Plasma Electrodynamics

1
Macroscopic Plasma Behaviors

• Physics of plasma fluid
• Plasma diamagnetism
• Braginskii equations
• Plasma waves
• Landau damping

2
Physics of plasma fluid

• Fluid velocity not the same with the velocity
  of guiding centers

• Pressure balance

In a steady-state,
for s e, i
For complete plasmas,
Using Amperes law,
For straight magnetic field,

• Pressure tensor

Force due to pressure tensor,
For anisotropic pressure,
3
Physics of plasma fluid

• Frozen in magnetic field

In a perfectly conducting fluid,
Magnetic flux through each surface moving with
the fluid is constant and consequently that the
magnetic flux can be thought of as frozen-in to
the fluid and moving with it.

• Polarization

In a perfectly conducting fluid,
polarization current
dielectric constant
4
Plasma Diamagnetism
Plasmas in magnetic fields are naturally
diamagnetic, the gyro-orbits of the charged
particles being such as to reduce the field.
current
With a distribution of particle velocities,
negligible for low ?
Or, from pressure balance equation,
5
Plasma Diamagnetism
Full pressure balance
Current associated with the magnetic field Bs
arising from the stationary orbits
Current caused by the magnetic field gradient
drift
Then,
6
Fluid Equations
Equation of state
Continuity equation
Equation of motion
These fluid equations can be closed with the
approximated expression of isotropic pressure
term from the equation of state. For more
accurate estimation, these quantities can be
determined by solving the kinetic equation with
the collision term given by the Fokker-Planck
equation as following
7
Braginskii Equations
The distribution functions can be expanded about
a Maxwellian distribution.
In a uniform plasma,
for Maxwellian distribution, and
with ?f from small gradients and drift
velocities.
8
Continuity and Momentum Equations
Continuity equation
Momentum equation
Friction force
Thermal force
9
Stress Tensors
Stress tensor
In a strong magnetic field
Rate of strain tensor
Viscosity coefficients
10
Energy Equation
Electron heat flux
Heat exchange between ions and electrons due to
collisions
11
Plasma Waves
Solve Maxwells equations and linearized plasma
equations together
Perturbed quantities in the form
When the current density j is determined in terms
of E from the linearized plasma equations by
setting the variables in the form of AAo A1 .
For non-trivial solutions of E, dispersion
relation need to be satisfied. And the refractive
index is defined by nn
For free space without any plasma, j0
12
Plasma Oscillations
Linear perturbation theory
Longitudinal oscillations the plasma current
and the displacement current cancel.
13
Transverse Electromagnetic Waves
In a transverse electromagnetic wave, the plasma
current and the displacement current do not
cancel.
Electromagnetic Wave
Wave propagates only for
Cut-off density
14
Sound Waves
At low frequencies, ion motions should be included
negligible electron inertia,
Adding them together,
from quasi-neutrality
Equation of state for adiabatic processes
Sound speed
15
Dielectric Tensors for Low-Frequency Waves

• Linearized fluid equation of motion

• Linearized continuity equation

with the angle ? between k and Bo,

• Linearized current density

• Wave equation and dielectric tensor

• Dispersion relation with tensor notation

16
Cold-Plasma Wave Equation

• wave equation with tensor notation

electron and ion plasma frequencies
where
electron and ion cyclotron frequencies
17
Cold-Plasma Dispersion Relation

• Dispersion relation for cold-plasma

• Using S2-D2RL and cos2? 1-sin2? ,

R-wave

• For parallel propagation (? 0),

L-wave
X-wave

• For perpendicular propagation (? ?/2),

O-wave
18
Magnetohydrodynamic Waves
Shear Alfven wave wave vector parallel to the
magnetic field
Linearized MHD equations,
From these equations,
19
Magnetohydrodynamic Waves
Magnetosonic wave Plasma displacement lies in
the plane containing the wave vector and the
magnetic field. Need adiabatic equation for
pressure.
Linearized MHD equations,
From these equations,
20
Magnetosonic Waves
fast
fast
slow
slow
21
CMA(Clemmow-Mullaly-Allis) Diagram

• The CMA(Clemmow-Mullaly-Allis) diagram divides
  the plane into a number of regions such that
  within each region the characteristic topological
  forms of the phase velocity surfaces remain
  unchanged.
• Slow and fast waves
• The solid lines represent the principal
  resonances and the dashed lines the
  reflection(cutoff) points.
• The magnetic field increases in the vertical
  direction, the plasma electron density increases
  in the horizontal direction.

22
Summary of Plasma Waves

• R-Wave (k//Bo and E1?Bo) two pass bands

High frequency pass band, vp --gt c as ? --gt ? .
Whistler wave, becoming shear Alfven R-wave at
low frequency, vp --gt vA as ? --gt 0 .

• L-Wave (k//Bo and E1?Bo) two pass bands

High frequency pass band, vp --gt c as ? --gt ? .
Shear Alfven L-wave, vp --gt vA as ? --gt 0 .

• Langmuir oscillation (k//Bo and E1//Bo)

Zero group velocity Langmuir oscillation vp
undefined

• For finite temperature (k//Bo and E1?Bo)

Langmuir wave, vp --gt ?3 vte as ? --gt ? .
Ion sound wave, vp --gt cS as ? --gt 0 .

• O-Mode (k ? Bo and E1//Bo) one pass band

High frequency pass band, vp --gt c as ? --gt ? .

• X-Mode (k ? Bo and E1 ? Bo) three pass bands

High-pass region of X-mode, vp --gt c as ? --gt ? .
Mid-pass region of X-mode, vp c as ? ?p .
Compressional Alfven (Magnetosonic) wave, vp --gt
vA as ? --gt 0 .
23
Landau Damping
Linearized form of the electron Vlasov equation
Gausss law for the perturbed electric field
Fourier and Laplace transforms
Inverse transforms
Dispersion relation