Experimental Measurements of the Propagation of Large Amplitude
Shear Alfv n Waves
Walter Gekelman, S. Vincena, N. Palmer, P. Pribyl, D. Leneman, C. Mitchell, J.
Maggs, all at the Department of Physics and Astronomy, University of California,
Los Angeles•
Published in Plasma Physics and Controlled Fusion, 42, ppB15-B26 (2000)
Experiments on the edge plasma of Tokamaks have discovered magnetic fluctuations
which are highly correlated along the magnetic field, and are correlated with scale size
of the electron inertial length (δ = c/ωpe) across the field. They are, in all probability,
shear Alfv n waves. The FREJA, FAST, and Interball satellites have frequently
encountered density striations in the auroral ionosphere. These can be narrow, also on
the order of δ. Intense wave activity has been measured within these structures and
tentatively identified as shear inertial (VA > Vthe) Alfvén waves. These waves have
been studied in great detail in the Large Plasma Device (LAPD) at UCLA. The
plasma, which is 10m in length and 500 ion Larmor radii in diameter (He (λ ||≈ 2m),
Ar λ ||≈ 10m), 1.5 kG, 40 cm plasma diameter, n=1- 4.0x1012 cm-3, fully ionized)
supports, Alfvén waves. Our initial investigations, which will be briefly reviewed,
involved low amplitude (δBwave/B0 ≈ 10-4) shear waves launched by modulating a skin
depth size current channel and have examined the wave characteristics in the kinetic
(VA < Vthe) and inertial regimes, and in magnetic field gradients. Launching higher
power waves (δBwave/B0≥ 10-3) waves with the use of a helical antenna has extended
these studies. Both shear Alfvén waves (ω < ω ci ) and compressional Alfvén waves
have been investigated. Below fci the wave fields slowly spread across the
background magnetic field and the current associated with it forms a rotating spiral.
The higher power wave causes a localized density perturbation when δBwave/B0
exceeds 10-3 even when the wave propagates below the cyclotron frequency. The
perturbation is measured using Langmuir probes as well as laser induced fluorescence
(LIF) signal from Ar II ions. We present data of the wave propagation in which the
temporal history of the vector magnetic field was acquired at 20,000 spatial locations.
The data is used to calculate 3D wave currents, wave phase fronts and energy
propagation. In helium the wave pattern is more complex than in argon. We also
present the space and time evolution of the density perturbations associated with the
wavein an Ar plasma. LIF data was used to directly measure the ion motion in the
•
Work supported by the U.S Department of Energy, the Office of Naval Research and the National
Science Foundation.
1
electric field of the wave, ion polarization currents and the motion of the ions as they
form the density non-uniformities.
Introduction
Alfvén waves are the fundamental means by which information about local
changes in plasma current and magnetic fields are communicated within a
magnetoplasma. There are two very different modes of electromagnetic propagation at
low frequencies, a compressional wave in which density and field strength vary and a
shear wave in which only the direction of the magnetic field changes. The
compressional wave has been used in heating schemes for thermonuclear plasmas.
These waves may also play a significant role in edge plasma turbulence in fusion
devices. Correlation measurements1 of magnetic field fluctuations in Tokamaks have
revealed structures with transverse dimension of order of the electron skin depth across
the background magnetic field and much longer correlation lengths along it. In all
likelihood these are shear waves. They are a source of concern in fusion plasmas in
which energetic alpha particles may occur. These particles can destabilize toroidal
Alfvén modes, which, in turn, can ill affect the particle confinement. Alfvén were
observed early on in the solar wind, Coleman2 saw propagating MHD waves in the solar
wind satellite data and interpreted them as Alfvénic turbulence, and have been studied
extensively from then on3 . They are involved in the heating and energzation of solar
coronal loops4, and in the physics of extragalactic jets5. There has been a number of
interesting observations concerning shear Alfvén waves in the Earth’s auroal ionosphere
by the Freja and FAST satellites, both which have high temporal resolution, which
means they probe small spatial scale lengths. Louran et al.6 using Freja data for an
auroral crossing region observe an Alfvén wave (with phase velocity derived from a
measurement of E/B) of scale size of 200-800 m or a distance of order of the electron
δn
inertial length. A deep ( ≈ 0.70 ) density depression is simultaneously observed. The
n
authors conjecture is that a highly nonlinear Alfvén wave is responsible for the density
depletion. Electron skin depth sized inertial shear waves were observed by the FAST
satellite7. These were also observed in conjunction with deep density perturbations and
regions of electron current. The observed density perturbations were much larger than
theories17,18 they used could predict. These observations lead to many intriguing
questions. Is a significant component of the parallel electric field observed in the
auroral ionosphere due to Alfvén waves? Do the waves cause the field aligned density
2
perturbations, or are they refracted into and interact with pre-exisitng cavities? We
report a series of laboratory experiments on these issues.
Shear and Compressional Alfvén waves
In the MHD limit, for infinite plasma the Alfvén wave has two distinct
branches, which we will call the shear mode and fast mode. The shear mode only
exists below the ion cyclotron frequency and the fast mode both above and below it.
The fast mode, which is sometimes called the compressional wave, since it is
accompanied by a density perturbation has the same phase velocity across and along B,
ω
B
, where ρm is the mass
and in the MHD limit its dispersion relation is
= VA =
4πρm
k
density. The shear wave on the other hand propagates only along the magnetic field
ω
= VA . The experiments discussed involve the shear
and its dispersion is given by
k||
wave for the most part. It has been long recognized that the shear wave can develop a
transverse wavenumber and, therefore, a parallel electric field. If the parallel phase
velocity of the wave is within the electron distribution function Landau damping is
important and these waves are best treated using a kinetic formalism. The dispersion
ω2
(1) ε|| ( k − 2 ε ⊥ ) = − k 2⊥ ε ⊥
c
2
||
with: (2) ε ||
ω 2pe 2
− ω 2p + (1 − e − λ + I 0 (λ + ))
= − 2 ζ Z ′(ζ ) ; (3) ε ⊥ = 2
ω
ω − Ω 2+
λ+
relation is8:
Here ωp+ and Ω+ are the ion plasma frequency and gyrofrequency, and where ζ = ω/k||a
and λ+ = (k⊥a+/Ω+)2. The average electron thermal speed is denoted by a = (2Te/me)1/2
with Te the electron temperature, measured in ergs, and me the electron mass. The ion
thermal velocity a+ is defined similarly with T+ and M+ denoting the ion temperature
and mass. Z'( ζ ) is the derivative of the plasma dispersion function with respect to ζ,
and I0 is the modified Bessel function of order zero. In a plasma in which Te > Ti , in
the regime where vA < a, the electrons respond adiabatically to the wave, and the
dispersion may be written as
3
ω2
ω
= v 2A ( 1 − ϖ 2 + k 2⊥ ρs2 ) , ϖ =
, ρ s = cs / Ω + , cs = (Te / M + )1/ 2
2
k ||
Ω+
Here ρs is the ion sound gyroadius, and cs the ion sound, or ion acoustic wave speed.
( 4)
The correct terminology for this branch of the wave is the kinetic shear wave. This
wave has a parallel electric field given by
− k || k ⊥ ρs2
− k A k ⊥ ρs2
(5) E|| =
E
=
E⊥
⊥
1− ϖ2
(1 − ϖ 2 )(1 − ϖ 2 + k 2⊥ ρs2 )1/ 2
When the wave phase velocity is much large than the electron thermal velocity, the
electron plasma beta, βe = (
8πnkT
B2
),
is much smaller than the mass ratio (the reverse is
true in the kinetic case) and the dispersion relation becomes:
ω2
(6) 2
k ||
=
v 2A (1 − ϖ 2 )
c
,δ=
2
2
ω pe
(1 + k ⊥ δ )
This wave is properly termed the inertial shear Alfvén wave and the electron inertial
length, δ, appears in the dispersion relationship, rather than ρs . In the limit
ω << ω ci , k ⊥ = 0 , these both reduce to the MHD limit. In some MHD formulations of
these waves, both δ and ρs appear in the same dispersion relation. This is incorrect
because they represent opposite limits, and the fact that distribution functions are not
part of MHD.
The dispersion relation alone is not enough to determine the three dimensional
structure of these waves. This is governed by the radiation properties of the antenna, or
disturbance that produced these waves. We have studied the propagation of shear
Alfvén waves in both the kinetic and inertial regimes when the waves were launched by
fluctuating current filaments with transverse dimension of order δ9. The “antenna” was
a small circular Cu mesh aligned with its normal along B0, the background magnetic
field, and of transverse dimension δ. The experimental setup for both these waves, and
the larger amplitude ones to be discussed later is shown in figure1. The waves
launched using the disk exciter, in both the kinetic and inertial case had λ|| >> λ⊥ and
moved across the background magnetic field at a small angle ≈ 10 . The wave magnetic
field is zero on axis, rises rapidly as a function of radial distance from the exciter and
then decays. In the inertial case there is a radial location beyond which all the wave
current is zero and the wave magnetic field drops as 1/r. In the kinetic regime the radial
dependence may include secondary maxima10, and a propagation cone is not as clearly
defined, but the magnetic field drops off outside the region of main current flow.
Figure 2a is a schematic of the disk exciter and a conical region in which most of the
parallel current is confined. A sketch of the radial dependence of the magnetic field
4
averaged over several wave periods is shown on a plane at an axial distance from the
exciter. Figure 2b shows an experimental measurement of the wave field in plane with
one axis parallel to the background magnetic field. The wave is linear with Bwave-max= 21
B
mG, wave− max = 2.1 × 10 −6 . Here the plasma was Helium and λ|| = 2.7m .
B0
Figure 1. Experimental setup. A timing circuit locked to a primary clock, which controls the
discharge, triggers and arbitrary waveform generator. This outputs a programmed waveform,
which is amplified, goes through a transformer and then fed to a disk exciter or helical antenna.
Magnetic fields are measured with a three-axis probe. The signal is amplified and sent to a
bank of digitizers (up to 5 GHz /channel). Temporal records for each position are written to a
computer disk.
5
Figure 2 a) A schematic of the cone-like propagation of shear Alfvén waves. In the inertial case
the parallel wave currents are confined within an expanding cone with axis aligned along the
background magnetic field. On any transverse plane the magnetic field has a null point on axis.
r b) Data of the wave magnetic field in a plane containing the background magnetic field
B = B0 kˆ . The wave is in the inertial regime, and the disk exciter is located 94 cm, on axis, to
the rear of plane. The black stripe is a 2D version of the |B(r)| line in 2a. The pattern is
symmetric about r=0
6
Helical Antenna:
The largest peak-to-peak current that can be drawn to a disk exciter, in theory, is
the electron saturation current. When this is exceeded probe rectification sets in and the
current waveform is no longer sinusoidal. To produce a large amplitude Alfvén wave
with a disk exciter the following method was used. A positive pulse is applied to the
exciter and electron current is drawn. The rf wavepacket used to launch the shear wave
is used to modulate the D.C. current. For the disk exciters the radius was of order of half
c
the collisionless skin depth ( δ =
≈ 5. mm ), and the current drawn ( I p − p = je − sat A )
ω pe
places an upper bound on the wave magnetic field. What is more troublesome is that a
channel of depleted density always forms whenever a large DC current is drawn to an
electrode in a magnetoplasma. This occurs because electron heating and depletion of
electrons within the channel causes a change in the plasma potential, which acts to
drive ions out. The density drops, and because the system has inductance, the electron
drift velocity goes up to maintain the current. This results in further heating, which
reinforces the process until a steady state is reached11.
To avoid these difficulties, an antenna with a large (400 App) circulating r.f.
current was employed. The antenna was shielded from the plasma and fed by a floating
transformer as shown in figure 1. The system used the antenna inductance as part of a
resonant circuit, which had to be modified for use at different frequencies. The antenna
consists of two helical Cu wires one half of a wavelength (45 cm) long for a shear
f
= 0.76 , B0 = 1.5kG . The ends were joined with a thin
Alfvén wave in helium at
fci
ring of Cu, which was 10 cm in diameter. In Argon, the ion cyclotron frequency is 1/10
that of He, the parallel wavelength is 10m, and the antenna is, for all practical purposes,
straight. Since the dimensions of this exciter were much large than δ, we did not expect
to see "Alfvén cone" like radiation patterns. A sinusoidal rf burst lasting 40 µsec with
antenna current or 600 A p-p was applied.
Data of the vector wave magnetic field was acquired on 15 planes with their
normal parallel to the background magnetic field. Data at 2048 time steps ( δt =400 ns )
and 1025 locations, spaced 1 cm apart on a rectangular grid, was acquired on each
plane. An example is shown in figure 3. Unlike the patterns from a disk exciter, there
are clearly two current channels roughly 4 cm in diameter and 12.6 cm apart. The wave
fills almost the entire plasma column in a plane orthogonal to B0. The largest wave
δB
magnetic in the volume is was 1.6 Gauss, which corresponds to wave ≈ 10 −3 , where
B0
7
me
= 3.8 × 10 −3 . The pattern is intriguing because, when viewed in time, it appears
M Ar
to be a rotating spiral. The wave vectors in the center are due to the constructive
interference of the fields from the two current channels. This reminiscent of interference
patterns previously observed in a different tsituation12 which involved the intereference
of two Alfvén waves.
Fig 3. Data on a plane 2.64 meters from the end of the antenna. The magnetic vector field is
color coded in accord with the field strength. Data was acquired at each location an arrow is
drawn. The background magnetic field is directed into the page.
The enhanced magnetic field between the current channels does not appreciably change
orientation as the wave goes by, but at every half wavelength, when the field become
small, the currents rotate in the electron diamagnetic direction (clockwise in this view).
The spatial morphology of the wave is shown in the next figure 4. The largest field, in
red, occurs at the wave maxima. The spiral pattern is clearly visible. The wave currents
flow where the field is small. This is clearly seen at dz=3.96. Here the currents flow in
the two elliptical blue patches next to the field maxima at approximately 2:00 and 7:00.
The pattern in Helium is much more complex and consists of a higher m mode
8
involving up to five current channels. These patterns will be discussed in far more detail
in a future publication.
r
Fig 4. The transverse scalar magnetic field B⊥ =
Bx2 + By2 of a shear Alfvén wave. Six of the
15 planes on which data was acquired are shown here. The spatial extent (dx=41cm, dz=25 cm)
is the same as that in the previous figure. The data is acquired at t=173 µsec (8.5 wave periods)
after the turn on of the rf.
In Ar the currents associated with this wave are not very complicated. They are shown in
the next figure 5. The wave current is calculated from the volume vector data set using
r 1
r
j=
∇ × B . At the low frequency of this wave the displacement current is negligible.
4π
Current flowing towards the observer is shown in red, that flowing away in blue. The
field-aligned current is carried by the electrons and cross-field current by the ions via the
polarization drift. The long stretches of field aligned current reflect the10-m parallel
v
wavelength of this wave. This wave is in the kinetic regime with the ≈ 2 . Since the
VA
cyclotron frequency in Ar is it is low ( fci = 57 kHz, the damping is relatively large,
ν
γ = collision ≈ 20 , which is reflected in observed wave amplitudes which drop by a factor
ν wave
of two, 6.3 meters from the antenna. Nevertheless the antenna produces what is
considered a large amplitude shear wave. For example, the wave magnetic field in these
9
experiments is in 1-2 orders of magnitude larger than the linear waves studied in the disk
exciter experiments.
10
Fig 5. Currents of a shear Alfvén wave launched by a helical antenna. The wave magnetic field on a
plane orthogonal to the background magnetic field is shown to guide the eye.
These waves are associated with spatially localized density perturbations. The process,
which leads to the perturbation, is not yet fully understood; the data is presented and then is
δn
discussed in a subsequent section. Figure 6 shows the density perturbation
, on three
n
planes as well as the vector magnetic field and its magnitude. The magnetic field was
measured on a 41cm x 25 cm grid (1025 locations). At each position, each component of the
vector field was averaged over 10 shots. The magnetic field amplitude is largest in the center
and the regions where the wave current channels are clearly visible. The density perturbation,
which is localized to the central region, was measured on a 25 X 41 position grid with 2.5 mm
spacing. The transparent plane is a surface of background density, when the wave is not
present. The density displayed here was derived from the ion saturation current drawn to a
1mm2 ( Vbias = -66 V, kTe = 6.5 e.V.) Langmuir probe. In a separate experiment the Langmuir
current voltage curve was rapidly (10 µsec), swept at several times and it was verified that
there was no significant change in the electron temperature in the region and that the ion
saturation current was a measure of the density. In this picture the ion data was smoothed over
a half wave period to eliminate high frequency noise. In a temporal sequence of this data one
sees the density on the closest plane (dz = 0.66m) is affected first, and then see disturbance
propagate along the device. One can also see the density on the closest plane oscillate at the
wave frequency with an amplitude of order 5-10% as well. As the sinusoidal oscillations
progress the “DC” density perturbation grows.
11
Figure 6. Plasma density perturbation in three planes at a time t = 224 µsec after the antenna is turned on. The
largest density perturbations are 10%. The vector magnetic field, along with its magnitude displayed on the
bottom are shown for dz=3.3 meter. The grid spacing in the x and y directions is 1 cm.
The axial propagation of the density perturbation can be estimated by measuring the time difference
between the onset of the density perturbation on the first plane at dz=0.66m and the furthest, dz=3.3m.
This time difference is ≅ 270 µsec and the propagation speed of order 104 cm/sec, or 25% of the ion
acoustic speed. As the density perturbation is spatially non-uniform and is not invariant along z this is
speed is a rough estimate. Examination of the temporal waveforms at different axial and radial
locations show different parts of the density perturbation seem to arrive at different speeds, and in
some cases these may exceed the ion sound speed. After the tone burst is over the density
perturbation gradually goes away and peaks and valleys of the perturbation wander around the
transverse planes ( at v⊥ < cs ) as they decay. The ion motion has been directly measured using Laser
Induced Fluorescence (LIF) with a tunable dye laser using the ArII spectral line at 611.492 nm13. The
laser light was introduced to the plasma with an optical fiber, which illuminated a volume of 1 mm3.
The observed signal, from the decay of a metastable state (3d2G9/2), is observed at 460.957 nm by a
second fiber capped with a grin lens and positioned at aright angle to the first. If one assumes that the
metastable density is directly proportional to the ion density this measurement yields the ion
12
distribution function. We record the LIF signal as a function of time (t = 819 µs, dτ = 1.6 µs) at 20
wavenumbers spanning the ion line.
13
Figure 7 Laser Induced Fluorescence measurements in the center of the density perturbation and 5 cm to the
side. Shown is the ion distribution function as well as the frequency of oscillation of fi(v)
The light was originally digitized at 200 ns/sample and then smoothed with a running average over 8
samples. The signal amplifier, however has a 100 kHz bandwidth which gives an effective time
resolution of 10 µs. The data contains information on the ion temperature as well motion of the ion
distribution function in velocity space, and along the probe laser beam in real space. Figure 7a shows
the density perturbation on a 4cm X 4cm plane 3.3 m from the exciter (this was a different data run
from the one shown in figure 6). The magnetic field pattern on this data run was measured as well,
and found to be the same as that in figure 6. The laser intersected the plane at (x,y)=(0,0) , the center
of the plane and at (x,y)=(5,0), a location 1 cm to the right of the edge. Figure 7b shows the ion
distribution function at an instant of time after the perturbation has developed. While the wave is
present the ion distribution function is observed to rock back and forth in velocity space about v=0,
and the amplitude of these oscillations is used to determine the AC drift velocity; in this case
vdrift
= 0.18 , vthi is the ion thermal speed for 1.14 eV. ar.gon. A time series of the motion of a point
vthi
near the peak is Fourier analyzed and the frequency spectrum is displayed in fig. 7b. The frequency
spectrum has a sharp peak at 49 kHz that of the Alfvén wave. Further away from the center of the
wave pattern, fig 7b, we observe slighter hotter ions with a smaller AC drift velocity. Also present at
x=5 cm is a DC component of the ion drift. This corresponds to motion of ions involved in the
formation of the density perturbation moving at vion = 0.12vthi = 2 × 10 4 cm/s = .05 cs . The LIF data
may be used estimate the perpendicular electric field of the wave. There are two components of the
E
1 dE
ion drift the EXB drift, vEXB = ⊥ and the ion polarization drift, v p =
. The latter is
B0
ω c B dt
responsible for the closure of the Alfvén wave currents across the magnetic field as shown in figure 5.
The two are out of phase and if magnetic field measurements are made at the point where LIF data is
taken they can be distinguished from one another. This will be done in future experiments in which
we will do LIF in an entire plane. Since f=0.86fci in this experiment these drift velocities will be
comparable in magnitude. For the drift velocity in 7b, E⊥ = 0.45 V/cm . One may use equation 5 to
predict the parallel electric field of these waves, E|| = 11 mV/cm . Finally a double-sided Langmuir
probe with two faces, each 1 mm2, was placed in the wave current with the normal to its face along
the machine axis and parallel/antiparallel to B0. If the probes are biased in the electron saturation
regime the difference in current to them is the net field aligned current. The current and voltage to
this probe is shown in figure 8 as it is swept from ion saturation to electron saturation current over
five cycles of the wave. The parallel current measured at this location was j|| = 0.5 A/cm 2 p-p . The
current measured with this method is spatially non-uniform; it is close to zero in the central region
between the two current channels (figure 5) and as large as 0.7 A/cm2, which corresponds to A-C
density perturbations of order 5%.
14
Figure 8. I-V characteristic curve of the difference current to a small double-sided probe place in the Alfvén
wave current. Th voltage to each face is swept from -45V to +80 V over a 200 µs interval. The wave period
is 20 µsec.
Discussion of results
Shear Alfvén waves come in different “flavors” depending upon the method of excitation. Low
amplitude shear waves excited by electron skin depth scale oscillating currents form cone-like
structures, which slowly spread across the magnetic field. They produce no significant density
perturbation and the wave magnetic field is transverse unless f ≅ fci 14, The magnetic field of these
waves is localized to the center of the plasma column. The wave produced by the helical antenna fills
the entire plasma column in the transverse direction and although it obeys the same dispersion relation
its spatial pattern and associated current system are very different. The wave magnetic field was
15
bwave
b
≅ 10 −3 ) and low wave ≅ 10 −6 amplitudes and the wave field looks the
B0
B0
δn
same (figure 3,4) in both cases. At the higher amplitude a density perturbation as high as
≈ 15%
n
developed in the region of the largest wave magnetic field. There is no observable DC current or
electron heating associated with this wave. Because of the importance of Alfvén waves in the auroral
ionosphere and their observation in, presumably, field aligned density cavities, several calculations of
ponderomotive force effects have been recently published15,16,17,18,19. These calculations, most of which
δn
are for the inertial shear wave, predict incredibly small perturbations ( ≈ 10 −6 ) for our plasma
n
20
parameters. Shukla and Stenflo have calculated the ponderomotive force of a kinetic shear Alfvén
wave and found that the parallel electron ponderomotive force drives density depressions and the
parallel ion ponderomotive force produces density enhancements, both of which are observed in this
experiment (figure 6). This theory, which neglects the compressional branch of the wave, predicts a
one- percent perturbation for our experimental parameters. We have observed with probes and LIF,
that close (dz < 3.3m < λ) to the exciter, a significant AC density oscillation exists while the wave is
measured at both high (
present. At this frequency the compressional wave is cut off since the column diameter is much less
than the wavelength. This does not mean that the compressional wave does not exist, it is evanescent.
The imaginary part of the wavenumber should scale as the reciprocal of the plasma column diameter
(40 cm) and is easily observable at 66 cm from the antenna, the lowest plane in figure 6. To test for
effects arising from the compressional wave we have extended these measurements to above the ion
cyclotron frequency and observed density perturbations, which seem to get smaller at increasing
frequency. The wave magnetic field pattern is dramatically different above the ion cyclotron
frequency. We have also applied a current pulse to the helical antenna and observed density
perturbations. A pulse contains a spectrum of frequencies and can launch field line resonances21 as
well as evanescent compressional waves. These measurements will be reported in a forthcoming
publication, but we take them as evidence that the compressional wave plays a role in the production
of the density perturbation. Finally Hollweg22 has done a fluid calculation for the AC density
perturbations due to the kinetic Alfvén wave using a dispersion relation which mixes the electron
inertial length and the ion sound gyroradius, but considers compressional effects. It predicts an AC
perturbation of order 14%, which is in accord with our observation. The mechanism for formation of
a DC density perturbation, and any feedback mechanism between the large AC density fluctuations
and the ponderomotive force, must be determined with future experiments and theory.
16
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22
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2
17