PHYSICS OF PLASMAS 22, 113101 (2015)
Experimental study of subcritical laboratory magnetized collisionless
shocks using a laser-driven magnetic piston
D. B. Schaeffer,1,a) E. T. Everson,1 A. S. Bondarenko,1 S. E. Clark,1 C. G. Constantin,1
D. Winske,2 W. Gekelman,1 and C. Niemann1
1
Department of Physics and Astronomy, University of California—Los Angeles, Los Angeles,
California 90095, USA
2
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Received 12 August 2015; accepted 20 October 2015; published online 3 November 2015)
Recent experiments at the University of California, Los Angeles have successfully generated
subcritical magnetized collisionless shocks, allowing new laboratory studies of shock formation
relevant to space shocks. The characteristics of these shocks are compared with new data in which
no shock or a pre-shock formed. The results are consistent with theory and 2D hybrid simulations
and indicate that the observed shock or shock-like structures can be organized into distinct regimes
by coupling strength. With additional experiments on the early time parameters of the laser plasma
utilizing Thomson scattering, spectroscopy, and fast-gate filtered imaging, these regimes are found
C 2015 AIP Publishing LLC.
to be in good agreement with theoretical shock formation criteria. V
[http://dx.doi.org/10.1063/1.4934983]
I. INTRODUCTION
Space and astrophysical collisionless shocks are rich and
complex phenomena that occur in many environments,
including the solar wind, coronal mass ejections, and supernovae remnants. Unlike hydrodynamic shocks, collisionless
shocks dissipate energy through collective electromagnetic
effects that occur on length scales far shorter than the classical
mean free path. These shocks form as a super-magnetosonic
plasma flows through an ambient plasma and so can often be
modeled as a magnetic piston interacting with a magnetized
background plasma.
Besides their ubiquity in space environments, collisionless shocks are of general interest for several reasons. While
these shocks have been studied remotely and in situ by
spacecraft for decades, those systems are difficult to diagnose or largely steady-state, and so have focused on properties of pre-formed shocks (see Refs. 1 and 2 for reviews on
terrestrial space and astrophysical shocks, respectively). As a
result, the microphysics of shock formation has been largely
left to theoretical efforts. With the recent discovery3 that
Earth’s bow shock can reform, there has been renewed interest in understanding shock formation. Additionally, several
physical effects are still not well understood, including the
mechanism by which entropy is generated in the shock front4
and the process by which ions are accelerated to extremely
high energies.5
By providing greater control over relevant parameters
and reproducibility, appropriately scaled laboratory experiments can thus contribute to an understanding of collisionless
shocks, despite orders of magnitude differences in scale.6,7
Moreover, experiments can help validate complex computational codes and complement spacecraft measurements. In the
past five decades, much work has been done on collisionless
shocks in the laboratory. Strictly perpendicular shocks were
successfully created using h-pinches, but those shocks did not
a)
Electronic mail: dschaeffer@physics.ucla.edu
1070-664X/2015/22(11)/113101/9/$30.00
separate from the imploding magnetic field (e.g., piston).8–11
In contrast, work by Refs. 12–14 achieved some success in
creating a shock-like structure that separated from the piston
by combing a laser-produced plasma with a h-pinch. Much
work has also been done on sub-Alfvenic laser-driven plasmas
in an external magnetic field,15–19 but these experiments cannot produce shocks. Given the difficulty of generating shocks
with lasers, more recent works have utilized a Z-pinch20 or
field-reversed configuration (FRC) plasma guns21 to study
collisionless shocks.
Previous efforts22,23 have been unsuccessful in generating shocks because there are several criteria that must be
simultaneously fulfilled by scaled laboratory shock experiments to be of relevance to space and astrophysical environments.24 The magnetic piston must drive a super-Alvenic
magnetic pulse through a magnetized ambient plasma long
enough for instabilities to grow in the pulse front. This also
implies that the system size must be large enough for a
shocked ambient ion moving at the shock speed to complete
one gyro-orbit. The mean-free-path of a shocked ambient ion
must also be much larger than the system size to avoid classical collisions. To ensure sufficient coupling between the
piston and ambient ions, hybrid simulations have predicted
that the piston width must be larger than an ion inertial
length25 and the piston radius must be larger than a piston
ion directed gyroradius.26 Together, these criteria simultaneously require a high-energy piston (so that the pulse is superAlfvenic) and a highly magnetized ambient plasma (so that
the piston couples to the ambient plasma) while keeping the
energy-density ratio b ¼ Episton ð8p=B2 Þ > 1.
By using a high-energy laser to drive a magnetic piston
through a large-scale magnetized ambient plasma, experiments at UCLA have recently demonstrated the formation of
a quasi-perpendicular, magnetized collisionless shock.27,28
Because the flexibility of the laser allows non-perpendicular
geometries, and because the piston is only temporarily tied
to the shock, these experiments allow more control over
22, 113101-1
C 2015 AIP Publishing LLC
V
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shock formation and evolution than the original ones on
h-pinches.
Building on the success of generating a collisionless
shock, this paper is concerned with the general aspects and
characteristics of forming a subcritical magnetized collisionless shock in the laboratory. New data taken in different formation environments and consistent with 2D hybrid
simulations show the relationship of subcritical shocks to
non-shocks and pre-shocks, allowing observed shock structures to be newly correlated with various experimental
regimes. Additionally, measured features from shocks in various stages of formation allow a new analysis on how well
theoretical shock formation criteria encapsulate shock formation physics. These shock criteria are also used to quantify the observed experimental regimes.
The paper is organized as follows. Section II provides
additional background information on sub-critical shock formation. Section III describes the experimental setup. Section
IV details the experimental results, which are then discussed
in Section V. Section VI summarizes important findings.
II. BACKGROUND
A shock is created when a flow moving faster than the
local sound speed encounters an obstacle. The obstacle
serves to slow down the flow. Because the oncoming flow is
faster than the sound speed, information about the obstacle
cannot propagate quickly enough upstream to directly affect
particles. To compensate, a region is established that acts to
slow the incoming flow at the expense of increased entropy.
In other words, the shock converts the high ram, low thermal
pressure flow upstream (the region that shock has not
reached) to a low ram, high thermal pressure flow downstream (the region the shock has passed). In a collisionless
shock, the transition region (shock layer) is much smaller
than the classical mean free path. For a comprehensive theoretical treatment of collisionless shocks, see Refs. 35–37.
Here, the relevant sound speed is the magnetosonic speed
c2ms ¼ v2A þ c2s —where vA is the Alfven speed and cs is the
sound speed—and, for the plasmas of interest, can usually be
approximated as cms vA . The magnetosonic speed cms ðhB Þ
can be further written as a function of the shock wave propagation angle hB relative to the magnetic field. As a result, shocks
can be classified by the angle hB as quasi-perpendicular
(p=4 < hB p=2) or quasi-parallel (0 hB < p=4). The
experiments detailed here then involve the formation of quasiperpendicular, fast mode shocks.
Using the magnetohydrodynamic (MHD) equations for a
single-fluid plasma, and assuming a planar shock, relationships
between the upstream (ahead of the shock and denoted subscript 1) and downstream (behind the shock and denoted subscript 2) values of density n, velocity v, magnetic field B, and
pressure P can be derived. These are known as the RankineHugoniot (RH) jump conditions, and describe the global (far
from the shock layer) conservation of energy across the shock.
For a strictly perpendicular shock (hB ¼ p=2) and assuming an
ideal gas equation of state,37,38 the jump conditions show that
the magnetic field and density downstream of the shock
increase by the same amount as the velocity decreases. For the
marginal b 1, quasi-perpendicular, low-Mach number
Phys. Plasmas 22, 113101 (2015)
plasmas in the experiments, these compression ratios
(B2 =B1 ; n2 =n1 ; v1 =v2 ) approximately scale as the Alfvenic
Mach number MA, as seen in Fig. 1.
Shocks can be further classified by their criticality, a
measure of how well a shock can maintain itself through the
anomalous (since there are no classical collisions) resistive
dissipation. Criticality is expressed in terms of a critical Mach
number Mc, where M < Mc is subcritical and M > Mc is
supercritical. A supercritical shock is unable to maintain itself
solely through resistive dissipation. Mc depends on the plasma
b and hB, but the largest possible value is Mc 2:76 (Mc
increases as hB increases or b decreases).36,39 For the experiments detailed here, the shocks are generally subcritical.
In addition to the expected shock features, the formation
process places limits on the conditions under which a collisionless shock will form. In the shock frame, the upstream
ions with flow v0 must, of course, be super-Alfvenic
(MA ¼ v0 =vA > 1). The collisional mean free path between
shocked ions must be larger than their gyroradii (kii;s > qa;s ,
where qa;s ¼ vs =Xci;s ; Xci;s ¼ Za eBs =ma c; vs v0 =MA , and
Bs MA B0 is the compressed magnetic field for low-Mach
number shocks). In order to set up a cross-ramp potential, the
shocked ions must be sufficiently magnetized,6 i.e., their gyroradii must be less than the system size (l ¼ D0 =qa;s > 1).
This is equivalent to the condition that the upstream ions have
sufficient time to establish the cross-ramp potential
(T ¼ D0 =vs > X1
ci;s ). The system size D0 is defined as the distance from the target to the edge of the (suitable) ambient
plasma.
Shocks can be further (though incompletely) characterized by a coupling parameter that helps distinguish when
a shock will form. The parameter was originally derived by
Bashurin et al.41 using the equal charge radius R
¼ ð3Nd Zd =4pZa na Þ1=3 . An alternate, empirically motivated
version using the equal mass radius is given by
2 R2M
RM
md Z a
¼
¼
;
(1)
qd qa
qd
ma Zd
FIG. 1. Compression ratio (ratio of downstream to upstream value) versus
Alfvenic Mach number MA for a strictly perpendicular (hB ¼ p=2) planar
shock with lines of constant b ¼ 8pnT=B2 . For b 1, the compression ratio
scales as the Mach number for MA ⱗ3.
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Schaeffer et al.
where
RM ¼ ð3Nd md =4pma na Þ1=3 ; qd ¼ v0 md c=ðeZd B0 Þ;
qa ¼ v0 ma c=ðeZa B0 Þ, v0 is the piston speed, and the subscripts refer to debris and ambient ions, respectively. Note
that this parameter was derived strictly for large-Mach number shocks, so its application is approximate. Hybrid simulations26 suggest that ⲏ 0:5 is necessary for shock formation.
III. EXPERIMENTAL DESIGN
The experiments were carried out on the Large Plasma
Device (LAPD)29 using the Raptor high-energy laser30 at
UCLA. The LAPD is a large scale (18 m long by 11 m) facility that provides a well-characterized and highly reproducible magnetized ambient plasma. The plasma is steady-state
(10 ms), quiescent and current-free, and customizable in both
background magnetic field (0.2–1.8 kG) and ambient gas fill
(H2, He, Ne, Ar). A BaO-coated Ni cathode generates the
main 160 cm plasma (na 2 1012 cm3), while a second
LaB6 (lanthanum hexaboride) cathode generates a smaller
120 cm core plasma (na 2 1013 cm3) roughly centered
on the main plasma (see Fig. 2(b)). While this arrangement
naturally leads to an inhomogeneous ambient plasma density
profile, previous experiments and simulations31 have shown
that the resulting density gradient enhances piston-ambient
coupling and reduces shock formation time. The ambient
plasma has a typical electron temperature Te ¼ 6 eV and ion
temperature Ti ¼ 1 eV. The background field is oriented axially along the machine and defines ^z , with x^ oriented horizontally perpendicular to the field and y^ oriented vertically
(see Fig. 2(a)). The Raptor laser is a kJ-class laser (1053 nm,
25 ns) used to drive the magnetic piston.
A typical experimental setup is shown in Fig. 2. A graphite or plastic (high-density polyethylene C2H4) target was embedded in a magnetized ambient H or He plasma 30 cm from
the machine center axis. The target was translated and/or
rotated every shot for a fresh surface. Since the laser-ablated
plasma is always directed normal to the target surface, the target can be rotated relative to the background field to study different shock geometries. For quasi-perpendicular shocks, the
target normal was oriented along x^, resulting in a laser blowoff across the background magnetic field. The Raptor laser
(250 J) was focused through a 1.8 m focal-length lens to a
0.5–2 mm2 spot (0:8–40 1011 W/cm2) on the target at a 30
angle relative to the target normal.
FIG. 2. Schematic layout of an experiment on the LAPD. (a) Coordinate system of the experiment. (b) Cartoon cross-section of the LAPD plasma in a
two-cathode configuration.
Phys. Plasmas 22, 113101 (2015)
The evolution of the magnetic field was measured with a
variety of differentially wound bdot probes.32 Single-axis
bdots measured the ^z component of the field up to 60 cm
from the target along x^. Individual probes were positioned in
1 cm increments with 1–3 shots taken per position, while a
“5-tip” probe—containing 5 bdot cores spaced 1 cm apart—
was incremented in 5 cm steps with generally 1 shot per position. Additional tri-axis bdots measured all three components
of the field up to 10 m from the target along ^z . The probe signals were sent though custom-built 150 MHz differential
amplifiers and coupled to fast (1.25 GHz) 10-bit digitizers.
Data characterizing the ablated and ambient plasma
were measured with Thomson scattering, spectroscopy, and
fast-gate filtered imaging, as detailed elsewhere.33
IV. EXPERIMENTAL RESULTS
To illustrate the characteristics of subcritical shock formation in a laboratory setting, several representative experimental runs with a range of laser energies, intensities,
background magnetic fields, and ambient plasma conditions
were selected, each carried out on the LAPD. A summary of
the runs is given in Table I. In general, each run took magnetic flux measurements over multiple laser shots along x^, the
axis parallel to the debris blowoff and perpendicular to both
the target surface and background magnetic field. All measurements are of the ^z component (along the field) of the magnetic field.
The runs were chosen to emphasize distinct experimental
regimes defined by their dominant observed features: weakly
coupled systems with shallow and weakly compressed magnetic pulses comparable to the sub-Alfvenic regime; moderately coupled systems with strongly compressed but shallow
pulses; and strongly coupled systems with both strongly compressed and steep pulses. For each run, space-time plots of
magnetic field amplitude and corresponding spatial and temporal lineouts are provided. Wherever possible, data are compared to vacuum (no ambient plasma) conditions. Since shock
formation strongly depends on ambient density, ambient density profiles are also provided. In the following, the ion inertial
TABLE I. Summary of collisionless shock experiments. E0 and I0 are the
average laser energy and intensity, respectively, for each run. na is the peak
ambient ion density, X1
ci is the ambient ion gyroperiod, c=xpi is the ambient
ion inertial length, qd is the debris (laser-ablated) plasma ion gyroradius, and
vA is the Alfven speed. Note that while Run 3 used a plastic (C2H4) target,
measurements and simulations show no significant effect from the H debris.
However, the plastic target allowed much higher laser intensities than a
graphite (C) target.
E0 (J)
I0 (1011 W/cm2)
Target material
Ambient plasma
B0 (G)
na (1012 cm3)
X1
ci (ls)
c=xpi (cm)
qd (cm)
vA (km/s)
Run 1
Run 2
Run 3
19 6 1
5
C
Heþ1
275
561
1.5
20
35
134
90 6 5
1.4
C
Hþ1
200
561
0.5
10
32
195
190 6 10
12
C2H4
Hþ1
300
12 6 1
0.3
7
31
189
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Schaeffer et al.
length c=xpi and ion gyro-frequency X1
ci are given in terms
of ambient ion conditions as listed in Table I, and unless otherwise stated, distances are relative to the target and times are
relative to laser ablation.
A. Weak coupling regime
Fig. 3 shows results from Run 1 (see Table I). In the figure, a diamagnetic cavity is seen expanding from the target,
reaching a maximum size of 20 cm at 0:9 ls. Soon after,
the cavity begins to collapse due to magnetic diffusion. Also
seen is a magnetic compression initially moving at the same
speed as the cavity (v0 330 km/s, MA 2:7). Later, the
magnetic pulses separate from the cavity as the cavity slows
down.
This run lies in the weak coupling regime, in which the
piston fails to transfer significant energy to the ambient
plasma. While super-Alfvenic for most of its lifetime, the
resulting magnetic field structures are very similar to the
sub-Alfvenic features seen in Refs. 17, 18, and 34. As is typical in sub-Alfvenic cases, at early times (t ¼ 200 ns or
0:1 X1
ci ), the ramp of the compression is shallow and ends
at the cavity edge (i.e., no pulse width, see Fig. 3(b)). This is
due to the penetration into the ambient environment of
FIG. 3. Results from shock experiment Run 1 (see Table I). (a) Contour
streak plot of normalized magnetic field. Also shown are spatial and temporal lineouts (dashed). (b) Spatial lineouts from (a), including at maximum
compression (solid). The ambient density profile is also shown (gray overlay). (c) Temporal lineouts from (a) containing maximum compression. The
results indicate no shock or pre-shock is formed, consistent with a weak coupling regime.
Phys. Plasmas 22, 113101 (2015)
magnetic field from the cavity, as well as the decoupling of
faster debris ions that carry some magnetic compression with
them. The magnetic pulse reaches a peak compression of
Bz =B0 ¼ 1:360:1 at t ¼ 549 ns (0:4 X1
ci ), but its leading
edge remains shallow. The pulse also begins to broaden due
to the difference in speeds between the cavity (MA ¼ 2:7)
and leading (MA ¼ 3:7) edges. By t ¼ 800 ns (0:5 X1
ci ),
however, the compression remains broad but weakly compressed with a shallow ramp. Unlike the following runs, the
ambient density remains largely uniform (though lower)
throughout this run due to the use of just the larger-diameter
BaO cathode.
B. Moderate coupling regime
Run 2 (Fig. 4) is typical of the moderate coupling regime. Like Run 1, Run 2 has at early times (t ¼ 500 ns or
1:3 X1
ci ) a compression with a shallow ramp ending at the
cavity edge (see Fig. 4(b)). Unlike Run 1, however, the compression continues to grow until reaching a maximum value
Bz =B0 ¼ 1:760:1 at t ¼ 1.1 ls (1:9 X1
ci ) and x ¼ 29 cm
(0.9 6 0.1 qd). At this time, the compression is also moving
at MA ¼ 1:860:1 (see Fig. 4(a)). Additionally, comparing
FIG. 4. Results from shock experiment Run 2 (see Table I). (a) Contour
streak plot of normalized magnetic field. Also shown are spatial and temporal lineouts (dashed). (b) Spatial lineouts from (a), including at maximum
compression (solid). The ambient density profile is also shown (gray overlay). (c) Temporal lineouts from (a) containing maximum compression and
corresponding vacuum profile. The results indicate that a pre-shock formed
but did not have sufficiently uniform ambient density to steepen and separate
from the piston.
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the magnetic field profile with and without an ambient
plasma indicates a strong ambient effect (see Fig. 4(c)). The
compression with ambient plasma is stronger and has a
steeper temporal ramp (but comparable breadth). The compression’s leading edge also arrives faster, while its cavity
edge is slower, indicating coupling with the ambient plasma.
However, as the magnetic pulse reaches its maximum compression, the ambient density falls significantly (see Fig.
4(b)). This causes the local Alfven speed to increase and the
pulse to become sub-Alfvenic. As a result, the compression
fails to further steepen or separate and instead dissipates.
C. Strong coupling regime
Results from Run 3 (Fig. 5) indicate a strong coupling regime. Like the previous run, Run 3 shows signs of moderate
coupling at t ¼ 280 ns (0:6 X1
ci , see Fig. 5(b)). The leading
edge of the compression moves at MA ¼ 1:560:1 with a magnetic compression Bz =B0 ¼ 1:660:1, but has a shallow ramp.
However, the compression continues to grow until reaching a
maximum value at t ¼ 708 ns (1:9 X1
ci ) and x ¼ 29 cm
(0.9 6 0.1 qd), at which point the pulse has a considerably
Phys. Plasmas 22, 113101 (2015)
different structure. The compression has increased to Bz =B0
¼ 2:060:1 and moves at MA ¼ 2:160:1. The ramp of
the compression, though, has steepened to a width
d ¼ 0:560:2 c=xpi . Furthermore, the width of the pulse has
increased to D ¼ 1:460:2 c=xpi , showing a significant separation of the front of the compression from the cavity edge.
This can also be seen in the difference between the speeds of
the front (MA ¼ 2:160:1) and back (MA ¼ 1:660:1) of the
compression (see Fig. 5(a)).
Like the moderate coupling run, comparison to shots
taken in vacuum under the same conditions indicates that the
above features are tied to the ambient plasma (see Fig. 5(c)).
At x ¼ 37 cm (1.2 6 0.1 qd), the magnetic pulse in vacuum
has a much shallower ramp. Though the width of the pulse in
vacuum is broader than seen in previous runs, it is, nonetheless, much narrower than the corresponding pulse with ambient plasma. Furthermore, the cavity moves more slowly with
an ambient plasma (350 vs. 390 6 20 km/s), consistent with
energy being coupled to the ambient ions, while the leading
magnetic compression moves more quickly (530 vs.
420 6 10 km/s), consistent with a pulse that is being carried
by accelerated ambient ions.
Similar to Run 2, by t ¼ 940 ns (2:5 X1
ci ), the compression has entered a region of significantly reduced ambient
density. This causes both the Alfven speed and ion inertial
length to increase, decreasing and smearing out the leading
edge of the magnetic compression.
V. DISCUSSION
FIG. 5. Results from shock experiment Run 3 (see Table I). (a) Contour
streak plot of normalized magnetic field. Also shown are spatial and temporal lineouts (dashed). (b) Spatial lineouts from (a), including at maximum
compression (solid). The ambient density profile is also shown (gray overlay). (c) Temporal lineouts from (a) containing maximum compression and
corresponding vacuum profile. The results indicate that a shock formed but
did not have sufficiently uniform ambient density to fully separate from the
piston.
From the considerations of Section II and experimental
conditions, a quasi-perpendicular, fast mode, low-Mach
number, subcritical shock is expected with a shock layer
thickness on the order of an ion inertial length and magnetic
field compression similar to the Mach number. Fig. 6 shows
a diagram of the process by which the shock is expected to
form (in terms of the magnetic field, though the process
should be similar for the ambient density). Initially, a diamagnetic cavity and compression at the cavity edge are
formed by the piston. The compression moves superAlfvenically, and the leading ramp of the compression is
much larger than an ion inertial length due to faster ions slipping through. Later in time, a shock begins to form as
piston-accelerated ambient ions overrun unperturbed ions.
As the piston loses energy, a large compression width develops as the shock separates from the slowing cavity edge. The
leading compression ramp also steepens. Late in time, the
shock has separated from the piston, with a well-defined
shock ramp and downstream region. Additional details on
the shock formation process can be found in Ref. 27.
These features allow the various experimental regimes
of Section IV to be related to shock formation. In the
weakly coupled regime, no shock forms. Indeed, in Run 1,
the magnetic compression (Bz =B0 ¼ 1:3) and Mach number
(MA ¼ 2:7 3:7) are inconsistent with the RH conditions.
Instead, the resulting magnetic pulses are most similar to
the sub-Alfvenic regime.
In the moderately coupled regime, a pre-shock can
form. A pre-shock has sufficient initial conditions to begin
shock formation (unlike the weak coupling case), but fails to
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Phys. Plasmas 22, 113101 (2015)
FIG. 6. Diagram of the subcritical shock formation process with a magnetic piston. At t1, a super-Alfvenic piston forms a diamagnetic cavity and magnetic
field compression at the cavity edge. The width of the compression’s leading ramp is much larger than an ion inertial length c=xpi . In the lab frame, the compression is moving in the þx direction. At later times (t2 > t1 ), a shock begins to form as piston-accelerated ambient ions overrun unperturbed ions. The forming shock (still moving super-Alfvenically) separates from the piston as the piston loses energy. The shock compression goes like the Mach number, and the
shock ramp is on the order of an ion inertial length. Late in time (t3), the shock has fully formed and separated from the piston.
fully steepen due to a lack of space, time, and/or sufficient
ambient density (pre-shocks were also observed in Ref. 22).
This is supported by Run 2, where the magnetic compression
(Bz =B0 ¼ 1:7) and Mach number (MA ¼ 1:8) are consistent
with the RH conditions for low-Mach number shocks, and
the initial conditions suggest the beginning of shock formation. The lack of sufficient ambient density (due the sharp
drop at the edge of the LaB6-generated plasma), however,
causes the pulse to dissipate before a shock can form.
This is further supported by 2D hybrid simulations of
Run 2 initialized to the same experimental conditions (see
Fig. 7). The simulations utilize a 2D3V collisionless magnetostatic hybrid code in two Cartesian spatial dimensions with
three-dimensional fields and velocities. Additional details
can be found in Refs. 26 and 40. The simulations do not
include the laser-target interaction, but focus on a plasma
expanding out from a planar target perpendicular to a uniform magnetic field into an ambient plasma at experimental
conditions. In the simulation, an ambient plasma with a
Gaussian density distribution centered 2:5 c=xpi from the
initial debris distribution is superimposed on a uniform lowdensity plasma covering the entire simulation domain. The
debris plasma consisted of a cloud of Cþ5 ions that expanded
out conically at MA 1:8. In Fig. 7(b), the simulation shows
the formation of a magnetic compression consistent with the
RH jump conditions. However, it also shows that the ambient density, while compressed from piston coupling, fails to
remain tied to the magnetic compression due to insufficient
magnetization. This is also seen in Fig. 7(c), where there are
no reflected ambient ions in phase space at the leading edge
of the magnetic pulse.
Finally, systems in the strongly coupled regime can support full shock formation, as seen in Run 3. The magnetic
pulse seen in Run 3 is not only consistent with the RH conditions but also has a steep leading edge on the order of the ion
inertial length that is beginning to separate from the cavity.
This can also be seen in Fig. 8, in which key features of the
magnetic pulse are tracked in time. As expected in shock formation, the magnetic pulse compression, steepness, and
broadness increase in unison, reaching a maximum value at
approximately the same time. For similar reasons as Run 2,
FIG. 7. 2D hybrid simulation of Run 2 initialized to the same experimental
conditions. At initial time, the ambient density is inhomogenous, with a uniform low-density component over the whole domain and a high-density
Gaussian core centered at x ¼ 25 cm (2.5 c=xpi ). The debris cloud (Cþ5) is
concentrated at ðx; yÞ ¼ ð0; 0Þ in a small disk (1 ¼ 0:25 c=xpi ) with
Gaussian intensity. (a) Contour streak plot of Bz at y ¼ 0 cm. Also shown
(dashed) is the time at which (b) and (c) are calculated. (b) Magnetic (solid),
ambient density (dot-dash), and debris density (dot-dot-dash) profiles at late
time showing the formation of a magnetic compression consistent with the
RH jump conditions, but an insufficiently magnetized ambient density compression. The original (t ¼ 0) ambient density profile is also shown (gray
overlay). (c) Phase-space plot of ambient ions (dots) at late time. A lack of
reflected ions is consistent with the failure to form a shock.
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Phys. Plasmas 22, 113101 (2015)
FIG. 8. Measured and simulated features of shock formation as a function of
time for Run 3. All profiles are scaled to the range [0, 1]. (a) Peak magnetic
field amplitude Bz. (b) Ratio of magnetic pulse width Dk to ramp width Dx.
The pulse width is defined as the distance between the first and last position
where the field reaches 50% of its maximum. The ramp width is defined as
the distance between the edge of the pulse and the position where the field
reaches 10% above background. (c) Magnetic ramp steepness DBz =Dx,
defined as the change in peak field over the ramp width. For each plot, the
corresponding simulated profile (dash dot) is also shown. As the measured
field reaches peak amplitude (vertical dashed), so do the pulse width and
ramp steepness, consistent with shock formation. The simulated profiles
have similar behavior, though the simulated pulse width tends to broaden
later in time. For both measured and simulated profiles, after reaching their
maxima, their values dissipate as the pulse enters a low-density regime.
though, the pulse fails to fully separate from the cavity as it
runs out of the high-density region and dissipates.
The results from Run 3 were also compared to 2D
hybrid simulations initialized to the same experimental conditions (see Fig. 9). The simulations used a two-component
ambient plasma, with one uniform low-density plasma covering the entire simulation domain, and another high-density
Gaussian component centered 4 c=xpi from the initial debris distribution. The debris plasma consisted of a cloud of
Cþ5 ions that expanded out conically at 675 at MA ¼ 2:65.
As can be seen in Figs. 9(a) and 9(b), the formation of a lowMach number collisionless shock is also reproduced in the
simulations. In particular, a magnetic and ambient density
compression consistent with the jump conditions is seen in
Fig. 9(b), while a small population of reflected ambient ions
from the shock front is seen in Fig. 9(c). The simulations
also show the dissipation of the shock as it leaves the highdensity ambient plasma (see Fig. 9(a)), consistent with experimental observations.
It is of further interest to determine whether the shock
criteria outlined in Section II can accurately capture the differences between these experimental regimes, which would
indicate that these criteria are encapsulating the physics of
laboratory shock formation. It would also allow the experimental regimes, previously defined by their dominant
observed features, to be quantified by shock formation
parameters.
In order to calculate the criteria, in particular, , it is necessary to have information on the initial state of the laser
plasma (piston). This is estimated from the results found in
Ref. 33. Here, laser-target interactions were simulated using
the 1D HELIOS radiation-magnetohydrodynamic code42 that
provided a range of particle number and charge state values
FIG. 9. 2D hybrid simulation of Run 3 initialized to the same experimental
conditions. At initial time, the ambient density is inhomogenous, with a uniform low-density component over the whole domain and a high-density
Gaussian core centered at x ¼ 30 cm (4 c=xpi ). The debris cloud (Cþ5) is
concentrated at ðx; yÞ ¼ ð0; 0Þ in a small disk (1 ¼ 0:25 c=xpi ) with
Gaussian intensity. (a) Contour streak plot of Bz at y ¼ 0 cm. Also shown
(dashed) is the time at which (b) and (c) are calculated. (b) Magnetic (solid),
ambient density (dotted-dash), and debris density (dot-dot-dash) profiles at
late time showing the formation of a low-Mach number shock, with magnetic and density compressions consistent with the RH jump conditions. The
original (t ¼ 0) ambient density profile is also shown (gray overlay). (c)
Phase-space plot of ambient ions (dots) at late time. A small population of
reflected ions at x ¼ 30 cm is consistent with the formation of a shock.
for various laser intensities. The simulation results were
benchmarked by the measured density and temperature profiles and were found to be in good agreement after scaling to
3D. Additionally, the numerical results compared well to
laser-based empirical scaling laws.43,44 The total number of
ablated particles was then calculated from the experimental
laser intensity, and, by using both the measured intensity and
expansion speed, interpolated average ablated charge states
for that speed were estimated. Together with the experimental ambient conditions, this allowed a calculation of the equal
mass radius RM, debris gyroradius qd, and ambient gyroradius qa, and hence .
The shock formation criteria for the experimental results
are summarized in Table II. As can be seen, only Run 3 satisfies all formation conditions, consistent with the measurements from Section IV C that indicate a collisionless shock
formed. While a super-Alfvenic expansion, sufficient coupling, and high b were achieved in most cases, the most
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Schaeffer et al.
Phys. Plasmas 22, 113101 (2015)
TABLE II. Criteria necessary for shock formation for runs from Table I.
MA ¼ v0 =vA is the Alfvenic Mach number, where v0 is the shock speed (generally the piston expansion speed). ¼ R2M =qd qa is the coupling parameter,
where RM ¼ ð3Nd md =4pma na Þ1=3 ; qd ¼ v0 md c=Zd eB0 , and qa ¼ v0 ma c
=Za eB0 . The magnetization parameter l ¼ D0 =qa;s is defined with respect to
the shocked ambient ions, where D0 is the system size, qa;s ¼ vs =Xci;s ;
Xci;s ¼ Za eBs =ma c; vs v0 =MA , and Bs MA B0 is the compressed magnetic
field for low-Mach number shocks. kii;s is the shocked ambient ion-ion collisional mean free path, b ¼ 1=2 na;s ma v2s =8pB2s , and na;s MA na .
Condition
MA > 1
> 0:5
l>1
kii;s =qa;s > 1
b>1
Run 1
Run 2
Run 3
2.8
0.04
0.8
1
0.7
1.3
0.9
0.7
1
1.3
1.9
0.6
1.8
1
1.9
difficult criterion to satisfy was sufficient magnetization l.
This in turn appears to be most sensitive, not to the ambient
density or magnetic field (though increasing both helps), but
rather to the amount of space available with suitable ambient
conditions. In Run 2, a shock appeared to be forming, but the
ambient density profile collapsed too close to the target and
no shock formed. In contrast, in Run 3, the ambient density
remained high long enough for the shock to form, though it
collapsed before the shock could separate from the piston.
Since Mach number and collisionality criteria were easily satisfied, the above results can also be organized using
just an l parameterization, as shown in Fig. 10. It can be
seen that the experiments (and others not included in the
FIG. 10. Location of experimental runs (circles) in an -l parameter space.
Additional runs (squares) not included in the text are also shown. The
dashed lines correspond to theoretical limits for the and l parameters. The
runs in quadrant II ( > 0:5 and l > 1) correspond to shocks and, as can be
seen, also satisfy both shock formation criteria (the strong coupling regime).
In contrast, runs in quadrants III and IV (l < 1) do not correspond to shocks
and fail to satisfy either the magnetization criteria (quadrant III) or both the
magnetization and coupling criteria (quadrant IV). The runs in quadrant IV
( < 0:5 and l < 1) are primarily simple magnetosonic pulses with no
shock-like features, which is consistent with poor coupling and magnetization (the weak coupling regime). The runs in quadrant III ( > 0:5 and
l < 1), though, are shock-like but fail to fully form, consistent with a piston
that couples energy into the ambient plasma but that does not have enough
space to steepen into a shock (the moderate coupling regime). Note that continuing to increase or l will eventually cause MA < 1.
text) fall into distinct regions which correspond to the
observed weak ( < 0:5; l < 1), moderate ( > 0:5; l < 1),
and strong ( > 0:5; l > 1) coupling regimes. While most of
the experimental runs were carried out with different ambient conditions, Fig. 10 also shows that the generation of different shock features is highly reproducible, depending on
which values of and l are chosen. This indicates that both
parameters are capturing much of the physics of piston coupling () and shock formation (l).
VI. SUMMARY
Laboratory experiments combining a laser-driven magnetic piston with a large, pre-formed magnetized ambient
plasma have allowed a unique investigation into collisionless shock formation not readily available to spacecraft.
Magnetic field measurements indicate that a subcritical,
quasi-perpendicular, magnetized collisionless shock was created, and was beginning to separate from the piston before
dissipating in a low-density ambient plasma. The 2D hybrid
simulations reproduced these features. In combination with
the above result, additional shock experiments in which no
shock or a pre-shock was formed were found to be consistent
with theoretical predictions of collisionless shock formation
and shock structure. Furthermore, these experiments could
be organized by weak, moderate, or strong coupling according to their magnetic profiles: weak coupling corresponding
to weakly compressed super-Alfvenic pulses; moderate coupling corresponding to well-compressed but un-steepened
pulses; and strong coupling corresponding to fully formed
shocks.
Multiple shock formation criteria were investigated,
revealing that the most important criteria, in addition to
being super-Alfvenic (MA > 1), were the coupling > 0:5
and magnetization l > 1 parameters. Within these parameters, the most important experimental consideration for
shock formation was the amount of space (D0) over which
ambient conditions were relatively constant. Consistent with
theory, shocks only formed in those experimental conditions
that satisfied all shock formation criteria. Using the results of
laser plasma characterization studies to calculate shock formation criteria, the experiments were found to lie in distinct
regions of an -l parameter space, which agreed well with
the observed shock formation regimes of weak coupling
( < 0:5; l < 1), moderate coupling ( > 0:5; l < 1), and
strong coupling ( > 0:5; l > 1). This indicates that the and l parameters are capturing most of the physics of laboratory shock formation. Future work will continue to characterize shock evolution, as well as transition to quasi-parallel
shock geometries.
ACKNOWLEDGMENTS
We would like to thank the staff of the Basic Plasma
Science Facility (BaPSF), Z. Lucky, and M. Drandell for
their help in carrying out these experiments. This work was
supported by DTRA under Contract No. HDTRA1-12-10024, DOE under Contract Nos. DE-SC0006538:0003 and
DE-NA0001995, and NSF Award No. 1414591. The
experiments were performed at the UCLA BaPSF, supported
by the DOE and NSF.
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