Fall Term 2003
Plasma Transport Theory, 22.616
LITTLE Problem Set #3
Prof. Molvig
Passed Out: Sept. 25, 2003
DUE: Oct. 2, 2003
1. Moment Equation Structure: Develop the structure of moment equations (for a single
species) that follow from the kinetic equation,
µ
¶
q
1
∂
∂f
+ v · ∇f +
E+ v×B ·
f =0
∂t
m
c
∂v
assuming the distribution is close to Maxwellian,
f ' fM =
n
(2πT /m)3/2
Ã
m (v − V)2
exp −
2T
!
In particular show that the density moment gives,
∂n
+ ∇ · nV = 0
∂t
that the momentum moment gives,
¶
µ
¶
µ
1
∂
V + V · ∇V = −∇P + qn E + V × B − ∇ · π
mn
∂t
c
where the tensor, π, is the stress moment resulting from deviations from Maxwellian,
Z
¢
¡
π ≡ d3 vmvv f − f M
and the energy moment can be expressed as,
µ
¶
∂ 3
1
nT + mnV 2 + ∇ · Q = qnV · E
∂t 2
2
with, Q, the total energy flux,
5
1
Q = mnV 2 V+ nT V + q + π · V
2
2
This energy equation contains both internal and hydrodynamic flow energies. Eliminate the
hydro flow energy by subtracting, V·Momentum Equation, from energy equation to yield,
µ
¶
µ
¶
5
∂ 3
nT + ∇ ·
nT V + q − V · ∇P = −π : ∇V
∂t 2
2
which can also be manipulated into a form for the temperature alone,
µ
¶
∂
3
n
+ V · ∇ T + P ∇ · V = −∇ · q −π : ∇V
2
∂t
1
Finally show that with zero viscous stress, π =0, and zero heat flux, q =0, that this is
equivalent to the adiabatic equation of state,
D ³ −5/3 ´
Pn
=0
Dt
implying that P n−5/3 is constant following the fluid.
2