Download ESS154_200C_Lecture7_W2016

ESS 154/200C
Lecture 7
Physics of Plasmas
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ESS 200C Space Plasma Physics
ESS 154 Solar Terrestrial Physics
M/W/F
10:00 – 11:15 AM
Geology 4677
Instructors:
C.T. Russell (Tel. x-53188; Office: Slichter 6869)
R.J. Strangeway (Tel. x-66247; Office: Slichter 6869)
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1/29
2/1
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Topic
Instructor
A Brief History of Solar Terrestrial Physics
CTR
Upper Atmosphere / Ionosphere
CTR
The Sun: Core to Chromosphere
CTR
The Corona, Solar Cycle, Solar Activity
Coronal Mass Ejections, and Flares
CTR
The Solar Wind and Heliosphere, Part 1
CTR
The Solar Wind and Heliosphere, Part 2
CTR
Physics of Plasmas
RJS
MHD including Waves
RJS
Solar Wind Interactions: Magnetized Planets YM
Solar Wind Interactions: Unmagnetized Planets YM
Collisionless Shocks
CTR
Mid-Term
Solar Wind Magnetosphere Coupling I
CTR
Solar Wind Magnetosphere Coupling II;
The Inner Magnetosphere I
CTR
The Inner Magnetosphere II
CTR
Planetary Magnetospheres
CTR
The Auroral Ionosphere
RJS
Waves in Plasmas 1
RJS
Waves in Plasmas 2
RJS
Review
CTR/RJS
Final
Due
PS1
PS2
PS3
PS4
PS5
PS6
PS7
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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• Maxwell’s equations
– Gauss’s Law
• E is the electric field
• r q is the charge density
• e 0 is the electric permittivity of free space (8.85 x 10-12 Farad/m)
– Absence of magnetic monopoles
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B is the magnetic field
– Faraday’s Law
– Ampere’s Law
•
j is the current density,
•
m 0 is the permeability of free space, m0 = 4p ´10-7 H/m
• c is the speed of light, c  1 00
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The Motion of Charged Particles
• Equation of motion
m
dv
dv
= q ( E + v ´ B) + Fg + m
dt
dt c
– Fgstands for non-electromagnetic forces (e.g., gravity) – often ignored
–Last term is change in momentum from collisions – important in the
ionosphere
• SI Units
–mass (m) - kg
–length (l) - m
–time (t) - s
–electric field (E) - V/m
–magnetic field (B) - T
–velocity (v) - m/s
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• B acts to change the motion of a charged particle only in
directions perpendicular to the motion.
– Set electric field equal to zero, define perpendicular direction with
respect to magnetic field:
dv ^ q
= v^ ´ B
dt m
– Taking time derivative of this equation:
d 2 v ^ q dv ^
q2
q2 B2
=
´ B = - 2 B ´ ( v ^ ´ B) = - 2 v ^
2
dt
m dt
m
m
– This is the equation of motion for a simple harmonic oscillator:
d2 v^
2
=
-W
v^
2
dt
qB
where W is the particle gyro-frequency, W =
, with units of
m
radians/sec.
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• Solution is circular motion dependent on initial conditions.
Letting the magnetic field define the z-axis of a cartesian coordinate
system then from the x-component of the momentum equation:
dvx dt = ±Wvy
with the upper sign corresponding to a positively charged particle.
• Assuming vx = -v^ sin (Wt ), then
.
• On integration we find
where x0 are y0 constants of integration.
• Equations of circular motion with angular frequency  (cyclotron
frequency or gyro frequency). Upper signs are for positive charge, lower
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
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• Radius of circle - cyclotron radius or Larmor radius (rL) or gyro radius
(rg).
v^ = r gW
rg =
mv^
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
2
1
dv d ( 2 mv )
F×v = v×m =
= qv × ( v ´ B) = 0
dt
dt
Only the electric field can energize particles
Particle energy remains constant in absence of E
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• Next Step – Include an Electric Field
dv q
= ( E + v ´ B)
dt m
– For parallel motion:
mdv|| dt = qE||
– This implies parallel electric fields are localized or of short duration,
otherwise particles would be accelerated to relativistic energies. In
general assume E|| = 0 , and hence v|| is constant.
–
E^ has a major effect on motion.
• As a particle gyrates it moves along the electric field and gains energy
• Later in the gyration it losses energy.
• This causes different parts of the orbit to have different radii - the orbit
doesn’t close on itself
• Results in a net “ExB” drift
• No charge dependence, therefore no currents
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• Take time derivative of momentum equation:
d 2 v ^ q dv ^
q2
q2
2
=
´
B
=
E
+
v
´
B
´
B
=
E
´
B
v
B
(
)
(
)
^
^
2
2
2
dt
m dt
m
m
• Assume v^ = v ^ + v E , where v ^ is a time-varying term and v E is
constant. Then:
d2 v^
q2 B2
= - 2 v^
2
dt
m
which is the same equation for a uniform magnetic field, resulting in
gyration, and
vE =
E´B
B2
• This is the ExB drift, which is independent of mass and charge. All
particles drift with this velocity.
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Ions and electrons gyrate in the
opposite sense, but ExB drift in
the same direction, with the
same velocity
Trajectory depends on
gyrational speed versus ExB
speed
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•
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’=(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 = 440 km/s
• 1eV electron = 600 km/s, or a 2.5 keV electron has a velocity of c/10
– Gyration:
• Proton gyro-period: 66 s * (1nT/B)
• Electron: 28 Hz * (B/1nT)
– Gyroradii:
• 1keV proton in 1nT field: 4600 km*(m/mp)1/2*(W/keV)*(1nT/B)
• 1eV electron in 1nT field: 3.4 km *(W/eV)*(1nT/B)
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•
Any force capable of accelerating and decelerating charged particles can cause an
average (over gyromotion) drift:
ö
dv q æ F
= ç + v ´ B÷
dt m è q
ø
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This is functionally the same as the momentum equation with E replaced by F/q.
vg =
mg ´ B
qB 2
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Example, gravity:
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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 , rg. This
leads to the concept of guiding center motion.
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• The Concept of the Guiding Center
– Separates the motion of a particle into motion perpendicular (
)v
and
) vto|| the magnetic field.
^ parallel (
– To a good approximation the perpendicular motion can consist of
a drift ( v D ) and the gyro-motion ( v ^ )
v = v|| + v^ = ( v|| + v D ) + v^ = v gc + v^
– 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
-1
such that rg l <<1 and time scales  such that (Wt ) <<1
– Closely related concept – first adiabatic invariant (or magnetic
moment), as this is related to particle gyration.
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• Time-varying magnetic fields – magnetic moment
– Time-varying magnetic field results in induction electric field
around gyro-orbit:
2prg Ej = -prg2
¶B
¶t
– Positively charged particles move in the same direction as the
azimuthal electric field, negatively charged particles in the
opposite direction. So both increase in energy for increasing
magnetic field:
q ¶ B mv^ ¶ B
¶ v^
m
= q Ej = rg
=
¶t
2
¶ t 2B ¶ t
– Or, defining the perpendicular energy as W^ = 12 mv^2 ,
¶W^ W^ ¶ B
¶ æ W^ ö ¶m
=
, i.e., ç ÷ =
=0
¶t
B ¶t
¶t è B ø ¶t
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• Magnetic moment – current loop
– The current in the loop defined by one gyration of the particle =
q/, where  is the gyro-period, or, in vector notation:
– Dipole moment of a current loop is given by IA, where A is the
area of the current loop:
– Negative sign indicates that particles in a plasma are diamagnetic;
their magnetic moment opposes the applied field.
– Magnetic moment conserved for slowly time-varying magnetic
fields. Also the case for magnetic fields with large scale spatial
variation.
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• Magnetic mirror
– Consider cylindrical geometry, from
:
1 ¶
¶B
r Br ) = (
r ¶r
¶z
– Therefore:
d Br » -
rg ¶ B
v ¶B
»- ^
2 ¶z
2W ¶ z
– Azimuthal component of momentum equation gives
q v^v|| ¶ B
dv^
v|| ¶ B
m
= - q v||d Br =
=m
dt
2W ¶ z
v^ ¶ z
since
– Hence
d ( m B) dW^
¶B
dB
dm
=
= mv||
= m , and = 0
dt
dt
¶z
dt
dt
–  is conserved for both temporal and spatial changes in B.
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• Magnetic mirror – force on a magnetic dipole
– Same cylindrical geometry
– Again
rg ¶ B
v ¶B
d Br » »- ^
2 ¶z
2W ¶ z
– Axial component of momentum equation gives
q v^2 ¶ B
dv||
¶B
m
= q v^v||d Br = = -m
dt
2W ¶ z
¶z
– Hence force on a magnetic dipole is:
– Derived for axial field and gradient, but this is a general result.
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• Inhomogeneous magnetic fields cause grad-B drift.
– If B changes over a gyro-orbit rg will change.
– rg gets smaller when particle goes into stronger B,
results in a drift perpendicular to grad-B.
– Use general force drift to derive grad-B drift
– From force on magnetic dipole:
– From generalized drift, grad-B drift given by:
– vG depends on charge, yields perpendicular currents (but see later,
when we discuss MHD).
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• Curvature of the magnetic field also causes a drift
– The curvature of the magnetic field line introduces a drift motion.
– As particles move along the field they undergo centrifugal acceleration.
– Centrifugal force (rc is radius of curvature – positive outwards):
Fcf = r̂c mv||2 / rc = 2W||r̂c / rc
– In terms of the magnetic field:
– Hence
– Both gradient and curvature drift can also be derived explicitly using Taylor
expansion of the particle momentum equation.
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•
At Earth’s dipole
vG , vC are in the same direction and comparable:
– ~ 1RE/min=100km/s for 100keV particle at 5RE, at 100nT
vG,C
æ 5RE öæ 100nT öæ W ö
» 1RE / min [ ~ 100km / s] × ç
֍
֍
÷
è r øè B øè 100keV ø
– The drift around Earth is: 0.5hrs for the same particle at the same location
æ r ö æ B öæ 100keV ö
t G,C = 0.5h ç
÷ ç
֍
÷
è 5RE ø è 100nT øè W ø
2
•
At Earth’s magnetotail current sheet vG , vC are opposite each other:
– Curvature drift dominates at center of the current sheet, gradient drift
dominates further away from center of current sheet
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• The first adiabatic invariant – again
mv2
 
 const .
B
1
2
– As a particle moves to a region of stronger (weaker) B it gains (loses)
perpendicular energy.
– Conservation of  results in betatron acceleration if the magnetic field
changes in time (slowly).
– Pitch angle is defined as sin 2 a = W^ W
– Particles with pitch angle less than sin 2 alc = B Bmax will be lost from
the mirror – loss cone (Bmax is the maximum field in the mirror).
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• The second adiabatic invariant – particle bounce motion
– The integral of the parallel momentum over one complete bounce
between mirrors is constant (as long as B does not change much
in a bounce).
J=
ò p ds
||
Since
p||2
= W - m B = m ( Bm - B)
2m
Then
J = 2mm ò ( Bm - B) ds = const.
1/2
• 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, v|| will increase
• Fermi acceleration
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• The total particle drift in static E and B fields is:
• From conservation of 
• For equatorial particle (W|| = 0) and static electric field
– Particle conserves total (potential plus kinetic) energy
• Rewriting centrifugal force in terms of a potential ( rcw = v||, v||rc = a J ):
• Bounce-averaged motion for particles with finite J
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– Particle’s equatorial trace conserves total energy
– 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 does
not change much in the time
required to drift around a planet the
magnetic flux F = ò B× dS inside the
orbit must be constant.
– Note it is the total flux that is
conserved including the flux within
the planet.
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• Drift paths for equatorially mirroring (J=0) particles, or
• Drift paths for bounce-averaged particles, equatorial traces
in a realistic magnetosphere.
Note, corotation
electric field ignored
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• Limitations on the invariants
–

is constant when there is little change in the field’s strength over a
cyclotron path.
ÑB
1
<<
B
rg
– All invariants require that the magnetic field not change much in the time
required for one cycle of motion
1 B
1

B t

where  is the cycle period.
  ~ 106 103 s (electrons) ; 103 1 s (ions)
 J ~ 0.1 s 10 s (electrons); 1 s 10 m (ions)
  ~ 10 m 10 hrs
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