
– Proton Density: 5 - 7 cm-3
– Proton Temperature 1-10eV (105-106 K)
Shaping Earth’s Magnetosphere
• Earth’s magnetic field is an
obstacle in the supersonic
magnetized solar wind flow.
• Solar wind confines Earth’s
magnetic field to a cavity
called the “Magnetosphere”
Auroral Displays: Direct Manifestation of
Space Plasma Dynamics
Societal Consequences of Magnetic Storms
• Damage to spacecraft.
• Loss of spacecraft.
• Increased Radiation Hazard.
• Power Outages and
radio blackouts.
• GPS Errors
Stellar wind coupling to planetary objects is ubiquitous in astrophysical systems
SUBSTORM
RECURRENCE: MERCURY: 10 min
EARTH: 3.75 hrs
JUPITER: 3 weeks
Magnetized wind coupling to stellar and galactic systems is common thoughout the Universe
ASTROSPHERE
Mira (a mass shedding red giant)
and its 13 light-year long tail
GALACTIC CONFINEMENT
In this class we study the physics that enable and control such planetary and stellar interactions
Introduction to Space Physics I-III
• Reading material
– Kivelson and Russell Ch. 1, 2, 10.5.1-10.5.4
– Chen, Ch. 2
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.
The Motion of Charged Particles
• Equation of motion
dv
m
qE qv B Fg
dt
• 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.
• 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.
mvx qvy B
mv y qvx B
qv y B
q 2vx B 2
2
vx
c vx
2
m
m
q 2v y B 2
2
vy
c vy
2
m
|q|B
c
m
• Solution is circular motion dependent on initial
conditions. Assuming at t=0: vx 0; v y v
vx v sin(ct )
and
v y v cos(ct )
v
x x0
cos( c t )
c
v
y y0
sin( ct )
c
– Equations of circular motion with angular frequency
c (cyclotron frequency or gyro frequency). Above
signs are for positive charge, below signs are for
negative charge.
• If q is positive particle gyrates in left handed sense
• If q is negative particle gyrates in a right handed sense
• Radius of circle ( rc ) - cyclotron radius or Larmor
radius or gyro radius. v
c
c
m v
c
qB
– 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 = 1.6X10-19
Joules (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
dv d ( 12 m v2 )
F v m
v
qv (v B) 0
dt
dt
Only the electric field can energize particles!
Particle energy remains constant in absence of E !
• The electric field can modify the particles motion.
E 0 but B still uniform and Fg=0.
– Assume
– Frequently in space physics it is ok to set E B 0
• Only E can accelerate particles along B
• Positive particles go along E and negative particles go
along E
• Eventually charge separation wipes out E
–
E has a major effect on motion.
• As a particle gyrates it moves along E and gains energy
• Later in the circle it losses energy.
• This causes different parts of the “circle” to have different radii it doesn’t close on itself.
EB
uE
B2
• Drift velocity is perpendicular to E and B
• No charge dependence, therefore no currents
Z
Y
X
• Assuming E is along x-direction, B along z-direction:
vx qEx c v y
v y c v x
vx c2 v x
Ex
vy ( v y )
B
2
c
• Solution is:
v x v sin( c t )
Ex
v y v cos( c t )
B
• In general, averaging over a gyration:
EB
dv
m
qE q v B 0 v
V E B
2
dt
B
•
Note that VExB is:
– Independent of particle charge, mass, energy
– VExB is frame dependent, as an observer moving
with same velocity will observe no drift.
– The electric field has to be zero in that moving frame.
• Consistent with transformation of electric field in moving frame: E’=g(E+VxB).
Ignoring relativistic effects, electric field in the frame moving with V= VExB is E’=0.
•
Mnemonics:
– E is 1mV/m for 1000km/s in a 1nT field
• V[1000km/s] = E[mV/m] / B [nT]
– Thermal velocities (kT=1/2 m vth2):
• 1keV proton = 440km/s
• 1eV electron = 600km/s
– Gyration:
• Proton gyro-period: 66s*(1nT/B)
• Electron: 28Hz*(B/1nT)
– Gyroradii:
• 1keV proton in 1nT field: 4600km*(m/mp)1/2*(W/keV)*(1nT/B)
• 1eV electron in 1nT field: 3.4km *(W/eV)*(1nT/B)
•
Any force capable of accelerating and decelerating charged particles can
cause an average (over gyromotion) drift:
FB
dv
m
q v B q( F / q) 0 v
VF
dt
qB2
– e.g., gravity
mg B
ug
qB2
– If the force is charge independent the drift motion will depend on the sign of the
charge and can form perpendicular currents.
– Forces resembling the above gravitational force can be generated by centrifugal
acceleration of orbits moving along curved fields. This is the origin of the term
“gravitational” instabilities which develop due to the drift of ions in a curved
magnetic field (not really gravity).
– In general 1st order drifts develop when the 0th order gyration motion occurs in a
spatially or temporally varying external field. To evaluate 1st order drifts we have
to integrate over 0th order motion, assuming small perturbations relative to c, rL
• The Concept of the Guiding Center
–
Separates the motion ( v ) of a particle into motion
perpendicular ( v ) and parallel ( v ) to the magnetic field.
– To a good approximation the perpendicular motion can
consist of a drift ( vD ) and the gyro-motion ( v )
c
v v v v vD vc vgc vc
– 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 very
useful for distances l such that c l 1 and time scales t
such that ct 1 1
• Inhomogenious magnetic fields cause GradB drift.
– If B changes over a gyro-orbit rL will change.
– c rL m v gets smaller when particle goes into stronger B.
qB
– Assume B is along Y
+
– Average force over gyration:
-
B
B zˆBz ( y ) zˆ ( Bz 0 y z 0 )
y
Fx 0
Fy qvx Bz ( y ) qv sin( c t )(Bz 0
Y
Z
x
B=Bzz
B
v
sin( c t ) z 0 )
c
y
B
v
1 m v Bz 0
Fy q sin 2 ( c t ) z 0
.
c
y
2 B y
2
2
2
1 m v
F
B
2 B
– uB depends on charge, yields perpendicular currents.
uB
B
B
B
B
2
2
1
1
m
v
2 m v
2
qB3
qB3
• The change in the direction of the magnetic field
along a field line can also cause drift (Curvature drift).
– The curvature of the magnetic field line introduces a drift
motion.
• As particles move along the field they undergo centrifugal
acceleration.
mv 2
F
Rc
Rˆ c
nˆ
(bˆ )bˆ)
• Rc is the radius of curvature of a field line (
Rc
where
B
ˆ
b , nˆ is perpendicular to B and points away from the center
B
of curvature, v is the component of velocity along B
uc
m v B (bˆ )bˆ
2
2
qB
m v B nˆ
2
Rc qB2
• Curvature drift can also cause currents.
•
At Earth’s dipole uB , uc are same direction and comparable:
–
uB , uc are ~ 1RE/min=100km/s for 100keV particle at 5RE, at 100nT
5R 100nT W
uB 1RE / min ~ 100km / s E
r B 100keV
–
The drift around Earth is: 0.5hrs for the same particle at the same location
t B ,C
•
r
0.5h
5RE
2
B 100keV
100nT W
At Earth’s magnetotail current sheet, uB , uc are opposite each other:
–
Curvature dominates at Equator, Gradient dominates further away from equator
• Parallel motion: Inhomogeniety along B (║B)
– As particles move along a changing field they experience force
• Parallel force is Lorenz force due to small dB perpendicular to nominal B (dW║)
• Force along particle gyration is due to dB/dt in frame of gyrocenter (dW┴)
• Total particle energy is conserved because there is no electric field in rest frame
– Consider 1st order force on 0th order orbit in mirror field
• B change due to presence of ║B must be divergence-less (B=0) so:
Z
Y
–
–
–
–
X
In cylindical coordinates gyration is: v v ; r r ; ct
The mirror field is: B 0; B 1 r (rBr ) r Bz z 0 Br 1 2 r Bz zr 0
The Lorenz force is: Fr q(v Bz ); F q(vr Bz vz Br ); Fz q(v Br )
From gyro-orbit averaging: Fr q v Bz F 0; ( Br vr 0) r 0
Fz 1 2 q v r Bz z r 0 1 2 m v2 / B Bz z
1 m v2
– Defining:
2 B
we get: F (B s) II B
– This is the mirror force. Note that is conserved during the particle motion:
dz
r
d
B
dvII
B s
dB
d m vII2
dB
F vII
vII m vII
s
dt
s t
dt
dt 2
dt
d
m v2
dB
d m v2
dB
d
dB
d
W
B
0
dt
2
dt
dt 2
dt
dt
dt
dt
dr
• Another way of viewing
– As a particle gyrates the current will be
q
I
Tc
where Tc
2
c
2
v
A rc2 2
c
IA
v2
q c
2 c2
m v2
2B
– The force on a dipole magnetic moment is
Where
dB
F B
dz
bˆ
I.e., same as we derived earlier by averaging over gyromotion
In a Magnetic Mirror:
• The force is along B and away from the direction of
increasing B.
• Since E 0and kinetic energy must be conserved
1
2
mv2 12 m(v 2 v2 )
a decrease in v must yield an increase in v
2
• Particles will turn around when B 12 mv
• The loss cone at a given point is the pitch angle below
which particles will get lost: sin 2 i Bi / Bm 1/ Rm
• Time varying fields: B
– As particle moves into changing field or
– As imposed/background field increases (adiabatic
compression/heating)
• An electric field appears
affecting particle orbit
B
E
t
• Along gyromotion, speed v┴ increases,
but μ is conserved:
dl
d
v m (v ) qE v qE , where : dl line _ elem ent
dt
dt
m v2
dB
d
qEdl q ( E )dS q
dS
dt
2 gyration gyro orbit
over _ surface
surface
0 for _ i
0 for _ e
0 for _ i
0 for _ e
m v2
dB
d
q
rL2 dB
dt gyration
2 gyration
d B dB d 0
const.
• Flux through Larmor orbit: =rL2B=(2 m/q2) remains constant
• Time varying fields: E
– See Chen, Ch 2
• Maxwell’s equations
– Poisson’s
Equation E
• E is the electric field
0
• is the charge density
• 0is the electric permittivity (8.85 X 10-12 Farad/m)
– Gauss’ Law
(absence of magnetic monopoles)
B 0
• B is the magnetic field
– Faraday’s Law
B
E
t
– Ampere’s Law
1 E
B 2
0J
c t
• c is the speed of light.
• Is the permeability of free space,
0
• J is the current density
0 4 107 H/m
• Maxwell’s equations in integral form
Note: use Gauss and Stokes theorems; identities A1.33; A1.40 in Kivelson and Russell)
Q
1
E E dS E dV dV E dS
A
0
0V
0
V
A
B 0 BdS 0
A
B
B
E
E dl ( E ) d S
dS
C
t
t
t
E
d
l
A
t
1 E
1
B 2
0 J B dl ( B) dS c 2 E dS 0 J dS
C
c t
t
1
CB dl c2 t E dS 0 I
– dS is a unit normal vector to surface: outward directed for closed
surface or in direction given by the right hand rule around C for
is magnetic flux through the surface.
– dl is the differential element around C.
open surface, and
• The first adiabatic invariant
B
–
E says that changing Bdrives E (electromotive
t
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.
m v2
const.
B
1
2
– 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).
• For a coordinate in which the motion is periodic the action integral
J i pi dqi constant
is conserved. Here pi is the canonical momentum: p mv qA
where A is the vector potential).
• First term:
mv ds 2 mv
• Second term:
4 m q
2 m2v2
qA ds q A dS qB dS qB 2 m q
• For a gyrating particle:
2m
J1 p ds
q
Note: All action integrals are conserved when the properties of the
system change slowly compared to the period of the coordinate.
• 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).
m2
m2
m1
m1
J p|| ds 2 m v|| ds 2 2m
J 2 2m
m2
m1
W B ds
Bm B ds cons.
• Bm is the magnetic field at the mirror point
– Note the integral depends on the field line, not the particle
– If the length of the field line decreases, u|| will increase
• Fermi acceleration
• The total particle drift in static E and B fields is:
EB 1 2
u D u EB uB uc
2 m v
2
B
EB
B B
B (b ) B
W
2
W
||
B2
qB3
qB3
rˆc B
EB
B B
2
W
||
B2
qB2
qRc B 2
ˆ
r
B
B B
2 c
m
v
||
3
2
qB
qRc B
• For equatorial particle in electrostatic potential (E=-):
B (q ) B (B) B (q W )
uD
qB2
qB2
qB2
– Particle conserves total (potential plus kinetic) energy
• Bounce-averaged motion for particles with finite J
B (q (r ) W ( , J , r ))
vD
qB2
– Particle’s equatorial trace conserves total energy
• Drift paths for equatorially mirroring (J=0) particles, or
• … for bounce-averaged particles’ equatorial traces
in a realistic magnetosphere.
– As particles bounce they also drift
because of gradient and
curvature drift motion and in
general gain/lose kinetic energy
in the presence of electric fields.
– If the field is a dipole and no
electric field is present, then 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 aplanet
the magnetic flux B dS
inside the orbit must be constant.
– Note it is the total flux that is
conserved including the flux
within the planet.
• Limitations on the invariants
–
is constant when there is little change in the field’s
strength over a cyclotron path.
B
1
B
c
– All invariants require that the magnetic field not change much in the
time required for one cycle of motion
1 B
1
B t
t
where
t is the cycle period.
t ~ 10 10 s
6
3
t J ~ 1s 1 min
t ~ 10m 10hrs
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 dvdr dvx dvy dvz dxdydz
• The number of particles in a phase space volume is f (r , v , t )dvdr
where f is called the distribution function.
– The density of particles of species “s” (number per unit
volume)
ns (r , t ) f s (r , v , t )dv
– The average velocity (bulk flow velocity)
u s (r , t ) v f s (r , v , t )dv / f s (r , v , t )dv
– Average random energy
1
2
2
ms (v u s )
1
2
2
ms (v u s ) f s (r , v , t )dv / f s (r , v , t )dv
– The partial pressure of s is given by
ps
2
(
ns
N
1
2
2
ms (v us ) )
where N is the number of independent velocity components
(usually 3).
– In equilibrium the phase space distribution is a Maxwellian
distribution
2
12 ms v u s
f s r , v As exp
kT
s
where As
ns m 2 kT
3
2
• For monatomic particles in equilibrium
1
2
2
ms (v us )
NkT / 2
• The ideal gas law becomes
ps n s kTs
– where k is the Boltzman constant (k=1.38x10-23 JK-1)
• This is true even for magnetized particles.
• The average energy per degree of freedom is:
Eaverage
1
1
2
ms (v us ) 2 kT / 2 ms vthermal
/2
2
2
– for a 1keV, 3-dimensional proton distribution, we mean:
kT=1keV, get Eaverage=1.5keV, and define vthemal=(2kT/mp)1/2=440km/s
– for a 1keV beam with no thermal spread
kT=0 and V=440km/s
– Other frequently used distribution functions.
• The bi-Maxwellian distribution
12 ms v u s
'
f s r , v As exp
kT s
3
1
exp
2
1
2
2
ms v us
kTs
– where As As Ts2 T sT s2
– It is useful when there is a difference between the distributions
perpendicular and parallel to the magnetic field
'
• The kappa distribution
2 1
ms v us
f s r , v As 1
E
Ts
1
2
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
• What makes an ionized gas a plasma?
– The electrostatic potential of an isolated charge q0
q
4 0 r
– The electrons in the gas will be attracted to the ion and will reduce the
potential at large distances, so the distribution will differ from vacuum.
– If we assume neutrality Poisson’s equation around q0 is
e
2 r
(ni ne )
0
0
– The particle distribution is Maxwellian subject to the external potential
1
f (v ) A exp ( mi ,e vi2,e q ) / kTi ,e ni ,e n expq / kTi ,e
2
• assuming ni=ne=n far away
– At intermediate distances (not at charge, not at infinity):
e kT 1
– Expanding in a Taylor series for r>0 for both electrons and ions
en e
e e 2 n
r
1
1
0 kTe
kTi 0 kT
1 1 1
T Te Ti
2
r
qeD
4 0 r
•The Debye length ( D ) is
1
2
TeTi
0 kT
D 2 ; T
Te Ti
ne
where n is the electron number density and now e
is the electron charge. Note: colder species dominates.
•The number of particles within a Debye sphere
3
4
n
D
needs to be large for shielding to occur
N
D
3
(ND>>1). Far from the central charge the electrostatic
force is shielded.
• The plasma frequency
– Consider a slab of plasma of thickness L.
– At t=0 displace the electron part of the slab by d e <<L and
the ion part of the slab by d i <<L in the opposite direction.
d de di
– Poisson’s equation gives
E
en0
0
d
– The equations of motion for the electron and ion slabs are
d 2d e
me
eE
2
dt
d 2d i
mion 2 eE
dt
e 2 n0
e 2 n0
d 2d d 2d e d 2d i
2 2 (
)d
2
dt
dt
dt
0 me 0 mion
– The frequency of this oscillation is the plasma frequency
p 2 2pe 2pi
2
pe
e 2 n0
0 me
2
e
n0
2pi
0 mion
– Because mion>>me
p pe
• Useful formulas:
1
2
T
D 7.4m ; T (eV ), n(cm3 )
n
3/ 2
T
N D 1.7 109 1/ 2 ; T (eV ), n(cm3 )
n
f pe 8.9kHz n ; n(cm3 )
f pi f pe / sqrt(mi / me ) f pi / 43 210Hz n ( for protons)
• A note on conservation laws
– Consider a quantity that can be moved from place to place.
– Let f be the flux of this quantity – i.e. if we have an element of area
then
f d A
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 s is the density of the substance then the total amount in the
volume is
s dV
V
– The rate at which material is lost through the surface is
d
s dV f dA
dt V
S
s
– Use Gauss’ theorem
V t f dV 0
–
f dA
S
s
f
t
An equation of the preceeding form means that the quantity whose
density is s is conserved.
dA
• Magnetohydrodynamics (MHD)
– The average properties are governed by the basic
conservation laws for mass, momentum and energy in a
fluid.
– Continuity equation
ns
ns u s S s Ls
t
– 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, s ms ns , s dr M s where Ms is
the total mass of s and dr is a volume element (e.g. dxdydz)
M s
M s
( s u s )dr
s u s ds
t
t
where ds is a surface element bounding the volume.
– Momentum
equation
s us
t
susus ps qs E J s B s Fg ms
u s
s (
u s u s ) ms us ( S s Ls ) ps qs E J s B s Fg ms
t
q
n
where qs
s s is the charge density, J s qs ns us is the
current density, and the last term is the density of nonelectromagnetic forces.
– The operator ( u s ) is called the convective derivative
t
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.
us us
– 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
( u) 0
t
– Add the momentum equations and use me<<mp
Momentum
u
( u u ) p J B Fg m
t
• Energy equation
1 2
2
( 2 u U ) [( 12 u U )u pu q ] J E u Fg m
t
where q is the heat flux, U is the internal energy
density of the monatomic plasma (U nNkT / 2) and
N is the number of degrees of freedom
–
q adds three unknowns to our set of equations. It is usually
treated by making approximations so it can be handled by
the other variables.
– Many treatments make the adiabatic assumption (no change
in the entropy of the fluid element) instead of using the
p
2
energy equation
u p cs ( u )
g
t
or t
p const.
2
where cs is the speed of sound cs
g p and g c p cv
cp and cv are the specific heats at constant pressure and
constant volume. It is called the polytropic index. In
thermodynamic equilibrium g ( N 2) N 5 / 3
• Maxwell’s equations
B
E
t
B 0 J
– B 0 doesn’t help because
( B)
E 0
t
– There are 14 unknowns in this set of equations - E, B, J , u, , p
– We have 11 equations.
• Ohm’s law
– Multiply the momentum equations for each individual species
by qs/ms and subtract.
1
1 me J
J s {( E u B) pe J B 2 [ ( Ju )]}
ne
ne
ne t
where J qs nsus and s is the electrical conductivity
s
– Often the last terms on the right in Ohm’s Law can be dropped
J s ( E u B)
– If the plasma is collisionless, s may be very large so
E uB 0
B
– Combining Faraday’s law ( E
), and
t
Ampere’ law ( B J ) with J s ( E u B)
0
B
2
(u B) m B
t
where m 1 s 0 is the magnetic viscosity
• Frozen in flux
– If the fluid is at rest this becomes a “diffusion” equation
B
2
m B
t
– The magnetic field will exponentially decay (or diffuse) from a
conducting medium in a time t D L2B mwhere LB is the system size.
– On time scales much
shorter than t D
B
(u B)
t
– The electric field vanishes in the frame moving with the fluid.
– Consider the rate of change of magnetic flux
d d
B
B nˆdA
nˆdA B (u dl )
A
A
C
dt dt
t
– 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 u dl is the area swept out per unit time
– Use the identity A B C C A B and Stoke’s theorem
B
d
(u B) nˆdA 0
A
dt
t
– 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.
• Magnetic pressure and tension
2
FB J B 10 ( B) B B 2 0 ( B ) B 0
since A B B A A B
–
2
B
pB
20
A magnetic pressure analogous to the plasma
pressure ( p )
–
p
B2
2 0
A “cold” plasma has 1 and a “warm”
plasma has 1
– In equilibrium J B p
• Pressure gradients form currents
Frozen-in Theorem Recap – 2
Rate of change of line element:
dr'
r2'
r1'
U1dt
r1
r1' = r1 + U1dt
r2' = r2 + U2dt
U2dt
dr
r2
Taylor Expansion:
U2 U1 dr U1
dr r2 r1 dr dt dr U1
i.e.,
Dd r
dr U
Dt
Frozen-in Theorem Recap – 3
Rate of change of magnetic field:
E B t ,
E UB 0
B t U B U B B U B U U B
DB
B U B U
Dt
Mass Conservation:
t U 0,
i.e., t U U D Dt
B
DB
B D
B
B U
, i.e., D Dt U
Dt
Dt
c. f .
Dd r
dr U
Dt
• Magnetic pressure and tension
FB J B 10 ( B) B B2 20 (B )B 0
since
–
A B B A A B
pB B
2
A magnetic pressure analogous to the plasma
20
pressure ( p )
–
p
B2
2 0
A “cold” plasma has
has
– In equilibrium
1
J B p
• Pressure gradients form currents
1 and a “warm” plasma
– The second term in
[( B ) B]
–
ˆ
b B B
0
2
bˆbˆ B
J B
0
can be written as a sum of two terms
ˆ
b B B
0
(B
2
ˆ
ˆ
0 )b b
2 0 cancels the parallel component
of the B
term. Thus only the perpendicular component
20
of the magnetic pressure exerts a force on the plasma.
2
–
2
ˆ
n
B
(B
)bˆ bˆ (
) is the magnetic tension and is
0
0 RC
2
directed antiparallel to the radius of curvature (RC) of the
field line. Note that nˆ is directed outward.
•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
p p0 sin(kx t )
–A sinusoidal wave can be described by its frequency
and wave vector k . (In the organ pipe the frequency is
f and 2 f . The wave number is k 2 ).
B(r, t )
B0 cos(k r t) i sin(k r t )
B(r, t ) B0 exp{i(k r t )}
• The exponent gives the phase of the wave.
The phase velocity specifies how fast a feature
of a monotonic wave is moving
v ph
k
2
k
• 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
v g
k
• The phase and group velocities are calculated
and waves are analyzed by determining the
dispersion relation
(k )
• When the dispersion relation shows asymptotic
behavior toward a given frequency, res, vg goes to
zero, the wave no longer propagates and all the wave
energy goes into stationary oscillations. This is called a
resonance.
• MHD waves - natural wave modes of a
magnetized fluid
– Sound waves in a fluid
• Longitudinal compressional oscillations which propagate
at
1
2
p g p
cs
1
2
1
2
• c g kT and is comparable to the thermal speed.
s
m
– Incompressible Alfvén waves
• Assume s , incompressible fluid with B0 background
field and homogeneous
z
B0
J
•
b, u
y
x
Incompressibility u 0
• We want plane wave solutions b=b(z,t), u=u(z,t), bz=uz=0
• Ampere’s law gives the current
b ˆ
J 10
i
z
• Ignore convection (
u )=0
u
p J B
t
• Sincep 0, J y 0 and J z 0 the x-component of
x
momentum becomes
u x
p
( J y B0 by J z ) 0
t
x
u uˆj
E uB0iˆ
• Faraday’s law gives
E
b ˆ
u ˆ
j x ˆj B0
j
t
z
z
• The y-component of the momentum equation
u y
B b
1
becomes
J B 0
t
0
0 z
• Differentiating Faraday’s law and substituting the ycomponent of momentum
2b
2u B 2 2b
2
B0
2
t
zt 0 z
2
2b
2 u
CA 2
2
t
z
1
where
B2 2
is called the Alfvén velocity.
C A
0
• The most general solution is b b( z C t ) . This is a
A
disturbance propagating along magnetic field lines at the
Alfvén velocity.
• 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 0
T 0
pT p0 p
BT B0 b
uT u
JT J
ET E
– Continuity
0 ( u )
t
u
1
p ( B0 ( b ))
– Momentum 0
t
0
p
2
– Equation of state p ( ) 0 Cs
– Differentiate the momentum equation in time, use Faraday’s
law and the ideal MHD condition E u B0
b
( E ) (u B0 )
t
2
u
2
Cs ( u ) C A ( ( (u C A ))) 0
2
t
B
where C A
1
( 0 ) 2
– For a plane wave solution
u ~ exp[i(k r t )]
– The dispersion relationship
between the frequency ( ) and the
propagation vector (
k ) becomes
2u (Cs2 CA2 )(k u)k (CA k )[(CA k )u (CA u)k (k u)CA ] 0
This came from replacing derivatives in time and space by
i
t
ik
ik
ik
– Case 1 k B0
2
2
u (Cs CA )(k u )k
2
k
• The fluid velocity must be along and perpendicular to B0
k
B0
• These are magnetosonic waves
u
v ph
2
2 12
k ( ) (C s C A )
k
k B0
– Case 2
(k 2CA2 2 )u ((Cs2 CA2 ) 1)k 2 (CA u)CA 0
•
A longitudinal mode with u k with dispersion relationship
Cs
k
(sound waves)
• A transverse mode with
(Alfvén waves)
k
k u 0
CA
and
• 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
VA=2CS
k and B0
B0
VA
S
F
I
Phase Velocities