PHYSICS OF PLASMAS 18, 055705 (2011)
Structure of an exploding laser-produced plasmaa)
A. Collette1,b) and W. Gekelman2
1
Laboratory for Atmospheric and Space Physics, UCB 392, Boulder, Colorado 80309, USA
Department of Physics and Astronomy, University of California, Los Angeles, Los Angeles,
California 90095, USA
2
(Received 24 November 2010; accepted 15 February 2011; published online 15 April 2011)
Currents and instabilities associated with an expanding dense plasma embedded in a magnetized
background plasma are investigated by direct volumetric probe measurements of the magnetic field
and floating potential. A diamagnetic cavity is formed and found to collapse rapidly compared to
the expected magnetic diffusion time. The three-dimensional current density within the expanding
plasma includes currents along the background magnetic field, in addition to the diamagnetic
current. Correlation measurements reveal that flutelike structures at the plasma surface translate
with the expanding plasma across the magnetic field and extend into the current system that
C 2011 American Institute of
sustains the diamagnetic cavity, possibly contributing to its collapse. V
Physics. [doi:10.1063/1.3567525]
I. INTRODUCTION
The expansion of dense plasmas across magnetic fields is
a topic of intense interest across a wide variety of disciplines
within plasma physics, with applications to solar,1 magnetospheric,2,3 and astrophysical4 phenomena. In particular, subAlfvénic expansions with length scales comparable to an ion
gyroradius are known to exhibit complex behavior, including
the formation of current systems that locally reduce the background field (a diamagnetic cavity or “magnetic bubble”) and
large [ðDn=nÞ 1] field-aligned density striations. Spaceborne chemical releases2,3 to study plasma expansion within
Earth’s magnetosphere found large-scale structuring and wave
activity.5 Fast photography of expanding plasmas in laboratory experiments6–8 found flute modes that grow rapidly at the
plasma surface; the origin of these instabilities has been the
focus of theoretical investigation.9,10 Laboratory work also
indicates that an expanding plasma can transfer energy to a
background plasma in the form of Alfvén waves,11,12 and that
the presence of a surrounding plasma may affect how the
expansion evolves.8
The behavior of sub-Alfvénic expansions has been studied under a variety of experimental conditions. Perhaps the
most dramatic experiment was conducted during March and
May of 1985 as part of the Active Magnetospheric Particle
Tracer Explorers (AMPTE) effort; in an effort to construct
and measure an “artificial comet,” two separate barium
releases were carried out in Earth’s magnetotail and
observed via ground-based (photographic) and space-borne
diagnostics. The barium was released with an initial (radial)
speed and photoionized rapidly compared to the time scale
of the experiment, resulting in a plasma shell expanding
across the magnetic field. At the surface of the expanding
plasma, a current system formed that reduced the magnetic
field within the plasma, a configuration referred to as a
“magnetic bubble” or diamagnetic cavity. While the cavity
a)
Paper GI3 2, Bull. Am. Phys. Soc. 55, 109 (2010).
Invited speaker.
b)
1070-664X/2011/18(5)/055705/8/$30.00
formation was expected, photography revealed unexpected
field-aligned density striations at the plasma edge, accompanied by oscillations in the magnetic field.5 The nature of
these structures became the subject of experiments at NRL,9
LLNL,6 and multiple theoretical works. A common feature
of experiments that exhibit this type of structuring is that the
diamagnetic cavity collapses quickly compared the magnetic
diffusion time s l0 rR2 (where r is the plasma conductivity and R the cavity radius). The effect of both microturbulence and the observed large-scale structures on the collapse
of the cavity remains an open question.
An additional question is the motion of the structures; in
other words, whether they are static structures or waves travelling along the plasma surface. In the experiment described
by Ripin et al.,13 a gated optical imager (GOI) was used to
observe the plasma at four different times during the same
shot. The images showed that, during the nonlinear phase of
the instability, the flutes acquire a curl in the electron diamagnetic direction. It could not be determined from that
experiment whether the observed curl was related to rotation
of the flutes along the cavity surface or movement of the
flute tips, although witness plates7 placed close to the expansion produced nonblurred impressions, suggesting that the
rotational speed, if any, was small considered on the time
scale of the expansion. The aims of this work with respect to
the surface instabilities are therefore to determine how they
evolve over time, including any rotational behavior, and to
investigate any effect they have on the dynamics of the diamagnetic cavity and its collapse.
Another topic that has received limited study is the structure of the currents, which form within the diamagnetic cavity.
Previous laboratory experiments have observed cavity formation and associated currents perpendicular to the background
magnetic field. However, the three-dimensional (3D) arrangement of this current system has never been directly measured.
We know from work by van Zeeland et al.14,15 that when the
expansion takes place within a magnetized plasma, background
electrons travel along field lines into the expanding plasma,
shorting out space-charge fields formed by the expanding ion
18, 055705-1
C 2011 American Institute of Physics
V
055705-2
A. Collette and W. Gekelman
Phys. Plasmas 18, 055705 (2011)
cloud. Energetic electrons, which in a vacuum would otherwise
remain confined in the expanding plasma by space-charge
effects, can escape. The result is a current system that extends
into the background on scales large compared to the expanding
plasma, which after a brief period becomes the current system
of a shear Alfvén wave. Although these wave currents span
time scales longer than the lifetime of the diamagnetic cavity,
significant electron currents parallel to the field exist outside
the cavity on the expansion timescale, hinting that the current
system within the plasma may have three-dimensional
structure.
II. EXPERIMENTAL SETUP
The Large Plasma Device (LaPD) (Ref. 16) provides a
unique experimental environment in which to study the phenomenon of plasma expansion within a magnetized background. LaPD provides a physically large magnetized
plasma compared to the expanding plasma, which can additionally support extended current systems such as the Alfvén
waves observed in Ref. 14. It also supports high-rep-rate
experiments (1 Hz), which vastly simplifies the use of probe
diagnostics, enabling the detailed measurement of quantities
in 3D using a small number of movable probes. The highly
reproducible nature of the background plasma makes LaPD
particularly suited to this form of investigation.
We investigate the cross-field expansion of a laser-produced plasma via direct volumetric probe measurement of the
magnetic field, along with fast photography. Figure 1 shows
the experiment configuration. The coordinate system used
here has the laser spot at ðx; y; zÞ ¼ ð0; 0; 0Þ; z points toward
the LaPD cathode. The cathode is coated with barium oxide to
a diameter of 60 cm; a 1 Hz pulsed bias between it and a
mesh anode 50 cm away produces the LaPD background
plasma. A uniform 600 G magnetic field oriented along þ^z
fills the device. Within the LaPD environment, we ablate a
solid carbon target with a 1.2 J, 10 ns laser pulse. The laser
enters the machine in the þx direction, resulting in a plasma
that expands across the background field B0 ¼ B0 ^z in the x
direction.
The expanding plasma is observed by an intensified
charge-coupled device (ICCD) camera outside the end window in Fig. 1, viewing in the þz direction. Figure 2 shows a
time series of the expanding plasma, composed of photographs taken during successive experiments at increasing
FIG. 2. Fast photographs of expanding plasma (10 ns shutter speed). Quantity
displayed is log-scaled intensity. Each frame is from a different LaPD shot.
time delay. Probe measurements are accomplished by one
three-axis magnetic-coil probe and one four-tip Langmuir
probe configured to measure the ac floating potential. Both
were mounted on an in-plasma probe drive17 capable of sampling cutplanes in XY. The Z-location is adjustable by means
of a sliding plate upon which the probe drive system is
mounted. Both probe heads are 1.5 mm in size, compared to
a maximum cavity diameter of about 4 cm.
III. THREE-DIMENSIONAL CURRENTS
As discussed in Ripin et al.13 and elsewhere, an expanding
plasma in the regime of interest creates a local depletion in the
magnetic field, as the initial kinetic energy of the laser ions
expanding outward in r is transferred into a current system in h
[equivalently,
into
the
magnetic
field
energy
Ð
EB ¼ ð1=2l0 ÞjBj2 dV]. This process sets natural time and
length scales for the system, which are the interval between
laser incidence and the time of peak EB (the “time to peak diamagnetism” sd ) and the observed radius of the magnetic
depression at this time RB . An estimate for the relation between
the speed v? of the outgoing ions and the cavity radius R can
be derived from energy balance arguments, assuming initial
energy U0 (with no losses over time) and that the entire background field B0 is depleted across a spherical region:
1 2
2pB20 3
Mv? ðtÞ þ
R ðtÞ ¼ U0 ;
3l0
2
FIG. 1. Experimental setup.
(1)
ð2=3Þ
which implies a maximum Rðsd Þ ¼ ð3l0 U0 =2pÞ1=3 B0
.
055705-3
Structure of an exploding laser-produced plasma
Phys. Plasmas 18, 055705 (2011)
FIG. 3. (Color online) Z-component of magnetic field over time
(z ¼ 10 mm).
The magnetic field BðtÞ of the cavity is investigated by
direct probe measurement, using the magnetic-coil probes and
microprobe drive system. Data were acquired in a series of 2D
cutplanes oriented perpendicular to the background magnetic
field, forming a volumetric survey of the expanding plasma.
At each spatial location, an ensemble of timetraces recording
dB=dt were collected. These timetrace ensembles were then
integrated, averaged together, and combined to form 2D maps
of Bðx; y; tÞ. With the laser focus located at ðx; y; zÞ
¼ ð0; 0; 0Þ (plasma expanding in the x direction), cutplanes
of B were acquired at z ¼ 10, 15 and 20 mm. The volume of data acquired forms a rectangular slab with limits
(in millimeters) of 10 x 55, 20 y 20, and 20
z 10. The resolution of the data in X and Y is 1 mm.
Figure 3 shows the effect of the diamagnetic cavity, in
the cutplane located at z ¼ 10 mm. A series of timesteps is
shown over the course of the expansion. The cavity initially
expands in x and y, reaches a maximum radius, and begins to
translate across the field. A vertical asymmetry is evident in
the magnetic field; the width of the field gradient at the bottom
of the cavity is several times greater than at the top. A field
enhancement is also visible outside of the expanding cavity.
Measurement of the time-varying three-component magnetic field Bðx; y; z; tÞ over a volume allows the direct computation of the current density Jðx; y; z; tÞ ¼ 1=l0 ðr BÞ.
Figure 4 shows (a) B and (b) the derived J in the three measured cutplanes, at the time of peak diamagnetism
sd ¼ 470 ns. The dominant feature in (a) is the expelled
magnetic field within the cavity, reaching a peak value of
approximately 60% of the 600 G background magnetic field.
FIG. 4. (Color online) Magnetic field (a) and current density (b) in the three
cutplanes measured through the expanding plasma, at the time of peak diamagnetism (470 ns).
Likewise, the current across the background field in (b) is that
which sustains the magnetic depletion. However, careful examination of the current density reveals that sizable currents
also exist along the background field. Figure 5 shows the
z-component of the current density at z ¼ 10 mm and
t ¼ sd . The black contour marks Jz ¼ 0. The data show
that currents exist along z with magnitude comparable to
those which sustain the diamagnetic cavity (on the order of
300 A/cm2 in z vs 600 A/cm2 in x y). In addition, the direction of these currents flips as one moves from the center of
the cavity (^z), through the diamagnetic current layer (þ^z),
to the outside (^z again). The result is a helical current system, shown in Fig. 6.
IV. CROSS-FIELD MOTION AND ENERGY
It is possible to track the edge of the diamagnetic cavity
as it moves across the field, both by photographic observations
with the fast camera and by the magnetic-field data. For the
magnetic-field measurements, the definition of the cavity edge
is that used in Ref. 18; a line is drawn tangent to the magneticfield profile Bz ðxÞ at the inflection point and the point where
this line crosses Bz ¼ 0 is considered the edge of the cavity.
055705-4
A. Collette and W. Gekelman
Phys. Plasmas 18, 055705 (2011)
FIG. 7. Cross-field speed, as measured from fast photography (crosses) and
magnetic-field measurements (solid line). Vertical line marks sd ; dashed line
shows deceleration of 1:5 1013 cm/s2.
FIG. 5. (Color online) Z component of the current density in the cutplane at
z ¼ 10 mm, at the time of peak diamagnetism (470 ns). Black contour
marks Jz ¼ 0.
Figure 7 shows the cross-field speed v? computed from
the magnetic-field time series, compared to the speed vx of
the leading edge of the plasma computed from fast photography. Since the magnetic-field data span a greater time window than the fast photography, it is possible to estimate the
speed at earlier times. The peak initial speed of the cavity
edge is 1:0 107 cm/s. This value agrees well with the value
of 8 106 cm/s computed from fast photography, as
expected since the outer edges of the ion shell must be
located outside the edge of the cavity. The behavior of v?
over time broadly follows that of the leading edge of the
expansion. Interestingly, the speed is approximately constant
at 1 107 cm/s from 150 to 300 ns (0:3sd to 0.6 sd ), at
which point it declines, following the plasma-edge speed.
The concept behind the scaling relation of Eq. (1) is the
transfer of energy from the expanding laser ions into the magnetic field; volumetric measurement of the magnetic field
allows direct computation of the energy stored in jBj2 as a
function of time. The three cutplanes represent a grid of magnetic-field measurements Bðx; y; zÞ; an estimate for the magnetic-field energy in the measured region is assembled by
summing over
P each cell of volume dV to obtain EB ðtÞ
¼ ð1=2l0 Þ x; y; z jBðx; y; z; tÞj2 dV .
Figure 8 shows this quantity, computed from all three
components of B over the volume sampled by the probe.
While this does not represent the full axial extent of the cavity, it is assumed to be a representative section that captures
the time behavior of the energy. An interesting feature of
Fig. 8 is the rate of decrease in magnetic energy after sd . Faraday cup measurements by van Zeeland et al.15 suggest that
for an ablation of this energy, on the order of 1015 carbon
ions are ejected, which yields a lower limit on the density
(assuming they uniformly fill a sphere with 2 cm radius at
sd ) of n ¼ 3 1013 cm3 . Conservatively estimating the
Spitzer resistivity with an electron temperature of 10 eV
(roughly twice the background temperature) yields
g ¼ 1:8 105 X m. The time expected for the magnetic
FIG. 6. (Color online) Three-dimensional currents (streamlines) within the
diamagnetic cavity. Target is shown at right; laser is incident from the front.
Magnetic field is shown (red arrows) along with an isosurface of
Bz ¼ 290G ¼ 0:48B0 .
Ð
FIG. 8. Magnetic energy EB ¼ 1=2l0 jBj2 dV associated with currents in
the expanding plasma as a function of time, from the volume sampled by the
probe. Dashed lines show measurement uncertainty.
055705-5
Structure of an exploding laser-produced plasma
field to diffuse one cavity radius (tdiff ¼ l0 R2 =g) is therefore
27 ls, much longer than the observed decay time tdecay of
about 400 ns. Likewise, assuming an average current of
J ¼ 600 A over the 400 ns time of collapse from a peak
energy of 5 mJ, we can estimate the effective resistivity
g ¼ 2 103 X m, nearly 2 orders of magnitude larger than
that expected.
V. INSTABILITIES
A striking feature visible in Fig. 2 is the formation of
structures at the surface of the expanding plasma. The nature
of these structures and their relationship to anomalously fast
magnetic-field penetration into the cavity is the subject of
ongoing study; an example can be found in Ripin et al.13 We
investigate this phenomenon experimentally, first by fast
photography and then by direct measurement via movable
probes, which measure magnetic field and ac floating
potential.
The ability to collect large ensembles of images with the
fast camera at a particular time delay allows the use of statistical techniques on the photographs collected. Figure 9
shows the variable part of each image, formed by subtracting
the average intensity of the ensemble at each timestep from a
single photograph. In other words, first let Pi ðtÞ be the ith
photograph of N, all taken at time t. Then, the quantity displayed in Fig.
P 2 is P0 ðtÞ, and that displayed in Fig. 9 is
P0 ðtÞ 1=Nð N1
i¼0 Pi ðtÞÞ. Consequently, only the structures
that vary from shot to shot are visible. This technique makes
FIG. 9. Variable component of fast photographs vs time.
Phys. Plasmas 18, 055705 (2011)
visible the extent of the flutes, which extend into the plasma
shell far beyond the outer boundary visible in Fig. 2.
Signals associated with the cavity expansion were measured directly with probes. The configuration is largely identical to that used for the magnetic field and current density
measurements; the in-plasma probe drive system discussed
earlier measures both magnetic field and floating potential
versus time at successive locations within a 2D cutplane
located at z ¼ 10 mm. An additional reference probe
records only the floating potential versus time.
The principal drawback of the fast photography as
employed here is that it cannot follow specific structures
over time. Probe signals have the opposite problem; they
have very good time resolution (up to the probe limit of
about 50 MHz in this experiment), but can only measure signals at a point. The behavior shown in Figs. 4 and 5 is composed from the ensemble average over many timetraces
collected at each spatial location. However, the individual
timetraces display oscillatory behavior, an example of which
is shown in Fig. 10. These oscillation bursts vary in both frequency and phase with respect to the start time of the experiment. Consequently, they are not visible in the ensembleaveraged data.
The oscillations are therefore investigated using a probeto-probe correlation technique, similar to that used in Refs. 19
and 20. The mechanism is the investigation of the phase
between signals on two probes, one fixed and one moving, as
the moving probe samples a 2D plane in XY. Even if the process that creates signals on both probes does not have constant
phase with respect to the start time of the experiment, the relative phase between the signals on the two probes should
depend only on their separation. In this way, a 2D map can be
constructed of the phase patterns associated with whatever
process being observed. If the signals are first Fourier-decomposed, it is possible to investigate spatial patterns associated
with behavior at different frequencies.
Denoted as Ai; k ðx; yÞ and Bi; k ðx; yÞ, the Fourier coefficients for the ith trace in the ensemble of N, collected by the
moving and fixed probes, respectively, when the moving
probe samples the location (x, y). The average of the ensemble has already been removed from each trace before the
FFT is computed, as with the photographs above. Then, for
FIG. 10. Oscillation bursts.
055705-6
A. Collette and W. Gekelman
Phys. Plasmas 18, 055705 (2011)
our ensemble of N pairs of timetraces, each of length T, the
quantity
N 1 1X
2
jAi; k jjBi; k jeið/k; A /k; B Þ
GAB; k ¼
(2)
N i¼0 T
is an estimate for the cross-spectrum at frequency index k.
Information on the 2D structure of the process lies in the
phase difference /k; A /k; B considered over the x y plane
sampled.
This correlation procedure was carried out for the case
in which both probes A and B measured the ac floating
potential. Figure 11 shows the region sampled. The X and Y
resolution was 1 mm; an ensemble of N ¼ 40 timetraces
from each probe was collected at each spatial location. The
reference probe was located at z ¼ 13; x ¼ 27; y ¼ 20.
The 2D spatial patterns associated with the first six nonzero
frequencies at the time of peak diamagnetism (t ¼ 470 ns)
are displayed; the quantity plotted is the real part of the
cross-spectral function jGAB ðx; yÞj cos ðhAB ðx; yÞ. Solid contours show
the boundaries within which the coherency
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c GAB = GAA GBB 0:3, a threshold established by considering the value of c expected to arise by chance with the
ensemble size and window length used in the analysis.
Dashed lines are contours of DBz =B0 ¼ 0:1 associated with
the diamagnetic cavity, and are provided as a reference.
Two features are evident from Fig. 11. First, the orientation of the phase fronts at all frequencies is vertical within the
area sampled by the probe. Second, the dominant wavelength
of the pattern decreases with the increasing frequency. It is
possible to quantify this relationship by performing a 1D FFT
in the spatial domain across lines of constant y in the region
where the phase fronts appear. The resulting xðkÞ relationship
can provide hints as to the origin of the oscillation.
FIG. 11. (Color online) 2D phase patterns associated with the signals in
floating potential.
FIG. 12. Frequency-wavelength relationship of structures. Contour plot
shows power (linear scale) associated with the 1D Fourier transform in x of
the structures displayed in Fig. 11. Dashed line is a linear fit at
f kx ¼ 5:5 106 cm/s.
Figure 12 is a contour plot demonstrating this relationship for the data displayed in Fig. 11. It is computed from
1D power spectra in x along the lines y ¼ 22… 18. The
data suggest a simple relation of the form x ¼ kv. The overplotted line shows the result of a weighted least-squares linear fit to the data. In this picture, the phase velocity of the
oscillation bursts in the x direction is simply the value of v
derived from the plot.
Interestingly, this derived slope of the line f kx
¼ 5:5 106 cm/s is quite close to the cross-field speed t?
of the magnetic cavity at the same time. Figure 13 shows
the value of f kx , computed using this procedure over the
course of the expansion, by windowing the probe signals in
time. The conclusion drawn from Fig. 13 is that the source
of the floating-potential oscillations is the translation of
static structures past the diagnostics; the structures are moving with the expanding plasma as it crosses the magnetic
field, with xreal << kv? . For comparison, the estimated
cross-field speed v? as measured by following the magnetic-field profile is also displayed. Interestingly, while the
two speeds roughly match at early times, there is substantial
FIG. 13. Effective cross-field speed computed via f kx for floating/floating
correlations (solid) and floating/B-field correlations (dashed). The cross-field
speed estimated from the magnetic-field measurements is also shown
(dashed-dotted). Vertical bars show the resolution limit in the Fourier transforms ðDvÞmin ¼ Df Dkx .
055705-7
Structure of an exploding laser-produced plasma
FIG. 14. (Color online) 2D phase patterns from correlation between B_ and
floating potential (compare Fig. 11).
disagreement for t sd . It should be pointed out that the
values for vx and v? are computed by following the cavity
along with line y ¼ 0; what Fig. 13 shows is that, at later
times, the phase speed of the flutes at the bottom of the cavity does not match the cross-field motion of the bubble
front. One possibility is that the structures have some phase
speed in the h direction around the cavity. Comparing to the
fast photographs of Fig. 9, we see that the purely radial
flutes visible at sd become curved as time goes on.
Experiments by Ripin et al.,7 in which the pattern of the
flutes was sampled on a witness plate close to the expansion,
showed similar curvature but no direct evidence for rotation
of the flutes. Another possibility is that the difference in
cross-field speeds at late times is partially responsible for the
breakup of the ring structure seen in Fig. 2 starting at 630 ns.
Correlation measurements were also carried out between
a moving magnetic-field probe and a fixed floating-potential
probe, following the previously established procedure. Due
to the mounting arrangement of the magnetic-field probe, the
area sampled was offset by Dx ¼ 4 mm, Dy ¼ 12:5 mm;
the limits of the plane sampled by this probe are therefore
7 x 51 and 35 y 5. The spatial patterns
associated with each frequency are shown in Fig. 14. Like
that of the floating-potential correlations, the wðkÞ relation
for this case is linear; it is plotted in Fig. 13 alongside the
previous one.
A new feature of Fig. 14, compared to Fig. 11, is a phase
change where the pattern enters the current system at the
edge of the diamagnetic current. Over the vertical range
measured, the total phase shift from the vertically oriented
pattern is roughly p. This pattern does not appear in the case
of the correlation between the floating probes. Comparing
with Fig. 4(b) indicates that the flutes are displaced in the
direction of electron motion in the current layer. Since the
quantity in Fig. 14 represents the correlation between density
and magnetic field, the result is consistent with the existence
of inhomogeneities within the current layer, which match the
period of the density striations. A long-standing question
regarding the behavior of sub-Alfvénic expansions is the
rapid collapse of the diamagnetic cavity; various explanations have been advanced, including theories for anomalous
resistivity based on microturbulence at the cavity edge. That
the structures on the cavity surface are correlated with variations in the diamagentic current layer suggests that they may
play a role in the current’s decay and the subsequent collapse
of the cavity.
The main difficulty of comparing these observations
to theory is that we cannot measure the instability in the
Phys. Plasmas 18, 055705 (2011)
linear regime. As soon as the flutelike structures are visible (Fig. 9, t ¼ 390), their spacing along the bubble
surface (k) is comparable to both their length and the
length scale of the density gradient. Likewise, the magnitude of the flute intensity in the fast photographs dI=I,
which can be used as a rough proxy for the density, is
about 0.5 at this time.
We find the smallest discernible wavelength in the fast
photographs of Fig. 9 at t ¼ 390 ns with k 2 mm. The
time also matches up well with the beginning of the deceleration phase as plotted in Fig. 7. In the most basic description
of the large-Larmor radius instability,21 the growth rate is
simply governed by the deceleration of the expanding
plasma g and the length scale of the of the density gradient
Ln ¼ ½ð1=nÞð@n=@xÞ1 . We can derive a value for g by following the time behavior of v? ðtÞ; the dashed line plotted in
Fig. 7 shows the fit result of g ¼ 1:5 1013 cm/s2. The density gradient width can likewise be estimated from fast photography at the time of interest, it is 5 mm. The growth
rateffi
pffiffiffiffiffiffiffiffiffi
obtained in Ref. 21 for kLn >> 1 is cLLR ¼ kLn g=Ln .
Obviously, this description is not physical as k ! 1; one of
the assumptions of the theory is that kqe << 1. However, it
represents a starting point for comparison of the observed
structures to theory. For the observed values of g and Ln , this
gives c ¼ 8:5 107 rad/s, giving a growth time of s ¼ 80 ns.
This value is large compared to the interframe period of the
photograph series, which places an upper bound on the
growth time of 20 ns. It is not particularly surprising that this
model fails to accurately predict the growth rate, given the
limits of the theory and our inability to resolve a period of
linear growth; this is also the case in, e.g., the experiments
by Dimonte and Wiley.6 In addition, the length scales of the
both the flute modes and the current layer, both on the order
of c=xpe ¼ 12 mm for estimated n ¼ 1013 1014 , coupled
with the rapid onset of the flute modes (s < 20 ns, compared
to an electron gyro-period of 1–2 ns), underscores the probable importance of electron dynamics in the expansion.
Although we did not attempt computer modeling of the
expanding plasma, a simulation with access to these phenomena (as opposed to fluid electrons) would doubtless
prove interesting.
VI. CONCLUSION
We have performed direct probe-based investigation of
the currents within an expanding laser-produced plasma and
of the flutelike features that exist on its surface. The currents
within the expanding plasma are found to be complex,
including a component of the current density along the background magnetic field, whose sense varies as a function of
radius from the center of the diamagnetic cavity to the outside. Due to the limited volume sampled in z, the extent to
which these currents are related to the outgoing fast electron
bursts observed in Ref. 14 is unclear, except to note that the
current densities are substantially higher, at about 300 A/cm2
as opposed to 5–10 A/cm2 , and comparable with the diamagnetic current density of 600 A/cm2 .
Correlation measurements of the ac floating potential at
the cavity edge reveal the flutelike density striations are
055705-8
A. Collette and W. Gekelman
attached to the expanding plasma and move with it across
the background magnetic field, with at most a small rotational component. The rapid formation of these structures
hinders comparison to theory; we are unable to observe a period of linear growth, either via fast photography or probe
techniques.
The cavity is also found to collapse quickly compared to
the expected magnetic diffusion time; simple estimation indicates the effective resistivity in the expanding plasma is at least
100 times the expected (Spitzer) resistivity for these conditions. Rapid collapse has been repeatedly observed for expanding plasmas in the regime of interest, but despite a number of
ideas including microturbulence, a definitive explanation is
still lacking. Correlation has been demonstrated between the
floating potential and magnetic field variations at the cavity
edge. Two-dimensional mapping of this effect shows that the
structures penetrate into the diamagnetic current layer, a finding that strongly suggests that they affect the behavior of the
diamagnetic current system and contribute to its collapse.
ACKNOWLEDGMENTS
We would like to acknowledge S. Vincena and others
for many useful discussions, along with the expert technical
assistance of M. Drandell, Z. Lucky, and M. Nakamoto.
Work performed at the Basic Plasma Science Facility was
with support from the Department of Energy and the
National Science Foundation.
Phys. Plasmas 18, 055705 (2011)
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