Experimental
observation
Walter Gekelman,
of Alfvhn wave cones
David Leneman, James Maggs, and Stephen Vincena
Department of Physics, Universiv of California at Los Angeles, Los Angeles, California 90024-1696
(Received 16 May 1994; accepted 22 August 1994)
The spatial evolution of the radial profile of the magnetic field of a shear Alfven wave launched by
a disk exciter with radius on the order of the electron skin depth has been measured. The waves are
launched using wire mesh disk exciters of 4 mm and 8 mm radius into a helium plasma of density
about 1.0X lo’* cme3 and magnetic field 1.1 kG. The electron skin depth ~=c/w,,~ is about 5 mm.
The current channel associated with the shear Alfvin wave is observed to spread with distance away
from the exciter. The spreading follows a cone-like pattern whose angle is given by tan 0=k,S,
where k, is the Alfven wave number. The dependence of the magnetic profiles on wave frequency
and disk size are presented. The effects of dissipation by electron-neutral collisions and Landau
damping are observed. The observations are in excellent agreement with theoretical predictions
[Morales et al., Phys. Plasmas 1, 3765 (1994)]. 0 1994 American Institute of Physics.
I. INTRODUCTION
Alfvkn waves are of fundamental importance in many
laboratory and space plasmas. When ions are strongly magnetized these waves communicate information about changes
in electrical current configurations and magnetic field topologies. In space plasmas Alfvin waves play key roles in many
naturally occurring interactions. For example, changes in the
aurora1 current magnitude and spatial configuration, or
changes in the magnetospheric configuration, involve propagation of information by Alfven waves. Alfvin waves may
play a fundamental role in the generation of magnetospheric
substorms and the coupling of current systems and transport
of energy between the magnetotail and ionosphere. In the
solar environment energy is transported from the photosphere through the solar atmosphere and into the solar wind.
The absorption of Alfvdn wave energy may play a key role in
heating the solar corona. In laboratory plasmas Alfvdn waves
are present in all fusion plasmas as background oscillations.
Heating schemes for both electrons and ions using Alfven
waves have been proposed and tried. The importance of Alf&n waves in the dynamics of a burning tokamak plasma are
still being assessed.
Alfvdn waves are a low frequency phenomena. In cold
plasmas, at frequencies below the ion cyclotron frequency,
the Alfvin wave dispersion relation has two branches, the
compressional wave and the shear wave. The compressional
wave is isotropic and characterized by fluctuations in both
the magnetic field strength and plasma density. The shear
wave is highly anisotropic, propagating along the ambient
magnetic field direction and, to first order, is characterized by
fluctuations in the direction, but not magnitude, of the magnetic field. Their existence, in the context of magnetohydrodynamic fluid theory, was predicted by Alfven in 1942.’
The first experimental evidence for the existence of Alfvdn waves appeared ten years later. A standing wave of the
appropriate frequency was observed by Bostick and Levine*
in a pulsed toroidal device. Subsequent experiments in
toroida13 and linear devices4 measured the wave phase velocity from the phase shift of detected wave magnetic fields, and
verified its dependence on magnetic field strength and denPhys. Plasmas 1 (12), December
1994
sity. The dispersion of the shear wave was measured by Jephcott and Stockers in a cylindrical column. The results compared favorably with a theory by Woods6 which included
boundary conditions and collisional damping by neutrals.
Despite their importance, comparatively few basic laboratory experiments have been done on these modes.7 The
primary reason for this is that the study of Alfvdn waves
requires large, dense, highly magnetized plasmas. Therefore,
most of the basic work on these waves has been done in
either research tokamaks or arc discharge plasmas. Both of
these plasma devices have drawbacks for studying the basic
properties of Alfvin waves; the geometry of the tokamak
magnetic field is inherently complex, and the large neutral
fill pressures of arc discharges result in inherently collisional
plasmas. Despite these difficulties many of the basic properties of the Alfvin modes have been studied and verified in
these plasmas. The properties of bounded Alfven modes have
been studied in fusion related and basic physics
experiments.* Recent studies of both the shear and compressional mode were done in a narrow, partially ionized
column.’ Wave propagation and polarization were measured
but damping by neutral particles was found to significantly
change the dispersion from the fully ionized case.
One aspect of Alfvdn wave propagation which has not
received much attention is the plasma response to a localized
excitation. We report here on a series of experiments on the
propagation of the shear mode launched by a small disk exciter. The fundamental scaling parameter in these experiments is the ratio of the size of the exciter to the collisionless
skin depth. For excitation by sources with size on the order
of the skin depth the propagation characteristics of the shear
wave change dramatically. The wave energy does not propagate directly along the ambient magnetic field. The radiofrequency (RF) current channel spreads radially in a pattern
contained within the Alfvdn cones. Our results are compared
to the theoretically predicted magnetic field patterns contained in a companion paper.14 A related experiment using a
localized source was performed by Cross in a linear
machine” and Borg and Cross in a small tokamak.” These
experiments differ from those reported here in that they were
performed in a highly collisional regime.
1070-664X/94/1 (12)/3775/9/$6.00
0 1994 American Institute of Physics
3775
Downloaded 13 Sep 2003 to 128.97.30.131. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
II. EXPERIMENTAL ARRANGEMENT
RF ampbfier
These experiments were conducted in a machine designed to study space plasma physics phenomena. The
LAPD (LArge Plasma Device) at UCLA’* (University of
California at Los Angeles) is a flexible, low maintenance
device with a plasma column up to 80 cm in diameter and 10
m in length. The vacuum system consists of three, electrically isolated stainless steel chambers with 128 radial ports
and a dozen end ports and has a base pressure of 5X 10m7
Ton: The vacuum chambers are surrounded by 68 pancake
magnets. The magnet power supply configuration allows for
a variety of magnetic field profiles ranging from uniform to
mirror geometry. The experiments reported here were performed in a uniform 1.1 kG magnetic field.
The plasma is produced by a DC (direct current) discharge driven by an oxide (BaO) coated cathode. This type
of plasma is quiescent (&zln=5%), and oxide coated cathode sources are stable for long periods of time (~6 weeks)
giving plasma discharges which are very reproducible from
shot to shot. The plasma used in these experiments was a He
plasma with a column diameter of 25 cm and density
1.0X lO’*/cm”. The discharge is pulsed at 1 Hz to allow for
efficient signal averaging and data processing. The discharge
typically lasts 5-10 ms after which the discharge pulse is
terminated. The termination phase of the pulse lasts about
0.5 ms during which the negative bias on the cathode falls
slowly, as does the plasma electron temperature but the
plasma density remains nearly constant. When the pulse is
completely terminated the temperature falls rapidly and the
density begins to decay.
At the LAPD laboratory we have achieved very highly
ionized helium plasmas (percentage of ionization =90%)13
during the active discharge, but these experiments were conducted in the termination phase of the discharge when the
electron temperature was about 4 eV. The neutral pressure,
within the plasma column, at the time of the measurements is
not known. However, the largest possible value of the neutral
collision frequency, based on the neutral fill pressure, is
350.0 l&z.
Figure l(a) shows the experimental setup for launching
and detecting the shear Alfvin wave. The radially movable
exciter is a circular disk of wire mesh (transparency=S%)
attached to the inner conductor of a coaxial cable. The outer
conductor is isolated from the plasma and the support is
made of glass. The antenna is mounted on a bellows to allow
for positioning in a transverse plane.
To excite the waves, a phase locked RF tone burst is fed
to the disk exciter through a broadband amplifier and isolation transformer. As shown in Fig. 1, the signal ground is
referenced with respect to an electrically floating Cu plate
terminating the plasma column. The modulated grid potential
drives a field aligned RF current which excites Alfven
waves. The tone burst was launched 120 p after the active
discharge, a time when the electron density is still high but
plasma magnetic noise is low.
The wave magnetic field was measured with three orthogonal magnetic induction loops. Each probe consists of
two coils, oppositely wound, so that any electrostatic pickup
could be subtracted out. Each coil consists of 58 turns,
3776
Phys. Plasmas, Vol. 1, No. 12, December 1994
/-
o*tbt(mw uth?e
FIG. 1. Schematic of experimental setup showing the plasma source, radially movablemagneticloop and disk exciter. The radial probe was inserted
through several ports along the device (z) axis. The insert on the bottom
shows the discharge current as a function of time. The plasma current decays
rapidly in the 0.5 ms interval during which the discharge is terminated.
During this interval the plasma density remains nearly constant.
wound on a spool with radius 2.5 mm and thickness 3 mm.
The received signals were subtracted, amplified, and subsequently digitized. The magnetic probes were calibrated using
a long straight wire modulated at frequency 300 kHz.
Ill. THEORY
The details of a theory of the magnetic field pattern radiated by a small disk exciter at frequencies below the ion
cyclotron frequency are presented in a companion paper.14
However, for convenience, we summarize its main conclusions here. A detailed comparison of the theoretical predictions with experimental findings follows.
The magnetic field radiated by a disk exciter in a cylindrical geometry is assumed to be azimuthally symmetric.
This symmetry implies that the associated electric field has
both a radial component and axial component. The Fourier
amplitude of the axial electric field is related to the radial
amplitude as follows,
.&k,
,z) = - &
;
I
&(k,
,z)
s
where k, is the perpendicular wave number and k, = l/S. For
a disk exciter with radius a, comparable to the skin depth,
the Fourier amplitudes of the electric fields contain kL values
on the order of k, and the axial electric field is approximately
kA/ks times the radial field. This axial electric field drives
electron currents along the magnetic field. The excitation of
these parallel eIectron currents is the mechanism by which
the disk exciter couples to the plasma.
The parallel wave number of the shear Alfven wave in a
cold plasma (i.e., in the inertial regime) is given by
kl;=k;(
1 +k:S*),
Gekelman
ef al.
Downloaded 13 Sep 2003 to 128.97.30.131. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
=1.>50&
'
8‘m m
bisk Exciter’
= 1.8 kA, B = 1.1 kG 100 us afterglow
exciter
FIG. 2. Schematic diagram illustrating the edge of the Alfven cone pattern.
For r > redsethe pattern falls off as I/r. The cone angle 4=taK’(k,S)
is of
order one degree in the experiment. The inner cone crosses the field line
through the disk center at an axial distance, z=al(k,6)
after which the
pattern is hollow. The data shown in subsequent figures are always at axial
positions larger than the cone crossing location.
where kA is the Alfvin
Ii:= &v~(
wave number
1 - ,2#)“2
(3)
The corrections to the standard MHD
with v~=B2/47rnMi.
(magnetohydrodynamic) shear Alfvdn dispersion relation
(i.e., ki=ki)
due to the nonzero value of 8, cause the wave
energy to spread perpendicular to the magnetic field. In the
general case (i.e., with a nonzero electron thermal velocity),
the dispersion relation is given by
(4)
where [=wlkllU, V is the electron thermal velocity, and Z’(L)
is the derivative with respect to argument of the plasma dispersion function.
With azimuthal symmetry the magnetic field has only a
component in the azimuthal direction, B=B,(~,~)c!@. The
spatial dependence of the radiated magnetic field is given by
an integral expression involving the first-order Bessel function Jt ,
Bdr,z)=
2
Q)
dk,
I 0
sin k,a
k
I
Jl(k,r)exp[ikll(k,)z].
(5)
The dependence of parallel wave number on perpendicular
wave number [i.e., klfk,)] is found by solving the dispersion
relation [either Eq. (2) or (4)].
The plasma currents associated with the shear wave are
contained within a conical structure which is the superposition of cones emanating from all points around the edge of
the disk. This structure is illustrated in cross section in Fig. 2.
The radial position of the outer edge of the cone is given by
rdge=kA&+a.
(f-5)
At any fixed axial position away from the disk exciter, the
radial profile of the magnitude of the wave magnetic field
has three characteristic features. First, the field is always zero
at the disk center. Second, it increases with radial distance
away from the disk center and reaches a peak value. The
radial location of the peak value increases with axial distance
away from the exciter. Third, upon reaching the position,
redge (the location of the outer Alfvdn cone), the field decreases as l/r. The l/r decrease outside the Alfvin cone
,:-.
,,“‘\
.:/“\
‘-..,
..,...‘-‘-dv, ,.,: ‘\ ;’ ‘.,
‘ii ,,,/ \ ,:i >.‘\,,,./ ‘.\.,,\,,,...^“‘L - ...,.
L,
...A ‘kc,:I” ....
;...’
L
!
z= 157 cm
z= 252 cm
____-..-..
z= 346 cm
f = 240,OOkHz
I I
I a I I I..
I
I , I.
I
10
5
15
20
25
0
30
35
40
tmle (“5)
FIG. 3. Alfvin tone
diameter disk exciter
seen to increase with
ity can be estimated
bursts received at three axial distances from an 8 m m
(located at z =O). The phase shift in the wave is clearly
increasing distance from the exciter. The phase velocfrom this diagram and is of order 8X IO7 cm/s.
indicates that all wave currents are contained within the
cone. At a fixed radius larger than r&se (i.e., outside the
Alfvdn cone) the magnetic field has the same value at all
axial locations. This means that the l/r portion of two radial
profiles taken at different axial locations should overlap.
IV. EXPERIMENTAL
RESULTS
Tbe first task is to establish that shear Alfvin waves can
be launched in the LAPD device using a wire mesh disk
exciter. The dispersion of the launched waves can be measured by using a phase-locked tone burst and measuring the
received signal at various axial locations. In experiments
measuring the wave dispersion the launched waves should be
linear. For the disk exciter used in these experiments it was
observed that harmonic generation and distortion of the received signal occurred when the applied peak-to-peak voltage exceeded 40 V. These experiments were performed in the
linear regime at Vpp = 15 V. The magnitude of the received
signal varied spatially, but no component exceeded a value of
40 mG for a ratio of wave magnetic field to background field
10-5.
of aBIB-4X
The wave phase velocity along the magnetic field may
be obtained from observing the change in wave phase as a
function of axial position. Figure 3 shows the temporal variation of the received wave signal at three axial locations in the
device. The phase velocity of the wave can be directly obtained from such observations by measuring the rate at which
a point of fixed phase (e.g., a wave crest) moves along the
magnetic field. The dispersion relation can then be obtained
by measuring the phase velocity of the wave at several frequencies.
Figure 4 compares the measured dispersion of waves
launched in the LAPD with the predicted dispersion values.
The expected dispersion of the shear Alfven wave (for
Phys. Plasmas, Vol. 1, No. 12, December 1994
Gekelman et al.
3777
Downloaded 13 Sep 2003 to 128.97.30.131. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
= 0.75
= 240.0
25
kHz
20
7
mm 15
g
P 3.0x105
:
s
r:
-
2.0x105
-
1.ox1oJt
z"
L:
10
.,,f,.,
I
I
200
.
I
400
Wavelength
I
r
600
(cm)
,
1
800
,
,
j
r/a
FIG. 4. Measured shear Alfven dispersion. The data was acquired in a
separate experiment with a 50 cm diameter plasma column. The launched
wave was a tone burst as in Fig. 3. An axially movable magnetic loop was
used to measure the wave field. The solid curve is a plot of Eq. (3) for a
density of 1.0X 10” cme3. The dashed curves are for densities of 0.9X 10”
cme3.
and 1.1X10'*
,I’.
I
’
”
+
3778
Phys. Plasmas, Vol. 1, No. 12, December 1994
I
”
I
= 280.0
-
’
kHz
_._._._._
& = 252
cm
12
16
a
4
0
I
= 157 cm
-dz
k,=O) is given by Eqs. (2) and (3). The launched waves
closely match the predicted dispersion in a plasma with density 1.0X lOI cmW3. No accurate measure of the plasma density was available, but the density measured using a Langmuir probe was found to be 8.0X10” cmm3. The measured
dispersion shown in Fig. 4 demonstrates that shear Alfven
waves are launched from the disk exciter.
The main objective of these experiments was to measure
the radial profile of the magnetic field radiated by a disk
exciter. For this purpose, the radial dependence of the time
derivative of the magnetic field radiated by a copper wire
mesh disk exciter, oriented with its normal along the magnetic field, was measured at various axial locations. A phaselocked tone burst was fed to the disk antenna through a
broadband amplifier. The center frequencies of the tone
bursts used in these experiments were 240,280 and 320 kHz.
The ion cyclotron frequency was, f,i=420 kHz. The three
components of the radiated magnetic field signal were measured using a triaxial magnetic induction loop probe inserted
radially into the cylindrical plasma chamber. The magnetic
loops were oriented so that one loop measured axial magnetic field and the other two loops were positioned to pickup
the transverse field. Measurements were taken every 2.5 mm
along a radial line, starting from the field line through the
center of the disk. The disk center was located using a small
electron beam. Two disk sizes (4 and 8 mm radius) were
used. Radial scans were taken at three axial locations (157,
252, and 346 cm from the disk location) for the 8 mm disk,
and two axial locations (157 and 252 cm) for the 4 mm disk.
At each axial location, measurements were obtained by
changing the frequency of the emitter at each radial location,
taking a ten shot average of the received signal, and then
changing radial position to repeat the process.
The magnitude of the measured magnetic field as a function of radial position is presented in Figs. 5 and 6. In pro-
j
20
16
12
8
4
0
20
r/o
m 10
F
I
,,
,,.’
P.,,,, l % _._.. ‘\.
%.__
- .-._
,!
L._
,l”
,’
5
,,
‘.- .-.-.-._. _,__
-‘.,_._.
,,+..;
~
00
0
.,
d’
1..
:\
4
8
12
16
20
r/a
FIG. 5. Radial magnetic fietd profile jBB(r)l for a 4 m m radius disk exciter,
p =0.75, at two axial locations. The parameter p = a/ S and dz denotes the
axial separation between exciter and receiver. The figures are for frequencies: (a) 240 ~Hz, flf,i=O.57, (b) 280 ~Hz, f/fCi=0.67, and
320 kHz,
flf,,=O.76.
(c)
Gekelman et ai.
Downloaded 13 Sep 2003 to 128.97.30.131. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
= 240.0
-
2
0
kHz
dz = 157
4
cm
a
6
10
r/a
“E
m 20
ET
I
2
0
4
a
6
10
r/a
40
"
( '.
'
4
"
"
I"'
= 1.51
fp = 320.0
"
kHz
dz
_._._._._dz
=
=
157
V. COMPARISON
cm
252 cm
-_..- .._ dz = 346 cm
0
0
.I.!*,,I...I(.(1%..
2
4
6
a
10
r/a
FIG. 6. Radial magnetic field profile III,(r)/ for 8 mm radius disk exciter,
p = 1.5 at three axial locations. The figures are for frequencies (a) 240 kHz,
flf,,=O.57,
(b) 280 kHz, f/fci=0.67,
and (c) 320 kHz, f/fci=0.76.
The
boxes in (c) denote the radial positions at which the hodograms presented in
Fig. 7 were taken.
Phys. Plasmas, Vol. 1, No. 12, December
1994
cessing the data, zero levels (which were very small -5%)
were subtracted out. Wave amplitudes were determined from
the received signals using a least-squares fit to sine waves of
the same frequency as the input signal.
The data exhibit several general features. The measured
magnetic field is nearly zero at a radial position which corresponds to the middle of the disk and rises to a peak value.
For each frequency, the rate of rise in magnetic field amplitude (i.e., the slope of B vs r) decreases with increasing axial
distance. After attaining its peak value the field may exhibit
some fluctuations, but at large radial distance the field shows
a llr decrease in amplitude. For each frequency and disk
size, the amplitude of the field in the l/r regime is nearly
identical for all axial locations. For a fixed frequency, the
location of the initial peak in amplitude moves out in radius
with increasing axial distance. At a fixed axial location there
is a tendency for the location of the peak amplitude to move
out in radial position for increasing frequency.
The polarization of the shear wave may be ascertained
by plotting hodograms of the data. Figure 7 shows
hodograms [B,(t) vs I,]
taken at three radial locations
(rla=O, 2.8, and=5.0) in the radial magnetic field profile
and 5.0
shown in Fig. 6(c). At the radial locations rla=2.8
the magnetic field is linearly polarized in the vertical direction as shown in Figs. 7(b) and 7(c). Data acquired along a
radial scan in the vertical direction (i.e., perpendicular to the
radial scans shown in Figs. 5 and 6) show the magnetic field
to be linearly polarized in the horizontal direction. These
measurements are consistent with an azimuthal polarization
for the radiated magnetic field as assumed in the theory.
Close to r=O, the wave appears to be circularly (or in some
cases elliptically) polarized as shown in Fig. 7(a). This effect
is likely due to the finite size (r=2.5 mm or 1.5R,,) of the
magnetic probe which averages signals over its detection
area.
WITH
THEORY
The observed radial profiles of magnetic field can be
compared with those predicted by theory using Eqs. (4) and
(5). Equation (4) is used to find the functional dependence of
the parallel wave number on perpendicular wave number
[i.e., k&k,)] and this result is then used in Eq. (5) to numerically compute the radial magnetic field profile.
In the theory the amplitude of the magnetic radial profile
is set by the current flowing to the disk exciter, I,. In comparing theory to observation the amplitude of the theoretical
profile is treated as a variable. In the comparisons shown
here, the amplitude of the theoretical profile is chosen so that
the l/r portion of the observed and theoretical curves match.
The choice of amplitude determines the theoretical value of
the current flowing to the disk. The current flowing to the
disk was measured using a calibrated magnetic current
probe. The measured current is consistently lower than the
theoretical value, indicating that the disk antenna couples
better to the plasma than in the theoretical model. The agreement is best at the highest frequency (the measured current is
about 80% of the theoretical value at 320 kHz) and worst at
Gekelman
et a/.
3779
Downloaded 13 Sep 2003 to 128.97.30.131. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
I 0.00
< t < 18.00~~
= 0.75
= 320.0
kHz
- dz = 157 cm
. . theory vs I’
Neutral Collisions
op-i ,,,,),.,,,,
0
4
-
~~_
t.,.I,,.j
8
12
16
20
r/a
10.00
-<
t < 18.00~s
ff
r
1 j
$;
I
:
j i
dz = 252 cm
... .... . theory vs I’
Neutral Collisions
,; j,
:
/ ,i;
,$
I
,;;:
r/a =
2.8
-30 ~I.........,.......J,........,.........I.........~
-20
-10
0
10
-30
Bx
20
30
0
4
8
12
16
20
r/a
FIG. 8. (a) The radial profile for the smaller exciter at axial position z= 157
c m and 320 kHz excitation frequency (solid line) is compared to theory with
only electron-neutral collisions Tire best match is for P= vlw=O.4. (b) The
radial field profile 95 c m farther
the exciter than shown in (a) is also
compared to theory with only collisions. At this axial position the experimental observations are best matched with P- 1. In both cases the collision
frequency needed to fit the data corresponds to higher neutral gas pressure
than was injected into the machine.
from
-30
~........,..,......~-...........,,~
-10
-20
-30
0
10
20
30
Bx
A. Effects of collisions
FIG. 7. Plots of B,(t) vs B,(t) (hodograms) at the three radial positions
indicated in Fig. 6(c). The hodograms are taken for 8 ,U (2.6 wave periods)
during &hemiddIe of the tone burst. At r>a the wave is linearly polarized in
the azimuthal direction. The overall spatial pattern indicates that the wave is
azimuthally polarized; E is in milligauss.
the lowest frequency (measured current about 60% of theoretical at 240 kHz).
7‘he other parameters that can be varied in the theory are
the collision frecpKncy and the electron temperature. First we
discuss the results of varying the collision frequency to obtain a fit, then the effects of varying the electron temperature
and finally the results of varying both parameters.
3780
Php. Pftlsmas, Vol. 1, No. 12, December 1994
Figure 8 shows the comparison of theoretical radial profiles and the observed profiles for the 4 m m radius disk at
320 l&z frequency for various values of normalized collision frequencies (r=v/w). At the axial position z= 157 cm
[Fig. 8(a)] the peak in the theoretical prediction is at larger r
value than the observed peak and the initial slope of the
predicted pattern is somewhat smaller than the observed
slope. The best fit to the observed peak amplitude would be
obtained using a value of r=O.4, but the theoretical profile is
broader than the observed profile, At the axial location
z =252 cm [Fig. 8(b)] the predicted pattern with l?=O.4 does
not match the observed pattern, but for I’= 1.0 the peak location and initial slope closely match the observed values.
Clearly, the observed radial profile patterns cannot be
Gekelman et a/.
Downloaded 13 Sep 2003 to 128.97.30.131. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
1 ’ ,.,..
25
,
P = 0.75
f = 320.0
251
:...:
’ ’ ’
’ ’
-dz
or.,.l...l,..l’,““,1
a
4
20
16
12
0
4
a
bb)
1.
1
’
of,‘.
0
/‘.‘.“, .,.,,,,
,-.
1
I
”
x
P = 0.75
f = 320.0
4.0
15 -
7
I
1 ,
4
-
I,
‘I
1 *
12
a
12
16
20
FIG. 10. (a) Comparison of radial field profiles to theory for the small disk
at frequency 280 kHz with both Landau and collisional damping included.
The larger curve (z = 157 cm) is best fit for T,=O.5 eV. and a plasma density
of 1.0X10’* cme3. The three dotted lines illustrate the sensitivity of the
theoretical profiles to changes in plasma density. The smallest theoretical
profile corresponds to a density of n =2.0X 10” cme3, and the largest to a
density of n =5.0X10” cmm3. (b) At axial position z=252 c m (the lower
curves) the theoretical profiles at density n = 1.0X 10” cmm3 and three electron temperatures are compared to observation to indicate the sensitivity of
the calculation to changes in electron temperature. The upper curve corresponds to 3.6 eV, the middle curve to 8 eV, and the lowest curve to 18.5 eV.
The theory in all cases is for r=O.2, or a collision frequency of 350 kH2.
kHz
dz = 252 cm
theory vs v+,,. XlO’cm/s
Landau Damping
, * 1
= 157 cm
r/a
r/a
2o
’ ’ ’ ’ 1
kHz
dz = 157 cm
theory vs v,,,. XlO’cm/s:
Landau
Damping
0
’ ’ ’ ’
j
’
16
’
20
r/a
FIG. 9. (a) The radial experimental profiles shown in Fig. 8 are compared to
theory with Landau damping only. In this case the curves are best fit with a
different electron temperature at each location. (a) At axial location z = 157
c m the thermal velocity is about 4.5X107 cm/s (-1 eV), and (b) at r=252
c m the thermal velocity is about 1.1 X 10’ cm/s (-7 eV).
matched using one value of r. Furthermore, the values of r
required for matching are larger than the largest possible
value of r based on the neutral fill pressure of the chamber
(i.e., T-O.2 or v=350 kHz). Thus the observed profiles cannot be matched using a single, reasonable value of electronneutral collision frequency.
B. Effects of Landau damping
Figure 9 shows a comparison of predicted radial profiles
to the observed profile for the same case considered above.
At axial position z = 157 cm [Fig. 9(a)] the theoretically predicted profiles peak at smaller r values than the observed
peak and the initial slope is steeper than observed. The closest fit to the observed pattern is obtained for a thermal velocity of about 4.5X lo7 cm/s (a temperature of about 1 eV).
At axial location z=252 cm [Fig. 9(b)] the predicted profile
for thermal velocity 1.1 X lo8 cm/s (about 7 eV) closely fits
the observed profile. While the temperatures required to fit
the data are within the range of observed temperature, the
Phys. Plasmas, Vol. 1, No. 12, December 1994
profiles cannot be fit with a single value of temperature.
However, it appears the observed profiles can nearly be fit in
a warm collisionless plasma if an axial temperature gradient
is assumed.
C. Fitting with both collisions
and Landau damping
Figure 10 shows an attempt to fit the observed magnetic
profiles for the 4 m m disk at 280 kHz varying both collision
frequency and thermal velocity. In this case, however, the
value of the collision frequency is restricted to less than 350
kHz. The various theoretical profiles shown also indicate the
sensitivity of the fitted profiles to changes in plasma temperature and density. The best fit to the observed radial profile at axial location z = 157 cm is obtained for a thermal
velocity of 3 X IO7 cm/s (T, =0.5 eV). Three theoretical profiles are shown for this thermal velocity corresponding to
densities 5X IO” cme3, 1.0X 1012cmm3, and 2.0X 1012cmm3.
The theoretical profiles are relatively insensitive to density.
At the axial location z=252 cm, theoretical profiles are
shown for density 1.0X 10” cmF3 and three different thermal
velocities, 8X107 cm/s (T,=3.6 eV), 1.2X10’ cm/s (8 eV)
and 1.8X 10’ cm/s (18.5 eV). The three curves indicate the
sensitivity of the fit to changes in temperature. The best fit is
obtained for a thermal velocity near 1.0X10’ cm/s (6 eV).
The current drawn to the disk determined from the theoretical profiles is 100 mA. The measured current was 25%
smaller. The collision frequency for all profiles shown has
the largest allowable value, 350 kHz.
Figure 11 shows a comparison of theoretical predictions
Gekelman et al.
3781
Downloaded 13 Sep 2003 to 128.97.30.131. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
f = 280.0
-
dz = 157
upon the slope of the velocity distribution at the wave phase
velocity, a complete comparison between theory and measurement is not feasible without the measured velocity distribution.
kHz
cm
VI. CONCLUSlONS
::
i,..
:
OP...l...l...l...f.,.,
2
0
6
4
a
IO
r/a
FIG. 11. The radial magnetic profiles for the large disk at frequency 280
kHz are compared to theoretical profiles at all three axial locations. The
plasma density is 1 X IO” cme3 and the electron-neutral collision frequency
350 kHz. The electron temperature at the farthest axial positions (252 and
347 cm) is 8 eV, while the temperature at the axial position closest to the
source is 0.5 eV.
and the observed magnetic field profiles for the 8 mm disk at
frequency 280 l&z. The density is 1.0X lOL2 cme3 and the
collision frequency 350 kHz. Once again the observations
are best fit by choosing an increasing thermal velocity as the
axial distance from the exciter increases. However, the observed profiles at z=252 cm and 347 cm can be fit with the
same value of temperature, 8 eV. Furthermore, this value is
nearly the same used to fit the profile for the small disk for
280 kHz at z =252 cm (Fig. 10). It appears that at sufficient
distance away from the exciter a single value of electron
temperature gives good fits between theory and observation.
The theoretical value of the current drawn to the disk in this
case is 225 mA. The measured current was 15% smaller.
In general, the theoretical predictions reproduce the
overall features of the data. The location of the peak, the
initial slope, and the width of the peak are fairly well represented by theory. Also the extent of the f/r region of the
profiles is generally well predicted and clearly illustrates the
spreading of the current channels. However, the observed
magnetic field for the 240 kHz frequency does depart somewhat from a I/r decay, especially for the 8 mm disk. This
feature may indicate the presence of a reflected wave or generation of another mode.
The apparent axial gradient in temperature probably
arises because the theory uses Landau damping which is an
asymptotic expression for wave damping and should not be
expected to accurately predict dissipation within one wavelength of the source. The temperature required to fit the observed profiles does appear to approach an asymptotic value
as the distance from the source increases. The asymptotic
value (8 eV) of the temperature determined from fitting profiles is higher than that measured using a Langmuir probe (4
eV). However, the theoretical fits assume the velocity distribution is Maxwellian and the actual velocity distribution was
not measured. Since Landau damping depends sensitively
3782
Phys. Plasmas, Vol. 1, No. 12, December 1994
We have measured the radial profiles of the magnetic
field radiated by wire mesh disk exciters with radii of electron skin depth size as a function of frequency, axial position,
and disk size. We have compared the observations to theoretical profiles and studied the effects of electron-neutral
collisions and electron temperature. In general, the observed
profiles are in excellent agreement with those predicted by
theory in a warm plasma with electron-neutral collisions. In
particular we have verified the divergence of the radiated
current channel along Alfven wave cones. The disk exciter
drives an RF current carried by the waves which moves
across the magnetic field in a cone-shaped pattern. The cone
is filled with a wave traveling along the magnetic field at the
Alfven velocity. The cone angle is given by the expression
tan( 6) =( r- a)lz = kA& Outside the Alfven cone, we have
verified that the magnetic field falls off as l/r, so that the
field aligned current is entirely confined to the region inside
the cone. This spreading of the magnetic disturbance across
the magnetic field from exciters of skin depth size causes
cross-field energy transport and possibly perturbs the motion
of axially distant particles. In contrast, when the spatial extent of excitation currents become much larger than the skin
depth the radiated waves and energy are field aligned.
We have studied the radiation patterns in a warm plasma.
Electron-neutral collisions, in this experiment, were found
to be a small effect compared to the effects of nonzero electron temperature. Agreement with observation was obtained
using a theory based upon Landau damping at distances farther than one wavelength from the exciter. At closer distances the theory required a colder temperature. This effect is
attributed to the expression for Landau damping being an
asymptotic limit. The observations could not be explained
using electron-neutral collisions alone. Thus, we have indirectly verified the existence of a magnetic field aligned electric field for the radiated shear Alfvin wave through its interaction with the plasma electrons.
We have not studied the radiation patterns in a plasma in
which the interaction between the wave and plasma electrons
is negligible. Nor have we studied the radiation patterns at
distances sufficiently far from the exciter to observe the predicted “diffraction” pattern. Furthermore the coupling coefficient 21clac [Eq. (5)] appears to depend upon frequency.
These topics will be addressed in future experiments.
The phenomenon of current channel spreading along Alfvdn wave cones could have important consequences in several areas of plasma physics. One of these is the plasma
physics of the aurora1 ionosphere. Figure 12 shows the skin
depth as a function of altitude in a model ionosphere. It
ranges from 100 m at altitudes of a few hundred kilometers
to over a kilometer at altitudes above 5000 km. Current
channels of this skin depth size are commonly observed in
the aurora1 regions. If these currents vary on time scales less
than an Alfven period they would radiate waves that move
Gekelman et a/.
Downloaded 13 Sep 2003 to 128.97.30.131. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
Electron Skin Depth vs. Altitude
in a Model Ionosphere
10
-I
0
E
2000
The authors would like to acknowledge the many useful
discussions and ongoing collaboration with George Morales.
This work was supported by the Office of Naval Research under Grant No. ONR NOOO14-91-J-1172, and by the
National Science Foundation under Grant No. NSF-ATM9214000.
I
I
4000
6000
8000
Altitude (km)
FlC. 12. The electron skin depth (c/ape) as a function of altitude along a
magnetic field line in a model of the topside amoral ionosphere is shown as
a function of altitude. The skin depth increases with altitude due to decreasing electron density.
across as well as along the ambient magnetic field. If this
phenomena is not recognized, the origin and nature of magnetic field fluctuations detected by rockets or satellites could
be misinterpreted. Finally magnetic noise below the ion cyclotron frequency has been studied in tokamak edge
plasmas.*5 This “naturally” occurring noise may be related
to local current fluctuations or filimentation.‘6 If the sources
are localized and of order 8, this could impact energy transport in these machines.
Phys. Plasmas, Vol. 1, No. 12, December 1994
‘H. Alfvin, Nature 150, 405 (1942).
‘W. Bostick and M. Levine, Phys. Rev. 94, 815 (1952).
‘D. F. Jephcott, Nature 183, 1652 (1959).
4G. A. Sawyer, P. L. Scott, and T. F. Stratton, Phys. Fluids 2, 47 (1959); J.
M. Wilcox, F. I. Boley, and A. W. De Silva, ibid. 3, 15 (1960).
‘D. F. Jephcott and P. M. Stocker, J. Fluid Mech. 13. 587 (1962).
‘L. C. Woods, I. Fluid Mech. 13, 570 (1962).
‘R. A. Ellis, L. P. Goldberg, and J. G. Gorman, Phys. Fluids 3, 468 (1960);
D. G. Swanson, R. W. Gould, and R. H. Hertel, ibid. 7, 269 (1964). R.
Morrow and M. H. Brennan, Plasma Phys. 13, 75 (1971); G. Miiller,
Plasma Phys. 16, 813 (1974).
‘R. C. Cross and J. A. Lehane, Aust. J. Phys. 21, 129 (1967); Nucl. Fusion
7,219 (1967); J. A. Lehaue and F. J. Paolini, Plasma Phys. 14, 701 (1972);
G. Miiller, Plasma Phys. 16, 813 (1973); J. A. Lehane and F. J. Paoloni,
Plasma Phys. 14, 701 (1971).
9Y. Amagishi, K. Saeki, and I. J. Donnelly, Plasma Phys. Controlled Fusion
31, 675 (1989); Y. Amagishi, J. Phys. Sot. Jpn. 59, 2374 (1990).
‘OR. C. Cross, Plasma Phys. 25, 1377 (1983).
“G. G. Borg and R. C. Cross, Plasma Phys. Controlled Fusion 29, 681
(1987).
“W. Gekelman, H. Pfister, 2. Lucky, J. Bamber, D. Leneman, and J Maggs,
Rev. Sci. Instrum. 62, 2875 (1991).
13J. E. Maggs, R. J. Taylor, and W. Gekelman, Bull. Am. Phys. Sot. 36,
2415 (1991).
14G. J. Morales, R. S. Loritsch, and J. E. Maggs, Phys. Plasmas 1, 3765
(1994).
“S. J. Zweben, C. R. Menyuk, and R. J. Taylor, Phys. Rev. Lett. 42, 1270
(1979).
16B D. Fried, G. J. Morales, and R. J. Taylor, Bull. Am. Phys. Sot. 36,2499
(1991).
Gekelman et al.
3783
Downloaded 13 Sep 2003 to 128.97.30.131. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp