GEOPHYSICAL RESEARCH LETTERS, VOL. 28, NO. 5, PAGES 923-926, MARCH 1, 2001
Laboratory observation of Alfvén resonance
C. Mitchell, S. Vincena, J. Maggs and W. Gekelman
Physics Department, University of California, Los Angeles
Abstract. We present laboratory observations of shear
Alfvén wave resonances in a cylindrical plasma column.
Fourier data are presented which show the existence of many
standing eigenmodes in the shear Alfvén frequency range.
is presented
Harmonic frequency dependence on ambient B
for three resonances. A wavelet analysis is used to show the
lifetimes of the identified harmonic modes. Experimental Q
values for the first five eigenfrequencies are measured and
compared to theoretical values.
Introduction
Theory
The Alfvén Resonance, or Field Line Resonance (FLR),
has a nearly 30 year history. FLR has enjoyed attention
from both fusion and space plasma communities. Like the
results presented here, studies have been done in linear machines [Tsushima et al., 1982; Cekic et al., 1991], but the
focus of these experiments was on energy absorption. To
our knowledge, a clear spectrum of Alfvén resonances has
not been obtained in the laboratory.
FLR’s are of particular interest to the space community.
Of specific relevance to this paper are the FLR frequency
spectra observed by spacecraft. These spectra have been
used to infer equatorial ion mass densities [Poulter, 1984],
and to determine both plasma distribution and transport
properties on Ganymede [Volwerk et al., 1999]. Harmonic
values of FLR’s in earth’s magnetosphere [Singer et al.,
1982] and on Mercury [Russell, 1989] have also been deduced
from spectral data.
By using a broadband impulsive driver in a laboratory
plasma, we have excited a clear standing wave spectrum
in the Alfvén region. A simple recursive dispersion relation exhibits excellent agreement with experimental results,
and the validity of the FLR’s use as a plasma diagnostic is
strongly supported by our data.
Experimental Setup
Alfvén wave data presented here were obtained in the
LArge Plasma Device (LaPD) [Gekelman et al., 1991] at
UCLA. The LaPD is a 10m long, 1m diameter cylinder with
a pulsed plasma source. The plasma is helium, the magnetic
field is axial, and the bulk density variation is ≈ 5%
The field aligned antenna used in this experiment is a
solid copper cylinder 0.635cm in diameter. This diameter
corresponds to approximately two electron skin depths in
this experiment, δ = c/ωpe , where c is the speed of light,
and ωpe is the electron plasma frequency. The antenna and
corresponding experimental setup are shown in Figure 1.
Copyright 2001 by the American Geophysical Union.
Paper number 2000GL012165.
0094-8276/01/2000GL012165$05.00
A phase-locked, one-cycle sine wave burst modulated the
current drawn from the antenna at plasma floating potential.
The bursts were excited for ten different ambient magnetic
field values, ranging from 500G to 1400G at 100G intervals,
keeping the fundamental frequency of the driving impulse
value, 200
at half the ion cyclotron frequency. For each B
plasma shots were recorded. For each shot, the three components of the magnetic field were received by a probe 61.2cm
away from the antenna, digitized, and recorded on computer
disk [Mandrake and Gekelman, 1997].
The shear Alfvén wave’s propagation characteristics can
be broadly categorized as “inertial” or “kinetic,” depending
on whether electron inertial effects or finite thermal effects
dominate the Alfvén wave [Hasegawa, 1976]. The wave is
termed kinetic if β = 8πnkTe /B 2 me /MI , where n is
density, Te is electron temperature, B is magnetic field, me
is electron mass, and MI is ion mass. If β me /MI , the
wave is inertial. Kinetic Alfvén waves have perpendicular
group velocity in the positive k⊥ , and inertial waves in the
negative k⊥ direction [Streltsov and Lotko, 1995; Leneman
et al., 2000].
In this experiment, β >
∼ 2me /MI , so the wave is kinetic.
Furthermore, k⊥ ρs <
∼ 0.1, where ρs = cs /Ωci is the ion sound
gyroradius, and cs = (Te /MI )1/2 is the ion sound speed.
This means that finite ion sound gyroradius effects can be
neglected. Lastly, Te ≈ 6Ti , so ion gyroradius effects can
also be ignored. The dispersion relation given by [Gekelman
et al., 1997], for kinetic Alfvén waves thus becomes simply
ω/k = vA (1 − ω 2 /Ω2ci )1/2 .
(1)
This is the dispersion relation which will be used to predict
standing mode eigenfrequencies in the LaPD.
We can predict the natural Alfvén modes of the plasma
if we assume that λn = 2L/n, n = 1, 2, . . ., where L is the
length of the plasma column in the LaPD (8.95m). Solving
Equation 1 for the eigenfrequencies, fn , we find
vA
.
fn = 2
2
2
(4L /n2 + vA
/fci
)
(2)
A recursion relation for the nth eigenfrequency in terms of
the ith eigenfrequency can be obtained. Its only fundamental dependence in terms of plasma parameters is on fci :
2
).
fn2 = fi2 n2 /(i2 + (n2 − i2 )fi2 /fci
(3)
Qualitatively, Equation 3 implies that for a given i, fi , the
successive eigenfrequencies n, fn will get increasingly closer
as fn → fci , and this is indeed observed in the data. Experimentally, Equation 3 is used in the following way: we
look at a Fourier spectrum of wave magnetic field data acquired from the LaPD, pick a peak (fi ), count the number
923
924
MITCHELL ET AL.: LABORATORY OBSERVATION OF ALVÉN EIGENMODES
Overview of
Experimental Setup
Received
signal
R.F amplif ier
70 V
Exciter
dB/ dt
probe
Grid anode
B
o
He plasma
Plasma chamber
Oxide-coat ed ca thode
End plat e
field is 91.4 cm long. The distance from the grid anode to the endplate is
Figure 1. The portion of the exciter parallel to the B
8.95m.
of peaks preceeding it to determine the harmonic number,
(i), and calculate the fn corresponding to each n. As indicated previously, this technique eliminates the need to know
plasma paramaters (other than Ωci ), and therefore eliminates the uncertainty associated with their measurement.
Alternatively, the measured plasma density, magnetic field
strength, and plasma column length can be used in Equation
2 to predict fn .
An indication of the sharpness of the resonances is obtained by measuring their Q values, which are defined to
be 2π times the ratio of the time averaged energy stored
in the cavity to the energy loss per cycle [Jackson, 1975].
Theoretical Q values may be computed by solving the general dispersion relation in a uniform plasma for azimuthally
symmetric shear Alfvén waves [Gekelman et al., 1997]:
n2
= ⊥
n2
1− ⊥
(4)
between theory and experiment is clear. Even though the
experimentally obtained spectral peaks are not as clearly
resolved at the n = 8 and n = 9 harmonics, one can still
see that the vertical lines of the theoretical prediction intersect the measured spectrum at peak locations. Equation 3
predicts an infinite number of harmonics with decreasing
separation as f → fci , but only the first nine harmonics are
shown here since no peaks are resolved after n = 9.
Although Figure 2 displays data from only a single spatial point, a regularly spaced square (31.5cm × 31.5cm) grid
0 and comprising 1225 spatial
oriented perpendicular to B
points was acquired. These data showed that while the relative amplitudes of the various harmonics vary across the
column, the observed values of the resonance frequencies do
not.
The close agreement between the experimental results
and the recursive frequency relation in Equation 3 suggests
that FLR’s can be used as a diagnostic tool for density mea-
where and ⊥ are the parallel and perpendicular components of the dielectric tensor, and similarly with the indices
of refraction: n and n⊥ . Solutions of Equation 4 are obtained numerically by solving for the complex frequencies:
ωnm = ωR,nm + iωI,nm while keeping both the parallel and
perpendicular wavenumbers as fixed, real values. This calculation assumes that there are no energy losses at the radial
or axial boundaries; only within the plasma itself. The Q
values obtained in this manner are discussed below.
Experimental Results
An example of the observed frequency response which
displays the frequency spectrum of the x-component of the
field is shown in
wave magnetic field in a 1400G ambient B
Figure 2. The data have been corrected for probe frequency
response, and have been ensemble averaged. There is a clear
drop in energy before the 536 kHz cyclotron frequency shear
Alfvén wave cutoff. The power spectrum of the input pulse
is also shown in Figure 2 as the dashed line.
The predictions from Equation 3 are plotted as vertical lines over the power spectrum acquired experimentally
in Figure 2. For the first seven harmonics the agreement
Figure 2. The power spectra of the input signal and the detected probe response are plotted with normalized amplitudes.
The lowest five harmonics have Q = 2.8, 3.9, 5.8, 8.2, 10.2. The
lone vertical bar to the far right is located at the ion cyclotron
frequency, 538kHz.
MITCHELL ET AL.: LABORATORY OBSERVATION OF ALVÉN EIGENMODES
925
Wave Bx @ 1400 Gauss
Continuous Wavelet Spectrum
5e+05
4.5e+05
4e+05
Frequency
3.5e+05
n=5
3e+05
n=4
2.5e+05
n=3
2e+05
0
1.5e+05
n=2
1e+05
n=1
50000
0
0
2e-05
4e-05
6e-05
Time in seconds
Plate 1. The wavelet spectrum for the y-component of the wave magnetic field shows the duration of the resonances. The colors
indicate relative amplitudes, with magenta being the highest and red the lowest.
atomic species, and plasma disurment. Since ambient B,
mensions are typically known quantities in laboratory plasmas, Equation 2 can be used to solve for an axially averaged
ne . One simply reads the measured peak frequencies (fn )
off the power spectrum and uses λn = 2L/n to determine
density. The numerous Langmuir measurements required
to compare this prediction to an experimentally obtained
column-length density average were not taken in this experiment. However, this technique agrees to within experimental uncertainty with the average density measured across the
column using a 57GHz microwave interferometer.
field values
The data obtained for the other ambient B
look similar to the spectrum shown in Figure 2, but it becomes increasingly difficult to resolve the higher eigenfre fields. Also, for all the
quencies (n ≥ 6) at the lower B
magnetic fields investigated, the n = 1 peak in all cases departed the most from theoretical predictions. The harmonic
field for three separate eigenvaldependence on ambient B
ues is plotted in Figure 3. As expected from Equation 2,
the frequency of a given harmonic increases linearly with
external B.
The sharpness of the resonances, Qn , is obtained directly from the power spectrum of the wave data using
Qn = ωn /Γn , where ωn is the n-th resonant frequency and
Γn is the full-width of the resonance line at the half power
point. Experimental Q values obtained at a background
magnetic field of 1000G and were compared to theoretical
Q values. Under these experimental conditions, the wave
damping is primarily due to electron-ion Coulomb collisions
and electron Landau damping [Vincena, 1999]. Both of
these damping mechanisms enter only through the term
in Equation 4. This means that the damping, and hence the
Q values, will be sensitive to the perpendicular wavenumber.
A Bessel decomposition of the radial wave pattern yields
a spectrum of perpendicular wavenumbers with the two
most significant contributions coming from the two lowestorder modes. These two modes have k⊥ = 0.174cm−1 and
k⊥ = 0.319cm−1 . Theoretical Q values were computed for
both of these wave numbers. Both experiment and theory
show a trend of increasing Q for increasing frequency. This
is likely due to the relatively diminished effect of Coulomb
collisions for higher frequencies. Experimental Q values ran
from 2.5 → 9 over five harmonics, while theoretical values
varied from 6.9 → 20 for k⊥ = 0.319cm−1 . Higher Q values
Figure 3. Harmonic frequency dependence on external magnetic field value is plotted for n = 3,4,5. The frequency of each
field as indicated
harmonic increases linearly with background B
by the shear Alfvén wave dispersion relation Equation 1.
926
MITCHELL ET AL.: LABORATORY OBSERVATION OF ALVÉN EIGENMODES
were obtained for k⊥ = 0.174cm−1 . The poor agreement is
reasonable since the theoretically obtained Q values do not
include loss of wave power from reflection at the end of the
plasma column.
A wavelet analysis was performed to determine the lifespans of the harmonics. A sample wavelet spectrum is shown
in Plate 1. As suggested by the trend in Q values, the
wavelet analysis confirms that the higher harmonics manifest more cycles than the lower harmonics. For example,
by the time the n = 4 and n = 1 harmonics in Plate 1 drop
off to the same amplitude the n = 4 harmonic has oscillated
for 13.5 periods and the n = 1 harmonic for only seven. For
values investigated, the wavelet spectra
all ten external B
exhibits the same qualitative behavior, i.e. lower frequency
harmonics have a longer temporal but shorter cyclical lifetime than the higher frequencies.
Conclusions
A broad-band impulsive driver in a helium plasma succeeded in exciting a spectrum of standing shear Alfvén waves
in the LaPD at UCLA. A linear dependence between ambient magnetic field and harmonic number was observed, as
expected for the shear Alfvén wave. The higher eigenmodes
were more transient temporally, but rang for approximately
twice as many cycles as the lower modes.
The observed spectrum agreed remarkably well with a
simple recursion formula for standing shear Alfvén waves.
The close agreement supports the use of shear Alfvén wave
resonance frequencies as a diagnostic tool. Since the dispersion relation in Equation 4 assumes axially uniform plasma
parameters, the technique of matching the computed real
frequencies to those observed yields a method for determining an average axial density within the resonant cavity, be
it in the laboratory or in a planetary magnetosphere.
Acknowledgments. We are grateful to M. Kivelson, D.
Southwood, and G. Morales for useful discussions. Work supported by the Office of Naval Research.
Gekelman, W., S. Vincena, D. Leneman, and J. Maggs, Laboratory experiments on shear Alfvén waves and their relationship
to space plasmas, J. Geophys. Res., 100, 19,457-19,472, 1997.
Gekelman, W., H. Pfister, Z. Lucky, J. Bamber, D. Leneman,
and J. E. Maggs, Design, Construction, and properties of the
large plasma research device - The LaPD at UCLA, Rev. Sci.
Instrum., 62, 2875-2883, 1991.
Goertz, C.K., Kinetic Alfvén waves on auroral field lines, Planet
Space Sci., 32, 1387-1392, 1984.
Hasegawa, A., Particle acceleration by MHD surface wave and
formation of aurora, J. Geophys. Res., 81, 5083-5096, 1976.
Jackson, J. D., Classical Electrodynamics, 848 pp., John Wiley
& Sons, New York, 1975.
Leneman, D., W. Gekelman, and J. E. Maggs, Shear Alfvén wave
radiation from a source with small transverse scale length,
Phys. Plasmas, 7, 3934-3946, 2000. 2000.
Mandrake, L., and , W. Gekelman, Large-scale Windows 95based data acquisition system using LabVIEW Computers in
Physics, 11, 498-507, 1997.
Poulter, E.M., W. Allan, J.G. Keys, and E. Nielsen, Plasmatrough ion mass densities determined from ulf pulsation eigenperiods, Planet Space Sci., 32, 1069-1078, 1984.
Russell, C. T., ULF waves in the Mercury magnetosphere, Geophys. Res. Lett., 16, 1253-1256, 1989.
Singer, H. J., W. J. Hughes, and C. T. Russell, Standing hydromagnetic waves observed by ISEE1 and 2: radial extent and
harmonic, J. Geophys. Res., 87, 3519-3529, 1982.
Streltsov, A., W. Lotko, Dispersive field line resonances on auroral
field lines, J. Geophys. Res., 100, 19,457-19,472, 1995.
Tsushima, A., Y. Amagishi, and M. Inutake, Observation of spatial Alfvén resonance, Phys. Lett. A, 88:9, 457-460, 1982.
Uberoi, C., Alfvén waves in inhomogeneous magnetic fields, Phys.
Fluids, 15(9), 1673-1675, 1972.
Vincena, S. T., Propagation of the shear Alfvén wave from a skindepth-scale source into nonuniform plasmas, Ph.D. thesis, 130
pp., Univ. of Cal. at Los Angeles, 1999.
Volwerk, M., M.G. Kivelson, K.K. Khurana, and R.L.McPherron,
Probing Ganymede’s magnetosphere with field line resonances,
J. Geophys. Res., 104, 14,729-14,738, 1999.
W. Gekelman, J. Maggs, C. Mitchell, S. Vincena, Department
of Physics and Astronomy, University of California Los Angeles, Los Angeles, CA, 90068 (e-mail: gekelman@physics.ucla.edu,
maggs@physics.ucla.edu, mitchell@physics.ucla.edu, vincena@
physics.ucla.edu)
References
Cekic, M., B. A. Nelson, and F. L. Ribe, Alfvén wave heating of a
high-beta plasma column, Phys. Fluids B, 4:2, 392-403, 1991.
(Received August 4, 2000; revised November 2, 2000;
accepted November 14, 2000.)