Massachusetts Institute of Technology
Subject 6.651J/8.613J/22.611J
30 November 2006
R. Parker
Due: 8 December 2006
Problem Set 9
Problem 1.
In this problem we’ll develop a general (if a bit formal) derivation of the Vlasov equation.
As we discussed in class, conservation of particles requires
r r
D( fΔr Δv )
= 0,
Dt
where the symbol D/Dt means that the time derivative is to be taken along a particle orbit in phase
space. Specifically,
r r
r r r r r r
D( f )
f (r + v Δt , v + a (r , v , t )Δt , t + Δt ) − f (r , v , t )
= lim(Δt → 0)
.
Dt
Δt
a) Show that the RHS of this expression reduces to
r
∂f r
+ v ⋅ ∇f + a ⋅ ∇ v f
∂t
b) Now consider the term
r r
r r
r r
Δr Δvrr + vrΔt ,vr + arΔt ,t + Δt − Δr Δvrr ,vr ,t
D (Δr Δv )
= lim(Δt → 0)
Δt
Dt
As indicated, the rate of change of the phase space volume is to be calculated along a particle trajectory
in phase space. For small Δt along this trajectory, the particle orbit is simply given by
r r r
r ' = r + v Δt
r r r r r
v ' = v + a (r , v , t )Δt
r r
r r
These equations define a simple transformation of variables between the r , v and r ' , v ' coordinates.
Accordingly, a well-known mathematical result is that the volume elements are related by the Jacobian
of the transformation defined as the determinant of the matrix
∂x′j ⎤
⎡
⎢cij =
⎥.
∂xi ⎦
⎣
r r
D(Δr Δv )
and complete the derivation of Vlasov’s equation.
Use this result to calculate
Dt
Problem 2.
In deriving the fluid energy equation in class, we got to the following point:
r r m r 2
r t r⎞
r
3
5
∂ ⎛1
⎛1
⎞
2
2r
⎜ mnu + nkT ⎟ + ∇ ⋅ ⎜ mnu u + nkTu + Π ⋅ u ⎟ + ∇ ⋅ q − qE ⋅ nu = ∫ dv v ∑ C ( f , f β ) .
2
2
2
∂t ⎝ 2
⎠
⎝2
⎠
β
It was then stated that this equation could be transformed into the simpler form:
r
r
∂u
3 d kT
n
+ p∇ ⋅ u = −Π ij i − ∇ ⋅ q + Q ,
2 dt
∂x j
where the summation convention applies and
Q=
m r 2
dww ∑ C ( f , f β ) .
2∫
β
Show that the second form follows from the first by using the continuity equation, the result of dotting
r
the momentum equation with u , and the identity
t
t r
r
∂u
− u ⋅ (∇ ⋅ Π ) + ∇ ⋅ (Π ⋅ u ) = Π ij i .
∂x j
Problem 3.
Consider 1-D plasma oscillations proportional to exp(−iωt + ikx) in a hot plasma with a 1-D electron
distribution function given by
~
v
f e (v x ) = e
1
.
π v + ve2
2
x
For simplicity assume that k is real, but that ω could be complex.
a) Determine an algebraic dispersion relation for electron oscillations, assuming that the ions are
immobile.
b) Solve the dispersion relation obtained in a) for ω (k ).
c) Now assume that the ions have a distribution function given by
~
v
f i (v x ) = i
1
,
π v + vi2
2
x
while the electron distribution function is the same as in part a). Assuming ω / k << ve , determine
ω (k ) for ion acoustic waves.
Possibly useful integrals:
∞
vx
1
π ⎧ (ς − ive ) 2 ⎫
dv
=
−
∫ (vx2 + ve2 ) 2 vx − ς x 2ve ⎨⎩ (ς 2 + ve2 ) 2 ⎬⎭ Im ς > 0
−∞
∞
v
1
π
∫−∞ (vx2 +xve2 ) 2 vx − ς dvx = − 2ve
∞
∫ (v
−∞
2
x
⎧ ( v + iς ) ⎫
vx
1
Im ς > 0
dv x = π ⎨ 2e
2
2 ⎬
+ ve ) v x − ς
⎩ (ve + ς ) ⎭
2
x
⎧ ( v − iς ) ⎫
vx
1
Im ς < 0
dv x = π ⎨ 2e
2
2 ⎬
+ ve ) v x − ς
⎩ (ve + ς ) ⎭
∞
∫ (v
−∞
⎧ (ς + ive ) 2 ⎫
Im ς < 0
⎨ 2
2 2⎬
⎩ (ς + ve ) ⎭
∞
v x2
1
π ⎧ (ς − ive ) 2 ⎫
dv
=
−
∫ (vx2 + ve2 ) 2 vx − ς x 2ve ς ⎨⎩ (ς 2 + ve2 ) 2 ⎬⎭ Im ς > 0
−∞
∞
v x2
1
π ⎧ (ς + ive ) 2 ⎫
dv
=
−
∫ (vx2 + ve2 ) 2 vx − ς x 2ve ς ⎨⎩ (ς 2 + ve2 ) 2 ⎬⎭ Im ς < 0
−∞