Massachusetts Institute of Technology
Subject 6.651J/8.613J/22.611J
1 December 2005
R. Parker
Due: 8 December 2005
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.
Use the 1-D model of the SOL developed in class to calculate the temperature at the divertor plates for
Alcator C-Mod using the following assumptions and parameters:
Conduction limited regime ( nu is neglible)
Temperature equilibrated ( Te = Ti ≡ T )
Length of divertor channel l = 10 m
Temperature channel input = 50 eV
Power density at channel input = 500 MW/m2.
Sketch T (x) where x is distance along the divertor channel from the input.