PHYSICS OF PLASMAS 18, 055501 (2011)
The many faces of shear Alfvén wavesa)
W. Gekelman,b) S. Vincena, B. Van Compernolle, G. J. Morales, J. E. Maggs,
P. Pribyl, and T. A. Carter
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
(Received 21 December 2010; accepted 1 February 2011; published online 1 June 2011)
One of the fundamental waves in magnetized plasmas is the shear Alfvén wave. This wave is
responsible for rearranging current systems and, in fact all low frequency currents in magnetized
plasmas are shear waves. It has become apparent that Alfvén waves are important in a wide variety
of physical environments. Shear waves of various forms have been a topic of experimental research
for more than fifteen years in the large plasma device (LAPD) at UCLA. The waves were first
studied in both the kinetic and inertial regimes when excited by fluctuating currents with transverse
dimension on the order of the collisionless skin depth. Theory and experiment on wave propagation
in these regimes is presented, and the morphology of the wave is illustrated to be dependent on the
generation mechanism. Three-dimensional currents associated with the waves have been mapped.
The ion motion, which closes the current across the magnetic field, has been studied using laser
induced fluorescence. The wave propagation in inhomogeneous magnetic fields and density
gradients is presented as well as effects of collisions and reflections from boundaries. Reflections
may result in Alfvénic field line resonances and in the right conditions maser action. The waves
occur spontaneously on temperature and density gradients as hybrids with drift waves. These have
been seen to affect cross-field heat and plasma transport. Although the waves are easily launched
with antennas, they may also be generated by secondary processes, such as Cherenkov radiation.
This is the case when intense shear Alfvén waves in a background magnetoplasma are produced by
an exploding laser-produced plasma. Time varying magnetic flux ropes can be considered to be low
frequency shear waves. Studies of the interaction of multiple ropes and the link between magnetic
field line reconnection and rope dynamics are revealed. This manuscript gives us an overview of the
major results from these experiments and provides a modern prospective for the earlier studies of
C 2011 American Institute of Physics. [doi:10.1063/1.3592210]
shear Alfvén waves. V
I. INTRODUCTION AND THEORY
Alfvén waves are of fundamental importance in space
plasmas, astrophysical plasmas, and in earthbound efforts to
achieve thermonuclear fusion in confined magnetized plasmas. None other than Alfvén conjectured Alfvén waves to
exist in the solar corona in 1945,1 and many others have discussed them in this context since. Recently their motion in
the corona was observed in detail by imaging the velocity of
a FeXIII line.2 They were first seen in the solar wind in
19663 and are abundant there.4 The solar wind fluctuations
seem to be filamentary with large correlation lengths along
the magnetic field5 (>1.5 106 km) and shorter ones across
the field (105 km). The earth’s aurora is replete with Alfvén
waves, and they have also been observed to be filamentary,
the structures, sometimes as small as the electron skin
depth.6 Alfvén waves in the aurora have been seen in association with skin depth size density striations.7 One may also
consider time varying magnetic flux ropes as low frequency
shear waves. The importance of these waves in the auroral
region cannot be understated. Measurements taken by the
Polar satellite indicate that the downward flux of Alfvén
waves has enough energy to power the aurora.8 A review of
the import of Alfvén waves in astrophysical processes such
a)
Paper AR1 1, Bull. Am. Phys. Soc. 55, 20 (2010).
Invited speaker.
b)
1070-664X/2011/18(5)/055501/26/$30.00
as stellar winds, extragalactic jets, and quasar clouds is given
by Jatenco-Pereira.9
In 1942, Hannes Alfvén proposed the existence of
hydrodynamic waves.10 This was met with skepticism since
it was believed that any conducting fluid would immediately
short circuit wave electric fields. As plasma lore has it, physicists suddenly reversed their thinking when Enrico Fermi
agreed with Alfvén at a seminar at the University of Chicago.11 As with many discoveries, the importance of these
waves in space, solar, astrophysical, and finally magnetic
fusion plasmas took decades to become apparent. Alfvén
waves were not observed immediately after their existence
was predicted. The original fluid description gives a dispersion relation for the shear (or torsional) wave of frequency x
and field-aligned wavenumber kjj
x
B
¼ VA ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ;
kjj
4pnM
(1)
where B is the magnetic field strength, n the plasma density
and M the ion mass. The waves exist only below the ion cyclotron frequency and in simple MHD descriptions propagate
along the magnetic field. The other branch of these waves is
the fast or compressional wave. These propagate at the
Alfvén speed along and across the magnetic field and exist
above and below the ion cyclotron frequency. To launch
shear Alfvén waves in an experimental device several factors
18, 055501-1
C 2011 American Institute of Physics
V
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-2
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
must be taken into consideration. The low frequency of the
waves coupled to the requirement that ions must be magnetized pose a requirement that the ion Larmor orbit be small
enough and the plasma be dense enough for the waves to fit
in the experimental device. If the plasma is very dense the
parallel wavelength is short, but unless the plasma is hot the
waves will be damped by collisions. High temperature translates to larger gyroradii and therefore larger plasma sources
and raises survival issues for internal diagnostic probes. In
the 1940s laboratory plasmas suitable to study these waves
did not exist, and the first observation was made in mercury.12 One of the first experiments in which Alfvén waves
were glimpsed in an ionized gas was by Bostick and Levine.13 In this experiment standing waves, in the compressional branch were observed in a pulsed toroidal plasma.
Other early experiments14 involved pulsed collisional plasmas and verified that the wave propagates at the predicted
velocity, but these lacked detailed measurements of the wave
electric or magnetic field. From the 1980s to the present, basic science groups in Australia, Japan, and in the United
States have done experiments on Alfvén waves. In these
experiments some fundamental properties of the shear
Alfvén wave have been explored and their direct relevance
to the spacecraft measurements previously discussed has
emerged. At the same time a great deal of research on Alfvén
waves in thermonuclear plasmas occurred. The greater part
of the waves studied was bounded cavity modes launched by
antennas close to the chamber wall. More recently a great
deal of attention has been paid to toroidal Alfvén eigenmodes (TAE) modes in magnetically confined fusion plasmas.15
These are shear waves with a dispersion relation affected by
the toroidal geometry of the background magnetic field of
the Tokomak plasmas in which they are generated. TAE
waves have been observed in several fusion experiments and
are worrisome if they adversely affect the plasma confinement where they to grow to large amplitudes. The dispersion
relation16 for shear waves in a warm plasma with ion temperature much less than the electron temperature, TI Te is
more complicated than Eq. (1).
The regime of transverse scale lengths small compared
to the ion skin depth is of interest in problems related to electron acceleration, ion heating, and current modulation and
has a fast or compressional mode that is evanescent. Thus its
contribution to the shear mode can be separated. This results
in a highly accurate and compact dispersion relation
1
2 2
x pffiffiffiffiffi
c2 k?
e? 1 2
;
kjj ¼
x ejj
c
(2)
where ejj and e? are the parallel and perpendicular (to the
confinement magnetic field) components of the dielectric
tensor, respectively. Here c refers to the speed of light and
k? is the transverse wavenumber.
For collisionless, hot electrons
!
kD2 0
x
;
(3)
ejj ¼ 1 2 Z pffiffiffi
2kjj
2kjj ve
while for a single species of cold ions,
e? ¼ 1 þ
x2pe
x2pi
:
x2ce x2 x2ci
(4)
0
Here Z ðnÞ is the derivative of the plasma dispersion function,17 with n ¼ pffiffi2xk v and ve is the electron thermal velocity.
jj e
For cold ions and for a wave frequency not too close to
the ion gyro-frequency xci , the dispersion relation takes the
form
2
V
x2
0
2 2
d ;
(5)
Z ðnÞ A2 1 2 n2 ¼ k?
2
ve
xci
where d ¼ c xpe is the electron skin depth or electron inertial length.
This expression has two important limiting cases. For
n1 the wave is called the inertial shear Alfvén wave, and
the dispersion relation (5) reduces to
x2
¼
kjj2
VA2
x
1
xci
2 !
2 d2 Þ
ð1 þ k?
:
(6)
The inertial limit occurs when the Alfvén velocity is much
higher than the electron thermal speed; the wave is well outside the electron distribution function. The other limiting
case is the opposite, n21, the electrons are hot, and the
wave phase velocity is within the main body of the electron
distribution function. This mode is referred to as the kinetic
Alfvén wave. The real part of the dispersion relation is
x2
x2
2
2 2
(7)
¼
V
1
þ
k
q
A
? s ;
kjj2
x2ci
where qs ¼ cs =xci is the ion sound radius, with cs the ion
sound speed.
In both of these cases the shear wave has a parallel electric
field, which is important as it can lead to particle acceleration.
If the value of n is not in these two extreme limits, the wave is
termed to be in the intermediate regime, and the full plasma
dispersion relation must be used. This intermediate regime is
significant in that the wave phase velocity is close to the electron thermal velocity, and Landau damping can be strong.
These expressions for the dispersion relation in the various regimes are only part of the story because they describe
a single wave number. Since any real signal originates at a
source, a spectrum of wave numbers must be considered to
ascertain its spatial structure. Equation (1) is for a plane
wave propagating exactly parallel to the confinement magnetic field. In this idealized case the wave’s magnetic and
electric fields are both at right angles to the direction of propagation. The wave carries plasma current, but how it closes
is not apparent from Eq. (1). What is more realistic is the
consideration of waves associated with currents of finite
dimensions, e.g., currents whose transverse dimension is
comparable to the electron inertial length or ion sound gyroradius. If the radius of the current channel is r0, and the current density is j0 at the channel center, the wave magnetic
field is in the azimuthal direction and given by18
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-3
Many faces of shear Alfvén waves
4pr0 j0
Bh ðr; z; xÞ ¼
c
ð1
0
Phys. Plasmas 18, 055501 (2011)
J1 ðkr0 Þ
J1 ðkrÞeikjj ðkÞz dk;
k
(8)
in which the confinement magnetic field is in the z direction.
In this expression the relationship between k and k\ is mediated by the dispersion relation which in practice may require
the inclusion of collisional effects.
For the case of the inertial wave19 the spatial structure
predicted by Eq. (8) is governed by the ratio p ¼ rd0 . For large
values of p (narrow current channels) the theory predicts that
the parallel wave current lies within a cone, which emanates
from the edge of the exciter disk. Further away from the
source the cone spreads at an angle h given by
r r0
¼
tan h ¼
z
rffiffiffiffiffiffi
me x
M xci
"
1
x
xci
2 #12
:
(9)
Outside of the cone the wave current is zero, and the wave
magnetic field decays as 1/r. The wave current is confined
within the cone, which expands radially with distance
along the magnetic field away from the current source. Far
from the source the wave magnetic field in the center of
the cone vanishes. The magnetic field is in some sense
similar to that of a current carrying wire; however, here
the “plasma wire” expands along its axis. Unlike the field
of a wire the pattern exhibits a radial diffraction pattern far
from the source.
The behavior in the kinetic regime does not exhibit the
characteristics of the cone given in Eq. (8). Instead the wave
is confined to a maximum angle of propagation given by18
x 2 pffiffiffiffiffi
be ;
(10)
tan hmax ¼
xci 332
where be is the electron beta. The magnetic field pattern
exhibits radial oscillations within hmax which evolves into a
1/r behavior at large radial positions.
Figure 1 shows the spatial pattern in an x-z plane from
Eq. (5), here the magnetic field is in the z direction and the
center of the current filament is located at x ¼ 0.
II. EARLY ALFVÉN WAVE EXPERIMENTS
There were no laboratory plasma sources adequate to
perform Alfvén wave measurements at the time of their prediction. The first theory was for a conducting fluid in which
the particle’s gyroradii are assumed to be zero, and the conductivity assumed to be infinite. The initial experiments
were done in liquid metals. Lundquist20 used mercury, which
is a liquid at room temperature. Waves on the surface of
stirred magnetized column were observed on the surface
with a mirror. Lehnert21 used sodium that is 34 times more
conductive than mercury but has to be heated (Tmelt ¼ 98 C)
to the liquid state. The experimental apparatus is shown in
Fig. 2. The sodium column was “twisted” in the azimuthal
direction by the stirrer. At higher conductivity the magnetic
field is frozen into the fluid, and the shear (or torsional)
FIG. 1. (Color) Theoretical patterns of one component, By, of the Alfvén wave in the kinetic and inertial regimes. The waves propagate from left to right.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-4
Gekelman et al.
FIG. 2. From Lenhardt (1952). Liquid Sodium apparatus for studying
Alfvén waves. The key elements are a heater on the bottom (21) and a stirrer,
which, oscillated at 30 Hz (15) and a double probe that measured potential difference on the surface (20). The sodium was threaded by a uniform vertical
magnetic field of 0.1-1T. The height of the sodium column was 10 cm.
Alfvén wave emanates from the stirrer. This is illustrated in
the second figure from Lenhardt, Fig. 3.
Using a fluid theory, which included the fact that the
waves are bounded in the radial direction and collisional
damping due to the finite conductivity of the medium, Lenhardt predicted the potential difference on the surface probes
and compared it to the experimental results. This is shown in
Fig. 4. The conductivity of a highly ionized plasma is orders
of magnitude higher than that of liquid sodium, and as soon
as plasmas became available the laboratory study of Alfvén
waves switched media. The first measurement in laboratory
plasma was by Bostick and Levine in 1952.4 In this simple
experiment a gas in a toroidal tube was ionized with a pulse
applied to a coil wrapped around it. There was a background
toroidal magnetic field of 400 G. A floating Langmuir probe
recorded oscillations of order 104 Hz. The oscillations were
interpreted as a standing MHD wave and satisfied the relaffi, where L is the circumference of the torus
tionship f ¼ 2LpBffiffiffiffiffi
4pq
and q is the mass density. No waves were seen when the toroidal magnetic field was zero. The diagnostics were not
good enough to distinguish between a shear and a compressional wave (it was probably the latter).
In another early experiment by Jephcott22 which was
similar to that of Bostick, an antenna wound around a torus
was used to ionize a gas (Id ¼ 10 kA, td ¼ 200 ms). A background toroidal magnetic field of 3–14 kG was present. Two
external coils were used to produce an oscillating magnetic
field transverse to the background magnetic field. Magnetic
Phys. Plasmas 18, 055501 (2011)
FIG. 3. From Lenhardt. The cylinder is the Na column. When the liquid is
moved in the azimuthal direction the magnetic field entering the bottom is
twisted sideways.
probes picked up 10 G oscillations, which exhibited delay times
in accord with the Alfvén propagation speed. The experiment
was highly collisional, and no wave structure was measured.
In yet another early experiment by Sawyer et al.,23
hydrodynamic waves were observed in a linear discharge of
60 cm long and 15 cm in diameter. There was an axial magnetic field of 500 G. The plasma was produced with a 200 kJ
capacitor bank in relatively high (0.1–10 mm) pressure deuterium. Magnetic loop probes 3 mm in diameter, electrostatically shielded, and set in quartz tubes were used to measure
the azimuthal, radial, and axial component of the magnetic
field. A self-generated helical wave was measured and
detailed plots of current distribution were presented. The
wave magnetic field was proportional to p1ffiffiq, and the wave
seemed to travel at the Alfvén speed. The resulting wave
was deduced to be a mix of hydrodynamic waves traveling
in the axial and azimuthal directions. Although the wave
physics is quite complex, this is one of the first experiments
where detailed probe measurements were made.
Wilcox et al. performed a better experiment, which
was executed in a linear device.24 The plasma was produced
by a high-pressure (100 l, n0 ¼ 3.5 1015 cm3) discharge
between the ends of the device using a switch and a capacitor
bank. The wave was launched by applying RF from a ringing capacitor between the cathode of the plasma source, and the wall
of the device after the plasma was formed. The wave phase velocity was measured with a magnetic pickup probe as a function
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-5
Many faces of shear Alfvén waves
FIG. 4. Theory and experimental predictions by Lenhardt (1952). The ordinate is the potential difference due to the wave electric field and the abscissa
the inverse of the background magnetic field. The excellent agreement verified the existence of the shear wave.
of the background magnetic field using the time delay between
the applied sinusoid and the received signal on a magnetic probe.
It agreed with the predicted Alfvén speed. Wave attenuation was
also measured, and this roughly agreed with theory.
As time elapsed, laboratory experiments on Alfvén
waves and detailed comparison with theory improved.
Another experiment by Jephcott and Stocker25 was carried
out in a 2-m long, linear device in an argon plasma. The
wave was launched by applying an oscillating waveform,
generated in an LC circuit, to a ball electrode 50 cm from
one end of the device. The time delay between two magnetic
pickup probes 50 cm apart measured the phase velocity of
the waves. Double Langmuir probes were used to calculate
the density at the position of the magnetic probes, and the electron temperature (n ¼ 1 1015 cm3, Te ¼ 2.3 eV). The ion
temperature was estimated from a spectroscopic measurement
of Doppler broadening (Ti ¼ 0.2 eV). The conductivity was estimated from electric probe measurements. The plasma was collisional; the ion neutral collision frequency was much greater
than the wave frequency (vei ¼ 4.5 106 Hz, fwave ¼ 125 kHz).
The magnetic probes could be moved radially and the azimuthal component of the wave magnetic field, Bh(r) was measured. The data were compared to predictions of a theory of
bounded Alfvén waves by Woods,26 which included the effect
of neutral collisions. The comparison was excellent.
Experiments continued for the next thirty years, which
for the most part took place in arc type plasmas that were
collisional. Plasmas of this sort allowed one to operate at
14
3
high density (n
10 cm ) and therefore relatively shorter
1
wavelengths k / pffiffin . Higher density plasmas, which are
3
cold, become collisional ðvei / nT 2 Þ, and this must be taken
Phys. Plasmas 18, 055501 (2011)
into account in theoretical comparisons. If one raises the
temperature to circumvent these effects then internal probes,
which are the best means of getting detailed spatial information,
tend to evaporate. One might think that lowering the magnetic
field is the path to shorter wavelengths. There are two problems
with this. First of all shear waves exist below the ion cyclotron
frequency, which is directly proportional to B. When the wave
frequency is low collisional effects become worse. The ion gyroradius is inversely proportional to B and one must have a larger
diameter plasma column, ideally many ion gyroradii across.
Experiments which measured the wave dispersion,
where the plasma column was narrow enough to require Bessel function solutions in the radial direction, were done by
Wilcox,27 Swanson et al.,28 and Müller.29 Starting in 1980,
Amagishi and his colleagues did experiments on Alfvén
waves at the TPH arcjet30 at Nagoya University in Japan for
about ten years. The arcjet produced a 2 m long, 15 cm diameter (full width at half maximum ¼ 6 cm) helium plasma
(n ¼ 5 1014 cm3, Te ¼ Ti ¼ 4 eV, B ¼ 2.5 kG). At these
densities the Alfvén wavelength is of order 40 cm (f ¼ 390
kHz, fci ¼ 950 kHz), but the plasma was not completely ionized (70%) and the collision rate vei was higher than the
wave frequency by a factor of ¼ 104. In another experiment
the compressional and shear waves were both observed using
a Stix type coil31 having a spectrum of wave numbers32 with
the plasma selecting parallel wavelengths that could fit in the
machine. The waves were detected by a small (5 mm diameter, 100 turn) magnetic probe. The wave modes were bounded
in the radial direction as the arc plasma diameter was about 6
cm. Other experiments33 by this group observed mode conversion of fast waves launched by a strap antenna near the wall
of the machine, to shear waves on the Alfvén resonance layer
(position on the radial density gradient where the axial phase
velocity matched that of the shear wave). These and other
experiments have been reviewed by Gekelman.34
The theory outlined previously indicates that narrow
current channels fluctuating below fci will excite coneshaped wave field structures. In the experiments outlined
thus far the plasma column was too narrow and the plasma
too collisional to observe them. One of the first clear experimental observations of these cone like structures was made
by Borg et al.35 in the afterglow of a pulsed plasma where
the magnetic field and density was high (Baxis ¼ 0.5T, ne
^ 1 1013cm3) and the electron temperature was Te^ 10 eV.
A small rectangular antenna 2 cm high and 8 cm long was
aligned with the toroidal magnetic field and placed in the
center of the plasma. The wave was detected by magnetic
pickup probes located on the opposite side of the torus that is
180 away from the exciter. The spatial measurement along
with theory of the predicted pattern is shown in Fig. 5. What
is clear is that the wave magnetic field vanishes at the center
of the current channel. The experiment was in the inertial regime with VVtheA ’ 3, but since it was collisional the fine cone
structure due to large k\ could not be seen.
III. THE LARGE PLASMA DEVICE
The original 10 m long large plasma device (LAPD) at
UCLA (Ref. 36) has been in operation since 1990 and was
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-6
Gekelman et al.
FIG. 5. Measurement by Borg et al. (Ref. 25) of the magnetic field of a
shear Alfvén wave launched by an antenna at r ¼ 5 cm. B ¼ 0.8 T, f ¼ 550
kHz, in a toroidal plasma. The solid lines are theory with N the number of
times the wave circled the torus.
Phys. Plasmas 18, 055501 (2011)
Possible magnetic field geometries are uniform, linearly
increasing in z, mirror fields, multiple mirrors, and cusps.
The plasma is produced by a DC discharge between a
Barium oxide coated cathode38 and mesh anode. The bulk of
the plasma carries no net current and is therefore quiescent.
The plasma parameters are: He, Ar, Ne, H, Xn, dn
n 3%,
1010cm3 ne 2 1012cm3, 0.05 Te 10 eV, Ti 1 eV. The plasma is pulsed at 1 Hz with typical plasma duration of 15 ms. The machine can run continuously for approximately four months before the cathode has to be cleaned
and recoated. There are 450 access ports serviced by portable
pump-down stations, which allow introduction of probes,
antennas, additional cathodes, ion beams, electron beams,
etc., while the machine is running. Probes go through specially designed ball valves39 that allow computer controlled
probe drives to accurately move them on transverse planes
throughout the device.
IV. ALFVÉN WAVES FROM NARROW CURRENT
FILAMENTS
upgraded to a 22 long meter long device in 2001. The present
LAPD plasma column is 60 cm in diameter and 18m long.
The axial magnetic field is produced by 90 solenoidal magnets (Fig. 6) and may be varied over the range 300G B02
2.5kG. Ten power supplies generate current for the magnets therefore the axial magnetic field profile is variable.
The LAPD plasma is much less collisional
(C ¼ vxcoll 1) than the plasma in the experiment by Borg
et al. (C ¼ vxcoll 29). A fluctuating current filament in the
LAPD was used to generate shear waves and the Alfvén
cone patterns were verified and compared to theory with collisions and Landau damping.40 Another early measurement
FIG. 6. (Color) Plasma production in the LAPD device. An oxide-coated cathode heated to 900 C is placed 55 cm from a molybdenum mesh anode. A transistor switch (Ref. 37) is used to apply a voltage between the cathode and anode, the current is supplied by a 4 F capacitor bank. A bank of 90 magnets supplies
the axial DC magnetic field, B0z. The upper right insert is a photograph of a He plasma through an end window located radially near the wall of the machine.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-7
Many faces of shear Alfvén waves
Phys. Plasmas 18, 055501 (2011)
V. THE MORPHOLOGY OF SHEAR WAVES
LAUNCHED FROM VARIOUS ANTENNAS
FIG. 7. (Color) The wave magnetic field of a shear Alfvén wave launched
by a small mesh grid in a plane transverse to the background magnetic field
B0z. An image of the grid antenna to scale is shown, although the grid is on
a plane dz ¼ 1.54 m (1 parallel wavelength) from the data plane shown. The
background magnetic field is B0z ¼ 1.1 kG, xxci ¼ 0:77, Helium, Te ¼ 10 eV,
n ¼ 2.0 1012 cm3.
of the structure of Alfvén waves due to a skin depth size currents measured the perpendicular wavenumber of the wave
magnetic field.41 Figure 7 shows the magnetic field of a shear
wave in a plane launched by applying a tone burst of oscillating potential to a circular mesh grid. A small DC potential is
applied to the grid during the tone burst so that the oscillating potential modulates the electron current to it at the shear
wave frequency. The magnetic field is governed by Eq. (5).
In Fig. 7 one can clearly see that Bh reverses in the radial
direction, outward from the center of the current channel.
From the data k\; 1.26 cm1, kjj ¼ 0.41 cm1, the wave is in
the kinetic regime.
A key element of the shear wave, often overlooked in
the MHD description, is the closure of the wave current. For
plane waves this issue is swept away by declaring that the
currents close at infinity. This is not the case with waves
associated with current filaments. Figure 8 shows the data of
Alfvén wave currents and was acquired in a plane with one
axis parallel to the background magnetic field.42 The figure
is highly compressed, dz ¼ 7m, dr ¼ 9 cm. The waves spread
across the background magnetic field by 0.3 or about 11 cm
over the 18-m LAPD column. Electrons, which move along
the magnetic field, carry the parallel wave current but as the
electrons are highly magnetized (rce 100lm), the crossfield current is carried by ions via the ion polarization drift.
This has been verified by tracking the ion motion using
laser-induced fluorescence.43 The ion distribution function
was measured with a fiber optics probe through which a
laser beam from a tunable dye laser was injected; a small
lens and second fiber were used to detect the LIF signal.
About 1 mm3 of the plasma was imaged, and the probe
moved to a series of locations in the same data plane that
the wave fields were measured. Different components of the
distribution function transverse to the field were obtained
by rotating the probe. The ion distribution function was
recorded as a function of time at each position. The drift velocity of the ions was obtained by measuring the displace! !
ment of the peak of f ð r ; v? ; tÞ. Figure 9 shows the magnetic
field of the shear wave at two instants of time as well as the
ion drift displayed as red arrows. The wave pattern is different from that shown in Fig. 7, because in this case the
waves were launched with a helical antenna. Here there are
two parallel current channels, which rotate about one
another in time. The non-random part of the ion motion is a
FIG. 8. (Color) The measured current
system of a shear Alfvén wave (in the kinetic regime, b mMe ¼ 6:8). The current
source has a 0.5 cm radius. The centers
of the current vortices are located 1.25 m
k
apart in z, which is 2jj . The pattern is
invariant with rotation about its center
along the x-axis. Note that the data is
acquired on a plane in the LAPD device,
the picture is highly compressed along z.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-8
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
~ B
~ direction and in (b) taken 76ls
FIG. 9. (Color) Magnetic fields shown as streamlines and background Ar ion drift (red arrows). In (a) the ions drift in the E
later the ions are seen to close the current via the ion polarization drift. The experimental conditions: n ¼ 1.3 1012cm-3, Te ¼ 3.6 eV, Ti ¼ 0.8 eV, B0z ¼ 1200 G,
fwave ¼ 36 kHz. The experiment was in the kinetic regime.
~ B
~ drift (vEB ¼ cE? 2 ) and the ion
combination of the E
B0 ð1- Þ
dE?
c
¼ xxci .
polarization drift (vpol ¼ xci ð12 Þ dt ) where x
The wave fields were on the order of 0.5 G in the center
of Fig. 9(a) and 0.1 G at the edges of the “O-points.” The
wave fields are stronger in the center because the magnetic
fields of the oppositely directed axial currents, jz, in the two
channels add. Twice each cycle the ions drift from one current
~ B
~ drift
channel to another, closing the current. Since the E
is measured directly the wave electric field can be calculated
and was E\ ¼ 75.6 mv/cm, at a location dz ¼ 66 cm from the
exciter. This was compared to the theoretical value
ð1x 2 Þ
E?
VA
1 : with agreement to within 16%, the
Bwave ¼ c
2
2 þk? q2s Þ2
ð1x
largest uncertainty was in the measurement of k\. The parallel
wave electric field was not measured directly but can be estiik k q2
jj ? s
mated from Ejj ¼ ð1
2 Þ E? . In this experiment the parallel
x
electric field was about 1% of the perpendicular electric field.
This is not large but can have important consequences in
space where the parallel wavelengths are large and the plasma
parameters non-uniform along the direction of propagation.
Experiments have been done studying the waves in both
the inertial and kinetic regimes.42 One marked difference in
these cases is the cross-field group velocity, which changes
sign as in Eqs. (11) and (12)
12
vgr?
xk? d2 ¼
(11)
1 þ ½k? d
2 ; ðVA > ve Þ
vgrjj
VA
!32
2
vgr?
xk? q2s
x
2
1
¼
þ ðk? qs Þ
; ðV A < v e Þ
vgrjj
VA
xci
(12)
Here d is the electron inertial length and qs the ion sound
gyroradius. The change in phase velocity between the two
regimes was verified39 in measurements during the LAPD
when the electron temperature is high (kinetic regime) and in
the early afterglow plasma when the density has not changed
significantly but the electron temperature is lower by a factor
of about 10 (inertial regime). The two regimes were also studied in the LAPD with a specialized antenna44 shown in Fig.
10 designed to produce shear waves with a fixed k\ (Fig. 11).
The morphology of the shear wave depends on how it is
launched. An antenna designed to launch a shear wave with
arbitrary polarization was constructed using two perpendicular
FIG. 10. (Color) (left) Schematic diagram of the Iowa University spatial
waveform antenna for launching shear
waves with fixed k\. The antenna has 48
elements that can be biased with individual waveforms for phase control in the x
direction as shown in the left image.
(right) Measured data for one component, (By), of the wave is shown on the
right. The perpendicular wavelength for
this case is 2.5 cm. For each launched
value of k\ the parallel wavelength can
be measured and the dispersion relation
plotted (Ref. 45). This is shown in Fig.
11 for a wave in the inertial regime.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-9
Many faces of shear Alfvén waves
FIG. 11. (Color) Measured phase velocity and damping of a shear wave in
the inertial regime where k\de is the relevant scale length. Here de ¼ 0.61cm,
B0z ¼ 2.3 kG. The wave frequency was 380 kHz (xxci ¼ 0:43). The theoretical
dispersion relation, including damping, is drawn as a solid black line. The
dash curves are theory bounds reflecting uncertainty in plasma parameters.
multi-turn loops.46 The experimental setup is shown in Fig.
12. The rotating magnetic field antenna (RMF) coils are
driven by independent RF amplifiers capable of driving more
than 1 kA in each loop (for high power wave studies). The kinetic shear Alfvén wave was studied in the kinetic regime for
typical plasma parameters: (n ¼ 2.3 1012cm3, Te
¼ 6 6 1eV, Ti ¼ 1 6 0.5 eV, B0z ¼ 1kG, He).
In this experiment the wave had a maximum amplitude of 4
0:4%. Figure 13 shows isosurfaces of the helical
G, dBBwave
0
wave currents derived from the measured magnetic field. In this
case the shear wave has two side by side current channels, with
the current as always closing due to the ion polarization drift.
A Green’s function technique similar to the one
described in Sec. I was used to model the RMF antenna47
and a 3D MHD code48 was used to study the antenna properties above the ion cyclotron frequency. The agreement with
experimental measurement was excellent in both cases.
VI. FIELD LINE RESONANCES
Alfvénic field line resonances (FLRs) occur when the
waves are bounded by reflectors of some sort, and a standing
wave is established, much as that on a guitar string. The subject has nearly a 30-year history and problems facing the
Phys. Plasmas 18, 055501 (2011)
space plasma community motivated the initial studies. An
overview of the topic can be found in a report by Hasegawa.49 In the earth’s plasma these field line resonances are
observed as low frequency oscillations.50,51 FLRs have been
observed at the onset of auroral substorms,52 and there is
some suggestion that there may be a casual relationship
between the two.53 Spacecraft cannot map out the structure
of the waves, but the spectra derived from the time history of
magnetic measurements on spacecraft clearly reveal the discrete modes.54 This is shown in Fig. 14 where data from the
THEMIS satellite was used55 to acquire magnetic field data
in magnetopause boundary layer.
The first observation in a laboratory plasma to identify
standing Alfvén waves56 was done in the previously mentioned arcjet. A peak in the amplitude of Bh was observed at a
position several centimeters off the device axis. The location
2
2
of the peak matched the radial position having kx2 V 2 ¼ 1 xx2 ,
ci
jj A
where x is the applied angular frequency, and n ¼ n(r).
The LAPD device is well suited to study field line resonances. The first version of the LAPD device was 10 m long and a
spectrum of standing waves was introduced by applying a current pulse to a field aligned antenna two electron skin depths in
diameter. The spectral content of the pulse extended to the cyclotron frequency. The experiment was repeated at ten different
ambient magnetic fields, and spectra such as that in Fig. 15
The
frequencies of the FLRs are pre(left) were recorded.57 p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dicted to be: fn ¼ VA = ð4L2 =n2 þ VA2 =fci2 Þ with the second
term in the denominator arising from the ion cyclotron correction in Eqs. (6) or (7). The allowed modes have kn ¼ 2L
n where
L is the length of the plasma column. In this approximation the
k\ terms are neglected. Figure 15 indicates that waves up to the
eighth harmonic were detected with no signal measured above
the ion cyclotron frequency as expected. The Q of the resonant
system can be obtained from the spectrum, and it is 3.8 for the
lowest harmonic (n ¼ 2) and 10.2 for the highest (n ¼ 8).
The perpendicular wavenumber of the n ¼ 1 mode was
determined by a Bessel function decomposition of the spatial
magnetic field Fig. 15 (left) after interpolating the data into
polar coordinates.58 The value of hk? i was determined to be
hk? i ¼ 0.55 6 .025cm1. The measurement was compared to
a theoretical prediction of the wavelength by Streltsov59 with
agreement within experimental uncertainties. When the
initial wave is localized, it spreads across the magnetic field
due to its finite k\ and could not survive successive reflections. The plasma parameters in the ionosphere vary considerably. If one follows a field line, close to the poles, the
magnetic field is higher and the plasma is colder and denser,
while the opposite is true near the equator. The waves therefore go from the inertial to the kinetic and back to the inertial
regime along their trajectory. The perpendicular group velocity changes sign during the trip [Eqs. (11) and (12)] and in a
sense the average wave can remain close to a field line. The
spread and then refocusing of a shear wave in a large gradient was observed in an experiment when the waves propagated in a plasma with a large gradient in the plasma beta
along the background magnetic field.60
When the shear wave due to a current filament propagates across the magnetic field it may encounter density and
temperature gradients. This was observed in an experiment
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-10
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
FIG. 12. (Color) Experimental setup (a). The RMF antenna is aligned so the rotation axis of the induced magnetic field at the center of the antenna is parallel
to the direction of the background magnetic field B0. The data are collected with the use of 3-axis magnetic pickup probes (1mm cube). (b) The magnetic field
of the wave measured on a plane transverse to the background field 33 cm from the antenna. Data was acquired at 13 448 spatial locations and in figure (b) every 4th vector was drawn.
8pnTe
on the LAPD in which the electron beta, be ¼ M
m B20z varied
by a factor of four across the outer edge of the plasma column.61 The center of the plasma was kinetic, and the perpendicular group velocity was outward; there was a frequency
dependent radial position at which vgr\ ¼ 0, and the wave
became inward propagating (backward) in the outer (inertial)
region. The difference in radial propagation velocities caused
pileup of wave energy on a ring of radius between
1 be 2 which agreed with theory. Spatially localized
accumulation regions such as this could result in large amplitude waves, representing sources of electron acceleration due
to Landau damping.
VII. MAGNETIC BEACH
Due to their long parallel wavelengths and nearly fieldaligned propagation, shear Alfvén waves typically encounter
spatially varying regions of ambient plasma parameters. Perhaps the most important of these is the variation of the background magnetic field. When the local ion cyclotron
frequency becomes comparable to the oscillation frequency
of the shear wave, additional effects arise such as ion heat-
ing, the generation of an axial magnetic field component,
and the change from linear to left-hand circular polarization.
Stix62 detailed the cyclotron damping of Alfvén waves in the
case of a uniform background magnetic field; experimentally, some of the earliest works on Alfvén waves in nonuniform magnetic fields were performed by researchers63–65
studying the excitation of radial eigenmodes of a cylindrical
plasma. The wave propagated axially into a region of
decreasing magnetic field to the point where the wave frequency matched the local ion-cyclotron frequency—the socalled ‘‘magnetic beach.’’ Later, the propagation of shear
Alfvén waves and ion heating were also of great interest in
the study of mirror machines. For example, radio frequency
heating was studied in the THM-2 mirror machine66 using a
Stix-type coil exciter, and in the Phaedrus-B tandem mirror67
using a rotating field antenna68,69 which selectively excited a
global, m ¼ 1 eigenmode of the shear Alfvén wave and
was found to efficiently transfer energy to the ions.
The magnetic beach scenario was later revisited by Vincena et al.149 They used a small circular mesh to broadcast
shear Alfvén waves into a decreasing magnetic field as
shown in Fig. 16. Detailed measurements of the wave
FIG. 13. (Color) Isosurfaces of current
density at s ¼ 4.6ls after the wave
(xxci ¼ 0:54) is launched by the RMF
antenna described in the text. The surfaces begin 33 cm to the right of the
antenna and end approximately 8 m
away. Two rotating counter-propagating
helical current channels can be seen
~0z . As
flowing in the z direction along B
time advances the currents rotate in a
left-handed sense. The outer surface represents a current density of 0.25 A/cm2
and the inner surface a current density of
0.5 A/cm2. Red denotes current flow in
the positive z direction and the blue
surfaces current in the negative z direction. Some representative magnetic field
vectors are also shown.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-11
Many faces of shear Alfvén waves
Phys. Plasmas 18, 055501 (2011)
cates the location where the wave frequency equals the local
cyclotron frequency of the helium ions. The wave is efficiently absorbed here, as in Fig. 17. In addition to the previously discussed radial ion polarization and parallel electron
currents, the streamlines show a pronounced azimuthal component. This current arises from the relative slippage in
Er B0 drift velocities between the ions and electrons near
the ion cyclotron frequency and leads to an observed axial
component of the wave magnetic field.
VIII. ALFVÉN WAVE MASER
FIG. 14. Distribution of oscillation frequencies obtained from the estimation
of magnetopause motion by spline interpolation (Plaschlke et al., 2009).
magnetic fields showed that in addition to the ion cyclotron
damping, the wave parallel electric fields lead to significant
transfer of wave energy to the electrons via (equal) electron
Landau damping and electron-ion Coulomb collisions. Figure 17 displays the measured wave fields and WKB solutions
to the wave equation that delineate the damping effects by
ions and electrons. The best agreement with the data required
all three damping mechanisms. The increase in the wave amplitude for the ion damping only (red) curve is due to the
slowing down of the parallel group velocity as the wave
approaches the cyclotron frequency. This effect is overwhelmed when all three damping mechanisms are included
(blue curve).
The magnetic field measurements were also used to calculate the volumetric wave currents, as shown in Fig. 18.
This figure displays a snapshot in time of the shear wave as
it propagates from left to right into the magnetic beach.
Shown is the magnitude of the azimuthal magnetic field in
the x-z plane, and streamlines of wave current calculated
using Ampere’s law and the azimuthal symmetry of the
wave magnetic field. The magenta surface on the right indi-
Several theoretical studies70–72 have investigated the
concept of a natural Alfvén wave maser formed in the plasmas surrounding the earth. Some theories consider the maser
cavity formed by the magnetospheric plasma bounded by the
highly conducting, conjugate ionospheres. These relatively
narrow regions play the role of reflectors. The nonequilibrium, active medium that excites the wave has been associated with a population of energetic ions73 in the radiation
belt. Another class of Alfvén maser activity is considered to
occur at the lower altitudes. Due to the spatial variation of
the ionospheric plasma parameters, it has been identified in
various studies55,74–76 that an Alfvén resonator is naturally
formed between the lower E region of the ionosphere and a
region in the topside ionosphere at an altitude of about 104
km. This higher region acts as a partial reflector that allows
amplified signals to propagate into the magnetosphere.
Observations by satellite77 and rocket78 suggest that signals
consistent with resonant amplification of Alfvén waves are
naturally generated and corroborate ground-based measurements79,80 of the phenomena. The amplified signals correspond to waves that undergo multiple reflections within the
resonator. A modeling study81 has proposed that the strong
fluctuations associated with the partially standing waves in
the ionospheric resonator are responsible for driving the outflow of source ions from the ionosphere to fuel the magnetospheric plasma.
Various nonequilibrium sources have been considered to
excite the ionospheric Alfvén resonator. These include the
natural magnetospheric convection,55,82 electron beams,83
lightning discharges,84,85 and powerful HF radio signals.86 In
FIG. 15. (Color) (left) Spectrum of frequencies in the exciter pulse (dotted line), and observed spectrum measured by an in-situ magnetic pickup probe. (right)
Measured magnetic field for the lowest order mode n ¼ 1, showing the FLR is confined to the center of the device.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-12
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
FIG. 16. Side view of the original LaPD
showing the magnitude of the confining
axial magnetic field (at r ¼ 0) for the
magnetic beach experiments. Also indicated is the region in which detailed
measurements were obtained. The wave
travels from a region of high electron
Landau damping to a region of ion cyclotron absorption.
laboratory studies87,88 performed in the LAPD device, an
equivalent Alfvén resonator has been realized in which the
nonequilibrium source leading to spontaneous amplification
is a field-aligned current.
The LAPD studies have demonstrated that a steady-state
Alfvén wave maser can be realized in the laboratory under
controlled conditions. The configuration shares the essential
elements of lasers and maser devices that operate at optical
and microwave frequencies, namely, a resonant cavity with a
partially reflecting boundary and an active medium that provides amplification. Because of the peculiar properties of
shear Alfvén waves, these elements take a form different
from the optical cavities and inverted atomic populations
typically found in conventional lasers and masers. The resonant cavity used in LAPD consists of a cathode and a semi-
FIG. 17. (Color) Comparison of measured to theoretical wave magnetic
field as the shear Alfvén wave propagates into a decreasing magnetic filed.
Lower axis is in cm and upper is the ratio of wave frequency to local helium
ion cyclotron frequency. The red curve shows on ion cyclotron damping in
the model while the blue further incorporates electron Landau damping and
electron-ion Coulomb collisions. The electron damping channels are equally
effective. The antenna is located at z ¼ 0 cm.
transparent mesh anode that permits the amplified shear
Alfvén waves to propagate into a large plasma column. In
this region, where the magnetic field is uniform, the unique
properties of the maser signal can be exploited to study a
wide variety of interactions of relevance to magnetic confinement research and space plasma investigations.
An example of maser behavior near threshold conditions
is shown in Fig. 19. The bottom panel of this figure shows
the temporal behavior of the spontaneous fluctuations of the
transverse component of the magnetic field when the plasma
current barely exceeds a threshold value for a given strength
of the confinement field. The threshold value corresponds to
the condition when the return current in the device results in
an electron drift velocity that exceeds the phase velocity of
the shear mode (approximately given by the Alfvén speed).
The top panel displays an expanded view of the temporal
behavior (single shot) of the flare event over a small interval
of 0.1 ms. It shows that the flaring shear mode is nearly
monochromatic and remarkably coherent, in particular since
it grows out of spontaneous ambient noise. For plasma currents well-above threshold the maser signal achieves a
steady-state whose length is determined by the discharge
duration.
The Alfvén maser in the LAPD produces a highly coherent shear mode with dx/x 5 103 excited spontaneously
from ambient noise when a threshold plasma current is
exceeded. There is no need for external pumping to trigger
the maser action. The amplitude of the magnetic fluctuations
associated with the noise-grown signal attains a level
exceeding 1.5% of the confinement magnetic field. The frequency of the maser mode is continuously tunable by changing the strength of the confinement field. The cathode-anode
separation and the dispersion relation of the shear mode
determine its specific value. For the particular experimental
arrangement in the LAPD this corresponds to x ¼ 0:6xci . A
quantitative comparison of the measured threshold plasma
current, over a factor of 2 variation in confinement field,
shows that the amplification mechanism is the electron drift
velocity associated with the natural return current in the
plasma. The maser action exhibits a wide range of temporal
variability. Near threshold it consists of flare events whose
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-13
Many faces of shear Alfvén waves
Phys. Plasmas 18, 055501 (2011)
FIG. 18. (Color) Shear Alfvén wave
currents (streamlines) and magnetic field
magnitude (colored plane) at one instant
of time. The wave propagates from left
to right until it encounters the magenta
surface at the right where the wave frequency equals the local ion cyclotron
frequency (see also Fig. 16). Parallel
currents are carried by electrons, radial
currents are due to ion polarization
drifts, and azimuthal currents result from
the slippage of the electron and ion
E B drifts near the cyclotron
frequency.
statistics displays a non-Gaussian tail. For conditions well
above threshold the maser behavior reaches a steady state
whose duration is determined by the lifetime of the plasma
discharge. Mode transitions, from azimuthal mode number
m ¼ 0 to m ¼ 1, occur and are associated with slow changes
in the radial density profile and with changes in plasma parameters. Under steady-state conditions the radiation resistance associated with the Poynting flux emanating from the
resonant cavity modifies the discharge current. In spite of the
relatively large level of the signals, the related magnetic fluctuations do not exhibit a significant nonlinear distortion. This
may be expected because, at the maser frequency of 0.6 Xci,
the higher-harmonic modes are non-propagating. However,
ion saturation signals that are phase coherent with the maser
fluctuations indeed exhibit strong nonlinearity.
FIG. 19. Alfvén wave maser signal near threshold shows bursty behavior.
Inset: blow up of a single shot trace of Bx is highly coherent. Above threshold the maser signal reaches steady state.
In summary, it has been demonstrated that Alfvén wave
masers can be realized in the laboratory and their operation
can be quantitatively predicted by a differential equation
model.89 The maser is not only a useful laboratory tool for
investigating a wide variety of problems related to Alfvénic
interactions,90 but may also be an important process in space
and astrophysical plasmas. A discussion of such perspectives
has been given by Boswell.91
IX. ALFVÉN GAP MODES
When a shear wave propagates in a rippled magnetic
field, one with periodic variation, an interesting effect
occurs; the waves can be observed only in certain propagation bands. The effect is analogous to Bragg scattering in a
crystal in which standing waves are set up by absorption and
re-radiation of an electromagnetic wave by the valence electrons in a crystal lattice. The first paper by to discuss the gap
modes in a toroidal paper was published in 1978 by Pogutse
and Yurchencko.92 In a plasma the variation of the magnetic
field results in a variation of the Alfvén velocity, which in
93
turn causes a variation of the index
of refraction
2
16pni mi c
2pz
:
(13)
N2 ¼
1 þ 2g cos
2
Lm
ðBmax þ Bmin Þ
Here Lm is the mirror length, Bmax and Bmin the maximum and
minimum values of the magnetic field, ni the ion density, mi
Bmin Þ
the ion mass, and g the modulation amplitude g ¼ ððBBmax
.
max þBmin Þ
Figure 20 shows the experimental setup. Four magnetic
mirrors, created along the length of the LAPD and a small
disk antenna40 or a field aligned “blade” antenna, made a fluctuating current filament and launched a shear Alfvén wave.
The magnetic field of the waves was measured on a number
of transverse planes and lines along the z axis ofthe machine.
Figure 21 displays the frequency response clearly showing the Alfvén gap where there is little wave activity (for an
infinite system of mirrors there would be none). The gap
width for an infinite number of mirror cells is given by
VA
, where VA is the average Alfvén speed
Df ¼ gfBragg ¼ g 2L
M
which was measured by probes that detected the delay
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-14
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
FIG. 20. (Color) Experimental magnetic field setup of the magnetic mirrors in the LAPD. Here g ¼ 0.26 and Lm ¼ 3.5 m. The plasma parameters: n ¼ 1 1012
cm3, Te ¼ 6 eV, Ti ¼ 1 eV, He. The waves were launched at 50kHz f 180kHz.
between launch and reception at two axial locations. Figure
21(right) illustrates how the gap modes develop after the
Alfvén waves have had time to reflect from the ends of the
device. The gap spectral width was found to increase with
the mirror depth, M as predicted by theory. This experiment
is related to TAE modes in fusion devices.94–96 In a burning
plasma these modes could be destabilized by energetic alpha
particles. Fast ions expelled by these instabilities have damaged vacuum components in past experiments97 and are a
worry for experiments on the ITER device.
X. DRIFT-ALFVÉN WAVES AND TRANSPORT
In large magnetized plasmas in which pressure gradients
are present the diamagnetic drift causes the Alfvén wave to
become coupled to the electrostatic drift wave. The resulting
mode is known as the drift-Alfvén wave. It displays simultaneous fluctuations in density and transverse magnetic field.
These modes can be driven unstable by electron Landau
damping and/or collisions. When they reach large amplitude,
they play a key role in the cross-field transport of energy and
particles. Drift-Alfvén waves have been observed in density
and temperature striations produced under controlled conditions in the LAPD.98–100
The anomalous electron heat transport induced by driftAlfvén waves has been documented in experiments on the
temporal and spatial evolution of an electron temperature filament. The filament is created by injecting low voltage (less
than the ionization voltage) electrons into the afterglow helium plasma of the LAPD using a small (3 mm) source.
FIG. 21. (left) The wave energy density as a function of f/fBragg. The width of the dip at 1.05 agrees well with the prediction for Df. (right) Time history of the
running cross-covariance between the wave field and the current of the antenna that launched the wave at g ¼ 0.25. The grey and white contours show the
wave magnetic field. The first white curve indicates the time it takes for the wave to travel towards the detector probe one-way, the second curve the time it
takes for the wave to return to the probe after being reflected from the anode/cathode for the first time. The standing wave and “gap” mode appears after this.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-15
Many faces of shear Alfvén waves
FIG. 22. A schematic of the experimental setup used in creating and studying the temperature filament showing the location of the electron beam and
the probes used to diagnose the experiment.
Since heat conduction along the magnetic field is much
stronger than across, the small size of the source limits the
length of the filament so that it fits entirely within the LAPD
plasma column. The experimental set up is shown in Fig. 22.
It is observed that initial temperature transport is governed entirely by classical processes (Coulomb collisions),
but that later in time there is a transition to anomalous transport.101 This transition in the nature of transport occurs as
the result of the growth of drift-Alfvén waves due to the
pressure gradient associated with the spatially localized filament.102 The drift-Alfvén waves driven by the gradient are
observed96 to have azimuthal wave numbers ranging from
~ B
~ flows
m = 1 to m 6, as shown in Fig. 23. Convective E
driven by these waves create structures in the temperature,
and, once a threshold amplitude is crossed (vDrift > x/k\),
the behavior of the plasma flow becomes chaotic leading to
enhanced transport.
The key signature of the transition in transport behavior
is found in the power spectrum of the fluctuations. The
power spectrum changes from a line spectrum observed during the early classical transport stage when the drift-Alfvén
waves become linearly unstable to a broadband spectrum in
the anomalous transport phase when the waves achieve large
amplitude. The broadband spectrum is linear in a log-linear
plot indicating that the spectrum is exponential in nature. It
has been shown that the exponential spectrum arises from
Phys. Plasmas 18, 055501 (2011)
the random occurrence of Lorentzian-shaped pulses in the
time signal.103 The Lorentzian temporal shape observed in
the time signals arises from structures in electron temperature created from the flow driven by drift-Alfvén waves. Figure 24 shows the contour of temperature in a simulation104
consisting of two waves, one with m ¼ 1 and one with m ¼ 6.
The wave induced flows to create complex structures whose
temporal signals exhibit Lorenzian shaped pulses. There is a
strong indication that the exponential power spectrum and
Lorentzian pulses are a universal feature of E B driven
edge turbulence, as they are observed in a limiter experiment
at the edge of the LAPD plasma column involving density,
and not temperature fluctuations at a much larger perpendicular scale.
XI. NONLINEAR ALFVÉN WAVES
Nonlinear effects associated with Alfvén waves are important in laboratory, space, and astrophysical plasmas.
Large amplitude Alfvén waves can modify the background
plasma density through ponderomotive forces,105 which arise
due to spatial gradients in the wave energy density. Fieldaligned density cavities in the presence of strong magnetic
fluctuations have been observed in the magnetosphere by the
FAST satellite.106 Alfvén waves are in turn affected through
interactions with field-aligned density structures. Fieldaligned density cavities can act as waveguides for Alfvén
waves,107 channeling Alfvén wave energy. In magnetic confinement fusion devices, nonlinear processes associated with
multiple Alfven eigenmodes can lead to a significant
enhancement of fast ion transport.108 It has recently been
proposed that Alfvén waves with frequency below the cyclotron frequency can through a nonlinear process resonantly
heat ions,109 which could explain anomalous ion temperatures in magnetic confinement experiments.110
Nonlinear interactions between Alfvén waves are responsible for the cascade of energy in magnetohydrodynamic
(MHD) turbulence.111,112 In the incompressible MHD limit,
nonlinear interactions are only possible among counter-propagating shear waves, and an anisotropic energy cascade
FIG. 23. (Color) (top panel) Contours of fluctuations in ion saturation current show a steady
progression in wave number of the drift-Alfvén
waves driven by the pressure gradient. (bottom
panel) The temporal signal obtained at the location marked in the first contour plot indicates
that the frequency of the signal is nearly constant even though the m-number of the mode is
changing dramatically, indicating that the eigen-frequencies of this system are nearly
degenerate.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-16
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
FIG. 25. (Color) (top) Measured magnetic fluctuation spectrum during weak
multimode emission (dominant m ¼ 1 mode). To the right is a line plot of the
power spectrum at t ¼ 8 ms. (bottom) Measured magnetic and density (ion saturation current) fluctuation spectrum during strong multimode emission.
FIG. 24. (Color) (top panel) Contours of temperature produced by convection in the presence of two drift waves shows the development of structure
from the initially Maxwellian temperature distribution. (middle panel) Lorentzian-shaped decreases in temperature produce near the center of the filament (r ¼ 1.85 mm) produced by inward transport. (bottom panel)
Lorentzian-shaped increases in temperature produced in the outer regions of
the filament (r ¼ 3.85 mm) due to outward transport.
results.113,114 As this cascade proceeds, perpendicular scales
are reached where dispersion and kinetic damping become
important, and a cascade of kinetic Alfvén waves is
expected,115 and evidence for the transition to a KAW cascade is observed in solar wind measurements.116 With the
inclusion of dispersion and compressibility, a greater range of
nonlinear wave-wave interactions are possible among shear
Alfvén waves. This includes decay instabilities, such as parametric117 and modulational118 decay, as well as interactions
between co-propagating waves.
Studies of large amplitude shear Alfvén waves and
wave-wave nonlinear interactions have been carried out in
the LAPD.90 Large amplitude shear Alfvén waves are generated in LAPD using two sources: (1) the resonant maser
cavity defined by the anode and cathode in the source
region89,90 and (2) loop antennas. Using these wave sources,
the interaction between two co-propagating shear Alfvén
waves has been studied. In these experiments, the two
waves, launched at slightly different frequencies are
observed to beat together and nonlinearly drive a density
fluctuation at the beat frequency. The driven density pertur-
bation then scatters the Alfvén waves, resulting in the creation of sidebands. The first observation of this interaction
was made during simultaneous, spontaneous emission of
two frequencies (two azimuthal modes, m ¼ 0 and m ¼ 1) by
the Alfvén wave maser. Multimode spontaneous emission is
often observed early in time (mode-hopping as the plasma
profiles evolve),92 but in some circumstances the multimode
emission continues for longer periods of time. Figure
25(left) shows measured magnetic field and ion saturation
current power spectra during two discharges, the first with a
dominant single mode (m ¼ 1) and the second with sustained
multimode emission (both m ¼ 0 and m ¼ 1 are emitted
simultaneously). The observed magnetic fluctuation spectrum in the multimode case contains many sidebands around
the MASER emission frequencies and a low frequency fluctuation at the beat/sideband separation frequency (and harmonics). The response at the beat frequency is a nonresonant quasimode, having wave number consistent with
three-wave matching rules, but inconsistent with the dispersion of linear plasma waves.
This beat wave interaction has been used in LAPD to
couple to linear plasma modes. An experiment was performed in which two Alfvén waves were broadcast along a
filamentary density depletion on which a coherent unstable
mode was present.119 The beating Alfvén waves were found
to resonantly drive the instability when the beat frequency
was tuned to match the unstable mode frequency. More
interestingly, it was found that when the beat frequency was
tuned to resonantly drive nearby (in frequency) damped
modes, the original unstable mode was suppressed.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-17
Many faces of shear Alfvén waves
XII. ALFVÉN WAVES FROM DENSE EXPANDING
PLASMAS
The expansion of a dense plasma into a magnetized
background plasma is of interest in a variety of contexts. It is
of importance in the study of coronal mass ejections
(CME)120 and high altitude explosions.121 In the past there
have been chemical releases in the ionosphere,122 in which
barium was released (AMPTE (Ref. 123) mission). The barium rapidly became photoionized and expanded across the
geomagnetic field. Experiments at UCLA in which a dense
laser produced plasma (lpp) expanding into the background
LAPD plasma have studied the formation of a diamagnetic
cavity124 and generation of Alfvén waves in the background
plasma.125 A discussion of the relevance of these experiments to space plasmas has been reported.126 Figure 26
shows the setup for laser target experiments in the case
where two targets are struck simultaneously. We first briefly
discuss the case of a single expanding lpp. When the dense
lpp plasma is created in vacuum the electrons cannot stream
away along the existing ambient magnetic field because they
are held back by the ion space charge. The presence of a
dense enough background plasma allows electrons to move
away in localized streams and be replaced by drifting electrons from the much larger background plasma. The initial
burst of electrons generates the emission of whistler and
lower hybrid waves for s 400ns (Ref. 127). The lower frequency current system that is set up subsequently becomes
that of shear Alfvén waves. Figure 27 illustrates the current
systems and Alfvén wave formation process.128
Note that initially the current system is coaxial. The
electrons streaming away (red) are surrounded by those in a
return current (blue). At later times the current system
becomes complicated because the lpp is moving across the
magnetic field in the x direction and shedding current as it
goes. The magnetic field shown on the plane at dz ¼ 1.3 m is
associated with a shear wave that is just beginning to form.
Note the morphology of the wave bears a stringing resemblance to that generated by the RMF antenna (Fig. 12).
The interaction of the current systems from multiple
laser-produced plasmas brings an additional phenomenon
into play, magnetic field line reconnection.129 Colliding plasmas are relevant to astrophysical situations.130 Consider two
targets simultaneously struck by laser beams (Fig. 26). The
lpp’s jet across the magnetic field and collide but the relevant
part of the experiment of concern here is the interacting
current systems. The experiments were done in helium ðB0z
¼ 600 G; fci ¼ 229 kHz; n ¼ 3 1012 cm3 ; Te ¼ 4 eVÞ. The
laser power incident on each target was 5 1010 W/cm2. In
the initial stage, instead of a single diamagnetic bubble there
are two. The expelled magnetic field is in the axial direction,
but as the diamagnetic currents are helical there is a field
component in the transverse direction. When they collide the
anti-parallel transverse field components are forced together
and reconnection occurs. This is shown in Fig. 28. In the
case of colliding lpp’s formed by more powerful lasers (100200J) at the Rutherford laboratory,131 the transverse fields at
the bubble edge are on the order of 1 MG. Jetting of plasma
from the edge of the reconnection region in the Rutherford
Phys. Plasmas 18, 055501 (2011)
experiment was interpreted as the signature of reconnection.132 The data in that experiment was interpreted in terms
of the Sweet-Parker model133 (which is two-dimensional and
predicts the correct reconnection rate if multiplied by a large
situation dependent number), a turbulent resistivity 25 times
the classical Spitzer resistivity was assumed. It was conjectured that this could be due to ion acoustic turbulence. Ion
acoustic turbulence and enhanced resistivity were observed
in a reconnection experiment over 25 years ago.134 Another
high energy colliding lpp experiment was designed to model
young stellar objects and studied collimated flows produced
in the interaction.135 Plasma jets and shocks were observed.
These experiments are complicated, the diagnostics are difficult to set up, and “campaigns” consist of several shots.
These experiments had no background plasma or background
magnetic field although in astrophysical situations they are
usually present.
The magnetic field data in the two-target LAPD experiment was acquired with a three axis differentially wound
magnetic probe 1 mm in diameter over weeks of continual
operation at a rep rate of 1 Hz. The magnetic probe was computer controlled and stepped through a series of planes transverse to the background magnetic field. Magnetic fields
associated with the dense plasmas and wave propagation in
the background plasma are digitized (14 bit, 100 MHz) and
acquired at 16 000 spatial locations. An average of 10 shots
was stored at each location. The reconnection event at s
^ 400 ns, shown in Fig. 28, is accompanied by a burst of
current induced along the background magnetic field
(I ¼ 30A Ielec-sat). This is not the result of fast electrons
escaping the bubbles but is due to the inductive electric field
generated in this first reconnection event. The lpp current/
return current systems follow the initial reconnection event
and develop on an Alfvénic time scale. These currents
(s ¼ 5.25 ls) are shown in Fig. 29. The system is dynamic,
and the currents are observed to merge, break up into two or
four channels, and re-merge. Over time scales
1ls s 40ls, the reconnection is that of the magnetic
fields of shear Alfvén wave currents. The magnitude of currents of the lpp Alfvén
waves varies in space and time and is
of the order of J~wave 0:2A=cm2 . The magnetic field during one such event is shown in Fig. 30. The induced electric
field may be calculated using
~
~ind ¼ 1 @ A ;
E
c @t
~ ¼1
A
c
ð ~ 0
J ð~
r Þ
dV 0 ;
r ~
r 0j
j~
(14)
V0
where J is the volumetric current evaluated from the magnetic field data. The induced electric field is on the order of
2.5 V/m during the bursts of reconnection. If one ignores all
the other relevant terms in Ohm’s law, an extreme oversimplification, the resistivity is g ¼ 1.25 103X m, 32 times
higher than the classical Spitzer resistivity gSpitzer ¼ 4
105X m. This is a limit one must view with suspicion.
XIII. MAGNETIC FLUX ROPES
A magnetic flux rope is a bundle of magnetic field lines
with pitch varying with radius. One may alternatively
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-18
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
FIG. 26. (Color) Two laser beams (shown in red although they are 1 micron wavelength and not visible, s¼ 8 ns, 1.5 J) strike a pair of carbon targets in the LAPD
device which contains the LAPD background plasma (diameter indicated), 60 cm in diameter. The port spacing along the background magnetic field is 32 cm.
c
~ Red/blue represents
FIG. 27. (Color) Magnetic field data at t ¼ 390 ns after the target is struck. Solid planes are current density obtained from J~ ¼ 4p
! B.
electrons going in the –z/þz direction. Vector plot shows the perpendicular magnetic field. The target is shown to scale at its appropriate location relative to the
z ¼ 6 cm plane. The laser is incident along the x axis.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-19
Many faces of shear Alfvén waves
Phys. Plasmas 18, 055501 (2011)
FIG. 28. (Color) Magnetic field measured 400 ns after the targets are struck by two 1-J lasers. The data plane is 5 cm away from the targets. The spatial extent
of the data is Dx ¼ 22 cm, Dy ¼ 23 cm. (a) Transverse magnetic field from a head-on perspective. A magnetic X point is clearly visible in the center. The bubbles are moving towards each other and flux is annihilated in the center. (b) Another view of the same data showing that most of the magnetic field is in the z
direction. The background magnetic field points downwards.
consider a rope as a spiraling current system. To create a flux
rope, one needs current along a background magnetic field,
Bz, large enough to produce a significant Bh. Current systems
in magnetoplasmas are rarely steady state, and once they
change on time scales less than the ion cyclotron period the
flux ropes may be considered to be Alfvén waves. The sun is
host to arched magnetic flux ropes136 which can erupt to
become coronal mass ejections. CME’s are flux ropes that
emerge the sun and extend outwards for large distances.137
They can break away from the sun to become plasmoids,138
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-20
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
FIG. 29. (Color) Three-dimensional currents in the two-target lpp experiment derived from the volumetric magnetic field data set. The targets are drawn in the
background for reference. The dominant Alfvén wavelength is kP ¼ 2.62m. The current closes by ion polarization drift every half wavelength, the currents
close at dz ¼ 1.31 meters from the targets at s ¼ 5.25 ls after the targets are struck.
which can strike the Earth. It is also possible that they remain
rooted in the Sun when their leading edge arrives at Earth.
An early experiment on the interaction of low-current ropes
without a background plasma139 observed that they spiraled
~ forces and then merged.
around one another due to J~ B
Over distance and time the currents became field aligned in
accord with Taylor’s conjecture.140 When conditions are
such that the flux ropes violently bash into one another,
which can happen, for example, if they are kink unstable,
field line reconnection can occur in the space between them.
This was recently seen in an experiment done on the RSX
device at Los Alamos141 where the magnetic topology,
reconnection rate, and resistivity (larger than classical as
usual) were measured. In the same year an experiment at
UCLA on colliding flux ropes142 was done on the LAPD
device.
The UCLA flux rope experiment, however, had a background plasma capable of supporting Alfvén waves (He, n
^ 2.5 1012 cm3, Te^ 5 eV, Ti^ 1 eV). The currents of
the ropes were emitted from two LaB6 cathodes. The experiment generated a volumetric data set (12 data planes perpendicular to B0z ¼ 270 G, dz ¼ 64 cm, dx ¼ dy ¼ 0.3 cm, 32
400 spatial locations, dt ¼ 0.64 ls, time digitized 6.55 ms).
The ropes twisted about themselves, writhed about each
other, and were also kink unstable. The flux rope collisions
led to multiple reconnection sites in space and time. As there
was a strong background magnetic field, the reconnection
occurred in the absence of a magnetic null point just as in
the lpp experiment. Solar physicists have introduced the concept of a quasi-separatrix layer143,144 (QSL) to study reconnection in complex geometries and no magnetic null points.
The volumetric magnetic field data set enabled the evaluation of the QSL associated with the reconnection event. The
QSL region has a simple interpretation. Two closely spaced
field lines, which enter the QSL, wind up at very different
spatial separations at finite distances along the current channel. Outside the QSL neighboring field lines remain close to
each other. The QSL can be found by moving the footprint
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-21
Many faces of shear Alfvén waves
Phys. Plasmas 18, 055501 (2011)
FIG. 30. (Color) Magnetic field lines of a shear Alfvén waves produced 1.6 ls after the collision of two laser-produced plasmas. The targets are several meters in
the back of the picture. At this instant of time there are two strong and two weak current channels. The background magnetic field (not shown) is out of the plane
of the image. The inductive electric field is calculated from the time varying magnetic field and rendered as sparkles on the field lines (the brighter the sparkle the
higher the field). The largest induced fields, parallel to B0z, are of order 2.5 V/m. The reconnection region is in the center.
of a field line a small distance from its original position (x,
y) on a plane (z ¼ z0) and measuring where it winds up (X,
Y) on a plane far away (z ¼ z1).
Consider the quantity
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u" 2 2 2 2 #
u @X
@X
@Y
@Y
:
þ
þ
þ
N¼t
@x
@y
@x
@y
(15)
If N >> 1 then the field lines used to generate it are said to
be on a QSL. The concept of the QSL has been extended to
encompass situations in which there is a guide field.145 The
slip squash factor Q is defined by
N2
:
Q ¼ Bz ðz0 Þ
B ðz Þ
z 1
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
(16)
055501-22
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
FIG. 31. (Color) (a) Magnetic field lines started at the edge of the two, 2.5 cm diameter flux ropes. The current in each rope is 30A. The 270 G guide field is
included in the calculation. The grid lines occur at dx ¼ 2 cm, dz ¼ 30 cm. Note the field lines twist about themselves, and rotate about one another. The magenta surface is the QSL ¼ 200 surface which is also a flux tube. The yellow lines are the transverse magnetic field of the ropes dz ¼ 1.3 meters from their
source. Careful inspection of transverse field lines shows there are “X” type structures in the center. The blue arrows are electron current density. Note that at
the right current is flowing backwards, i.e., electrons are going towards the source of the ropes. (b) Lower right corner X point topology dz ¼ 6.6 m from start
of flux ropes. t ¼ 2.5 ms after the flux ropes are switched on.
The squashing degree may be used as a measure of 3D
reconnection.146 In the case of this experiment the guide field
is much stronger than the field due to the currents; it does not
vary in z, and Q ! N2. The minimum value of interest is
Q ¼ 2, much larger values denote more intense reconnection
events. The maximum value of Q peaks when the flux ropes
become close to one another in the course of their writhing
about and when reverse current sheets develop. The flux
ropes field lines and QSL are shown in Fig. 31.
The QSL surface shown in Fig. 31 indicates a region
(colored magenta) snaking between the field lines at the edge
of the ropes; somewhere within that region magnetic field
FIG. 32. (Color) Magnetic field lines of
three flux ropes (background field of 300
G included). Note the scale difference
between the transverse (Dx ¼ 13.2 cm)
and axial distances (Dz ¼ 4.79 m). The
flux ropes twist around one another and
collide in space and time. The reflective
surface is placed on the bottom to aid in
perspective.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-23
Many faces of shear Alfvén waves
Phys. Plasmas 18, 055501 (2011)
FIG. 33. (Color) Three-dimensional picture (anaglyph) of three flux ropes and the resultant QSLs formed (Q ¼ 20). To view this requires red/blue 3D glasses,
which can be purchased in a host of specialty shops. The speckled pattern in the back provides a reference at infinity.
FIG. 34. (Color) Shear wave Hydra.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-24
Gekelman et al.
Phys. Plasmas 18, 055501 (2011)
line reconnection occurs. To identify the location(s) one
must examine the detailed magnetic field topology. The right
inset in Figure 31 shows the transverse magnetic field in a
plane dz ¼ 6.6 m from the source of the flux ropes. Viewed
in time field lines enter the “X” point, break, and reconnect.
It is also possible to estimate the resistivity
by computÐ
~ B
~dV and the
ing the time rate of change of the helicity: V A
magnetic energy.147
2
dH ð
dt 1
; with M ¼
B2 dV:
g
dM
8p
2M
dt
and even for Alfvén waves the world was a simpler place.
The advances continue and as they do so the complicated and
sometimes beautiful world of Alfvén continues to unfold.
This publication is accompanied by stereo slide pairs
and stereo movies, which may be downloaded from the
Physics of Plasmas website (http://pop.aip.org/shear_alfven_waves).150 There is an accompanying text file with figure captions as well as instructions for viewing.
ACKNOWLEDGMENTS
(17)
The relevant quantities were calculated from the volumetric
data set.148 The volume-averaged resistivity varies in time
and jumps to a minimum of five times the Spitzer resistivity
when the flux ropes collide and there is a QSL. When there
is no reconnection the QSL disappears.
The next study involved three flux ropes produced using
three emitters 2.5 cm in diameter arranged in a triangle. A
volumetric data set was acquired (32 400 spatial locations,
volume 13.2 cm 13.2 cm 9.6 m, dx ¼ dy ¼ 3 mm, dz
¼ 63.9 cm, dt ¼ 0.64 ls, Tacquis ¼ 6.55 ms). The data at each
location is averaged over 12 plasma shots. Light fluctuations
recorded with a fast diode are used to synchronize the data
stream to the motion of the ropes. The flux ropes at one
instant of time are shown in Fig. 32.
There are more degrees of freedom in this case, and one
expects that more than one QSL can occur. This is indeed
the case as seen in Fig. 33, which is an anaglyph of the three
ropes shown in Fig. 32 as well as two QSLs. The rotational
motion of the ropes is reflected in an optical signal
(f 4kHz). A vacuum ultraviolet spectrometer tuned to a
303 nm He II line (40 eV) recorded oscillations interpreted
as electron temperature fluctuations, perhaps due to successive reconnection events. These experiments point to the fact
that multiple interacting flux ropes will have multiple QSLs
and the associated reconnection will be bursty and occur at a
host of spatial positions. It is certainly three-dimensional.
XIV. SUMMARY AND CONCLUSIONS
Since the prediction and discovery of shear Alfvén
waves it has been realized that they touch or even dominate
nearly every area of plasma physics from Astrophysics to
fusion. This overview has concentrated on a number of
aspects of the wave, which has emerged in recent carefully
controlled experiments. A detailed review of the wave as
observed in space is left to other monographs. What appears
as a variety of very different waves, in different circumstances (uniform plasmas, non-uniform plasmas, density striations, magnetic field gradients, multiple mirrors, flux ropes,
dense expanding plasmas…) are all different manifestations
of the shear wave. This is exemplified in Fig. 34.
Experimental sources, diagnostics and theory have
become increasing sophisticated over the years and in all
likelihood will continue to do so. Thirty years ago data at
best were x-y plots, simulations, if any, were one dimensional
The authors would like to thank our collaborators: Craig
Kletzing, Bill Heidbrink, Andrew Collette, Roger McWilliams, Hans Boehmer, Dennis Papadopoulis, Yuhou Wang,
Eric Lawrence, Y Zhang, and Shu Zhou, for their valuable
contributions. We also appreciate the expert technical support of the LAPD staff, Zalton Lucky, Marvin Drandell, and
Mio Nakamoto. The operation of the LAPD at the Basic
Plasma Science Facility (BaPSF) is supported by DOE/NSF.
Partial support for some aspects of the work was provided by
ONR.
1
H. Alfvén, Mon. Not. R. Astron. Soc. 107, 211 (1947).
S. Tomczyk, S. W. McIntosh, S. L. Keil, P. G. Judge, T. Schad, D. H.
Seeley, and J. Edmondson, Science 317, 1192 (2007).
3
P. J. ColemanJr., Phys. Rev. Lett. 17, 207 (1966).
4
M. L. Goldstien, D. A. Roberts, and W. H. Matthaeus, Ann. Rev. Astrophys. 33, 283 (1995).
5
W. H. Matthaeus, M. L. Goldstein, and D. A. Roberts, J. Geophys. Res.
95 (20), 675 doi:10.1029/JA095iA12p20673 (1990).
6
V. M. Chmyrev, S. V. Bilchenko, O. A. Pokhotelov, V. A. Marchenko, V.
I. Lararev, A. V. Strelsov, and L. Stenflo, Phys. Scr. 38, 841 (1988).
7
P. J. Louran, J. E. Wahlund, T. Chust, H. de Feraudy, A. Roux, B. Holback, P. O. Dovner, A. J. Eriksson, and G. Holmgren, Geophys. Res. Lett.
21, 1823 doi:10.1029/94GL01424 (1994).
8
J. R. Wygant A. Keiling, C. A. Cattell, M. Johnson, R. L. Lysak,
M. Temerin, F. S. Mozer, C. A. Kletzing, J. D. Scudder, W. Peterson,
C. T. Russell, G. Parks, M. Brittnacher, G. Germany, and J. Spann,
J. Geophys. Res. 106, 675 (2000).
9
Y. Jantenco-Pereira, Phys. Scr. T 60, 113 (1995).
10
H. Alfvén, Nature 150, 405 (1942).
11
C.-G. Fålthammar, IEEE Trans. Plasma Sci. 25, 409 (1997).
12
S. Lundquist, Nature 164, 145 (1945);S. Lundquist, Phys. Rev. 76, 1805
(1945).
13
W. H. Bostick and M. Levine, Phys. Rev. 94, 671 (1952).
14
W. Gekelman, J. Geophys. Res. 104, 14417 doi:10.1029/98JA00161
(1999).
15
W. Heidbrink, Phys. Plasmas 15, 056107 (2008).
16
S. Vincena, G. J. Morales, and J. E. Maggs, Phys. Plasmas 17, 52106
(2010).
17
B. Fried and S. D. Conte, The Plasma Dispersion Function (Academic,
San Diego, CA, 1961).
18
G. J. Morales and J. E Maggs, Phys. Plasmas 4, 4118 (1997).
19
G. J. Morales, R. Loritsch, and J. E. Maggs, Phys. Plasmas 1, 3765 (1994).
20
S. Lundquist, Nature 164, 145 (1949).
21
B. Lehnert, Phys. Rev. 94, 815 (1952).
22
D. F. Jephcott, Nature 183, 1652 (1959).
23
G. A. Sawyer, P. L. Scott, and T. F. Stratton, Phys. Fluids 2, 47 (1959).
24
J. W. Wilcox, F. Boley, and A. De Silva, Phys. Fluids 3, 15 (1960).
25
D. F. Jephcott and P. M. Stocker, J. Fluid Mech. 13, 587 (1962).
26
L. C. Woods, J. Fluid Mech. 13, 579 (1962).
27
J. W. Wilcox, F. Boley, and A. De Silva, Phys. Fluids 3, 15 (1960).
28
D. Swanson, R. Gould, and R. Hertel, Phys. Fluids 7, 269 (1964).
29
G. Müller, Plasma Phys. 16, 813 (1974).
30
Y. Amagishi and A. Tsushima, Plasma Phys.Controlled Fusion 26, 1489
(1984).
31
T. Stix, Waves in Plasmas (AIP, New York, 1992), p. 12.
32
Y. Amagishi, M. Inutaki, T. Inutake, and T. Akitsu, Jpn. J. Appl. Phys.
20, 2171 (1981).
2
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-25
33
Many faces of shear Alfvén waves
Y. Amagishi, Phys. Rev. Lett. 64, 1247 (1990);Y. Amagishi, J. Phys. Soc.
Jpn. 59, 2374 (1990).
34
W. Gekelman, J. Geophys. Res. 104, 14417 doi:10.1029/98JA00161
(1999).
35
G. G. Borg, M. H. Brennan, R. C. Cross, L. Giannoue, and I. J. Donneley,
Plasma Phys. Controlled Fusion 27, 1125 (1985).
36
W. Gekelman, H. Pfister, Z. Lucky, J. Bamber, D. Leneman, and J. E.
Maggs, Rev. Sci. Instrum. 62, 2875 (1991).
37
P. Pribyl and W. Gekelman, Rev. Sci. Instrum. 75, 669 (2004).
38
D. Leneman, W. Gekelman, and J. E. Maggs, Rev. Sci. Instrum. 77,
015108 (2006).
39
D. Leneman and W. Gekelman, Rev. Sci. Instrum. 72, 3473 (2001).
40
W. Gekelman, D. Leneman, and J. E. Maggs, Phys. Plasmas 1, 3775
(1994).
41
W. Gekelman, S. Vincena, D. Leneman, and J. Maggs, Plasma Phys. Controlled Fusion 39, A101 (1997).
42
D. Leneman, W. Gekelman, and J. E. Maggs, Phys. Rev. Lett. 82, 2673
(1999).
43
N. Palmer, W. Gekelman, and S. Vincena, Phys. Plasmas 12, 072102
(2005).
44
C. Kletzing, S. Bounds, J. Martin-Hiner, W. Gekelman, and C. Mitchell,
Phys. Rev. Lett. 90, 035004 (2003).
45
C. Kletzing, D. Thucks, F. Skiff, S. Bounds, and S. Vincena, Phys. Rev.
Lett. 104, 095001 (2010).
46
A. Gigliotti, W. Gekelman, P. Pribyl, S. Vincena, A. Karavaev, X. Shao,
A. S. Sharma, and D. Papadopoulos, Phys. Plasmas 16, 092106 (2009).
47
A. V. Karavaev, N. A. Gumerov, K. Papadopoulos, X. Shao, A. S.
Sharma, W. Gekelman, P. Pribyl, Y. Wang, B. Van Compernolle, and S.
Vincena, “Generation of shear Alfvén waves by a rotating magnetic
field,” Phys. Plasmas (in review).
48
A.V. Karavaev, N. A. Gumerov, K. Papadopoulos, X. Shao, A. S.
Sharma, W. Gekelman, A. Giglotti, P. Pribyl, and S. Vincena, Phys. Plasmas 17, 012102 (2010).
49
A. Hasegawa and C. Uberoi, The Alfven Wave (Technical Information
Center, US Department of Energy, Springfield, VA, 1982).
50
K. Takahashi and R. McPherron, Planet Space Sci. 32, 1343 (1984).
51
M. J. Engebretson, L. J. Zanetti, T. A. Potemra, and M. H. Acuna,
Geophys. Res. Lett. 13, 905 doi:10.1029/GL013i009p00905 (1986).
52
M. R. Lessard, M. K. Hudson, J. C. Samson, and J. R. Wygant, J. Geophys. Res. 104, 12361 doi:10.1029/1998JA900117 (1999).
53
J. C. Samson, D. D. Wallis, T. J. Hughes, F. Creutzberg, and T. J. Hughes,
J. Geophys. Res. 97, 8495 doi:10.1029/91JA03156 (1992).
54
H. J. Singer, W. J. Hughes, and C. T. Russell, J. Geophys. Res. 87, 3519
doi:10.1029/JA087iA05p03519 (1982).
55
F. Plaschke, K. H. Glassmeier, H. U. Auster, O. D. Constantinescu, W.
Magnes, V. Angelopoulos, D. G. Sibeck, and J. P. McFadden, Geophys.
Res. Lett. 36, LO2104 doi:10.1029%2F2008GL036411 (2009).
56
A. Tsushima, Y. Amagishi, M. Inutake, Phys. Lett. 88A, 457 (1982).
57
C. C. Mitchell, S. Vincena, J. E. Maggs, and W. Gekelman, J. Geophys.
Res. 28, 923 (2001).
58
C. C. Mitchell, J. E. Maggs, and W. Gekelman, Phys. Plasmas 9, 2909
(2002).
59
A. Strelsov and W. Lotko, J. Geophys. Res. 101, 5343 doi:10.1029/
95JA03762 (1996).
60
C. Mitchell, “Alfven Eigenmodes and Field Line Resonances,” Ph.D. dissertation UCLA, 2001.
61
S. Vincena, W. Gekelman, and J. E. Maggs, Phys. Rev. Lett. 93, 105003
(2004).
62
T. H. Stix, Phys. Fluids 1, 308 (1958).
63
T. H. Stix and R. W. Palladino, in Proceedings of the 2nd UN International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958,
Vol. 31, p. 282.
64
F. I. Bowley, J. M. Wilcox, A. W. DeSilva, P. R. Forman, G. W. Gordon,
W. H. Hamilton, and C. N. Watson-Munro, Phys. Fluids 6, 925 (1963).
65
J. A. Lehane and F. J. Paoloni, Plasma Phys. 12, 823 1970.
66
D. G. Swanson, R. W. Clark, P. Korn, S. Robertson, and C. B. Wharton,
Phys. Rev. Lett. 28, 1015 (1972).
67
A. Breun, P. Brooker, D. Brouchous, J. Browning, G. Butz, J. Conrad, E.
Dales, J. Ferron, R. Goulding, N. Hershkowitz, T. Intrator, C. Litwin, R.
Majeski, S. Maessick, B. Nelson, L. Peranich, H. Persing, J. Radtke, D.
Roberts, G. Severn, D. Sing, E. Wang, D. A. D’Ippolito, J. R. Myra, and
G. L. Francis, in 11th IAEA International Conference on Plasma Physics
and Controlled Nuclear Fusion Research, Kyoto, Japan, 13–20 November
1986, Vol. 2 of Nuclear Fusion Supplement 1987, pp. 263–271.
Phys. Plasmas 18, 055501 (2011)
68
Y. Yasaka, R. Majeski, J. Browning, N. Hershkowitz, and D. Roberts,
Nucl. Fusion 28, 1765 (1988).
D. R. Roberts, N. Hershkowitz, R. P. Majeski, and D. H. Edgell, AIP
Conf. Proc. 190, 462 (1989).
70
S. V. Polyakov, V. O. Rapoport, and V. Yu. Trakhtengerts, Fiz. Plazmy
9, 371 (1983).
71
P. P. Belyaev, S. V. Polyakov, V. O. Rapoport, and V. Yu. Trakhtengerts,
Geomagn. Aeron. 24, 2 (1984).
72
P. P. Belyaev, S. V. Polyakov, V. O. Rapoport, and V. Yu. Trakhtengerts,
Izv. Vyssh. Uchebn. Zaved. Radiofiz. 33, 408 (1990).
73
V. Yu. Trakhtengerts and A. Ya. Feldstein, J. Geophys. Res. 96, 19363
doi:10.1029/91JA00376 (1991).
74
B. J. Thompson and R. L. Lysak, J. Geophys. Res. 101, 5359
doi:10.1029/95JA03622 (1996).
75
R. L. Lysak, J. Geophys. Res. 104, 10017 doi:10.1029/1999JA900024
(1999).
76
O. A. Pokhotelov, D. Pokhotelov, A. Streltsov, V. Khruschev, and M.
Parrot, J. Geophys. Res. 105, 7737 doi:10.1029/1999JA900480 (2000).
77
M. Grzesiak, Geophys. Res. Lett. 27, 923 doi:10.1029/1999GL010747
(2000).
78
N. Ivchenko, G. Marklund, K. Lynch, D. Pietrowski, R. Torbert,
F. Primdahl, and A. Ranta, Geophys. Res. Lett. 26, 3365 doi:10.1029/
1999GL003588 (1999).
79
P. P. Belyaev, T. Bosinger, S. V. Isaev, and J. Kangas, J. Geophys. Res.
104, 4305 doi:10.1029/1998JA900062 (1999).
80
F. Z. Feygin, A. K. Nekrasov, K. Mursula, J. Kangas, and T. Pikkarainen,
Ann. Geophys. 12, 147 doi:10.1007%2Fs00585-994-0147-8 (1994).
81
C. C. Chaston, J. W. Bonnell, C. W. Carlson, J. P. McFadden, R. E.
Ergun, R. J. Strangeway, and E. J. Lund, J. Geophys. Res. 109, A04205
doi:10.1029/2003JA010053 (2004).
82
R. L. Lysak and Y. Song, J. Geophys. Res. 107, 1160 doi:10.1029/
2001JA000308 (2002).
83
N. G. Lekhtinen, G. A. Markov, and S. M. Fainstein, Izv. Vyssh. Uchebn.
Zaved. Radiofiz. 38, 312 (1995).
84
H. Fukunishi, Y. Takahashi, M. Sato, A. Shone, M. Fujito, and Y. Watanabe, Geophys. Res. Lett. 24, 2973, doi:10.1029/97GL03022 (1997).
85
A. I. Sukhorukov and P. Stubbe, Geophys. Res. Lett. 24, 829
doi:10.1029/97GL00807 (1997).
86
V. Yu. Trakhtengerts, Geomagn. Aeron. 29, 3 (1989).
87
J. E. Maggs and G. J. Morales, Phys. Rev. Lett. 91, 035004 (2003).
88
J. E. Maggs, G. J. Morales, and T. A. Carter, Phys. Plasmas 12, 013103
(2005).
89
G. J. Morales and J. E. Maggs, Phys. Plasmas 13, 052109 (2006).
90
T. A. Carter, B. Brugman, P. Pribyl, and W. Lybarger, Phys. Rev. Lett.
96, 155001 (2006).
91
R. Boswell, Nature (London) 425, 352 (2003).
92
O. P. Pogutse and E. I. Yurchencko, Nucl.Fusion 18, 1629 (1978).
93
Y. Zhang, W. W. Heidbrink, H. Boehmer, R. McWilliams, G. Chen, B.
N. Breizman, S. Vincena, T. Carter, D. Leneman, W. Gekelman, P.
Pribyl, and B. Brugman, Phys. Plasmas 15, 012105 (2008).
94
C. Uberoi, Phys. Fluids 15, 1673 (1972).
95
D. A. D’Ippolito and J. P. Goodbloed, Plasma Phys. 22, 1091 (1980).
96
C. Z. Cheng, L. Chen, and M. S. Chance, Ann. Phys. 161, 21 (1984);C. Z.
Cheng and M. S. Chance, Phys. Fluids 29, 3695 (1986).
97
H. H. Duong, W. W. Heidbrink, E. J. Strait, T. W. Petrie, R. Lee, R. A.
Moyer, and J. G. Watkins, Nucl. Fusion 33, 749 (1993).
98
J. E. Maggs and G. J. Morales, Phys. Plasma 4, 290 (1997).
99
A. T. Burke, J. E. Maggs, and G. J. Morales, Phys. Plasmas 7, 1397 (2000).
100
S. Vincena and W. Gekelman, Phys. Plasmas 13, 064503 (2006).
101
A. T. Burke, J. E. Maggs, and G. J. Morales, Phys. Rev. Lett. 81, 3659
(1998); A. T. Burke, J. E. Maggs, and G. J. Morales, Phys. Plasmas 7,
544 (2000).
102
J. R. Penano, G. J. Morales, and J. E. Maggs, Phys. Plasmas 7, 144
(2000).
103
D. C. Pace, M. Shi, J. E. Maggs, G. J. Morales, and T. A. Carter, Phys.
Plasmas 15, 122304 (2008).
104
M. Shi, D. C. Pace, G. J. Morales, J. E. Maggs, and T. A. Carter, Phys.
Plasmas 15, 122304 (2008).
105
C. C. Chaston, C. W. Carlson, R. E. Ergun, and J. P. McFadden, Phys.
Scr. T 84, 64 (2000).
106
C. C. Chaston, J. W. Bonnell, C. W. Carlson, and J. P. McFadden, J. Geophys. Res. 108, 8003 doi:10.1029/2002JA009420 (2003).
107
E.V. Mishin and M. Forster, Geophys. Res. Lett. 22, 1745 doi:10.1029/
95GL01442 (1995).
69
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp
055501-26
108
Gekelman et al.
E. D. Fredrickson, N. N. Gorelenkov, R. E. Bell, J. E. Menard, A. L.
Roquemore, S. Kubota, N. A. Crocker, and W. Peebles, Nucl. Fusion 46,
S926 (2006).
109
L. Chen, Z. H. Lin, and R. White, Phys. Plasmas 8, 4713 (2001).
110
D. A. Gates, N. N. Gorelenkov, and R. B. White, Phys. Rev. Lett. 87,
205003 (2001).
111
R. H. Kraichnan, Phys. Fluids 8, 1385 (1965).
112
C. S. Ng and A. Bhattacharjee, Astro Phys. J. 465, 845 (1996).
113
J. V. Shebalin, W. H. Matthaeus, and D. Montgomery, J. Plasma Phys.
29, 525 (1983).
114
P. Goldreich and S. Sridhar, Astro Phys. J. 485, 680 (1997).
115
G. G. Howes, W. Dorland, S. C. Cowley, G. W. Hammett, E. Quataert,
A. A. Schekochihin, and T. Tatsuno, Phys. Rev. Lett. 100, 065004 (2008).
116
S. D. Bale, P. J. Kellogg, F. S. Mozer, T. S. Horbury, and H. Reme, Phys.
Rev. Lett. 94, 215002 (2005).
117
A. Hasegawa and L. Chen, Phys. Rev. Lett. 36, 1362 (1976).
118
J. V. Hollweg, J. Geophys. Res. 99, 23431 doi:10.1029/94JA02185
(1994).
119
D. W. Auerbach, T. A. Carter, S. Vincena, and P. Popovich, Phys. Rev.
Lett. 105, 135005 (2010).
120
B. C. Low, J. Geophys. Res. 106, 25142 doi:10.1029/2000JA004015
(2001).
121
S. Colgate, J. Geophys. Res. 70, 3161 doi:10.1029/JZ070i013p03161
(1965).
122
P. A. Bernhardt, Phys. Fluids B 4, 2249 (1992).
123
G. Haerendel, G. Paschmann, W. Baumjohann, C. W. Carlson, Nature
320, 720 (1986).
124
M. A. Van Zeeland and W. Gekelman, Phys. Plasmas 11, 320 (2004);A.
Collette, W. Gekelman, A. Collette, and W. Gekelman, Phys. Rev. Lett.
105, 195003 (2010).
125
M. A. Van Zeeland, W. Gekelman, S. Vincena, and G. Dimonte, Phys.
Rev. Lett. 87, 105001 (2001).
126
W. Gekelman, M. A. Van Zeeland, S. Vincena, and P. Pribyl, J. Geophys.
Res. 108, A781 doi:10.1029/2002JA009741 (2003).
127
S. Vincena, W. Gekelman, M. A. Van Zeeland, J. Maggs, and A. Collette,
Phys. Plasmas 15, 072114 (2008).
128
M. A. Van Zeeland, W. Gekelman, S. Vincena, and J. Maggs, Phys. Plasmas 10, 1243 (2003).
129
W. Gekelman, A. Collette, and S. Vincena, Phys. Plasmas 14, 062109
(2007).
130
R. Ruytov, R. P. Drake, J. Kane, E. Laing, B. A. Remington, and W. M.
Wood-Vasey, AstroPhys. J. 581, 821 (1991).
131
P. M. Nielson, L. Willingale, M. C. Kaluza, C. Kamperidis, S. Minardi, M.
S. Wei, P. Fernandes, M. Notley, S. Bandyopadhyay, M. Sherlok, R. J. King-
Phys. Plasmas 18, 055501 (2011)
ham, M. Tatarakis, Z. Najmudin, W. Rozmus, R. G. Evans, M. G. Haines,
A. E. Dangor, and K. Krushelnick, Phys. Rev. Lett. 97, 255001 (2006).
P. M. Nielson, L. Willingale, M. C. Kaluza, C. Kamperidis, S. Minardi,
M. S. Wei, P. Fernandes, M. Notley, S. Bandyopadhyay, M. Sherlok, R.
J. Kingham, M. Tatarakis, Z. Najmudin, W. Rozmus, R. G. Evans, M. G.
Haines, A. E. Dangor, and K. Krushelnick, Phys. Plasmas 15, 092701
(2008).
133
E. N. Parker, J. Geophys. Res. 62, 509 doi:10.1029/JZ062i004p00509
(1957).
134
W. Gekelman, R. L. Stenzel, and N. Wild, J. Geophys. Res. 87, 101
doi:10.1029/JA087iA01p00101 (1982).
135
C. D. Gregory, J. Howe, B. Loupias, S. Myers, M. M. Notley, Y. Sakawa,
A. Oya, R. Kodama, M. Koenig, and N. C. Woolsy, AstroPhys. J. 676,
420 (2008).
136
B. Roberts, Physics of Magnetic Flux Ropes, Geophysical Monograph
Vol. 58, edited by C. T. Russell, E. R. Priest, and L. C. Lee (AGU Washington, DC, 1990), p. 113.
137
J. Chen, R. A. Howard, G. E. Brueckner, R. Santaro, J. Krall, S. E. Paswaters, O. C. St. Syr, R. Schwenn, P. Lamy, and G. M. Simnett, AstroPhys. J. Lett. 490, L191 (1997).
138
J. T. Gosling, Physics of Magnetic Flux Ropes, Geophysical Monograph
Vol. 58, edited by C. T. Russell, E. R. Priest, and L. C. Lee (AGU Washington, DC, 1990), p. 343.
139
W. Gekelman, J. Maggs, and H. Pfister, IEEE Trans. Plasma Sci. 20, 614
(1993).
140
J. B. Taylor, Phys. Rev. Lett. 33, 1139 (1974).
141
T. Intrator, X. Sun, G. Lapenta, L. Dorf, and I. Furno, Nat. Phys. 5, 521
(2009).
142
E. Lawrence and W. Gekelman, Phys. Rev. Lett. 103, 105002
(2009).
143
C. H. Mandrini, P. Démoulin, L. Driel-Gesztelya, B. Schneider, G.
Cauzzi, A. Hoffan, Sol. Phys. 168, 115 (1996).
144
E. R. Priest and P. Démoulin, J. Geophys. Res. 100, 23443 (1995);
P. Démoulin, Adv. Space Res. 27, 1269 (2006).
145
V. S. Titov, G. Horning, and P. Démoulin, J. Geophys. Res. 107, 1164
doi:10.1029/2001JA000278 (2002).
146
V. S. Titov, T. Forbes, E. Priest, Z. Mikic, and J. Linker, Astro Phys. J.
693, 1029 (2009); P. Démoulin, Adv. Space Res. 37, 1269 (2006).
147
M. Berger, Geophys. Astrophys. Fluid Dyn. 30, 79 (1984).
148
W. Gekelman, E. Lawrence, A. Collette, B. Van Compernolle, P. Pribyl,
M. Berger, and J. Campbell, Phys. Scr. T 142, 014032 (2010).
149
S. Vincena, W. Gekelman, and J. Maggs, Phys. Plasmas 8, 3884 (2001).
150
See supplementary material at http://dx.doi.org/10.1063/1.3592210 which
contains stereo image pairs, captions, and instructions for viewing.
132
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp