PHYSICS OF PLASMAS 14, 062109 共2007兲
Three-dimensional current systems generated by plasmas colliding
in a background magnetoplasma
W. Gekelman, A. Collette, and S. Vincena
Department of Physics and Astronomy, University of California, Los Angeles, Los Angeles,
California 90095
共Received 27 December 2006; accepted 24 April 2007; published online 26 June 2007兲
Results are presented from an experiment in which two plasmas, initially far denser than a
background magnetoplasma, collide as they move across the magnetic field. The dense plasmas are
formed when laser beams, nearly orthogonal to the background magnetic field, strike two targets.
The merging plasmas are observed to carry large diamagnetic currents. A reconnection event is
triggered by the collision and the electric field induced in this event generates a field-aligned current,
which is the first step in the development of a fully three-dimensional current system. After several
ion gyroperiods, the current systems become those of shear Alfvén waves. As local currents move,
small reconnection “flares” occur at many locations throughout the volume, but they do not seem to
affect the overall system dynamics. The data clearly show that the induced electric field is carried
though the system by shear Alfvén waves. The wave electric fields as well as local magnetic helicity
are discussed. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2741462兴
I. INTRODUCTION
The interaction of streaming dense plasmas with a background plasma is of interest in a variety of situations such as
Coronal Mass ejections,1 above atmospheric explosions,2 and
chemical releases,3 to name a few. A laboratory experiment
detailing the expansion of a dense laser-produced plasma
共lpp兲 and its relationship to phenomena in space has been
reported.4,5 The initial laboratory experiment discovered that
shear Alfvén waves are generated in the expansion,6 although
the laser pulse duration is far shorter than the period of the
waves. The Alfvén waves are generated by Cherenkov radiation by fast electrons, as established in a separate
experiment7 in which streams of fast electrons were generated. The outgoing electrons and the return currents of the
waves become the current system of the Alfvén waves.
The topic of the collision of dense plasma clouds embedded in a background plasma is fascinating because of the
large range of phenomena that can occur. The dense plasma
clouds carry significant energy and momentum and their collision can involve intense nonlinear interactions. The complex interactions involved in the collision can lead to unexpected consequences, such as long-lived structures that
subsequently dominate the behavior of the surrounding
plasma. Spatially localized regions of enhanced energy and
momentum density collide in highly turbulent plasmas, such
as those that occur in stellar coronas or stellar winds. A dramatic example from astrophysical plasmas is the case in
which fast “winds” generated in supernovae collide.8 There
are several reports of experiments investigating the collision
of very dense laser-produced plasmas. Collisionless shocks
were observed9 when two foils were irradiated with lasers
with 100 times the energy of the Nd-YAG laser available for
this experiment. There was no background plasma in that
experiment and the experimental time scale was hundreds of
picoseconds. In this experiment, the laser energy was too low
to create a collisionless shock. Experiments10 at laser inten1070-664X/2007/14共6兲/062109/13/$23.00
sities near 1014 W / cm2 show that colliding plasmas stagnate
if the ion-ion mean free path due to Coulomb collisions is
smaller than the density scale length. In this event, a fraction
of the kinetic energy of the collision is transformed to ion
heat. Ion temperatures of 15 keV were spectroscopically
measured in the stagnated plasma. Such experiments are relevant to the development of x-ray lasers.11
In the present experiment, colliding plasmas produce effects not seen in a single lpp expansion. An intense current
channel generated by magnetic field reconnection occurs
when the plasmas collide. This current as well as electrons
emitted from the dense plasma blobs generated a complex
three-dimensional Alfvénic current system not present in the
single target experiment. Small reconnection “flares” are embedded within the Alfvénic currents.
II. EXPERIMENTAL SETUP
The experiment is performed in the upgraded Large
Plasma Device12 共LAPD兲 at UCLA. The plasma is generated
by a direct current 共DC兲 discharge13 using a barium oxide
coated cathode. The plasma column is 17 m long, 60 cm in
diameter, and is highly ionized helium. The machine is well
diagnosed. It has 450 access ports, 65 of which have vacuum
interlocks that probes and probe drives can be attached to
while the machine is running. It is therefore possible to acquire data at thousands of locations on planes transverse to
the axial magnetic field and space the planes 32.0 cm apart at
65 axial locations. Such a data set would take months to
acquire. In these experiments, the magnetic field ranged from
600 to 1200 G and the plasma density ranged from 2 to 4
⫻ 1012 cm−3. The electron temperature of the background He
plasma was 4 eV and was measured with a swept Langmuir
probe. The ion temperature of an argon plasma under the
same conditions was measured to be 0.8 eV using laserinduced fluorescence. The helium ion temperature was measured with a Fabry-Perot interferometer under similar dis-
14, 062109-1
© 2007 American Institute of Physics
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062109-2
Gekelman, Collette, and Vincena
Phys. Plasmas 14, 062109 共2007兲
FIG. 1. Schematic of the experimental
setup 共not to scale兲. The cathodeanode spacing is 52.5 cm. The targets
are located 12.5 m from the anode and
4.1 m from the end of the plasma column at the left. The carbon targets are
6.5 cm apart. The inner diameter of
the vacuum vessel is 1 m and the
plasma diameter is 60 cm. The machine has 450 access ports and over 70
vacuum interlocks into which probes
may be inserted.
charge conditions to be 1 ± 0.5 eV. The ion temperature is
also consistent with measurements of the Alfvén wave speed
near the ion cyclotron frequency.14 The experimental setup is
shown schematically in Fig. 1. The sequence was that 共a兲 the
background plasma was switched on and a steady-state background plasma was generated after several milliseconds.
Next 共b兲 two carbon targets were simultaneously struck by
laser beams 10 ns in duration while the plasma discharge
was active. A 1.5 J Nd-YAG laser beam was split into two
beams and through a series of lenses and mirrors entered the
LAPD through two side ports hitting the targets at an angle
of 25 degrees to the surface normal. The laser power on each
target is approximately 5 ⫻ 1010 W / cm2. This entire sequence is repeated at 1 Hz. The plasma is terminated 5 ms
after the laser fires. Much of the data in this paper were
derived from three axis magnetic field probes sensitive to
ជ / dt兲. The probe signals were digitized with 14 bit,
共dB
100 MHz digitizers and numerically integrated. The motion
of the probes in planes transverse to the background magnetic field was accurate to 0.5 mm and positioning along the
axis of the device accurate to within a centimeter. The probes
were centered in the transverse planes using a transit and
fiduciary mark on the machine window.
Since the carbon targets become eroded from being
struck by the laser, both targets are rotated by computercontrolled stepping motors every five shots. After a complete
rotation, a second set of motors lowers the targets by 1 mm.
The laser spot size on the target is at most 0.5 mm.
The plasma diagnostics include three-axis differentially
wound magnetic pickup loops approximately 3 mm in diamជ / dt兲, Langmuir probes for deeter for a measurement of 共dB
termination of the plasma density and electron temperature, a
gated 共3 ns兲 CCD camera for imaging the colliding plasmas,
and a 100 GHz, 150 MHz bandwidth microwave interferometer for measurement of plasma density.
In the initial expansion, the large pressure gradients associated with the dense lpp give rise to diamagnetic currents,
which expel a significant part of the background magnetic
field. At this time, the configuration is termed a “magnetic
bubble.”15 The bubble radius, without the presence of collisions, is obtained by equating the magnetic field to particle
Ⲑ
energy and is Rb = 共 23 ␮0共Elpp / ␲B20兲兲 1 3 共this is derived for
expansion from a flat surface兲. Here Elpp is the energy of the
laser-produced plasma, which is approximately one-half the
laser energy per bubble. In our experiment, each bubble radius is about 4 cm, about 1/15 the plasma column diameter.
2
The expanding bubble experiences a deceleration 共共v⬜
/ 2Rb兲兲
and stops expanding at ␶ = 2Rb v⬜ , which is approximately
1 ␮s in the laboratory. Subsequent to bubble formation, the
lpp becomes magnetized and continues to expand along and
across the magnetic field. In these experiments, fast ions are
produced in the lpp and their Larmor radius 共18 cm CII,
9 cm CIII兲 is always greater than the bubble radius 共for
charge states less than 2兲. The lpp electrons have a small
gyroradius and the difference results in a vertical electric
ជ
field. This coupled to the background field produces an E
ជ motion that causes each lpp plasma to jet across the
⫻B
background magnetic field.5 This has been studied in previous experiments16 and in simulations.17 This transverse motion allows the two lpps to collide. In the experimental configuration described here, the two lpp bubbles collide in the
center of the machine before they collapse.
Figure 2 comprises two false color images taken with a
fast CCD camera 共3 ns exposure兲 looking down the axis of
the machine, 6 m from the lpp targets. Red corresponds to
light of higher intensity. The carbon targets are seen as blue
vertical bands on the left and right side of each image. The
targets are 6.5 cm apart center to center and each target is
2 cm in diameter. The background magnetic field was 600 G
and points into the page; the background plasma was helium.
Data for Fig. 2共a兲 were acquired 300 ns after the targets were
struck. The collision of the magnetic bubbles is clearly seen.
Data for Fig. 2共b兲 were acquired 3.32 ␮s after 共a兲. At this
time, the magnetic bubbles have completely collapsed and
the lpp plasmas have collided in a plane far forward of the
targets and the previous location of the bubbles. The photographs reflect two processes on two time scales. The first is
Ⲑ
FIG. 2. 共Color兲 Photographs of 共a兲 colliding lpp bubbles 共␶ = 300 ns兲 and 共b兲
colliding lpp plasmas ␶ = 3.62 ␮s. Red corresponds to the most intense light,
blue the least. Both pictures are on the same color scale. The light is unfiltered. The camera exposure time was 3 ns. B = 600 G. The targets were
struck at ␶ = 0.
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062109-3
Three-dimensional current systems...
Phys. Plasmas 14, 062109 共2007兲
FIG. 3. 共Color兲 A single 3 ns exposure of the colliding plasmas acquired
700 ns after the targets are struck. The filamentary structure, which varies on
a shot-by-shot basis, is clearly visible.
the collision of magnetic bubbles. These photographs were
averaged over 10 experimental shots, therefore any nonreproducible structure has been smoothed out. Figure 3 is a
single shot acquired at 700 ns 共using twice the laser energy兲.
Gradients in the light in Fig. 3 suggest that bubble turbulence exists. The filaments, which extend outward, are due
to surface instabilities to be discussed in a future work. They
resemble the Rayleigh-Taylor-like structures seen in earlier
experiments,18 which did not have a background magnetized
plasma. The three photographs were taken using unfiltered
light, but the use of carbon filters shows that none of the light
is due to neutral carbon; it is a mixture of CII and CIII ionic
light. Processes which occur on the slower microsecond time
scale involve current systems and wave generation, which
will be discussed in subsequent sections of this paper.
III. MAGNETIC FIELDS AND CURRENTS
Magnetic field is measured by three-axis magnetic
pickup loops. The signals are integrated digitally after the
data are acquired to yield B共t兲 at thousands of spatial positions. The magnetic probes detect very strong signals close to
the targets when a large fraction of the background field is
expelled from the magnetic bubbles. This is followed by signals generated by fast electrons rushing away from the interaction region, and finally by the magnetic signature of
Alfvén waves. The magnetic probes and associated amplifiers were calibrated using a Helmholtz coil and their frequency response with a network analyzer. The probe response was linear to 2 MHz. The probe signals are integrated
and the data present the field in Gauss.
At early times and close to the targets, the magnetic
fields are dominated by the diamagnetic field of the plasma
bubbles. This is shown in Fig. 4, which was taken in a second data run at twice the laser power when a second identical
laser was acquired. The magnetic fields measured at twice
the laser power are larger 共by roughly a factor of 2兲 but
topologically the same 共the bubble radius is slightly larger兲.
The diamagnetic currents of the individual bubbles have associated large magnetic fields. Most of the diamagnetic field
is in the axial direction but a residual, transverse field, pos-
FIG. 4. 共Color兲 Magnetic field measured 400 ns after the targets are struck
by two 1-joule lasers. The data plane is 4 cm away from the targets. The
spatial extent of the data plane is ␦x = 22 cm, ␦y = 23 cm. 共a兲 is a view
looking in the z direction 共facing away from the cathode兲. In this data plane,
a three-axis probe 1 mm in diameter with a linear frequency response to
100 MHz was used. A distance scale with a 5 cm interval is shown at the
upper left. 共a兲 shows the magnetic field from a head-on perspective. It accentuates the transverse fields and clearly shows a magnetic neutral sheet
between the plasma bubbles. 共b兲 is another view of the magnetic field in
which the negative z direction is upwards. The plasma bubbles have large
axial fields, directed against the background magnetic field.
sibly due to the nonspherical shape of the bubbles as well as
the effect of fast electrons streaming along the background
field, is clearly visible in Fig. 4共a兲. This transverse field has
the classic topology of a magnetic neutral sheet.19 The nearly
vertical magnetic fields have an opposite sense and are
forced together as the bubbles collide. The entire reconnection process lasts less than a microsecond, roughly half the
bubble lifetime. Inside the magnetic bubbles the background
field is greatly reduced 共it was not possible to place a probe
in the center of the bubbles兲. Between the bubbles, in the
region of the neutral sheet the axial magnetic field remains
600 G. The reconnecting flux drives a current along the axial
field between the magnetic bubbles. The magnetic field 470
ns later, due to this current, is shown in Fig. 5 on a plane,
which is 27.5 cm away from the plane in Fig. 4. The magnetic field structure shown in Fig. 4 is not evident at ␶
= 400 ns and at ␦z = 31.5 cm, in fact the magnetic disturbance
is very small. The first sign of the magnetic disturbance
shown in Fig. 4 arrives at ␶ = 50 ns after the targets are
struck, and corresponds to a velocity of 6 ⫻ 108 cm/ s, which
is larger than the Alfvén speed in helium. Whistler waves,
which can set up currents on time scales faster than the
Alfvén time, are also observed in this experiment and are the
subject of a separate publication.20
The current system evolves into a system in which cur-
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062109-4
Gekelman, Collette, and Vincena
FIG. 5. 共Color兲 Magnetic field at ␦z = 31.5 cm from the lpp acquired at ␶
= 870 ns after the laser is fired. The magnetic field data are recorded at 1 cm
spacing in the x and y 共transverse兲 directions. This is the same grid used in
all the following magnetic field planes. Here the x axis is horizontal and the
y axis is vertical. The background magnetic field points into the page. The
induced plasma current is of order 30 Amperes.
rents and magnetic fields are tangled. Earlier in time, the
current is due to fast electrons emanating from the plasma
bubbles. One such system is shown in Fig. 6 for ␶
= 1.12 ␮s and at ␦z = 94.8 cm. The current density is as large
as 0.55 A / cm2. This should be compared to the electron
saturation current in the background plasma, which is
27 A / cm2. The electron saturation current is the largest current which can be drawn from the background plasma. The
lpp current is a non-negligible fraction of this.
The field topology does not remain as simple as that in
Fig. 5. The two diamagnetic plasma bubbles have collapsed
after approximately 2 ␮s, but the dense plasmas continue to
move across the magnetic field until they collide. Fieldaligned electrons emanate from them and add to the complex
current system. The magnetic field due to the colliding lpp
current systems becomes fully three-dimensional, as illustrated in Fig. 7. Shown are data on three planes taken at ␶
= 10.74 ␮s after the targets were struck. The closest plane to
the targets 共left-hand side兲 shows the magnetic field of a
broad current channel. The current channel is elongated and
ជ on a plane ␦z = 95 cm
FIG. 6. 共Color兲 Plasma currents derived from ⵜ ⫻ B
and at ␦␶ = 1.12 ␮s. The red streamlines indicate electrons streaming away
from the lpp target area. The axial current density Jz is in A / cm2.
Phys. Plasmas 14, 062109 共2007兲
about to split into the two channels in the intermediate plane
while the topology of the magnetic field at ␦z = 285 cm indicates the presence of multiple current channels. This is a
dynamic process. The magnetic field changes on each plane
as a function of time and the data are best appreciated by
viewing a movie of the fields. Interesting features come and
go as the current system evolves.
Figure 8 shows the magnetic field lines on an x-y plane
located 94.8 cm from the targets within the 3D dataset at
time ␶ = 5.25 ␮s. Data were collected using a 1 cm grid spacing and at 16384 time steps at 10 ns intervals. The lines are
started at each position on the rectangular grid and followed
using a Runge-Kutta scheme. The field lines are colored using the local value of the magnetic field. The field lines indicate that there are two current channels 共one on the left and
one on the right of the plane兲 as well as a narrow current
channel in the center of the plane and below a line connecting the center of the two main channels. A magnetic null, or
“X” point topology, is visible above a line connecting the
centers of the two main current channels. The magnetic field
at the “X” point does not vanish because of the background
magnetic field of 600 G. The magnetic reconnection community commonly refers to this as the guide field. If it were
included in Fig. 8, the field lines would be highly elongated
in the z direction. Under these conditions, the helium ion
gyroradius is 3.4 mm and the ion sound gyroradius is
8.3 mm.
Figure 9 is a close-up view of a bundle of magnetic field
lines, which pass through the reconnection region above the
central current channel. In Fig. 9共a兲, which is an end-on
view, the field lines that connect to the right current channel
are visible. Figure 9共b兲 is an alternate view of the plane in
Fig. 9共a兲, which illustrates the three-dimensional nature of
the magnetic field near the reconnection region. The constant
background magnetic field is not included in these figures.
Its addition would highly elongate the structures.
The full spatial morphology of the magnetic field lines at
␶ = 5.25 ␮s is presented in Fig. 10. This is displayed on eight
data planes, each approximately a meter apart along the machine axis. The arrow in Fig. 10 points to the plane shown in
Figs. 8 and 9. The color bar in Fig. 10 shows that at this time
the magnetic field is actually larger farther from the interaction region. The magnetic fields are due to currents originating from the targets that are superimposed on the current
system of Alfvén waves radiated by electrons streaming
away from the targets. The Alfvén waves become the farfield radiation pattern of the lpp collision. At this time, the
complex fields near the target become associated with two
main current channels 共and a smaller one below them as seen
in Fig. 8兲, and these in turn merge into a single current channel. When the currents merge, the current density increases
and the magnetic field around the single current channel increases. The picture is a dynamic one. At other times, the
single current channel breaks up into multiple channels,
which merge in turn.
The frequency of the waves is time-dependent and is
best calculated from a wavelet analysis of the magnetic field
data. This is presented in Fig. 11. The data are acquired from
a single component of the magnetic field, Bx, 284 cm from
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062109-5
Phys. Plasmas 14, 062109 共2007兲
Three-dimensional current systems...
FIG. 7. 共Color兲 Magnetic field on three planes at t = 10.74 ␮s after the targets are struck. The lpp targets are 158 cm to the left of the leftmost plane. The
constant background magnetic field of 600 G is not shown in this picture. The background plasma is helium. The data planes are 29 cm on a side and the
magnetic field is measured at 1 cm spacing in the x and y directions. In this figure, the z axis has been compressed to allow for viewing of the three planes
simultaneously. The transverse planes are 29⫻ 29 cm.
the targets and averaged over the entire plane. Oscillations
are of shear Alfvén waves, some of which last for more than
60 ␮s. The frequency is displayed on a linear scale, and the
ion cyclotron frequency for the background helium plasma
共230 kHz兲 is indicated with a dotted line.
IV. MAGNETIC VECTOR POTENTIAL AND WAVE
ELECTRIC FIELD
The magnetic vector potential is calculated in two steps.
The current is calculated from the three-dimensional magᠬ . The inductive part,
netic field data using ᠬj = 共1 / ␮0兲 ⵜ ⫻ B
ជ / ⳵t兲, is neglected because the highest frequency of the
␧ 0共 ⳵ E
magnetic fields oscillates on the ion cyclotron frequency
time scale 共f ci = 230 kHz, He兲, and the currents are evaluated
long after the magnetic bubbles have vanished. The vector
potential is then evaluated from the currents using
ᠬ 共rជ兲 = ␮0
A
4␲
冕
V⬘
ᠬj共rជ⬘兲dV⬘
兩rជ − rជ⬘兩
.
共1兲
In this calculation, the currents were within a plasma
volume formed by a cross-sectional area of 841 cm2 and a
length of 252.8 cm along the magnetic field. This is by no
means the entire plasma column, but this volume contained
the bulk current generated by the lpp interaction on one axial
side of the targets. Throughout this volume, the ratio
共兩ⵜ · Bជ 兩 / 兩ⵜ ⫻ Bជ 兩兲 ⬍ 5%. In most of the volume it is less than
1%. We neglect the contribution when the current density
approaches the noise level 共5 mA/ cm2兲. The expansion
along the field is nearly symmetric both upstream and down-
stream of the target with respect to the background magnetic
field. The vector potential was used to calculate the induced
electric field as well as the magnetic helicity. Figure 12
shows the axial component of the vector potential, Az, at
␦z = 158 cm. In this case there are four current channels as
illustrated by the presence of the “O” points. The magnetic
“X” point is now in the center, in contrast to the location of
reconnection sites in Fig. 8. The number of current channels
in any data plane varies in space and time as the threedimensional currents merge and break up.
As time goes by, the left and the right current channels
merge and vector potential contours move toward the center
“X” point in from the sides and reconnect in the center. They
then move out to the top and bottom. In many experiments
designed to study reconnection, flux is forced into or out of a
region of space when time-varying currents are sent through
metallic conductors close by, thereby changing the flux in the
plasma.21 In this experiment, the magnetic “O” points are
produced by the current channels of Alfvén waves and electrons generated by the lpp. The flux change in the center
gives rise to an inductive electric field, Ez = −共⳵Az / ⳵t兲. As
Alfvén waves are electromagnetic, the inductive field becomes the far-field electric field of these waves. Ez is derived
from the data and shown in Fig. 13 for the same measurement plane in Fig. 8, ␦z = 94.8, where there are three current
channels. Note that the inductive field shown in Fig. 13共a兲
does not have three current channels or any evidence of neutral sheets. Figure 13共b兲 shows the inductive field 3.42 ␮s
later and has a striking similarity to the topology of Fig. 8.
The Alfvén transit time over 94.8 cm in the background
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062109-6
Phys. Plasmas 14, 062109 共2007兲
Gekelman, Collette, and Vincena
FIG. 8. 共Color兲 Projection of magnetic field lines onto the plane centered ␦z = 94.8 cm from the targets. The data were derived from vector magnetic field
measurements taken on five planes at distances ranging from 31.5 to 284.4 cm from the targets. The time is ␶ = 5.25 ␮s after the targets are struck. Magnetic
field data in each plane were acquired at 900 spatial locations on a rectangular grid with 1 cm spacing in the x and y directions. 共The scale is the same as that
in Fig. 5.兲 The background magnetic field 共which goes out of the plane of the figure兲 is not included. The field lines are colored according to the local magnetic
field in G. Two reconnection regions are visible above and below the central current channel. The field lines are three-dimensional, so some appear to cross
each other in this view.
plasma is 2.63 ␮s. One cannot pinpoint the time of arrival of
the correct topology from the data, but it strongly suggests
that the inductive fields are swept through the plasma by
Alfvén waves.
The time rate of change gives a parallel electric field of
order 2.5 V / m in the “reconnection” region 关Fig. 13共b兲, x
⯝ −2 , y ⯝ 7兴. This is, of course, much smaller than the
Dreicer runaway electric field22 of 350 V / m. Furthermore,
the displacement current estimated using the temporal
change of the highest induced electric fields in Fig. 13 is
orders of magnitude lower than the currents carried by
charges. We can compare Ez to the axial electric field of
Alfvén waves. For a rough check, we use the expression for
the relationship between the parallel and perpendicular electric fields for a single mode of a kinetic shear Alfvén wave,
ik储k⬜␳s2
E储
=
,
E⬜ 共1 − ␻
¯ 2兲
¯⬅
␻
␻
,
␻ci
␳s =
cs
.
␻ci
共2兲
Here cs is the ion sound speed and ␳s is the ion sound gyroradius.
The Alfvén speed in the background He plasma is VA
⯝ 共E⬜ / B⬜兲 ⬵ 3.6⫻ 107 cm/ s, where the subscript refers to
the wave field. The wave fields in the data plane of Fig. 6 are
B⬜ ⬇ 10−4 T, E⬜ ⯝ 36 V / m. The dominant wavenumbers determined by a spatial Fourier transform of the magnetic field
corresponding to Fig. 8 are k储 = 0.024 cm−1. Using Eq. 共2兲,
the parallel wave electric field is of order 4 mV/ cm. The
inductive electric field is by no means the total electric field.
The electric field in the fluid approximation is given by a
generalized Ohm’s law,
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062109-7
Three-dimensional current systems...
Phys. Plasmas 14, 062109 共2007兲
FIG. 9. 共Color兲 A field line bundle
composed of those field lines that pass
near the upper reconnection region
shown in Fig. 8. The color bar is the
same as that in Fig. 8. The background
magnetic field is in the z direction 共but
is not included in this bundle兲. 共a兲 is a
global view looking away from the
cathode 共plasma source兲. To help grasp
the three-dimensional nature of the
image, a semitransparent plane with
white grid lines on a black surface is
placed at ␦z = 95 cm. Because of the
transparency, field lines appear grayed
if they are beneath the grid lines and
are darker beneath the open areas.
Field lines above the plane are unaffected. The gray grid lines are spaced
at 5 cm intervals and are centered at
␦z = 95 cm. Data were acquired at
1 cm intervals everywhere in the
transverse plane. Note in 共b兲 that the
field lines span both sides of the grid,
and near the reconnection point have a
substantial out-of-plane component.
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062109-8
Phys. Plasmas 14, 062109 共2007兲
Gekelman, Collette, and Vincena
FIG. 10. 共Color兲 Three-dimensional magnetic field lines. The 600 G background field is not included in the calculation. The arrow points to the data plane
highlighted in Fig. 7. Twisting vortex-like structures near the targets eventually merge and settle down to the relatively simple patterns of torsional 共shear兲
Alfvén waves.
ជ = Bជ ⫻ uជ + ␩Jជ + 1 Jជ ⫻ Bជ − 1 ⵜ P.
E
ne
ne
共3兲
We have not measured most of the terms in 共3兲. In fact, if the
terms were known it would be possible to use 共3兲 to deter-
FIG. 11. 共Color兲 Bx共z = 285 cm, t兲. Contour plot of the wavelet transformation of the time evolution of the frequencies 关By共␻ , t兲兴 averaged over the
data plane at z = 285 cm. The ion cyclotron frequency for helium, f ci, is
shown as a dotted line.
mine the plasma resistivity, ␩. The resistivity is generally not
given by classical Coulomb collisions.
In a previous experiment,23 in which shear waves with
magnetic fields of one-tenth the magnitude of the fields observed in this experiment were launched with an antenna, the
parallel wave field inferred from LIF measurements was E储
⯝ 0.4 mV/ cm.
At the later time 共␶ = 8.67 ␮s兲 and in the center of the
upper reconnection region in the data plane shown in Fig.
13共b兲, the parallel electric field derived from ⳵Az / ⳵t is
270 mV/ cm, k⬜ = 0.65 cm−1, and the parallel electric field
predicted by Eq. 共2兲 is 2.6 mV/ cm. Much later in time 共␶
= 86 ␮s兲, Alfvén waves are still present with magnetic fields
of order 60–100 mG. The parallel electric field from ⳵Az / ⳵t
is of order 5 mV/ cm, much closer to that of the wave field.
If one uses the Spitzer resistivity and current densities at ␶
= 86 ␮s, the predicted parallel field is 0.5 mV/ cm. One concludes that the initial electron currents are entirely due to the
fast electrons from the target plasmas and by inductive electric fields from reconnection. Return currents are set up by
the plasma to space charge neutralize the plasma blobs. The
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062109-9
Phys. Plasmas 14, 062109 共2007兲
Three-dimensional current systems...
FIG. 12. 共Color兲 The magnetic vector potential Az in the vicinity of the “X” point taken at z = 158 cm. Figure 11共a兲 is a global view taken at ␶ = 9.8 ␮s. The
data plane, as the others, is 29 cm on a side. The three successive frames are taken at temporal intervals delayed by ␦␶ = 10 ns from that in Fig. 10共a兲. A portion
of a field line is colored magenta to illustrate a “reconnection” event. The laser is fired at ␶ = 0.
currents close to the lpp become near-field Alfvén wave current systems on a time scale of order of an ion gyroperiod
共4.4 ␮s兲. Several ion gyroperiods later and at distances
greater than a parallel Alfvén wavelength 共1.6 m兲, the electric fields derived from the measured magnetic field agree
reasonably well with theoretical prediction.
Reconnection mediated by the Alfvén wave current systems is not a key process after the collision of the lpp
bubbles. That is, it does not dominate the physics of this
interaction. There may be local bursts of particles and other
disturbances connected to the reconnection “nano flares” but
they are not a driver of secondary processes that could drastically change the state of the plasma. Although the reconnection sites are intriguing and fully three-dimensional, reconnection is a trivial consequence of the current systems
and the magnetic field topology.
V. ALFVÉN WAVES
It is instructive to examine the spectra of the magnetic
fields generated in the experiment. The temporal development of the x component of the magnetic field, derived from
a wavelet transformation of the data, is illustrated in Fig. 14.
One feature of the spectra is the long-lived component
below the ion cyclotron frequency for both locations. These
are shear Alfvén waves similar to what has been observed in
previous, single target lpp experiments. Closer to the target,
the wave spectrum extends above the ion cyclotron frequency at early times. The higher frequencies at ␶ ⱕ 10 ␮s
are associated with evanescent compressional Alfvén waves.
The compressional wave propagates symmetrically across
and along the background field with dispersion ␻k = VA. For a
wave frequency of 1.5 times the ion cyclotron frequency, the
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062109-10
Phys. Plasmas 14, 062109 共2007兲
Gekelman, Collette, and Vincena
FIG. 13. 共Color兲 共a兲 Inductive part of the electric field derived from the vector potential at ␶ = 5.25 ␮s, z = 94.8 cm. The dark blue areas indicate the highest
inductive field 共28.5 V / m兲 and are in the center of the current channels. The dashed contours are negative. 共b兲 The inductive fields at the same location
3.42 ␮s later. The pattern closely resembles the field lines of Fig. 8.
wavelength is 1.35 m, which is more than twice the column
diameter. The compressional wave is therefore cut off at this
frequency and propagates only above three times f ci. Wavelet
transforms of the Bz component have been performed and are
in accord with this. The dispersion relation for the shear
wave under these conditions is given by24
␻2
2 2
= VA2 共1 − ␻2 + k⬜
␳s 兲.
k2储
共4兲
Here cs is the ion sound speed 共cs = 1.2⫻ 106 cm/ s兲 and ␳s is
the ion sound gyroradius 共␳s = 5 mm兲. From Eq. 共4兲 it is apparent that the wave phase velocity parallel to magnetic field
becomes slower as the wave frequency approaches the ion
gyrofrequency. The scalloped shape of the spectra below f
= f ci is discernible in Fig. 14共a兲. The higher frequency waves
are less evident at greater distance from the interaction region, as is evident in Fig. 14共b兲. According to the dispersion
relation 关Eq. 共4兲兴, the lower-frequency waves travel faster
and therefore outrun the higher-frequency components. The
long-lasting component at 0.25 f ci is evident in panel 共a兲, but
at the farther location all of the higher-frequency components
have been damped away, although a burst of lower frequencies is seen at early times. This effect was also observed in a
microwave experiment in which a spectrum of Alfvén waves
was produced in the interaction.15 Most of the Alfvén wave
activity has ceased 40 ␮s after the target was struck; it is all
gone after 70 ␮s.
The Alfvén waves are initially generated by Cherenkov
emission from fast electrons produced when the targets are
struck by the laser. Alfvén waves were produced by the same
mechanism in an experiment in which high-power microwaves were focused on a resonant region in the LAPD density gradient.25 The shear wave has a current system in which
field-aligned currents are carried by electrons and cross-field
currents carried by ions via the ion polarization drift.14 The
return current of the wave also serves to neutralize the space
charge on the laser-produced plasma due to the escaping fast
electrons. Below the cyclotron frequency, all information on
changing plasma currents is broadcast by Alfvén waves in a
magnetoplasma. In effect, the complicated magnetic fields
and associated current system presented in the previous section are those of the Alfvén waves.
VI. MAGNETIC HELICITY
The time evolution of the magnetic fields is one in which
field lines become intertwined. This suggests that the magnetic helicity in this experiment varies in time. The magnetic
helicity is defined by26
FIG. 14. 共Color兲 Wavelet transform of the x component of the magnetic
field at 共x , y , z1兲 = 共0 , 0 , 32.5兲 in frame 共a兲 and 共x , y , z2兲 = 共0 , 0 , 94.8兲 in the
lower panel. The transform was taken with Morlet basis vectors and it enables one to view the time dependence of the spectra. The time is referenced
to when the two targets are struck. The targets are located at z = 0. Both
contour maps are on a linear scale. The amplitude in 共b兲 is multiplied by 2
relative to 共a兲 to bring out features as the wave amplitude is smaller farther
away. The largest wavelet amplitude is colored red; zero amplitude is dark
blue.
H=
冕
ᠬ ·B
ᠬ dV.
A
共5兲
V
This expression consists of two components, H
ᠬ ·B
ᠬ
= 兰 VA
wavedV + B0兰VAzdV, where B0 is the constant background magnetic field. The fields are integrated over a volume which includes all currents and is bounded by a flux
surface. In this experiment, the fields were not measured
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062109-11
Phys. Plasmas 14, 062109 共2007兲
Three-dimensional current systems...
FIG. 16. 共Color兲 Wavelet spectra of magnetic helicity in the measurement
volume previously described. Most of the magnetic entanglement is due to
Alfvén waves at 0.6f ci. The spectra are on a linear scale with red representing the highest value.
FIG. 15. Magnetic helicity in a volume of 2.12⫻ 105 cm3 where magnetic
field data were acquired. The laser is fired at ␶ = 0. 共a兲 is the magnetic
ជ · Bជ dV in which only the wave magnetic field is used in the intehelicity 兰A
ជ = Bជ
gral, and 共b兲 in which B
wave + B0z.
throughout the entire volume of the LAPD. The temporal
development of the magnetic helicity derived from the data
over a limited volume is shown in Fig. 15. Figure 15共a兲
contains the first term in which the magnetic field is that of
the Alfvén waves. The second part of Fig. 15 shows the sum
of both terms in Eq. 共5兲. The vector potential Az is due to the
waves and the second term is therefore wave-dependent as
well. In either case, the magnetic helicity is observed to oscillate. The largest helicity occurs in the first 20 ␮s of the
experiment when the Alfvén wave activity is greatest. Examination of Fig. 10, acquired 5.25 ␮s after the targets are
stuck 共and when the helicity in Fig. 10 is large兲, shows that
especially near the target the field lines are fully threedimensional and can link. The helicity dissipates over 70 ␮s
as the waves and their associated tangled currents disappear.
The dissipation is both due to electron Landau damping and
electron-ion Coulomb collisions.27 Although helicity can
change from flux linked to twist or writhe helicity,28the expression for it is given by Eq. 共5兲 in any event.
The magnetic helicity density is defined by
ᠬ ·B
ᠬ.
h=A
It is clear that in this experiment, the helicity is not conserved. There is no explanation, at this time, of its oscillatory
behavior. It is possible that helicity left and then returned the
plasma volume that was probed, and if measurements were
made throughout the entire device the helicity would not
oscillate, but simply decay. However, the frequencies of the
helicity oscillation correspond to those observed for the
shear Alfvén waves. It is possible that the field lines, which
are associated with the 3D current system of the Alfvén
waves, repeatedly tangle and untangle during the course of
the experiment.
As the magnetic helicity is both time- and frequencydependent, its behavior can also be displayed using a wavelet
spectra. This is shown in Fig. 16.
Figure 16 may be compared to Fig. 17, which is a graph
of the magnetic energy of the waves, WB = 兰 B2 2␮0 dV, in
the measurement volume. It contains the same oscillatory
behavior seen in the helicity.
The measurement volume is smaller than the experimental volume. It is possible that if the entire plasma volume
were mapped, the helicity would be conserved. Figure 16
strongly suggests that the helicity is associated with the
Alfvén waves. For simple waves generated by fluctuating
currents with cross field scale on the order of the skin
depth,30 the wave currents and wave magnetic fields are orthogonal and there is no helicity. The wave has a current
Ⲑ
共6兲
The temporal rate of change of the helicity density is governed by29
⳵h
ᠬ ⫻A
ᠬ + ␾B
ᠬ兲 = − E
ᠬ ·B
ᠬ.
+ ⵜ · 共E
⳵t
共7兲
FIG. 17. Magnetic energy, B2 / 2␮0, of the Alfvén waves in the measurement
volume.
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062109-12
Phys. Plasmas 14, 062109 共2007兲
Gekelman, Collette, and Vincena
parallel to the background magnetic field and a return current
ជ and ជj are parallel, there is no net
antiparallel to it. Since A
helicity. In the present experiment, the wave magnetic field
spirals as well as the wave current and the waves can carry
helicity.
The Alfvén wave velocity in the background He plasma
is 3.6⫻ 107 cm/ s. The wave transit time to the closest end of
the device is 19 ␮s, which is far too long for a reflected wave
to contribute to the helicity exchange between the wave and
fast electrons.
The maximum magnetic energy associated with the
waves in the measurement volume is 0.3 mJ. Because the
waves propagate upstream and downstream from the target,
one can approximate the total wave energy in the device as
0.6 mJ. The energy of the laser was 1.5 J with roughly half
going into kinetic energy of particles;31 therefore, of order
0.1% of the particle energy has gone into Alfvén waves in
this volume, which is only 6% of the plasma volume. The
entire energy content of the background helium plasma is
7.8 J. The measurement does not include the larger expelled
field of the plasma bubbles or the weaker wave fields at
distances more than 2.5 m away. The energy also exhibits a
100 kHz oscillation 共nearly half the ion gyrofrequency兲 in
the region where the helicity peaks.
VII. SUMMARY AND CONCLUSIONS
Two carbon laser-produced plasmas with initial density
that is orders of magnitude greater than an ambient helium
plasma were produced such that they collided in the center of
the plasma column. The collision process occurs in two
stages. In the first, two magnetic “bubbles” collide. The collision produces turbulent local magnetic fields. The morphology and wavenumber spectrum of this turbulence will be the
subject of a future paper. The collision of the lpps results in
an early magnetic reconnection event 共Fig. 4兲. This creates a
short-lived induced electric field parallel to the background
magnetic field and drives a large field aligned current. After
about a microsecond, the magnetic bubbles collapse but the
cross-field collision continues. In the final state, the two plasmas coalesce and stream along the magnetic field away from
the impact point.
The creation of the lpp is accompanied by jets of highenergy field aligned electrons. The electrons cause highfrequency waves in the lower hybrid and whistler regimes,
which will be discussed in detail in another publication. In
the first microsecond of the process, it seems that the initial
electron current is driven by magnetic field-line reconnection. A portion of the electron velocity distribution matches
the parallel phase velocity of shear Alfvén waves. After the
fast electrons radiate whistlers, they Cherenkov radiate
Alfvén waves. These are the waves seen from ␶ = 1 to 60 ␮s
after the targets are struck. The current system of the Alfvén
waves is fully three-dimensional and exhibits coalescence
and splitting of the currents in space and time. Recently,
three-dimensional effects have shown up in a reconnection
experiments with a very different geometry.32 In the SSX
experiment, reconnection is triggered in the coalescence of
two toroidal spheromak plasmas. Magnetic field lines close
to the center of a planar magnetic “X” point were observed
to wander out of the plane containing it. Three-dimensional
effects have also been observed in experiments where magnetic flux ropes interact and coalesce.33,34 Transitory reconnection regions were observed to exist between the flux
ropes. In this experiment, magnetic field line reconnection
sites pepper the plasma volume and are identified with the
Alfvén wave structures. In the experiment, the reconnection
is not driven by forcing together magnetic flux derived from
internal magnetic coils. The reconnection in this experiment
is self-consistent with the three-dimensional current systems
that are generated when the lpp dense plasmas form and
move. The currents are those of Alfvén waves in the background helium plasma. The background plasma is essential
in that it provides return current. Although some magnetic
flux is dissipated in the reconnection regions, it does little to
effect the overall energy balance. The waves also transport
the inductive component of the electric field as illustrated in
Fig. 13. The magnetic helicity is derived from the data and is
observed not to be conserved.
ACKNOWLEDGMENTS
The authors would like to thank Shreekrishna Tripathi,
Pat Pribyl, Bill Heidbrink, and George Morales for their useful comments. We are also indebted to Marvin Drandell, Mio
Nakamoto, and Zoltan Lucky for their expert technical
assistance.
The work was supported by the DOE-NSF partnership in
plasma physics under NSF Grant No. PHY–0408226. The
work was performed on the LAPD device, which is maintained as a Basic Plasma Science Facility under a DOE-NSF
cooperative agreement. The research support of A.C. was
performed under appointment to the Fusion Energy Sciences
Fellowship Program administered by Oak Ridge Institute for
Science and Education under Contract No. DE-AC0506OR23100 between the U.S. Department of Energy and the
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