j.p. chittenden- two and three-dimensional modelling of the different phases of wire array z-pinch...
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Two and Three-dimensional Modelling of the Different Phases of
Wire Array Z-pinch Evolution
Dr. Jeremy P. Chittenden,William Penney Research Fellow,
Plasma Physics Group, Blackett Laboratory,
Imperial College of Science, Technology and MedicinePrince Consort Road, London, SW7 2BZ, U.K.
tel. 44 207 594 7650, fax. 44 207 594 7658, email. j.chittenden@ic.ac.uk APS-DPP, Quebec 2000
J.P. Chittenden
Imperial College
In collaboration with
S.V. Lebedev, S.N. Bland, F.N. Beg, J. Ruiz-Camacho,A.Ciardi, C.A. Jennings, A.R. Bell, and M.G. Haines
from Imperial College,
With additional experimental data from
S.A. Pikuz, T.A. Shelkovenko,
from P.N. Lebedev Physical Instituteand D.A. Hammer from Cornell University
With grateful thanks for funding from
the AWE William Penney Fellowship scheme
Sandia National Laboratories
and the U.S. Department of Energy
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Assumption of rapid shell formation followed by 2D(r,z) MRT instability omits
plasma formation effects and other important 3D phenomena
APS-DPP, Quebec 2000
If we were to assume that the initial flow of current causes rapid and uniform explosion of the wires then analmost uniform cylindrical shell of plasma results. THIS DOESNT HAPPEN
100 120 140 160 180 200 2200
2
4
6
8
thin shell
0D model
spikes
innermost
bubble
Radiusinmm
Time in ns
The growth of the magneto-Rayleigh-Taylor instability is then responsible
for shell broadening which determines the X-ray rise-time. THIS IS NOTTHE ONLY EFFECT AND SOMETIMES ISNT IMPORTANT AT ALL.
Rise-time ~ shell thickness / velocity
THIS CANNOT EXPLAIN LOWWIRE NUMBER RESULTS
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Wire arrays cover a wide range of parameters
but exhibit the same physical processes
APS-DPP, Quebec 2000
Owl II,
6x20m Al, 7mm SATURN
64x15m Al, 17mm MAGPIE
64x15m Al, 16mm MAGPIE 16x15m Al, 16mm + 16x15m Al, 8mm
0 50 100 150 200 2500
5
10
15
20
Z
SATURN
SATURN
Long Pulse
MAGPIEOwl II
CurrentinMA
Time in ns
Z, 240x7.5m W, 40mm + 120x7.5m W, 20mm
A wide range of materials and
diameters are used
Total currents vary considerably
but currents per wire andinter-wire gaps are similar
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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MAGPIE wire array experiments show intrinsically 3D phenomena
with scales ranging from a few m to several mm
APS-DPP, Quebec 2000
Side-on laser schlieren,
r-z modulation
(m=0 like instabilities in each wire?)
End-on laser interferometer,
r- modulation
radial plasma streams
Side-on X-pinch X-ray back-lighter
reveals dense wire cores embedded
within the coronas
279ns At late times, structure
apparently resembles aglobal Rayleigh-Taylor
instability
For details on experiments seeDO2.007
MP1.084
WO2.006
Simultaneous laser schlieren showsrelative size of coronas
16mm
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Talk Outline
APS-DPP, Quebec 2000
Philosophy
Bench-mark 2D and 3D models in detail
against MAGPIE wire array data and
several single wire experiments.
Use these models to understand behaviour of similarexperiments at higher currents on SATURN, Z, X1..
Cannot model whole problem (3D + global & finescale structures) simultaneously. Therefore model
different phases separately and attempt to link them
0 50 100 150 200 2500
2
4
6
8
10
Nested array
interaction
Stagnation and
X-ray generatio
Global instability
development
Coronal merger,
mass injection and
precursor formation
Instability growth
in each wire plasma
Wire Initiation
(Plasma Formation)
Radiu
sinmm
Time in ns
Research Topics
1. 1D and 2D(r,z) cold-start single wire calculations :-
formation of the core-corona structure,
m=0 instability growth in individual wire plasmas.
2. 2D(r-) plane calculations:-how core-corona structure affects dynamics
radial plasma streams, coronal merger, precursor.
the physics of what controls the core ablation rate
3. A brief discussion of the physics of the precursor
4. 2D(r-) plane calculations of nested wire arrays :- momentum and current transfer during collision
how these determine which mode of implosion results
5. 3D simulations of a single wire in an array :-
origins of local and global perturbations differences in behaviour from single wires
structure and trajectory of implosion
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Plasma formation in wires depends on complex EOS and transport coefficients
APS-DPP, Quebec 2000
j2 dt exceeds energy budget to heat, melt, vaporise and ionise all material in wires within a few ns.However this energy is not deposited uniformly, formation of a plasma corona greatly reduces energy transfer
rate to cold, dense wire core, allowing it to survive until late times.
101
102
103
104
10-5
10-4
10-3
10-2
10-1
100
101
10 eV
3 eV
1 eV
0.3 eV
0.1 eV
0.03 eV
perfect gas
condensation
degeneracy
P
ressure(MBar)
Density (kg/m3)
0.1 1 10 100 100010
-8
10-7
10-6
10-5
solid /10
solid/3
solid
Melting point
Spitzer - like
Res
istivityinm
Temperature in eV
Modified Thomas-Fermi Equations of State
In condensed phase electron pressure is allowed to go
negative, so that total pressure is zero.This is an oversimplification, but appears to work.
Numerically such an EOS is a pain to use. However
after a few ns, core expansion is sufficient for it to beapproximated by a cold unionised gas.
Lee and Mores transport model
Modifications to transport remain important long after
modifications to EOS, not least because Ohmic heatingis found to be the dominant mechanism for energy
transfer to the core.
Considerable uncertainty remains over the resistivtiesaround 1-10eV [see GP1.066 M.P. Desjarlais]
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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1D cold-start MHD simulations show formation of core-corona structure
APS-DPP, Quebec 2000
Consider a single 15m Aluminium wire with ~1kA/ns current
0 100 200 300 4001E-4
1E-3
0.01
0.1
1
10
100
1000
Density(kg/m3)
0 100 200 300 400.01
0.1
1
10
100
1000
Te
Ti
Z*
Temperature
(eV)
0 100 200 300 4000.0
5.0x109
1.0x1010
1.5x1010
2.0x1010
2.5x1010
CurrentD
ensity(A/m2)
Radius in m
0 100 200 300 400.0
5.0x106
1.0x107
1.5x107
2.0x107
Eqn. of State
Perfect Gas
TotalPressure(Pa)
Radius in m
0 100 200 300 4001E-4
1E-3
0.01
0.1
1
10
100
1000
Density(kg/m3)
0 100 200 300 4000.01
0.1
1
10
100
1000
Te
Ti
Z*
Temperature
(eV)
0 100 200 300 4000.0
2.0x1011
4.0x1011
6.0x1011
8.0x1011
1.0x1012
1.2x1012
CurrentD
ensity(A/m2)
Radius in m
0 100 200 300 4000.0
5.0x108
1.0x109
1.5x109
2.0x109
2.5x109
Eqn. of State
Perfect Gas
TotalPressure(Pa)
Radius in m
10ns 25ns
Once vaporised core expands at roughly its sound speed.
Surface regions drop to low density and are readily
ionised.
Current gradually transfers from core to corona, which
heats and expands.
Core pressure
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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In 2D(r,z) cold-start simulations of a single wire,
pinching of the corona excites the m=0 instability
APS-DPP, Quebec 2000
0 1 20
1
2
3
4
> 1.
0.3 - 1.
0.1 - 0.3
0.03 - 0.10.01 - 0.03
0.003 - 0.01
0.001 - 0.003
0.0003 - 0.001
0.0001 - 0.0003
R axis in mm
Zaxisinmm
0 1 2
R axis in mm
0 1 2
R axis in mm
0 1 2
R axis in mm
0 1 2
R axis in mm
0 1 2
R axis in mm
20ns 25ns 30ns 35ns 40ns 45ns
Short wavelengths at early times give way to longer wavelengths as plasma expands so that / radius roughly constant
Current by-passes contorted path through flares and flows through narrow region just outside the core.
Initially necks fail to penetrate core which remains virtually unperturbed.
Since the core retains the majority of the mass, when the necks eventually penetrate to the axis, this represents a
dramatic increase in total perturbation amplitude.
Depletion of the core material in the region of penetration results in high temperatures and X-ray bright-spots
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Comparison to single wire data provides benchmark tests for
2D MHD code plus EOS and transport models therein
APS-DPP, Quebec 2000
Experimentat 51ns
Simulationat 51ns
Experimentat 85ns
Simulationat 85ns
0 20 40 60 80 100 1200.0
0.2
0.4
0.6
0.8
1.0
1.21.4
1.6
1.8Corona Min.
Corona Max.
Core Min.
Core Max.
Corona Exp.
Core Exp.
Radius(mm)
Time (ns)
For example comparison to laser probing and X-pinch radiography of
100m Al wires at Cornell [D. Kalantar and D. Hammer, Phys. Rev.Lett. 71, 3806 (1993)] allows simultaneous tests of wavelength and
amplitude of m=0 in corona plus core expansion.
Alternatively recent quantitative X-pinch radiography of low current
Al wires at Cornell [S.A. Pikuz and T.A. Shelkovenko] provides more
detailed test of core expansion -0.4 -0.2 0.0 0.2 0.40
20
40
60
80
100
120
140
160
ArealDe
nsity(g/cm
2)
Radius (mm)
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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3D behaviour of wires in arrays limits the application of
2D single wire calculations to scaling arguments
APS-DPP, Quebec 2000
M=0 instability in single wires
Amplitude and wavelength increase as
corona expands
Necks penetrate cores forming X-ray
bright-spots
Growth dependent on current per wire
0 1 20
1
2
3
4
R axis in mm
Zaxisinmm
0 1 20
1
2
3
4
R axis in mm
Zaxisinmm
24ns 36ns 48ns 24 x 25m Alon SATURN
64 x 15m Alon SATURN
148ns
Instability (or just modulation ?) in wires
in arrays
Amplitude, wavelength and size in
azimuthal direction are almost constant in
time
X-ray bright-spots not observed before
global implosion initiates
Growth is a weak function of current per
wire
~90% of mass remains in core
For low current per wire, instability
doesnt penetrate core, perturbation
amplitude remains small
For higher current per wire (N 1eV current transferred to
cores,
somejxB force applied directly
cores rapidly heat and expand
Trajectory similar to thin shell model T
R
Close
cores
jz
Reduces injection of material and
current between cores.Trajectory similar to thin shell model
T
R
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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The precursor plasma is an apparently stable, uniform and long-lived, 1D plasma.
APS-DPP, Quebec 2000
carbon aluminium tungsten
Gated soft X-ray images of
precursor indicate that
equilibrium radius is a strongfunction of material
Can be modelled in high resolutionwith 1D MHD
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
1.2 stationary precursor
flux through boundary
radially convergent streamDens
ityinKg/m3
Radius in mm0.0 0.5 1.0 1.5 2.0 2.5 3.00
10
20
30
40
50
60
TemperatureineV
Radius in mm
0.0 0.5 1.0 1.5 2.0 2.5 3.00
2
4
6
8
10
12
Zsta
r
Radius in mm
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-1.5x105
-1.0x105
-5.0x104
0.0
Vr
inms-
1
Radius in mm
Precursor lifetime > 100ns
Initial formation phase is collisionless.
Once collisonal, converges to a two
component equilibrium of high densitystationary precursor and lower densityconvergent radial plasma stream.
Pressure balanced by v2 of bombardingstream. Little or no current.
Kinetic energy delivered (v3A) is theroughly balanced by radiation losses.
Similar to a test developed to evaluatedifferent artificial viscosity formulations
in 1D hydrodynamics (W.F. Noh,J. Comp. Phys. 72, p78 (1987).
Density ratio between precursor and
stream ~[(+1)/(-1)]2. Data suggestsfor Al 5/3. For W precursor densitymuch higher 1.1
Ideal test-bed for opacity measurements,
X-ray laser experiments and
benchmarking radiation hydrodynamicscodes.
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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There are at least 3 different theoretical modes of nested wire array dynamics
APS-DPP, Quebec 2000
Hydrodynamic Collision (or Shell on Shell) Mode
60 70 80 90 100 110 12002
468
101214161820
Time in ns
Transparent Inner (or Current Transfer) mode
60 70 80 90 100 110 12002468
1012
14161820
Time in ns
Flux Compression (or Magnetic Buffer) mode
60 70 80 90 100 110 12002468
101214161820
Outer
Inner
Time in ns
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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2D(x,y) simulations reproduce collapse dynamics of nested arrays on MAGPIE
APS-DPP, Quebec 2000
Outer
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
90 ns
X Axis in cm
YAxisincm
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
150 ns
X Axis in cm
YAxisincm
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
200 ns
X Axis in cm
YAxisincm
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
230 ns
X Axis in cm
YAxisincm
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
240 ns
X Axis in cm
YAxisincm
Inner
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
90 ns
X Axis in cm
YAxisincm
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
150 ns
X Axis in cm
YAxisincm
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
200 ns
X Axis in cm
YAxisincm
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
230 ns
X Axis in cm
YAxisincm
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
240 ns
X Axis in cm
YAxisincm
16+16 x 15m Aluminium(equal length arrays)
Inner wires heated by bombarding plasmastreams from outer
Small fraction of current flowing throughinner array produces B between arrays
Compression of this flux by implosion of
outer produces sufficient current to driveinner ahead of outer.
160 180 200 220 240 2600
1
2
3
4
5
6
7
8
9Outer Sim.
Inner Sim.
Outer Expt.
Inner Expt.
Radiu
sinmm
Time in ns
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Model inner and outer arrays on Z separately, first calculate radial plasma
flux from outer array, then use this to bombard the inner array.
APS-DPP, Quebec 2000
240x7.5m tungsten wires on a 40mm diameter
-0.2
0.0
0.2
20 ns
-0.2
0.0
0.2
40 ns
-0.2
0.0
0.2
60 ns
Y(or)Axisin
mm
18.5 19.0 19.5 20.0
-0.2
0.0
0.2
X (or R) Axis in mm
70 ns
Similar features to lower wire number cases,
Dense wire cores retain most of the mass untilimplosions commences.
Low density corona swept around cores formingradial plasma streams.
At 75ns precursor stream extends down to 6mmand contains 20% of mass.
In 2D, the remaining 80% is in a 1mm wide shell
After this stage the plasma is largely
homogeneous in the azimuthal direction.
Use the flux through the LHS of the outer array
simulation to provide RHS boundary conditionsfor simulation of an inner array wire.
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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2D(x,y) simulations predict the implosion modes of nested arrays on Z
APS-DPP, Quebec 2000
Inner array of60x10.5 m W wires Plasma stream from outer of 240x7.5m W83 ns
8.0 8.5 9.0 9.5 10.0 10.5 11.0-0.50
-0.25
0.00
0.25
0.50
Y(or)Axisinmm
8.0 8.5 9.0 9.5 10.0 10.5 11.0-0.50
-0.25
0.00
0.25
0.50
Y(or)Axisinmm
98 ns
8.0 8.5 9.0 9.5 10.0 10.5 11.0-0.50
-0.25
0.00
0.25
0.50
Y(or)Axisinmm
8.0 8.5 9.0 9.5 10.0 10.5 11.0-0.50
-0.25
0.00
0.25
0.50
Y(or)Axisinmm
102ns
8.0 8.5 9.0 9.5 10.0 10.5 11.0-0.50
-0.25
0.00
0.25
0.50
Y(or)Axisinmm
8.0 8.5 9.0 9.5 10.0 10.5 11.0-0.50
-0.25
0.00
0.25
0.50
Y(or)Axisinmm
104ns
8.0 8.5 9.0 9.5 10.0 10.5 11.0-0.50
-0.25
0.00
0.25
0.50
Y(or)Axisinmm
8.0 8.5 9.0 9.5 10.0 10.5 11.0-0.50
-0.25
0.00
0.25
0.50
Y(or)Axisinmm
107ns
8.0 8.5 9.0 9.5 10.0 10.5 11.0-0.50
-0.25
0.00
0.25
0.50
X (or R) Axis in mm
Y(or)Axisinmm
8.0 8.5 9.0 9.5 10.0 10.5 11.0
-0.50
-0.25
0.00
0.25
0.50
X (or R) Axis in mm
Y(or)Axisinmm
Inner array wires see 20ns of low
density coronal bombardmentfollowed by main mass in 1mm
wide shell.
At first little expansion of inner
wires outer material streamsthrough, setting up bowshock.
Later bombardment by densermain mass heats each wire with
~100GW of kinetic flux.
Inner wires expand rapidlyallowing effective momentum
transfer. Compression of
magnetic flux carried by plasmastream effectively increases
momentum transferred.
Trajectory similar to hydro-
dynamic collision mode withreduced radiation at collision.
Trajectories consistent with
transparent inner mode require
30 wires in inner.
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Even with 30% amplitude perturbation on (z) with 0.5mm wavelength,apparent modulation is much less than in experiment
APS-DPP, Quebec 2000
50 ns
80ns
100ns
Side-on laser schlieren of Al arrays on MAGPIE show:modulation in corona from ~60ns
roughly constant amplitude (r+ - r-) and wavelength
3D MHD simulation shows:
initial modulation amplitude retained in core & corona
no apparent growth or change in modulation
no apparent difference in cross-section for differentaxial positions
Maybe this isnt an MHD instability at all ?
r
z
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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3D simulation of m=0 instability in ideal MHD equilibrium pinch:
growth rate agrees well with analytic theory
APS-DPP, Quebec 2000
Similar results have been obtained for m=1 instabilities [S.G. Lucek, private communication]
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Modulating core resistivity versus z, gives results similar to experiment
APS-DPP, Quebec 2000
150ns 220ns 240ns
7.0 7.5 8.0 8.5-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
R Axis in cm
ZAxisinmm
7.0 7.5 8.0 8.5-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
R Axis in cm
ZAxisinmm
7.0 7.5 8.0 8.5-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
R Axis in cm
ZAxisinmm
Lower core resistivity in centre, higher core resistivity at ends modulated core heating and ablation.During implosion wire core breaks, current penetrates inside wire array, cold core regions left behind.
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8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution
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Two and Three-dimensional Modelling of the Different Phases of
Wire Array Z-pinch Evolution
APS-DPP, Quebec 2000
Conclusions
2D cold-start models illustrate important processes involved in plasma formation phaseand provide model verification through comparison to single wire experiments.
Absence of 3D effects, however, severely limits their ability to predict the behaviour of
wires in an array.
2D(x,y) simulations show how the flow of material ablating from the core is redirected by
jB forces forming the radial plasma stream and the precursor.
Require better resistivity models to cover all array parameters.
2D(x,y) simulations of nested arrays model momentum transfer and magnetic flux
compression during collision.
All shots to date on Z have been hydrodynamic collision like, transparent inner mode
requires fewer wires on the inner and larger (>1.5mm) inter wire gaps.
Preliminary 3D modelling suggests MAGPIE data can be explained in terms of wire
breaking and not necessarily Rayleigh-Taylor.
Can this model be extrapolated to Z ?
Moreexperimental data is needed.
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