Download Single Particle Motion in a Magnetized Plasma

Single Particle Motion in a Magnetized Plasma
Bounce Motion
At Earth, pitch angles are defined by the velocity
direction of particles at the magnetic equator,
therefore:
•  Particles with 90° pitch angle (i.e. all energy is in
V⊥) are trapped at the equator.
•  Particles with 0° or 180° pitch angle are moving
straight along the field line.
•  Particles with too much parallel speed will mirror
or ‘bounce’ at very low altitudes and thus will not
remain trapped since they will collide with
atmosphere and be lost from the system.
Aurora observed from the
Space Shuttle
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•  Hence there is a minimum pitch angle for trapped
particles, defined as the ‘loss cone’ -- it is this loss
or ‘precipitation’ of particles into the atmosphere
that creates the aurora!
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Single Particle Motion in a Magnetized Plasma
Bounce Motion
A few examples of magnetically trapped plasmas:
After space storms, plasma particles can
be accelerated to extremely high energies
(millions of eV!), and are then trapped in
the Earth’s magnetic field in intense zones
of radiation: Van Allen Belts
The turbulent magnetic
field of the sun forms
loops of intensified B-field
strength, trapping the
plasma in the Sun’s
atmosphere and forming
prominences.
Solar Prominences (from NASA SOHO)
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Single Particle Motion in a Magnetized Plasma
Bounce Motion
A few examples of magnetically trapped plasmas:
Courtesy of the Swedish Solar
Observatory
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Solar flare observed by
TRACE
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Homework/Reading/Last thoughts…
Reading Assignment: Finish though the end of Chapter
2 in Baumjohann Treumann
Note in the homework, the questions as for plasma
frequency, fpe , which is in Hz. ωpe is in radians/s and
needs to be converted to Hz. (f = ω/2π)
Upcoming -- Drift motion, finish up plasma motion… and
onto Solar activity.
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Introduction to Space Plasmas
 The
Plasma State
What is a plasma?
Basic plasma properties:
Qualitative & Quantitative
Examples of plasmas
 Single particle motion in a magnetized plasma
Gyromotion
Bounce motion
⇒ Drift motion
 Generating electric currents in plasmas
 Fluid approximation for plasmas
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Introduction to Space Plasmas
Drift Motion
Guiding Center -- The center of the orbit of a charged
particle defined by the gyroradius (rg)
proton
electron
B
Guiding Center
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Introduction to Space Plasmas
Drift Motion
⇒ Qualitative
Assume a force on the charge perpendicular to the B-field
proton
F
B
* Force is charge independent, i.e. gravity
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Introduction to Space Plasmas
Drift Motion
⇒ Qualitative
Assume a force on the charge perpendicular to the B-field
proton
VF
F
B
* Force is charge independent, i.e. gravity
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Introduction to Space Plasmas
Drift Motion
⇒ Qualitative
Assume a force on the charge perpendicular to the B-field
electron
F
B
* Force is charge independent, i.e. gravity
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Introduction to Space Plasmas
Drift Motion
⇒ Qualitative
Assume a force on the charge perpendicular to the B-field
VF
electron
F
B
* Force is charge independent, i.e. gravity
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Introduction to Space Plasmas
Drift Motion
⇒ Qualitative
Assume a force on the charge perpendicular to the B-field
VF
electron
proton
VF
F
B
* Force is charge independent, i.e. gravity
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Introduction to Space Plasmas
Drift Motion
⇒ Qualitative
Oppositely drifting ions and electrons establishes currents!
VF
electron
proton
VF
j
j = e( n i v i − n e v e ) , since n e ≈ n i
j = en e ( v i − v e )
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Introduction to Space Plasmas
Drift Motion
General Force Drifts: Forces perpendicular to the
magnetic field will cause charged particles to drift in a
direction perpendicular to both the force and the B-field.
The drift motion is superimposed on the gyromotion,
which is why describing it in terms of the guiding center is
a useful simplification.
Force on a Gyrating Particle
VF
1 $ F B' 1 F × B
vF =
& × )=
ωg % m B ( q B2
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Introduction to Space Plasmas
Drift Motion: E x B drift
What about forces that are not charge independent?
⇒ F=qE
From our generic equation, the drift velocity would be in
the same direction for both + and - charges. Therefore, no
currents are generated by this drift motion.
Force on a Gyrating Particle
VF
1 $ F B' 1 F × B
vF =
& × )=
ωg % m B ( q B2
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Introduction to Space Plasmas
Drift Motion: E x B drift
For and Electric
Field perpendicular
to B:
F=qE
1 F ×B
q B2
1 qE × B E × B
vE =
=
2
q B
B2
vF =
Ion
E
VE
B
Electron
Modified from B. & T.
For a fun interactive demo, check out:
http://www.phys.hawaii.edu/~teb/java/ntnujava/emField/emField.html
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Whiteboard Notes
Introduction to Space Plasmas
Drift Motion: ∇B drift
What happens when a charged particle experience an
increasing magnetic field during its orbit?
mv ⊥
rg =
qB
€
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Introduction to Space Plasmas
Drift Motion: ∇B drift
What happens when a charged particle experience an
increasing magnetic field during its orbit?
mv ⊥
, F∇ = −µ∇B
qB
1 F ×B
vF =
q B2
1 µ∇B × B 1
2 B × ∇B
v∇ = −
= mv ⊥
2
q B
2
qB 3
rg =
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Introduction to Space Plasmas
Drift Motion: Magnetic curvature drift
What happens when a charged particle moves along a
curved field?
B
R
2
FR = mv|| c2
Rc
FR
1 F ×B
vF =
q B2
1 mv|| 2Rc × B
vR =
q Rc 2 B 2
Total magnetic drift motion:
B × ∇B
v B = v R + v ∇ = (v|| + v ⊥ )
2
ωg B
2
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Introduction to Space Plasmas
Drift Motion: Summary
There are several other drifts that occur from perpendicular
forces such as gravity, and slowly time varying E-fields
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Introduction to Space Plasmas
Drift Motion: Creating Currents
For drift motion that acts differently for + and - charged
particles, associated currents are generated by the
differential flux of oppositely charged particles, i.e.
For E x B drift, the + and - charged particles drift in the
same direction, so no associated currents.
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Introduction to Space Plasmas
Drift Motion: Creating Currents
Example: Earth’s Magnetosphere
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Introduction to Space Plasmas
Fluid Approximation for Plasmas
Describing the plasma population as a whole:
Single Particle Motion -- Individual particle dynamics
calculated for all particles in the system, computationally $$$.
Magnetohydrodynamics (MHD) -- Single conducting
fluid, only bulk (or average) parameters are tracked, assumes
plasma maintains local equilibria, which works for low
frequency waves in highly conductive fluids in magnetic fields.
Multi-fluid -- Separates electrons and ion species into
distinct but simultaneously tracked fluids, coupled through
electric and magnetic fields. Similar to MHD, but can track
differential mass effects and higher frequency waves.
Kinetic Theory -- Statistical approach, tracks the
evolution of particle distribution functions.
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Introduction to Space Plasmas
Things to Review
  Plasma
Criterion
Debye Length/Sphere, Plasma Parameter, Plasma
Frequency
  Single
Particle Motion
Lorentz Force Law, Gyromtion, Bounce Motion, Drift
Motion, Adiabatic Invariant, Pitch Angle
  Net
Plasma Motion
Currents, Guiding Center, Approaches for
describing plasma population as a whole:
MHD, Multi-fluid, Kinetic
  General
Maxwell’s Equations, Div, Grad, Curl,
Cross products
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