1
Electric Field Measurements of Directly Converted Lower Hybrid Waves at a
Density Striation
S. Rosenberg, W. Gekelman
Physics Department, University of California, Los Angeles
Geophys. Res. Letters, 25, 865 (1998)
Abstract. Previous experiments at the Large Plasma Device (LAPD) at UCLA provide
evidence that whistler waves incident on a field-aligned density striation will produce
lower hybrid waves via a linear mode-coupling mechanism [Bamber, et. al, 1994; Bamber,
et. al, 1995]. These experiments were limited in that only the magnetic fields of the
waves were measured. Recent experiments at the LAPD directly measure the whistler
and lower hybrid wave electric fields. The lower hybrid wave electric field amplitude is
0.8 times the incident electric field. These measurements are well within theory and
allow us to make some estimates about the conversion efficiency of the linear conversion
mechanism.
Introduction
Lower hybrid waves are the subject of renewed attention. Observations in the ionosphere
and space indicate that these waves may be involved with many physical processes. In
particular, observations have shown a high correlation between density depletions and
intense bursts of lower hybrid wave activity, so-called lower hybrid solitary structures
(LHSS). Rocket [Arnoldy et al., 1996; Kintner et al., 1996; Vago et al., 1992] and
satellite observations [Block et al., 1987] have concentrated on the association of lower
hybrid wave packets with transversely accelerated ions. More recent observations focus
on the structure of the LHSS [Dovner et al., 1994; Eriksson et al., 1994; Pinçon et al.,
1997]. Earlier work [Bell et al., 1983; Bell and Ngo, 1988, 1990] has shown that whistler
waves in the ionosphere can convert to lower hybrid waves in non-uniform plasmas.
An understanding of how whistler waves and lower hybrid waves interact with striations
would greatly benefit the community's understanding of these observations and give us the
opportunity to compare experimental results with theoretical predictions. The ability of laboratory experiments to provide high resolution spatial and temporal measurements makes them ideal
for performing a detailed study of these interactions. Up to now, the amount of experimental
work on whistler waves in striated plasmas has been limited. Therefore, we present further work
on whistler wave conversion to lower hybrid waves at a density striation. Our data show that
lower hybrid waves can be produced at a striation boundary by linear mode-coupling. Wave
electric fields were measured directly allowing us to compare the relative magnitudes of the
whistler and lower hybrid modes for the first time.
Description of Experiment
These experiments were performed in the LAPD at UCLA [Gekelman, et al, 1991].
This linear machine comprises a 10 m long, 1 m diameter vacuum chamber with 68
magnet coils capable of producing fields from 10 to 2500 G. The plasma is created by
pulsing an oxide-coated nickel cathode with respect to a grid anode 60 cm away. The
electrons emitted by the cathode ionize the background gas in the chamber, in this case,
Helium. We achieve a fairly uniform repeatable plasma; shot to shot density variations
across the bulk of the plasma are ≈ 5%. For this experiment, we used a background
magnetic
field of Boz=750 G and a plasma density outside the striation of n = 2 × 10 12
-3
cm . A phase-locked wave packet of 60 MHz whistler waves was launched into the
plasma. Note that the lower hybrid resonance frequency is flh=15 MHz. Table 1 contains
a list of relevant experimental frequencies and lengths and their relationship to the ionosphere
at about 1000 km. Obviously we can not create an identical plasma to the ionoshpere in
the lab. The physics of the interaction is governed by the scaled parameters. The LAPD
plasma compares favorably with the ionosphere in these paramters. numbers which, for
the most part, compare favorably with the ionosphre. The ratio of plasma frequency to
cyclotron frequency in the LAPD does not compare well to typical ionospheric plasmas.
However, this parameter does not affect the physics of whistler waves below fce/2.
[Strangeway, 1997]
2
Parameters and Ratios
Ionosphere
LAPD
f
5 - 30 kHz
60 MHz
flh
4.5 - 6 kHz
15 MHz
fc e
1 MHz
2.1 GHz
fpe
0.6 - 1 MHz
12.5 GHz
rci
6.5 m
0.3 cm
d
10 - 1000 m
6cm
λ ||
10 - 300 m
28 cm
λ⊥,lh
0.5 - 6.5 km
1 cm
f/flh
1-7
4.0
f/fce
0.005 - 0.030
.029
fpe /fce
0.6 - 1.0
5.95
d/rci
2 - 150
20
λ||/λ⊥,lh
15 - 300
28
Table 1. Typical Whistler-Wave Striation Interaction Parameters and Ratios in the Ionosphere and the
LAPD
f, flh , fce, and fce are the wave, lower hybrid, electron-cyclotron and electron-plasma frequencies, respectively;
rci is the ion Larmor radius; d is the effective diamter of the striation; λ || and λ ⊥,lh are the parallel and
perpendicular lower hybrid wavelengths.
Whistler waves were launched using a small (diameter = 20.0 mm) single-turn loop antenna.
The net current flows only in this small loop, since the current feed to the loop closes coaxially
(i.e., I = I θ̂ where θ̂ is the direction around the current loop). Therefore, the only source of
wave energy in the experiment is the loop itself. θ
This work uses a standard electrostatic dipole probe to measure directly the wave electric
field. This is particularly useful in measuring the electrostatic whistler wave (the lower hybrid
wave) since so little of that wave's energy is contained in its magnetic fields. This dipole probe
is approximately 8 mm from tip to tip ( ≈ 300λ Debye ). A 1 mm diameter silver sphere is soldered at
the end of each wire filament. The probe is oriented to measure the perpendicular electric field,
Ex. Dipole probes have their limitations, particularly when they are larger than a Debye length.
Most notably, it is not possible to make a measurement with an absolute calibration at the
frequencies in which we operate. Therefore, all electric field measurements are relative measurements, with all magnitudes normalized to that of the incident (launched) whistler wave. Density
measurements were made with a separate Langmuir probe; a 70 GHz microwave interferometer
provides an absolute calibration for the density measurements.
In this experiment, the striation was created by placing a small masking disk (diameter = 2.5
cm) between the cathode and anode. This paddle retards production of plasma along the field
lines which intersect the disk. The striation it creates has ∆n n ≈ 25% , d ≈ 2.5 cm (FWHM). A
plot of the density in an x-y plane that cuts through the striation is shown in Figure 1
1.5 x 10
3
12
1.0
ne
(cm-3 )
0.5
0.0
4
3
2
y (cm)
1
0
-1
-2
-3
-4
-10
-9
-8
-7
-6 -5
x (cm)
12
-4
-3
-2
-3
Density (× 10 cm )
1.5 1.3
1.1 0.9 0.7
0.5
0.3
0.1
Figure 1. Density as a function of position near the density depletion. The absolute amplitude of the density was
found with a 70 GHz interferometer and confirmed by comparing the observed whistler wave pattern with Greens
function calculations. The line plot is a cut across the plane at y=0. The origin is centered on the whistler antenna.
In order for the mode-conversion to occur, the gradient scale length must be on the order of the lower hybrid
perpendicular wavelength.
. Note that the depression extending up and to the right is created by the support structure for
the paddle. The experimental geometry is pictured in Figure 2.
When a whistler wave propagates into the striation, a number of new waves can be generated.
The new waves must satisfy the local dispersion relation and each wave must have the same
parallel wavenumber (since the density gradient normal is perpendicular to the background
field). Two dispersion curves are plotted in Figure 3, one for the density inside the striation and
one for the density outside. The incident k-vector is marked along with the possible k-vectors
for converted lower hybrid waves.
Electric field data are collected in the following manner: the dipole probe is placed at a series
of locations in the plasma. At each location, we digitize a time series of data representing the
time evolution of the field. Sixteen of these time series, each covering the same time period, are
digitized at each location, and then the probe is moved to a new location and the process is
repeated.
Electric field data were taken along a series of points that made up an x-z plane. The plane
extends approximately 20 cm along the background magnetic field and 13 cm across it. By
solving the dispersion relation, one can find the angle of propagation for whistler and lower
hybrid waves in the plasma. Based on this work, we expect to find whistler waves permeating
the entire measurement area; however, we expect to find lower hybrid waves confined to a
region closer to their source, the striation gradient region. Therefore the data were taken at two
spatial resolutions: where we expect to see short-wavelength lower hybrid waves, the dataaquisition grid has fine spacing (1 mm ⊥ Bo); where we expect to see only the longerwavelength whistler waves, grid spacing is more coarse (5 mm ⊥ Bo). Parallel to Bo, points are
4
spaced about 1 cm apart.
(a) Top View
x
z, Bο Data Grid
Whistler Antenna
Plasma Striation
Density Grid (in x-y plane)
90 cm
(b) Axial View
Striation-Making
Paddle
y
⊗Bο
x
Data Grid (in x-z plane)
Density Grid
Whistler Antenna
Figure 2. Schematic of the experimental layout in the LAPD. The shape of the measurement areas is controlled by the
fact that the plasma is accessed through radial ports in the vacuum chamber. The experimental origin is centered on the
whistler antenna. (a) The experimental layout as seen from above. The cathode, anode and the disk which creates the
striation are 3 m to the left. Waves are launched from the Whistler Antenna and impinge on the density striation. Electric
fields (Ex) were measured in the area labeled as Data Grid. (b) The experimental geometry as seen looking down a field
line. The density data were measured in the area labeled as the Density Measurement Grid. Waves were launched from
the Whistler Antenna which was offset from the striation in the x-direction. The electric field were measured in the Data
Grid which cut through a diameter of the striation.
n e = 2.5 × 1012 cm −3
.6
-1 )
(cm
|
n e = 5.0 × 1012 cm −3
.5
.4
.3
vg
kLaunched
k LH,T
k LH,R
.2
vg
.1
0
-15
-10
-5
0
5
10
15
k⊥ (cm-1 )
Figure 3. The dispersion curves for this experiment. We solve the dispersion relation and find the locus of accessible k || as a
function of k⊥. The dashed line represents the solution in the plasma outside of the striation; the solid line represents the solution
5
in the plasma in the middle of the striation. The white-tipped arrow marks the k-vector of the launched wave; the two blacktipped arrows mark the k-vectors of the measured lower hybrid waves (one transmitted, the other reflected). Note that in the
conversion process, k|| is unaffected.
Experimental Data
The total collection of all the time-evolving data at all positions forms an ensemble
allowing one to look at the data at all positions as a function of time. We begin the data
analysis by averaging the digitized time series at each point. The measured electric field
at one time step is shown in Figure 4a as a function of x and z. Next, we examine the
Fourier transformation of each line of data perpendicular to the background magnetic
field (in the x-direction). In k-space, we were able to identify the peaks which corresponded
to the two different modes, and confirm their relationship to theory (see Figure 3 and its
caption for a brief description of these relationships). We then applied digital filters
which were designed to include as broad an area in Fourier-space as possible; one set of
filters removed short-wavelength data, another removed long-wavelength data. The results
are displayed in Figure 4b and 4c.
75
z (cm)
80
85
90
95
-12
-8
-4
x (cm)
0
4
-12
-8
-4
x (cm)
0
4
-12
-8
-4
x (cm)
0
4
Figure 4. Ex(x,z) The amplitude has been corrected for the probe's wavelength response and has been normalized
to the incident amplitude. The data displayed here are taken on a grid with .1 cm spacing from -10 cm ≤ x ≤ -3.5 cm; .5
cm spacing from -3.5 cm ≤ x ≤ 2 cm; and 1 cm spacing in z. The experimental origin is centered on the whistler
antenna. (a) Averaged and unfiltered data. (b) The averaged data are filtered to remove short wavelengths. Whistler
waves are visible in the right hand part of the plot. The whistler waves are evanescent in the striation (the left part of
the plot, -4.5 cm ≥ x ≥ -7.0 cm). (c) The data are filtered to show short wavelength (~1 cm) lower-hybrid waves.
Short wavelength data are only available in the region where the data were taken on a fine grid. Therefore, there are no
short-wavelength data in the right side of the plane.
We observe the time-evolution of the electric field data by examining a series of time steps.
Keep in mind that each snapshot in time looks qualitatively the same. However, since whistler
and lower hybrid waves have very different behavior, particularly with respect to the sign of their
perpendicular phase velocity, one can gain insight about the two modes by investigating the
time-evolution of the data.
The long wavelength waves move in z and in x away from the loop-antenna source as one
would expect for a forward-traveling wave. Outside the striation, the phase fronts for the short
wavelength data move toward the striation boundary. This is expected since the lower hybrid
9
75
(a) Raw Data
(b) Long-Wavelength Data
(c) Short-Wavelength Data
z (cm)
80
Figure 4.
85
↑
90
95
-12
-8
-4
x (cm)
0
4
E
=−1
E inc
-12
-8
-4
x (cm)
0
4
-12
-8
E
=+1
E inc
-4
x (cm)
0
4
up
5
in the plasma in the middle of the striation. The white-tipped arrow marks the k-vector of the launched wave; the two blacktipped arrows mark the k-vectors of the measured lower hybrid waves (one transmitted, the other reflected). Note that in the
conversion process, k|| is unaffected.
Experimental Data
The total collection of all the time-evolving data at all positions forms an ensemble
allowing one to look at the data at all positions as a function of time. We begin the data
analysis by averaging the digitized time series at each point. The measured electric field
at one time step is shown in Figure 4a as a function of x and z. Next, we examine the
Fourier transformation of each line of data perpendicular to the background magnetic
field (in the x-direction). In k-space, we were able to identify the peaks which corresponded
to the two different modes, and confirm their relationship to theory (see Figure 3 and its
caption for a brief description of these relationships). We then applied digital filters
which were designed to include as broad an area in Fourier-space as possible; one set of
filters removed short-wavelength data, another removed long-wavelength data. The results
are displayed in Figure 4b and 4c.
75
z (cm)
80
85
90
95
-12
-8
-4
x (cm)
0
4
-12
-8
-4
x (cm)
0
4
-12
-8
-4
x (cm)
0
4
Figure 4. Ex(x,z) The amplitude has been corrected for the probe's wavelength response and has been normalized
to the incident amplitude. The data displayed here are taken on a grid with .1 cm spacing from -10 cm ≤ x ≤ -3.5 cm; .5
cm spacing from -3.5 cm ≤ x ≤ 2 cm; and 1 cm spacing in z. The experimental origin is centered on the whistler
antenna. (a) Averaged and unfiltered data. (b) The averaged data are filtered to remove short wavelengths. Whistler
waves are visible in the right hand part of the plot. The whistler waves are evanescent in the striation (the left part of
the plot, -4.5 cm ≥ x ≥ -7.0 cm). (c) The data are filtered to show short wavelength (~1 cm) lower-hybrid waves.
Short wavelength data are only available in the region where the data were taken on a fine grid. Therefore, there are no
short-wavelength data in the right side of the plane.
We observe the time-evolution of the electric field data by examining a series of time steps.
Keep in mind that each snapshot in time looks qualitatively the same. However, since whistler
and lower hybrid waves have very different behavior, particularly with respect to the sign of their
perpendicular phase velocity, one can gain insight about the two modes by investigating the
time-evolution of the data.
The long wavelength waves move in z and in x away from the loop-antenna source as one
would expect for a forward-traveling wave. Outside the striation, the phase fronts for the short
wavelength data move toward the striation boundary. This is expected since the lower hybrid
6
wave is a backward traveling wave perpendicular to Boz . Inside the striation, there is an interference pattern; phase fronts appear to move toward both striation boundaries.
In this experiment, the dipole probe length is of the order of the lower hybrid wave perpendicular wavelength. As a result, the amplitude we record for lower hybrid waves is smaller than
predicted by theory. The probe's sensitivity as a function of the wavelength it measures can be
calculated based on its geometry. When we take these geometrical factors into consideration, we
find that the lower hybrid wave electric field is 0.8 the incident wave amplitude. This agrees
with the previous paper [Bamber, et. al., 1995], which predicted the magnitude of the electric
fields for the lower hybrid wave would be 70 - 122% of the incident wave. Most of the energy in
the lower hybrid wave is in its electric field; only a fraction of the whistler energy is. Therefore
the measurements, which show the two modes' electric fields are very nearly equal, do not imply
that all whistler energy is converted to lower hybrid waves. In fact, the present observations
agree with Bamber's calculations and observations which show that the energy density of the
lower hybrid wave outside the striation is only about 10% of the incident energy density.
Conclusions
Electric field measurements confirm the conversion of whistler waves to lower hybrid
waves at a density striation. The amplitude of the converted waves and the conversion
efficiency are well within theoretical predictions. We are limited in our ability to discuss
the overall efficiency of the interaction, however, because we have not been able to
measure a Poynting vector at all points in space. In order to account for the total energy
balance of the system one would need to have electric and magnetic field data taken over
a three-dimensional volume. This is a large task which will be undertaken at a future
time.
Currently, much interest is being paid to lower hybrid wave evolution in striated plasmas.
Rocket measurements provide hints about how lower hybrid waves behave in the ionosphere.
Further experiments at the LAPD plan to address some of these questions. We can launch lower
hybrid waves from a slow-wave structure in the plasma. The study of the interactions of densitydependent lower hybrid waves with a variety of density striations is underway.
Acknowledgments. Particular thanks are due to Jim Bamber and Jim Maggs whose
help in understanding this interaction is invaluable. This work has been supported by
ONR and NASA.
References
Arnoldy, R., K. Lynch, P. M. Kintner, J. Vago, S. Chesney, T. E. Moore, and C. J.
Pollock, Bursts of transverse ion acceleration at rocket altitudes, Geophys. Res. Lett.,
19, 413 - 416, 1992.
Bamber, J. F., Gekelman, W. and Maggs, J. E., Observations of Whistler Wave Mode
Conversion to Lower Hybrid Waves at a Density Striation, Physical Review Letters,
73, 2990-2993 1994.
Bamber, J. F., Gekelman, W. and Maggs, J. E., Whistler Wave Interaction with a Density
Striation: A Laboratory Investigation of an Auroral Process, J. Geophys. Research,
100, 23,795 - 23,810, 1995.
Bell, T. F. and H. D. Ngo, Electrostatic waves stimulated by coherent VLF signals
propagating in and near the inner radiation belt, J. Geophys. Res., 93, 2599, 1988.
Bell, T. F. and H. D. Ngo, Electrostatic lower hybrid waves excited by electromagnetic
whistler mode waves scattering from planar magnetic-field-aligned plasma density
irregularities, J. Geophys. Res., 95, 149, 1990.
Bell, T. F., H. G. James, U. S. Inan, and J. P. Katsufrakis, The apparent spectral broadening
of VLF transmitter signals during transionospheric propagation, J. Geophys. Res., 88,
4813, 1983.
Block, L. P., C. G. Fälthammer, P. A. Lindqvist, G. Marklund, F. S. Mozer, A. Pedersen,
T. A. Potemra, and L. J. Zanetti, Electric field measurements of Viking: first results,
Geophys. Res. Lett., 14, 435 - 438, 1987.
Dovner, P. O., A. I. Eriksson, R. Boström, and B. Holback, Freja multiprobe observations
of electrostatic solitary structures, Geophys. Res. Lett., 21, 1827 - 1834, 1994.
Eriksson, A. I., B. Holback, P. O. Dovner, R. Boström, G. Holmgren, M. André, L.
Eliasson, and P. M. Kintner, Freja observations of correlated small-scale density
depletions and enhanced lower hybrid waves, Geophys. Res. Lett., 21, 1843 - 1846,
1994.
Gekelman, W. G., H. Pfister, Z. Lucky, J. Bamber, D. Leneman, and J. Maggs, Design,
construction, and properties of the large plasma research device -- The LAPD at UCLA,
Rev. Sci. Instrum., 62, 12, 2875 - 2883 1991.
Kintner, P. M., J. Vago, S. Chesney, R. L. Arnoldy, K. A. Lynch, C. J. Pollock, and T. E.
Moore, Localized lower hybrid acceleration of ionospheric plasma, Phys. Rev. Lett., 68
2448 - 2451, 1992.
Pinçon, J. L., P. M. Kintner, P. W. Schuck, and C. E. Seyler, Observation and analysis of
7
lower hybrid solitary structures as rotating eigenmodes, J. Geophys. Res., in press,
1997 .
Strangeway, R. J., L., Kepko, C. W. Carlson, R.E. Ergun, J. P. McFadden, W. J. Peria, G.
T. Delory, C. C. Chaston, M. Temerin, R. F. Pfaff, FAST Observations of Electrostatic
and Electromagnetic VLF Emissions in the Auroral Zone, IPELS conference, June 23
to June 27, 1997
Vago, J. L., P. M. Kintner, S. W. Chesney, R. L. Arnoldy, K. A. Lynch, T. E. Moore, and
C. J. Pollock, Transverse ion acceleration by localized lower hybrid waves in the
topside auroral ionosphere, J. Geophys. Res., 97, 16935 - 16957, 1992.
____________
S. Rosenberg and W. Gekelman, UCLA Department of Physics and Astronomy, Los
Angeles, CA 90095 (email: stever@physics.ucla.edu; gekelman@physics.ucla.edu)
(Received August 19, 1997; revised November 18, 1997;
accepted Jaunary 15, 1998.)