Massachusetts Institute of Technology
Subject 6.651J/8.613J/22.611J
17 November 2005
R. Parker
Due: 29 November 2005
Problem Set 8
Problem 1.
In this problem, we will calculate the damping of Alfvén waves using the cold plasma two-fluid model
including collisions between ions and electrons. The equilibrium plasma is considered to be
homogenous and without flow.
r
As usual, we assume that all variables take the form x = X exp(ik ⋅ r − iωt ) where X is a complex
constant. The linearized two fluid equations can then be written in the form:
r
r
r r
r r
− iωmαVα = qα E + qα Vα × B0 − mαν αα ' (Vα − Vα ' )
r
where n0 is the constant equilibrium density and B0 is a uniform magnetic field which we will take in
the z-direction. We will further assume that the ion charge is e corresponding to singly charged ions.
Note from momentum conservation mαν αα ' = mα ν
' α 'α .
a) The solution of these equations for the ion and electron velocities is rather complicated. However, as
a first step show that the ion and electron velocities satisfy the equations
i ) Vix =
ωci∗
mi
ωci∗
mi
qi
qi
− iω
iω
ν
V
ν ieVey
+
−
+
E
E
x
y
2
∗2
∗
2
∗2
∗
2
∗2
∗ ie ex
2
∗2
− ω + ωci mi
− ω + ωci mi
− ω + ωci mi
− ω + ωci mi∗
ωci∗
mi
ωci∗
mi
qi
qi
− iω
iω
ν V −
ν ieVex
Ey −
Ex −
ii ) Viy =
2
∗2
∗
2
∗2
∗
2
∗2
∗ ie ey
2
∗2
− ω + ωci mi
− ω + ωci mi
− ω + ωci mi
− ω + ωci mi∗
iii ) Vex =
ωce∗
me
ωce∗
me
qe
qe
− iω
iω
ν
V
ν eiViy
+
−
+
E
E
x
y
2
∗2
∗
2
∗2
∗
2
∗2
∗ ei ix
2
∗2
− ω + ωce me
− ω + ωce me
− ω + ωce me
− ω + ωce me∗
ωce∗
me
ωce∗
me
qe
qe
− iω
iω
ν V −
ν eiVix
Ey −
Ex −
iv) Vey =
2
∗2
∗
2
∗2
∗
2
∗2
∗ ei iy
2
∗2
− ω + ωce me
− ω + ωce me
− ω + ωce me
− ω + ωce me∗
where mi∗,e = mi ,e (1 + i
m
ν i ,e
) and ωci∗ ,ce = i∗,e ωci ,ce .
ω
mi ,e
For Alfvén waves we are interested in frequencies much less than the cyclotron frequency.
Accordingly, expand the denominators in these expressions using the relation
Ex, y
ν
ων
qe , i
ω 2 Ex, y
1
1
1
≈
(
1
+
)
≈
(1 + i ( ei ,ie + 2 2ei ,ie ))
E
x, y
∗2
∗
∗
∗2
2
ωce,ci
ωce,ci B ωce,ci
ω
ωce,ci
B
− ω + ωce,ci me,i
correct to first order in
ν
ω
and ei,ie . Then further approximate the electron terms using
ω
ωce,ci
1
ωce
ν ei
ων
ν
1
+ 2 2ei )) ≈
(1 + i ei ) .
ωce
ωce
ω
ω
(1 + i (
and notice that
ωce∗ ,ci
qe , i
ν e ,i E x , y
ω2
≈
(
1
+
(
1
+
2
))
E
i
x
,
y
− ω 2 + ωce∗2,ci me∗,i
ωce2 ,ci
ω
B
b) Using these approximations and consistently ignoring terms of order ω / ωce and ω 2 / ωci2 , show that
the above equations can be simplified to:
ν ie
ων
E
ων E y ων ie
ν
+ 2 2ie )) x + (1 + i 2 2ie )
− i 2 Vex + ie Vey
ωci
ω
ωci
B
ωci B
ωci
ωci
Ey
ν
ων
ων E
ων
ν
− iω
ii′) Viy =
(1 + i ( ie + 2 2ie ))
− (1 + i 2 2ie ) x − i 2ie Vey − ie Vex
ωci
ω
ωci
B
ωci B
ωci
ωci
ν E E y ν ei
iii′) Vex = ei x +
+
Viy
ωce B B ωce
ν E y E x ν ei
iv′) Vey = ei
−
−
Vix
ωce B B ωce
i′) Vix =
− iω
(1 + i (
to first order in the collision frequency, i.e., when only terms proportional to the collision frequency
are retained. In the absence of collisions, we can see from these expressions that the ion response
r r
r r
consists of polarization and E × B drift, while the electrons undergo simply the E × B drift.
Substituting the electron velocities into the ion equations and ignoring terms of order ν ei2 / ωce2 we get
ων ie E x
ων E
)
+ (1 + i 2ie ) y
2
ωci
ωci B
ωci B
E
ων
ων E
− iω
ii′′) Viy =
(1 + i 2 2ie )) y − (1 + i 2ie ) x
ωci
ωci
B
ωci B
i′′) Vix =
− iω
(1 + i 2
and back substituting into the electron equations yields
iii′′) Vex =
Ey
B
+
− iω ν ei E y
ωci ωce B
iv′′) Vey = −
E x iω ν ei E x
+
B ωci ωce B
In obtaining these expressions, only terms up through and including the first power of the collision
frequency are retained.
Then,
ων ie E x
)
ωci2 B
ωci
ων E
− iω
(1 + i 2 2ie ) y
Viy − Vey =
ωci
ωci B
Vix − Vex =
− iω
(1 + i 2
and the elements of the dielectric tensor are found to be simply
ω pi2
K ⊥ = 2 (1 + iγ ), K× = 0
ωci
where γ = 2
ων ie
.
ωci2
c) Use this result to calculate the damping rate of an Alfvén wave correct to lowest order in the
collision frequency.
Problem 2.
Using your favorite plotting software (Matlab, IDL, etc.) prepare a graph accurately depicting regions
of propagation and cutoff in the log n2 – log(ω/ωci) plane for two sets of Alcator C-Mod parameters: a)
ne=1020 m-3, B = 5 T and b) ne = 4x1020 m-3 , B = 5 T. In both cases assume that the ions are deuterium.
In the plots, set the abscissa range to -2 < log(ω/ωci) < 4 and the ordinate range to -2 < log n2 < 3.