The spatial evolution of the radial profile of the magnetic field
of a shear Alfvén 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 4mm and 8 mm radius into a Helium plasma of density
about 1.0 x 10^12 cm^-3 and magnetic field 1.1 KG. The electron skin depth 
is about 5 mm. The current channel associated with the shear Alfvén wave
is observed to spread with distance away from the exciter. The spreading follows
a cone-like pattern whose angle is given by 
,
where 
,
is the Alfvén 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 [companion paper: "Structure of Alfvén
Waves at the Skin-Depth Scale, G.J. Morales, R.S. Loritsch, J.E. Maggs.].
I. Introduction.
Alfvén 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 Alfvén waves play key roles in many naturally occurring
interactions. For example, changes in the auroral current magnitude
and spatial configuration, or changes in the magnetospheric
configuration, involve propagation of information by Alfvén waves.
Alfvén 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 Alfvén wave energy may play a key role in heating
the solar corona. In laboratory plasmas Alfvén waves are present in
all fusion plasmas as background oscillations. Heating schemes for
both electrons and ions using Alfvén waves have been proposed and
tried. The importance of Alfvén waves in the dynamics of a burning
Tokamak plasma are still being assessed.
Alfvén waves are a low frequency phenomena. In cold
plasmas, at frequencies below the ion cyclotron frequency, the
Alfvén 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
Hannes Alfvén in 1942(1).
The first experimental evidence for the existence of Alfvén
waves appeared ten years later. A standing wave of the
appropriate frequency was observed by Bostick and Levine(2) in a
pulsed toroidal device. Subsequent experiments in toroidal(3), and
linear devices(4) measured the wave phase velocity from the phase
shift of detected wave magnetic fields, and verified its dependence
on magnetic field strength and density. The dispersion of the shear
wave was measured by Jephcott and Stocker(5) in a cylindrical
column. The results compared favorably with a theory by Woods(6)
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 Alfvén 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 Alfvén 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 Alfvén modes have been studied and verified in
these plasmas. The properties of bounded Alfvén modes have been
studied in fusion related and basic physics experiments(8). Recent
studies of both the shear and compressional mode were done in a
narrow, partially ionized column(8) . 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 Alfvén 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 r.f. current
channel spreads radially in a pattern contained within the Alfvén
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(10) and Borg and Cross in a small Tokamak(11).
These experiments differ from those reported here in that they were
performed in a highly collisional regime.
II. Experimental Arrangement.
These experiments were conducted in a machine designed to
study space plasma physics phenomena. The LAPD (LArge Plasma
Device) at UCLA(12) (University of California, Los Angeles) is a
flexible, low maintenance device with a plasma column up to 80 cm
in diameter and 10 meters 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 5x10^-7
Torr. 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 (
n/n
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 1.0x10^12 /cm^3. The discharge is pulsed at one Hz to
allow for efficient signal averaging and data processing. The discharge typically
last 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. In the termination phase the plasma density remains
nearly constant, while the temperature falls slowly. 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 KHz.
Figure 1a shows the experimental setup for launching and
detecting the shear Alfvén wave. The radially movable exciter is a
circular disk of wire mesh (transparency = 55%) 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 additional vertical positioning.
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, 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.
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,
The parallel wave number of the shear Alfvén wave in a cold
plasma (i.e., in the inertial regime) is given by
With azimuthal symmetry the magnetic field has only a
component in the azimuthal direction,
. The spatial dependence of the radiated magnetic
field is given by an integral expression involving the first order Bessel
function J1,
The plasma currents associated with the shear wave radiation
are contained within cones emanating from all points around the
edge of the disk as illustrated ( in cross section) in Figure 2. The
radial position of the outer edge of the cone is given by
IV. Experimental Results
The first task is to establish that shear Alfvén 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 milliGauss
for a ratio of wave magnetic field to background field of B/B4
x10^-5.
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.
FIG 3. Alfvén tone bursts received at three axial distances from an
8mm diameter disk exciter (located at z=0). The phase shift in the wave is
clearly seen to increase with increasing distance from the exciter. The
phase velocity can be estimated from this diagram and is of order 8x10^7 cm/s.
Figure 4 compares the measured dispersion of waves launched
in the LAPD with the predicted dispersion values. The expected
dispersion of the shear Alfvén wave (for 
= 0) is given by Eqs. 2
and 3. The launched waves closely match the predicted dispersion
in a plasma with density 1.0 x 10^12 cm^-3. No accurate measure of
the plasma density was available, but the density measured using a
Langmuir probe was found to be 8.0 x 10^11 cm^-3. The measured
dispersion shown in Fig. 4 demonstrates that shear Alfvén waves
are launched from the disk exciter.
FIG 4. Measured shear Alfvén dispersion. The data was acquired in
a separate experiment with a 50cm 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 1.0x10^12/cm^3. The dashed curves are for densities of
0.9x10^12/cm^3 and 1.1x10^12/cm^3.
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 phase-locked tone burst was fed to the disk
antenna through a broad band amplifier. The center frequencies of
the tone bursts used in these experiments were 240, 280 and 320
KHz. The ion cyclotron frequency was, fci = 420 KHz. The three
components of the radiated magnetic field signal were measured
using a tri-axial 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 mm 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 8mm disk,
and two axial locations (157 cm 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 Figures 5 and 6. In processing 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.
Fig 5a,5b,5c.
FIG 5. Radial magnetic field profiles for a 4mm radius disk exciter,
p=0.75, at two axial locations. The parameter p=a/, and dz denotes the axial separation between
exciter and receiver. The figures are for frequencies: (a) 240KHz, f/fci=0.57,
(b) 280KHz, f/fci=0.67, and (c) 320KHz, f/fci=0.76.
Fig 6a,6b,6c.
FIG 6. Radial magnetic field profiles for an 8mm radius disk exciter,
p=1.5, at three axial locations. The figures are for frequencies: (a) 240KHz,
f/fci=0.57, (b) 280KHz, f/fci=0.67, and (c) 320KHz, f/fci=0.76. The boxes
in (c) denote the radial positions at which the hodograms presented in Fig 7
were taken.
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 1/r decrease in amplitude. For
each frequency and disk size, the amplitude of the field in the 1/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 [Bx(t) vs.
By(t)] taken at three radial locations (r/a = 0, 2.8, and = 5.0) in the
radial magnetic field profile shown in Figure 6 c. At the radial
locations r/a = 2.8 and 5.0 the magnetic field is linearly polarized in
the vertical direction as shown in figures 7b and 7c. 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 = 0, the wave
appears to be circularly (or in some cases elliptically) polarized as
shown in Figure 7a. This effect is likely due to the finite size ( r =
2.5 mm or 1.5 Rci ) of the magnetic probe which averages signals
over its detection area.
Fig 7a,7b,7c.
FIG 7. Plots of Bx(t) vs By(t) (hodograms) at the three radial positions
indicated in Fig. 6(c). The hodograms are taken for 8 microseconds (2.6 wave
periods) during the middle 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; B is in milligauss.
V. Comparison with Theory
The observed radial profiles of magnetic field can be
compared with those predicted by theory using Eqs. 4 and 5. Eq. 4
is used to find the functional dependence of the parallel wave
number on perpendicular wave number [i.e., 
] 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, I0. 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 1/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 the lowest frequency (measured current about 60 % of
theoretical at 240 KHz).
The other parameters that can be varied in the theory are the
collision frequency 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.
A. Effects of Collisions
Figure 8 shows the comparison of theoretical radial profiles and the observed
profiles for the 4 mm radius disk at 320 KHz frequency for various values
of normalized collisions frequency (
).
At the axial position z = 157 cm ( Fig. 8a) 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 
=
0.4, but the theoretical profile is broader than the observed profile. At
the axial location z = 252 cm (Fig. 8b) the predicted pattern with
= 0.4 does not match the observed pattern, but for
= 1.0 the peak location and initial slope closely match the observed values.
Clearly, the observed radial profile patterns cannot be matched using one
value of .
Furthermore, the values of
required for matching are larger than the largest possible value of
based on the neutral fill pressure of the chamber (i.e., 0.2
or 
=350 KHz). Thus
the observed profiles cannot be matched using a single, reasonable value of
electron-neutral collision frequency.
Fig 8a,8b.
FIG 8. (a) The radial profile for the smaller exciter at axial position
z = 157 cm and 320 KHz excitation frequency (solid line) is compared to
theory with only electron-neutral collisions The best match is for =0.4.
(b) The radial field profile 95 cm further from the exciter than shown
in 8(a) is also compared to theory with only collisions. At this axial
position the experimental observations are best matched with 1. In both cases the collision frequency needed to fit
the data corresponds to higher neutral gas pressure than was injected into the
machine.
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. 9a) 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 4.5 x 10^7 cm/sec
(a temperature of about 1 eV). At axial location z = 252 cm ( fig. 9
b) the predicted profile for thermal velocity 1.1 x 10^8 cm/sec (about
7 eV) closely fits the observed profile. While the temperatures
required to fit the data are within the range of observed
temperature, the profiles can not 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.
Fig 9a,9b.
FIG 9. 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 cm the thermal velocity is about 4.5x10^7
cm/sec (1eV),
and (b) at z = 252 cm the thermal velocity is about 1.1x10^8 cm/sec
(7eV).
C. Fitting with both collisions and Landau damping
Figure 10 shows an attempt to fit the observed magnetic
profiles for the 4 mm 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 3.0x10^7 cm/sec (Te = 0.5 eV).
Three theoretical profiles are shown for this thermal velocity
corresponding to densities 5x10^11 cm^-3, 1.0x10^12 cm-^3 and 2.0x10^12
cm^-3. The theoretical profiles are relatively insensitive to
density. At the axial location z = 252 cm, theoretical profiles are
shown for density 1.0 x 10^12 cm^-3 and three different thermal
velocities, 8.0x10^7 cm/sec (Te = 3.6 eV), 1.2 x 10^8 cm/sec (8 eV) and
1.8 x 10^8 cm/sec (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 1.0 x 10^8 cm/sec (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.
Fig 10.
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 Te = 0.5 eV.
and a plasma density of 1.0x10^12/cm^3. 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 = 2.0x10^12/cm^3, and the largest to a density of n = 5.0
x10^11/cm^3.
(b) At axial position z = 252 cm (the lower curves) the theoretical
profiles at density n = 1.0x10^12/cm^3 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 =.2, or a collision frequency of 350 KHz.
Figure 11 shows a comparison of theoretical predictions and
the observed magnetic field profiles for the 8 mm disk at frequency
280 KHz. The density is 1.0x10^12 cm^-3 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 z = 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.
Fig 11.
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.x10^12/cm^3 and the electron-neutral
collision frequency 350 KHz. The electron temperature at the
farthest axial positions ( 252 cm and 347 cm) is 8 eV, while the
temperature at the axial position closest to the source is 0.5 eV.
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 1/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 1/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 upon the slope of the velocity distribution at the wave
phase velocity, a meaningful comparison between theory and
measurement is not feasible without the measured velocity
distribution.
VI. Conclusions
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 Alfvén 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 Alfvén
velocity. The cone angle is given by the expression 
).
Outside the Alfvén cone, we have verified that the magnetic field falls
off as 1/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 non-zero electron temperature. Agreement with observation was
obtained using a theory based upon Landau damping at distances further 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 Alfvén 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
2I0/ac (Eq. 5) appears to depend upon frequency. These topics will be addressed
in future experiments.
The phenomena of current channel spreading along Alfvén wave cones
could have important consequences in several areas of plasma physics. One
of these is the plasma physics of the auroral 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 auroral regions. If these currents vary on timescales less
than an Alfvén period they would radiate waves that move 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(15). This "naturally"
occurring noise may be related to local current fluctuations or filamentation(16).
If the sources are localized and of order ,
this could impact energy transport in these machines.
Fig 12. 
FIG 12. The electron skin depth () as a function of altitude along a
magnetic field line in a model of the topside auroral ionosphere is
shown as a function of altitude. The skin depth increases with
altitude due to decreasing electron density.
Acknowledgements
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
ONR N00014-91-J-1172, and by the National Science Foundation
under grant NSF-ATM-9214000.
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Liner Device, Bull. Amer. Phys. Soc., 36, 2415 (1991).
14 companion paper: "Structure of Alfvén Waves at the Skin-Depth Scale,
G.J. Morales, R.S. Loritsch, J.E. Maggs.
15 S.J. Zweben,C.R. Menyuk, R.J. Taylor, Phys. Rev. Letts, 42, 1270 (1979)
16 B.D. Fried, G.J. Morales, R.J. Taylor, Bull. Amer. Phys. Soc, 36,2499 (1991).
(1)
.
For a disk exciter with radius a, comparable to the skin depth, the Fourier
amplitudes of the electric fields contain 
times the radial field. This axial electric field drives electron currents along
the magnetic field. The excitation of these parallel electron currents is the
mechanism by which the disk exciter couples to the plasma.
,
(2)
(3)
.
The corrections to the standard MHD
(Magneto-hydrodynamic) shear Alfvén dispersion relation (i.e.,
)
due to the
non-zero value of 
(4)
,
is the electron
thermal velocity and 
is the derivative with respect to the argument of the plasma dispersion
function.
(5)
(6)
(the location of the outer Alfvén
cone), the field decreases as 1/r. The 1/r decrease outside the
Alfvén cone indicates that all wave currents are contained within
the cone. At a fixed radius larger than 
