Plasmas as Fluids
At this point we need to use a number of basic equations that describe plasmas as fluids.
While it is possible to calculate these equations from first principles, using Maxwell’s
electromagnetic field equations and Maxwell’s velocity distribution (Man that guy did
everything!) the process is tedious, fairly difficult and very time consuming.
As the aim
of this class is to develop a basic feel for plasmas (wisdom rather than book smarts) in
this class we will simply assume that the equations are correct.
For those of you who feel
more adventuresome, notes going through those derivations are found at
//www.utdallas.edu/~goeckner/plasma_sci_class/
Please realize that those derivations
took ~ 4 class periods to complete.
The basic equations are follows:
Boltzmann’s Equation
df
dt
= 0 = ∇
ρ
φ
(
29 ∞ω+
∇
ω
φ
(
29 ∞α+
∂
φ
∂
τ
This is relatively easy to prove…
df
dt
=
∂
φ
∂ρ
δ
ρ
δτ
+
∂
φ
∂ω
δ
ω
δτ
+
∂
φ
∂
τ
= ∇
ρ
φ
(
29 ∞ω+
∇
ω
φ
(
29 ∞α+
∂
φ
∂
τ
= ω∞∇
ρ
φ
(
29 +
Φ
μ
∞∇
ω
φ
(
29
+
∂
φ
∂
τ
Zeroth moment of the Boltzmann Equation – The Equation of Continuity (Particle
conservation)
f
c
=
∇
ρ
∞
ν
ω
(
29 +
∂
ν
∂
τ
Moments are derived by multiplying by
v
moment
f(
v
) and integrating over all velocity.
Thus
what we are seeing is a measure of the ‘average’ of this particular parameter.
First moment of the Boltzmann equation – Momentum Conservation
This is also known as the fluid equation of motion
mn
∂ ω
∂
τ
+
ω
∞∇
ρ
ω
= ∆Μ
χ
μομεντυμ
λοστωια
χολλισιονσ
123

μ
ω
φ
χ
μομεντυμ
χηανγε
ωια παρτιχλεγαιν
/
λοσσ
1 2
4
3
4
 ∇
ρ
∞Π+
θν
Ε +
ω
∧ Β
(
29
Poisson’s Equation (which comes straight from Maxwell’s EM equations)
∇
2
F = 
r
e
=
e
e
n
e

n
i
(
)
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Boltzmann Density Relation
n
=
ν
0
εξπ
ε
Φ
κΤ
ε
Now we have the basic equations (There are a few higher moments that come into play
elsewhere that we will not have to deal with in this class.) we can begin to see how
plasmas react as a whole.
Remember plasmas have these ‘collective’ behaviors – which
we are going to explore now.
(We need to understand these behaviors so that we can
understand many of the diagnostics used in plasmas.)
The first behavior that we will explore is the ability of the plasma to shield out static
electric fields.
This behavior should make sense as any strong electric field in a plasma
will separate the negative and positive charge carriers.
The distance over which a field
can penetrate is known as the Debye length.
(Note that electromagnetic fields can also be
shielded out BUT the behavior is very different.
We will get to this soon.)
Debye Length
We can now calculate the Debye length – an effective length over which a plasma will
shield an electric field.
(The length is the 1/e distance for reducing a potential.)
First, we have Poisson’s equation
∇
2
F = 
r
e
=
e
e
n
e

n
i
(
)
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 Spring '07
 SMITH

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