1.
Introduction
The development and behaviour of shocks is fundamental to the evolution of many systems,
particularly in astrophysics, and many basic plasma physics issues require further investigation.
Raditative cooling, collisionality and magnetic fields all play a role to some extent, and the interplay of
these drives the observable phenomenon. Plasma flow interpenetration and the transfer and dissipation
of energy are determined initially by the collisionality of the flow, which in turn drives the evolution of the
interaction region. High streaming velocities relative to the local sound speed, magnetization of particles,
or streaming instabilities lead to the rapid localization of the mass density and the formation of shocks in
the flow. In an adiabatic shock, the pre- and post densities can be described by the Hugoniot jump
conditions for strong shocks, and for a ratio of specific heats, , of 5/3 for monatomic gas the density
increase across the shock is limited to a value of ( +1/ -1) ~4. Where radiative cooling is important, the
dissipation of energy by radiation loss results in an increase in the compression of the plasma across the
shock, which can be >10. In addition, the strong radiative cooling from high Z materials may lead to the
collapse of the shock thickness to small values, and the development of hydrodynamic instabilities under
such conditions has not been studied in detail previously. Indeed, the recent FESAC report on High
Energy Density Science [1] highlighted the area of highly compressible flows, and the need for
investigation of modified or new instabilities which may be discovered where very strong density,
ionization or entropy gradients are present.
The presence of magnetic fields has a strong influence on the behaviour of flows and shocks in
plasmas. In low density, high temperature collisionless plasmas magnetic fields are often responsible for
localisation of charged particles. In higher density, high temperature collisional plasmas, in the ideal
magneto-hydrodynamic limit, the magnetic field is entrained, or frozen into the fluid. Where flows collide
or converge this entrained magnetic flux resists compression and limits the density obtained in shock
fronts. If strong radiative cooling is also present, this leads to rapid dissipation of the heat generated at
the shock front leading to a reduction in temperature and an increase in resistivity. For colliding plasmas
with magnetic Reynolds number of order unity, a dynamical competition develops between flux
compression within the shock, and resistive diffusion and magnetic relaxation. The limited compressibility
induced by entrained magnetic field is also invoked, for example in the discussion of proto-stellar jets,
where the collision of plasma regions occurs and heating is observed, but with only a limited increase of
the plasma density (i.e. compression) in this region (e.g. [2]).
The combination of long particle mean free path and magnetic fields is common in space plasmas,
and dominates the evolution of several astrophysical systems. The physics of magnetic collisionless
shocks [3-5] is important in determining the non-linear evolution of plasmas including the Earth’s
magnetosphere and the solar corona [6, 7], and particle acceleration at such shocks is thought account
for much of the observed cosmic ray spectrum [8-10]. Basic issues in such systems remain untested in
the laboratory, such as the evolution timescale and final spatial discussion of the shock, whether
experimental systems are directly applicable to the space scenarios, the details of particle acceleration for
both electrons and ions. Additional well-constructed experiments to allow dissection of these processes in
alignment with simulation and observation work would be extremely valuable.
1
The focus of this proposal is to carefully evolve a test-bed for the study of shock formation and
evolution relevant to many of the above issues. This will be based on flexible, high shot-rate pulsed power
experiments following recent successful studies of bow-shock formation in z-pinch experiments and
promising initial work at UC San Diego. Experimental work will be supported by parallel computational
studies at Imperial College London, using the 3-dimensional Magneto-hydrodynamic (MHD) code
GORGON, which will be developed into hybrid kinetic-MHD code during the proposal timescale. One of
the anticipated outcomes of the project is the successful design of a magnetized, collisionless shock
system which will form the basis for a future proposal to study particle acceleration in such a shock.
2. Supersonic Plasma Flows from Ablating Wire Experiments
2.1 Background
Ablating wire experiments are driven with currents from to 10-100 kA with rise rates of 1011 – 1012
A/s. The current waveform is typically an approximately sin2(t) waveform peaking at ~100ns (i.e. the first
quarter wave period). The current rapidly ohmically heats the wires and creates a low density plasma
which expands from the wire surface. The low resistivity of this plasma means that much of the drive
current flows here once this is formed, heating and ionising the plasma, and a generic heterogeneous
structure forms which is observed in all ablating wire experiments: a cold dense wire core surrounded by
a hot low density corona which carries the majority of the drive current.
For isolated single wires, the plasma expansion and development of MHD instabilities is dominated
by the field local to the wire, Blocal. In presence of a global field, Bglobal such as in wire array z-pinches, the
same processes occur, but the global field has several effects. The plasma ablated from the wire cores
experience both a local pinching and a jwire x Bglobal force directed towards the system axis. As the low
density plasma is accelerated away from the wire position, mass is ablated from the wire core to
continually replenish the coronal material. The results is a continuous stream of plasma from each wire
(Figure 1) the time scale of which is determined by the current drive timescale.
16mm
Gated XUV Emission image (hn>30eV)
Wire
positions
Stream of ablated
Wires material
( r ~1017 ions/cc, T ~ 15eV
V~150 kms -1, M~4)
Figure 1: (Left) Schematic of core-corona model of ablating wires, (center) time slice from a 3D MHD
GORGON [11] simulation showing ablated plasma stream in a cylindrical wire array z-pinch and (right)
axial gated XUV emission image of plasma streams for a W wire array on the 1MA MAGPIE generator
The rate mass is ablated from the wire, dm/dt, can be described by an analytical rocket model
developed by Lebedev et al [12]. For a wire array of radius R0, subjected to a fast rising current, I this
approximately given by:
2
Vabl
0 I 2
dm

dt
4R0
(1)
Here Vabl is a scaling constant representing the exhaust velocity with which plasma is ejected to the array
interior. This ‘ablation velocity’ has been inferred from experiments to be ~1.5 x 10 5 ms-1, and is taken as
a constant. The model assumes that acceleration of the plasma occurs in a narrow region close to the
wire core, and convection of current towards the array axis is minimal (Rem <1). The ablation velocity
parameter is relatively invariant with geometry and current drive level, and varies no more than a factor of
a few across large areas of phase space. Wires are metallic, and as a result resistivity and diffusion of
magnetic fields must be taken into account, however despite its simplicity, this model provides a good
comparison to both experiments ( e.g. [13-15] and simulations [16, 17].
An important detail of the ablated plasma flow, is the apparent non-uniformity in the axial direction
which appears as a quasi-periodic modulation (Figure 2). This modulation is observed at all current levels
and wire configurations, and the axial wavelength is determined primarily by the wire material [12]. It is
the combination of a local and global B-field geometry which causes this change in behaviour, and this
phenomenon has been the subject of a great deal of interest in wire array z-pinch research.
to array axis
49ns
Scale / mm
ANODE
CATHODE
Figure 2: Wire ablation flares in the presence of a global B-field: (left) radiograph from the 1 MA MAGPIE
generator, (center) laser Schlieren image of an x-pinch at 80kA from UCSD [18], and (right) mass density
slice through a 3D MHD simulation using the GORGON code [19]
The PI is highly active in providing experimental data on the ablation of wire systems against which
theories and simulations can be tested. The axial density contrast for this structure in W and Al wires has
recently been measured for both x-pinch and wire array experiments using a range of current levels and
rise-times. Radiography and interferometry measurements recorded a contrast of the order of ~2 [15, 20,
21] which does not vary strongly in space or time during an experiment. The flow can therefore be
considered quasi-uniform in the axial direction. Importantly, recent simulation work by Chittenden et al
[19] (PI for the Imperial College subcontract) using the GORGON code [11] demonstrated the
development of the ‘ablation flares’ in high resolution 3D MHD simulations [19]. The removal of mass by
the global Lorentz force eliminates the positive feed-back which drives the m=0 instability at the wire,
effectively stabilizing the axial structure. The modulation observed in the experiments can be produced
spontaneously from small scale thermal noise, which serves as an approximation to the wire core early in
3
time. This significant advance means that the details of the quasi-uniform flow from the wires in
experiments can be well-reproduced in simulations for the first time.
Through the combination of experimental, analytical and simulation data, the ablated plasma flow
can be well characterized for a given system. Typical parameter ranges are T e ~10-20 eV, ne~ <1020 cm-3,
z ~2-10, V~150 km/s, Mach number~2-10, Alfven Mach number ~1-2. Of particular note is that the flow
density is determined by the current applied to the wires. One can therefore specify the required density
for an experiment, calculate the required current, and then arrange the load or generator to provide the
correct value. This is typically achieved by dividing the current amongst a number of wires, or reducing
the charging voltage of the generator prior to the experiment. The plasma from ablating wires provides a
flexible, long lived, quasi-uniform, macroscopic flow for the recent and proposed shock studies discussed
below.
2.2 Bow-Shock Development in Radiatively-cooled Flows
Wire array systems have recently been used to examine the formation of hydrodynamic shocks.
This work was the subject of an invited talk at the 2009 American Physical Society, Division of Plasma
Physics (APS-DPP) meeting and was subsequently published [22]. The PI was closely involved in the
execution and analysis of this work and is a co-author on the paper. In a nested wire array, two concentric
cylindrical arrays are mounted between the electrodes. In Fig 3 a), the ‘inner’ array has a diameter of
8mm and the ‘outer’ array has a diameter of 16mm. The outer array has a large number of closely spaced
wires, and so the flows from each wire merge as they travel towards the axis, forming a quasi-uniform
flow in the azimuthal direction. This high Mach number flow is incident on the wires of the inner array,
which form a stationary obstruction.
a)
b)
Figure 3: a) Hardware set-up and cartoon demonstrating ablation streaming behaviour of wire arrays and
position of inner wire obstructions, and b) Gated XUV emission image showing bow shock formation
around inner wires.
For these experiments, carried out on the MAGPIE generator at Imperial College London, the total
drive current was 1 MA with rise-time of 260 ns. The majority of the current is carried by the outer array
for the duration of the experiments discussed below, and approximately 2% of the current is carried in the
4
inner array. Figure 3 b) shows an axial gated XUV pinhole camera image taken during the experiment,
showing approximately one quarter of the circumference of the inner array. Here darker regions denote
greater XUV emission (≈ greater plasma temperature). The outer array flow is incident from the top-left of
the image, and the inner wires appear white and angled on the image due to the perspective of the
diagnostic.
The ablated plasma flow is super-sonic (M ~10), super-Alfvenic (MA ~2) and as the flow reaches
the inner wires, a bow-shock develops around each. The shock position is denoted by the increase in
emission at these locations. Secondary shocks are also observed radially inward of the inner arrays
(towards the bottom right of the image in Figure Shocks a)). These structures remain for extended times
(> 100ns), and the bow shock angle is stationary over this period. This is a result of the slow timescale of
the change of the mass ablation rate (~100ns) compared to the flow transit time across the shock (~1ns),
and so the upstream flow is quasi-static. In addition, the ion-ion mean-free-path in the upstream flow is
<1m, and so is highly collisional on the shock length
scale of ~100m. Much of the broad dynamics of the
experiments were reproduced in simulations using the
3D MHD code Gorgon and this allowed an examination
of the density jump across the shock front and the local
sound and Alfven speeds relative to the flow velocity
(Fig. 4).
For a fixed configuration, increasing the radiation
loss rate from the downstream plasma reduces the
pressure here, and hence the shock angle will be
reduced. This was examined experimentally through
the use of tungsten wires in the array, rather than the
aluminum used for Figure 3. The resultant opening
angle was reduced from ~40° to ~15. It was also
hypothesized that the current, and hence B-field,
associated with the inner wires contributed to the shock
formation process. A simple test of this was to
disconnect the inner wires entirely from the current
drive to ensure the inner wires behaved purely as a
hydrodynamic obstruction. In this case, the shock angle
was again reduced with reference to the shocks in Fig
3. The small amount of current through the inner wires
(~1.2 kA at peak current) may provide a magnetic
Figure 4: Gorgon simulations of bow shocks
pressure sufficient to balance the kinetic (rv2) pressure
formed in nested wire arrays:
in the flow. Estimates suggested that this would occur
at a radius of 6 m from the inner wire position; comparable to radius the inner wires of 7.5 m. It is
therefore feasible that the magnetic field plays a role in the bow-shock formation, however it is also
5
possible that the small amount of current in the inner wires leads to a small expansion which also plays a
role.
These experiments demonstrated that the ablated plasma flow from ablating wires provides an
interesting test-bed for shock formation in radiatively cooled flows. Whilst the role of magnetic fields at the
obstruction was investigated, the exact role of the magnetic pressure could not be determined. The
proposed project will build upon and greatly expand this initial work, to include an accurate evaluation of
the role of the B-field at the target, and direct measurement of the density profile at the shock front. In
addition, the collisionality of the upstream flow will be reduced to give an ion mean-free path which a
greater than the shock spatial scale. In this case, the interaction of the flow with a magnetic pressure at
the target will produce a collisionless shock.
2.3
Suitability of the System for the Shock Studies Proposed
The plasma flow system to be used in this project has a number of useful properties and also some
advantages over previous schemes. In addition some of the conditions in the flow are readily altered
using simple means to examine shock formation in different conditions. The GenASIS pulsed power
generator to be used for the proposed work is a new linear transformer design on loan from Sandia
National Laboratories, and has previously been used by the PI to examine the ablation phase in wire
arrays, plasma jet production and x-pinch plasmas [20, 23]. The maximum current level which drives the
plasma flow can be varied by changing the charge voltage of the machine, which has been demonstrated
to deliver between 150 and 210 kA in ~150 ns. A second variable which can be used to change the flow
density is the wire number of the load. If a single wire is used, this takes all the drive current, and the
density profile can be calculated accordingly from the rocket ablation model (equation 1, and discussed
for a specific geometry below). If more wires are used, one of these is placed to drive the shock
experiments, and the remaining wires serve simply to reduce the current in the main wire (by inductive
splitting), and so reduce the ablated plasma density. In this way the
density arriving at a target position can be adjusted generate both
collisional and collisionless flow conditions.
jxB
j wire
Laser
interferometry
To demonstrate this we include calculations for a relatively
new array configuration called the inverse wire array z-pinch
Bglobal
originated at Imperial College London (see figure 5) [17]. Here a
central extended cathode (red) is used and the wires (black) are
strung concentrically to this at larger radius to connect to the anode
plane (grey). The current density in the wires interacts with the high
magnetic field produced around the cathode pin, to give a radially
RADIAL VIEW
Figure 5: Schematic of the
inverse wire array z-pinch
outwards flow.
In this geometry the plasma flow is accelerated away from the central hardware, and so provides
excellent diagnostic access to regions where a target may be placed to study shock formation. The rocket
ablation model can be extended to this geometry to estimate the flow density, given in equation 2 which
can then be used to calculate the ion-ion mean free path for given flow parameters (equation 3) [24]:
6
0 Z

r  R0 
n e L( r , t )  
I
(
t

)

2
Vabl 
4Vabl
N wires m p 
 perp 
2
(2)
2
3
mion
vabl
8Z 4 e 4 nion ln   / 2
(3)
From these, the flow density arriving at a target and the local ion-ion mean free path, mfp, can be
estimated as a function of time for a given geometry. For example, the array radius R 0 is taken as 4mm,
and a target is placed 6mm from this (at a total of 10mm from the system axis). The flow velocity is taken
as the ‘ablation velocity’ discussed early and is set at 1.5x105 ms-1. The plots in Fig 6 examine the areal
electron density, and mfp/L ratio where L is the characteristic size of the system, taken as 1 mm here.
The plot shows the variation of these parameters as a function of a time at the target position for four
cases: i) a single aluminum wire at 200 kA, ii) single tungsten wire at 200 kA iii) One of 4 tungsten wire
flows at 170 and iv) one of 8 aluminum wires at 150 kA . The collisionality difference between Al and W is
primarily in the ion mass and charge state which directly affect the mean free path, Al always having a
shorter mfp. The atomic number also changes the radiative cooling rate, so it is important to determine
which parameters can be independently adjusted.
Figure 6 a) shows the areal electron density calculated from equation 2 for an inverse wire array
load on GenASIS at UCSD. Close to peak current, the flow density varies only by a factor of 2 over
>120ns and is within a factor of 10% of the peak value for ~50ns. The shock upstream conditions can be
considered static on shock crossing timescale of ~5 ns for a 1 mm shock width (an upper estimate).
Indicated on the plot is the approximate range of densities quantifiable by the laser interferometer system
(laser = 532nm), which shows that at peak current all initial stream densities are measureable, and
importantly that the upper limit allows an order of magnitude increase to be measured since strong
density increases (i.e. >>4) are expected at the shock front. If necessary, the lower limit can be extended
by the used of Moiré deflectometry which decreases this limit by an order of magnitude [25].
Areal Electron Density / cm
-3
Ratio of mean free ion path to experimental
length scale (l=1mm)
Plasma density range diagnosable
by laser interferometry
1E18
1E17
1E16
1E15
W, single wire at 200 kA
W, 4 wires at 170 kA
1E14
Al, single wire at 200 kA
Al, 8 wires at 150 kA
W, single wire at 200 kA
W, 4 wires at 170 kA
1000
Al, single wire at 200 kA
Al, 8 wires at 150 kA
100
Collisionless
10
1
Collisional
0.1
1E13
50
100
150
200
250
50
300
100
Time / ns
150
200
250
300
Time / ns
Figure 6: Analytical plots showing the variation of a) areal electron density, b) ion-ion mean free path and
mfp/L as a function of time at a target position
The ratio of the ion-ion mean free path to the characteristic system size is given in Figure 6 b). If
single wires for Al and W are used a full generator charge, the target interaction is collisional for >100ns
7
in both cases. If the machine charge is reduced and the number of wires increased, the mean free path is
greater than the characteristic scale length throughout the entire experiment. These calculations
demonstrate that experiments can be designed to investigate the effect of radiative cooling in both
collisional and collisionless systems, where the flow density and shock structure can be directly quantified.
The use of a low current (<1 MA used in the previous study) compact generator is therefore ideal for
examining low collisionality shocks.
As discussed above, the upstream flow conditions are well known, and for example electron
density can be measured directly in the experiments. The agreement with simulations is generally very
good (e.g. [17]) and so values of the sound speed and Alfven velocity at an obstruction location can be
derived with confidence. It should be noted that whilst density profiles are well constrained, there currently
exists very little experimental data regarding the radial and axial temperature profile. Such data are taken
almost exclusively from the simulation work to enable calculations of plasma parameters (e.g. [16]). Part
of the proposed project aims to address this point directly, which will enable more accurate interpretation
of quantitative diagnostic methods such as interferometry where knowledge of the ionization state is vital.
The velocity in the flow is relatively invariant, but if the radiative cooling of the flow is changed, the
radial temperature profile and hence sound speed profile is also changed. This is achieved by using
different wire loads; for example changing from Al to W wires significantly increases the radiative cooling
loss rate. For a fixed geometry, the Mach number at the target can be changed from ~2 to ~10. Similarly,
for a given load, the placement of the target can determine the upstream Alfven Mach number since this
varies slightly with radial position (see figure 9). This is typically of order 2 for larger displacement of
target from the wire, but can be reduced to ~1 if the separation is reduced.
In addition to the above, the system outlined here has several inherent properties which are
advantageous in both the formation of the shock itself and the development of associated physical
processes. Many of these are discussed by Drake et al [26] in the context of designing a collisionless
shock system for a laser experiment. The flow is maintained for ~ 100ns, and this provides adequate time
for the development of turbulence at the shock, which is of order of a few ns for the parameter space here
[27]. The flow is directed by the jxB force into a large vacuum region, and so the location of experimental
walls can be maintain far from the experiment. The plasma beta also can be varied. Radiative cooling
determines the temperature as a function of propagation distance and so the thermal pressure can be
varied by changing the obstruction position, whilst the magnetic field at a target can be maintained
relatively constant (see below).
It is important to note that many of the controls of the flow parameters discussed above are simple
changes to the load of system geometry. This enables rapid investigation of the parameter range
available, and also makes the project amenable to direct involvement of both graduate and
undergraduate students. One disadvantage with high atomic number materials plasmas, is that larger Bfields are required to achieve an ion gyro-radius which is less that the system or shock front scale size.
However, the fields generated by pulsed power experiments, even using relatively low current levels, can
be significant. Taking 100kA in a rod of 1mm diameter gives a magnetic field strength of ~40 T (~400 kG).
This is sufficient to give a gyro-radius for a tungsten ion travelling at the ablation velocity of less than the
8
system characteristic spatial scale of ~1mm. For lower Z elements (e.g. Al) magnetization of the ions
occurs at lower field strengths, which are easily achievable.
Whilst several experiments and reviews are published in the use of laser facilities [28-34] and theta
pinches [35-37] for the possible generation of collisionless shock regimes this study represents the first
application of ablating wire plasmas in this area. The properties of the ablated plasma flows outlined
above strongly suggest a test-bed for hydrodynamic and magnetized shocks can be established, and that
a realistic attempt to generate a collisionless shock is possible. In addition, the strong computational
support and development of a hybrid - MHD particle code means experimental results can be closely
examined with the use of high resolution simulations.
2.4
Initial Studies at UCSD
To examine the proposed UCSD set-up, an
experimental undergraduate project was carried out in
summer 2010, supervised by the PI, which showed some
promising initial results. The inverse wire array z-pinch
geometry similar to that in Figure 5 was constructed and
tested using the one of the cases above, a single wire
tungsten load at 200 kA, and so a hydrodynamic
-2
could be demonstrated in each case. The first was a
simple
planar
target,
which
allows
analytical
approximation to a 1-dimensional shock. An example
interferogram and areal electron density lineout from this
experiment is given in Figure 7. The spatial resolution of
the laser image is ~20 m and the interferogram can be
resolved throughout much of the image. The areal density
lineout shows a rapid and a ten-fold increase is reported
over the quasi-planar shock formed. It is important to note
that an independent measure of the temperature was not
Areal Electron Density (x10) / cm
preliminarily examined to determine if shock formation
18
collisional interaction was expected. Two targets were
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
1000
2000
3000
4000
Distance from Wire position / microns
Figure 7: Single wire flow incident on a
planar target: (upper) schematic, (center)
interferogram, and (lower) electron
density lineout showing shock
possible in this experiment, and so the increase in ionization across the shock could not be quantified.
The increase in mass density is likely not as high as the increase in electron density.
The second geometry was a simple, easily diagnosable version of the nested wire array shock
region discussed in section 2.2. Here a 0.5mm diameter cylindrical target is placed through the plasma
flow to provide an obstacle. Simultaneous interferograms and dark-field Schlieren images are shown on
Fig 8, along with a line-out from the Schlieren image along wire-to-target line. The target is oriented into
the plane of the image, and continues approximately 1cm either side of the flow region. Note that the
vertical structure from the target is the support and is away from the plane of incidence of the flow. A clear
bow shock is formed around the obstacle and again a strong increase in density is observed. The
Schlieren image shows the region of strong electron density gradients (i.e. the shock front) is ~400m
9
across. This is likely to be an upper estimate since imperfections in the target orientation will give some
integration over the flow width.
Simulations of these experiments
were carried out by the Imperial College PI,
200
resistive
magneto-hydrodynamic
Contrast / arb. units
using an explicit, parallel version of the 3D
(MHD)
code GORGON [11]. The MHD equations
are solved on a three-dimensional grid in
either Cartesian or cylindrical geometry
the
single
fluid
Wire
150
100
50
0
0
using
Shock
Target
approximation.
1
2
3
4
Scale / mm
However, the ion and electron components
Figure 8: Bow shock formation experiments at UCSD:
of the plasma are allowed to be out of
(left) simultaneous interferogram and Schlieren image,
thermodynamic equilibrium with respect to
and (right) lineout through the shock in the Schlieren
each other and their
relative energy
image showing shock position and width.
equations are solved separately. The internal energy and the pressure are related by an equation of state,
and the average ionization state of the plasma, Z, is calculated from either a Saha model or an averageion Thomas-Fermi model. Radiation effects are included through either a single group radiation transport
algorithm, or an optically thin loss model. Simulations include the full electrode configuration for coupling
to the circuit model, along with a range of simulated diagnostics (XUV framing images, radiography,
radiated power) for direct comparison to data. A GORGON simulation for a 2-wire inverse wire array zpinch on the GenASIS generator is presented in Figure 9. This initial simulation shows reasonable
agreement with the electron densities recovered in the flow from experiments, and allows analysis of the
radial Mach and Alfven Mach number.
Figure 9: GenASIS exploder sims Demonstration of exploder calcs for GenASIS with plot of sonic
and Alfven mach numbers vs radial direction.
3. Proposed Program Overview
The program will investigate the effect of strong radiative cooling, entrained magnetic fields and
collisionality on the formation of shock in plasma flows. At each stage, simulation work at Imperial College
will parallel the experimental work. In this way an active dialogue between the two can be maintained,
and allows details or issues identified in the simulation work to be rapidly implemented in the experiments.
The proposal is based on previous published work, and successful previous and on-going
collaborations between the PI, co-PI and the Imperial College PI which have centred on the experimental
10
and computational analysis of ablating wire systems. As demonstrated above both the experimental and
simulation components of the program have been provisionally examined and so the proposed work
could start immediately if funded. The training of 2 graduate students in the experimental and
computational study of shock formation will be a fundamental part of the program.A detailed description of
the proposed research throughout a 3 year timescale is given below and a Program Timetable is given in
Section 5.
3.1 Proposed Experimental Studies and Computational Code Development
The program will progress from investigating hydrodynamic (collisional) only shock systems, to
collisional shocks with a significant B-field, and finally to collisionless, magnetized shocks. In each case
the effect of radiative cooling will be significant and can be varied by the choice of wire material. The
results from collisional shocks in Year 1 will provide a basis for comparison to the inclusion of magnetic
fields in Year 2, and these in turn will provide a comparison for the effect of collisionless interaction in
Year 3. At UCSD, several novel experimental arrangements will be implemented to give high diagnostic
access to the shock formation region. These experiments will use over-massed configurations, in that the
wires do not break on the timescale of the interaction, and simply act as a mass source for the plasma
flow. The generator can be rapidly reloaded and can conduct up to 10 experiments per day. Once an
experimental design is completed data acquisition rates are high and a substantial examination of simple
systems can be carried out over few months. At Imperial College, the graduate student and PI will be
responsible for the development of computational models to investigate the effects of magnetization and
ion mean free path on the structure and width of radiatively cooled shocks in supersonic flows.
The primary diagnostics currently available on the GenASIS
generator are laser interferometry and Schlieren, gated XUV emission
Wire
Load
Pinhole
imaging and x-ray power measurement. These can provide the gross
Emission
image
dynamics of the systems and quantify the electron density as a
function of space and time. Typically, laser backlighting and gated
XUV emission imaging are independently recovered along different
Laser image
lines of sight through the plasma. The direct comparison of, for
Figure 10: Schematic of
example, an electron density map from an interferometer image to a
co-linear laser and XUV
gated emission image is therefore not possible in this case. However,
emission imaging
it is possible to examine both these properties along a single line of
sight with a suitable experimental arrangement. A laser interferometry image of a plasma can be reflected
from a mirror in which a pinhole is mounted, and the self emission from the plasma can then be imaged
onto a gated camera (Figure 10). The pinhole diameter of 50-100m limits the transmission of low energy
light through diffraction and so the laser light will not impinge on the gated emission detector. The
reflected laser beam is imaged into a CCD camera as normal, the only difference being that the pinhole
position causes a small section at the center of the beam to be lost. At all positions away from this point,
the laser image and the emission image are taken along the same light of sight and simple fiducial
arrangement allow direct comparison of the images. The electron density from an interferogram can be
directly spatially correlated to the emission image. This arrangement has been constructed and is under
test at UCSD, and would be available for the proposed work. No funding is requested for this diagnostic.
11
During the first year, three experimental arrangements will be examined to facilitate a broad study
of the formation of radiatively cooled shocks in ablating wire plasma systems. In each system the shock
region is likely to be static or quasi static, which may simplify a detailed study of the shock, and the effect
of radiative cooling will be assessed by the use of different wire materials (eg, Al, Fe, W). Multiple frame
laser interferometry timed to image the initial collision will be used to determine both the pre– and postshock density as well as the propagation velocity of the plasma flow. In addition, a time-integrated grazing
incidence XUV spectrometer will be constructed to allow radially resolved spectra to be recovered, and
hence infer the temperature profile. The PI has previous experience operating this device for ablating wire
experiments [15], and this will be a valuable addition to the diagnostic suite throughout the proposal
timescale. The first experimental geometry (Fig 11a) is the same arrangement shown in the previous
section, and a thorough study of the evolution of this system will be conducted using different wire
materials to vary the radiative cooling rate.
Wire core
jxB
j wire
Laser
interferometry
Bglobal
Ablated Plasma
Stream
Interaction
Shock
Post-shock
stream
Generator Axis
jxB
jxB
Blocal
Current out
Current in
Figure 11: The three proposed experimental set-ups for radiatively cooled hydrodynamic shock formation
in Year 1: a) bow-shock formation, b) obliquely colliding flows and c) counter propagating flows
The second experimental arrangement will use a modification of the first to examine the interaction
of supersonic planar flows at small incidence angles (Fig 11b). If two wires are placed close to each other
the radial acceleration of the plasma flow also experiences a mutual attractive force. The resultant flows
from the two wires will therefore interact at an oblique angle a short distance from the initial wire position.
The strength of the local jxB force can be varied by changing the wire spacing and this provides a
mechanism to systematically vary the inter-flow collision angle. The basic dynamics of such a system was
studied in wire array geometry by the PI [38], but here investigation of the plasma flow interaction is of
interest. Specifically, the interaction of the stream will generate an oblique shock, and the downstream
flow velocity will be a function of the incident angle of the flows. These measurements then allow an
estimation of the downstream Mach number as the angle of incident of the interaction flows is varied. The
third experimental arrangement will examine counter-propagating flows (Figure 11c). Wires can be
mounted such that current passes through the wires in series, with close return current post providing a
jxB force to the common axis of the wires. No global magnetic field is induced, and current will not be
advected to the common axis providing a magnetic field free interaction. For these experiments using
collisional flows, the established Gorgon resistive MHD code will be used to model the ablating wires and
the acceleration of material. The magnetic Reynolds number can be established as a function of distance
from the wire and used to determine the correct position for ideal MHD collision experiments. The detailed
atomic physics models presently under development for inclusion into the Gorgon code will be used to
establish the correct level of radiative cooling and shock compressibility for these simulations.
12
During the second year, a magnetic field will be introduced to examine the limit of compressibility
generated by the additional pressure in the shock region. The bow shock case can be modified by
connecting the original target to the current drive and allowing the current to split between this and the
ablating wire (Fig 12a). The ablation flow impinges on the B-field at the target, which can be varied by
modifying the diameter of the target and current through it. The B-field configuration shown in Fig 12a is
azimuthal, but this can be simply altered to give a B-field in the vertical direction by redesigning the target
to pass horizontally through the plasma plane, as in the test experiments discussed in section 2.4
Wire core
Laser
interferometry
j wire
jxB
dI /dt
jtarget
Ablated Plasma
Stream
Magnetized
Shock
Generator Axis
jxB
jxB
Blocal
Btarget
Current out
Current in
Figure 12: Experimental set-ups for shock formation with significant embedded magnetic fields in Year 2:
a) bow-shock formation, b) obliquely colliding flows and c) counter propagating flows
For the obliquely interacting flows, the extent to which current is advected in the flow is determined
by the wire positioning [38]. Closer spaced wires create a greater jxB force from the field encompassing
both the wires and the resultant force causes the shock region to form close to the wires. This, in turn
allows a greater current to flow here and the B-field induced will affect the shock formation. In addition,
wire material may play a role in this, and cylindrical wire arrays formed from Ni in particular have shown
greater current convection downstream [39]. The counter-propagating flows case can be altered to
include a B-field, simply by changing the wire mounted to a more typical wire array arrangement. If the
wires are mounted in parallel from cathode to anode, the current flow defines a global magnetic field and
hence a common axis as in the case for wire array z-pinches (Figure 12c). Once plasma initially reaches
the axis, the lack of inductive shielding of the axis by the wires allows a significant amount of current, and
hence magnetic field, to be advected in the plasma flow here.
Experimentally determining the magnetic field strength inside shock structures is a difficult process.
However, the gross observables can be obtained in a similar way to the hydrodynamic work in Year 1,
and interpretation of the compression at the shock front can be achieved by comparison to accurate
simulation of the plasma conditions. In addition, miniature magnetic probes [40] can be placed
immediately above and below the shock region to assess the field outside the shock region. These data
can taken from positions relatively far from the shock and used as constraints in the simulations work, to
be carried out by the Imperial College graduate student. This will examine the effectiveness of flux
compression with applied magnetic fields and the details of the magnetic field strength and distribution.
This process is a fundamental part of the proposed studies, and close collaboration of the experimental
simulation and theory teams will enable an accurate interpretation.
In the third year, testing with the three configurations will be carried out in the collisionless regime.
Here the flow density will be decreased by altering the generator change or array wire number, as
discussed in Section 2.3. Experiments will be designed to satisfy the collisionless criterion (mfp > shock
13
scale length) based on the actual shocks examined in Years 1 and 2. The Imperial College student and PI
will be responsible for the development of particle in cell and hybrid models which can be used to model
shock formation in these collisionless shocks. The gradual transition of a collisionless shock into a
collisional shock was previously studied at Imperial College using a hybrid (fluid electron, kinetic ion)
model of the precursor plasma in a wire array Z-pinch [24]. This work has been subsequently extended to
investigate the effect of magnetic fields on high energy, kinetic ion species within burning thermonuclear
plasmas [41] and in relativistic electro-magnetic PiC codes used to model plasma formation in
magnetically insulated transmission lines. These models make use of the Monte-Carlo method for
Coulomb collision in hybrid plasma models developed by Sherlock [42]. Adaptation of these models to
modelling these experiments will allow the trajectories of ions with large mean free paths within the
magnetic field compressed within the shocked region to be solved self consistently with the flux
compression established by the plasma motion. The effective increase in shock width as a result of
increasing mean path and the localisation of the shock by magnetization of a collisionless plasma can
then be studied and the resulting shock structures compared to directly to experimental observations in
the different regimes. At the end of the project timescale, the code developments and extensive
comparison to experiments will have established a solid platform for shock studies using pulsed powerdriven ablating wire systems. This advance will allow the trajectories of energetic electrons injected from
an external source to be followed through a measured magnetized shock region, enabling. the
computational design of a future experiment to measure particle acceleration within magnetized
collisionless shocks.
4.
Intellectual Merit and Broader Impact of the Proposed Studies
Intellectual Merit: The proposed program has significant potential in several areas related to the call,
primarily Flows in Plasmas, their Interaction and Interpenetration and Plasmas in Magnetic Fields, but
also in, Turbulence and Structure in Plasmas, Advanced Methods for Plasma Modeling and Simulation,
and Astrophysical and Solar Plasmas and related areas. The program will carry out an extensive
experimental and computational investigation of shock formation, and will be directed to investigate the
effect of three crucial parameters: radiative cooling, on shock formation of embedded magnetic fields on
fluid compressibility and collisionality. The interplay of these physical processes investigated are
fundamental to many plasma systems, particularly in astrophysical plasmas including the Earth’s
magnetosphere, supernovae and cosmic ray acceleration.
The use of ablating wire systems represents a novel platform for radiatively cooled, magnetized
and collisionless shock studies. The results obtained will significantly extend current understanding in the
areas investigated by providing a systematic approach the analysis of each parameter. The long
timescales possible will allow a thorough investigation of the formation and final state of shock structures,
which is not possible on impulse driven systems using high power lasers, and so the studies proposed
here provide a complementary approach to previous and current work in this area. The experimental
campaign will be supported by a 3D MHD code which will be developed into a fully MHD-kinetic hybrid
code capable of providing detailed analyses of the shock formation and evolution. In addition, the
14
program will form the basis for future experiments to examine the effect of the shock details on particle
acceleration in the laboratory.
Broader Impact: The development of the XUV spectrometer proposed will a valuable addition to the
laboratory capabilities at UCSD, and significantly extend the range of data that can be recovered from
experiments. The program will train two graduate and one undergraduate student in this wide range of
experimental methods which are applicable to many areas of plasma and HED science, as well as
providing an opportunity to experience computational plasma physics and the inherent cross-over
between experiment and simulation. Data resulting from the research program will be broadly
disseminated by publication in scientific journals, and presentation at international and national
conferences and workshops. Graduate and undergraduate students will be encouraged to present results,
particularly at UCSD Graduate and Undergraduate research Expos and events such as the HEDP
Summer School organized by LLE at Rochester. Presently, the PI supervises the experimental research
of three graduate and two undergraduate students in the HEDP laboratory, and regularly provides
guidance for summer students. In additional, the PI supervised a project at UCSD as part of the Princeton
National Undergraduate Fellowship program in 2009, and each year proposes an undergraduate
research project in the HEDP laboratory. Students are regularly authors or co-authors on publications and
presentations from the group.
5. Program Timetable
Year 1: Hydrodynamic Shocks
1. Design and construction of time-integrated flat field XUV spectrometer
2. Examine hydrodynamic (collisional) shock formation in bow-shock, counter-propagating flows and
obliquely propagating flow systems
3. Simulate all three experimental geometries using GORGON, including detailed atomic routines
for radiation loss
4. Analysis and publication of results
Year 2: Effect of B-field on plasma compressibility
1. Purchase of MCP and completion of time-resolved XUV spectrometer
2. Examine effect of significant B-field on compressibility on hydrodynamic (collisional) shock
formation in two of the experimental systems investigated in Year 1
3. Simulations of experiments to assess B-field at shock
4. Begin implementation of kinetic routines in GORGON code
5. Analysis and publication of results
Year 3: Low collisionality magnetized shocks
1. Examine magnetized systems from Yr 2 using collisionless flows
2. Completion of kinetic routines in GORGON and simulations of experiments
3. Analysis and publication of results
4. Computational design of particle acceleration experiments
15
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