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Massachusetts Institute of Technology
Subject 6.651J/8.613J/22.611J R. Parker
17 November 2005 Due: 29 November 2005
Problem Set 8
Problem 1.
In this problem, we will calculate the damping of Alfvn waves
using the cold plasma two-fluid modelincluding collisions between
ions and electrons. The equilibrium plasma is considered to be
homogenous and without flow.
As usual, we assume that all variables take the form )exp(
tirikXx = r
where X is a complex
constant. The linearized two fluid equations can then be written
in the form:
)( ''0 VVmBVqEqVmirrrrrr
+=
where n0is the constant equilibrium density and 0Br
is a uniform magnetic field which we will take in
the z-direction. We will further assume that the ion charge is
ecorresponding to singly charged ions.
Note from momentum conservation
''' mm = .
a) The solution of these equations for the ion and electron
velocities is rather complicated. However, asa first step show that
the ion and electron velocities satisfy the equations
eyie
i
i
ci
ciexie
i
i
ci
y
i
i
ci
cix
i
i
ci
ix Vm
mV
m
miE
m
qE
m
qiVi
+
++
+
++
=
22222222)
exie
i
i
ci
cieyie
i
i
ci
x
i
i
ci
ciy
i
i
ci
iy Vm
mV
m
miE
m
qE
m
qiVii
+
+
+
+
=
22222222)
iyei
e
e
ce
ceixei
e
e
ce
y
e
e
ce
cex
e
e
ce
ex Vm
mV
m
miE
m
qE
m
qiViii
+
++
+
++
=
22222222)
ixei
e
e
ce
ceiyei
e
e
ce
x
e
e
ce
cey
e
e
ce
ey Vm
mV
m
miE
m
qE
m
qiViv
+
+
+
+
=
22222222)
where )1(,
,,
eieiei imm +=
and ceciei
ei
cecim
m,
,
,
,
= .
For Alfvn waves we are interested in frequencies much less than
the cyclotron frequency.
Accordingly, expand the denominators in these expressions using
the relation
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B
Ei
B
EE
m
q yx
cice
ieeiieei
cice
yx
cicecice
yx
ie
ie
cice
,
2
,
,,
,
,
2
,
2
,
,
,
,
2
,
2))2(1(
1)1(
11
+++
+
correct to first order incice,
and
ieei,. Then further approximate the electron terms using
)1(1
))2(1(1
2
ei
cece
eiei
ce
ii +++ .
and notice that
B
EiE
m
q yxie
cice
yx
ie
ie
cice
cice ,,
2
,
2
,
,
,
2
,
2
,))21(1(
++
+
b) Using these approximations and consistently ignoring terms of
order ce/ and , show that
the above equations can be simplified to:
22 / ci
i) eyci
ieex
ci
iey
ci
iex
ci
ieie
ci
ix VVi
B
Ei
B
Ei
iV
+++++
=
222)21())2(1(
ii) exci
ieey
ci
iex
ci
iey
ci
ieie
ci
iy VViB
Ei
B
Ei
iV
+++
=
222)21())2(1(
iii) iyce
eiyx
ce
eiex
VB
E
B
EV
++=
iv) ixce
eixy
ce
eiey
V
B
E
B
EV
=
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
consists of polarization and BErr
drift, while the electrons undergo simply the BErr
drift.
Substituting the electron velocities into the ion equations and
ignoring terms of order22
/ ceei we get
i)B
Ei
B
Ei
iV
y
ci
iex
ci
ie
ci
ix )1()21( 22
+++
=
ii)B
Ei
B
Ei
iV x
ci
iey
ci
ie
ci
iy )1())21( 22
++
=
and back substituting into the electron equations yields
iii)B
Ei
B
EV
y
ce
ei
ci
y
ex
+=
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iv)B
Ei
B
EV x
ce
ei
ci
xey
+=
In obtaining these expressions, only terms up through and
including the first power of the collisionfrequency are
retained.
Then,
B
Ei
iVV x
ci
ie
ci
exix )21( 2
+
=
B
Ei
iVV
y
ci
ie
ci
eyiy )21( 2
+
=
and the elements of the dielectric tensor are found to be
simply
0),1(2
2
=+= KiKci
pi
where2
2ci
ie
= .
c) Use this result to calculate the damping rate of an Alfvn
wave correct to lowest order in the
collision frequency.
Problem 2.
Using your favorite plotting software (Matlab, IDL, etc.)
prepare a graph accurately depicting regionsof propagation and
cutoff in the log n
2 log(/ci) 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(/ci) < 4
and the ordinate range to -2 < log n2< 3.
