Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution 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 grateful thanks for funding from the AWE – William Penney Fellowship scheme Sandia National Laboratories
With additional experimental data from S.A. Pikuz, T.A. Shelkovenko, from P.N. Lebedev Physical Institute and D.A. Hammer from Cornell University Dr. Jeremy P. Chittenden, William Penney Research Fellow, Plasma Physics Group, Blackett Laboratory, Imperial College of Science, Technology and Medicine Prince Consort Road, London, SW7 2BZ, U.K. tel. 44 207 594 7650, fax. 44 207 594 7658, email.
[email protected]
and the U.S. Department of Energy
APS-DPP, Quebec 2000
Assumption of rapid shell formation followed by 2D(r,z) MRT instability omits plasma formation effects and other important 3D phenomena
If we were to assume that the initial flow of current causes rapid and uniform explosion of the wires then an almost uniform cylindrical “shell of plasma” results. THIS DOESN’T HAPPEN
Radius in mm
8 6 4
spikes innermost bubble
2
thin shell 0D model
0 100 120 140 160 180 200 220
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 NOT THE ONLY EFFECT AND SOMETIMES ISN’T IMPORTANT AT ALL.
Rise-time ~ shell thickness / velocity THIS CANNOT EXPLAIN LOW WIRE NUMBER RESULTS APS-DPP, Quebec 2000
Wire arrays cover a wide range of parameters but exhibit the same physical processes
Owl II, 6x20µm Al, 7mm φ
SATURN 64x15µm Al, 17mm φ
MAGPIE 16x15µm Al, 16mm φ + 16x15µm Al, 8mm φ
MAGPIE 64x15µm Al, 16mm φ 20
Z
Current in MA
15
10 SATURN
SATURN Long Pulse
5 Owl II
0
0
50
100
MAGPIE
150
200
250
Time in ns Z, 240x7.5µm W, 40mm φ + 120x7.5µm W, 20mm φ
A wide range of materials and diameters are used
Total currents vary considerably but currents per wire and inter-wire gaps are similar APS-DPP, Quebec 2000
MAGPIE wire array experiments show intrinsically 3D phenomena with scales ranging from a few µm to several mm Side-on laser schlieren, r-z modulation (m=0 like instabilities in each wire?)
279ns
End-on laser interferometer, r-θ modulation radial plasma streams
Side-on X-pinch X-ray back-lighter reveals dense wire cores embedded within the coronas
At late times, structure apparently resembles a global Rayleigh-Taylor instability For details on experiments see DO2.007 MP1.084 WO2.006 16mm
Simultaneous laser schlieren shows relative size of coronas APS-DPP, Quebec 2000
Talk Outline Philosophy
Research Topics
Bench-mark 2D and 3D models in detail against MAGPIE wire array data and several single wire experiments.
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.
Use these models to understand behaviour of similar experiments at higher currents on SATURN, Z, X1..
2. 2D(r-θ) plane calculations:how core-corona structure affects dynamics radial plasma streams, coronal merger, precursor. the physics of what controls the core ablation rate
Cannot model whole problem (3D + global & fine scale structures) simultaneously. Therefore model different phases separately and attempt to link them Wire Initiation (Plasma Formation) Instability growth in each wire plasma Coronal merger, 10 mass injection and precursor formation
3. A brief discussion of the physics of the precursor 4. 2D(r-θ) plane calculations of nested wire arrays :momentum and current transfer during collision how these determine which mode of implosion results
Radius in mm
8
Global instability development
6
Nested array interaction
4
Stagnation and X-ray generation
2
0
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
0
50
100
150
200
250
Time in ns
APS-DPP, Quebec 2000
Plasma formation in wires depends on complex EOS and transport coefficients ∫ ηj2 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. -5
10 10
0 -1
10
-2
10
-3
10
-4
10
-5
10 eV 3 eV perfect gas
1 eV 0.3 eV
condensation
0.1 eV 0.03 eV
10
1
10
2
10
3
10
4
3
Density (kg/m ) 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 be approximated by a cold unionised gas.
ρsolid /10 ρsolid /3
degeneracy
Resistivity in Ωm
Pressure (MBar)
10
10
1
-6
10
ρsolid -7
10
Spitzer - like Melting point
-8
10
0.1
1
10
100
1000
Temperature in eV Lee and More’s transport model Modifications to transport remain important long after modifications to EOS, not least because Ohmic heating is found to be the dominant mechanism for energy transfer to the core. Considerable uncertainty remains over the resistivties around 1-10eV [see GP1.066 M.P. Desjarlais] APS-DPP, Quebec 2000
1D “cold-start” MHD simulations show formation of core-corona structure Consider a single 15µm Aluminium wire with ~1kA/ns current
1E-3 100
200
300
400
3
Te Ti
1 0.1
0.01 0 7 2.0x10
Z 100
200
10
1.5x10
10
1.0x10
10
5.0x10
9
0.0
0
Total Pressure (Pa)
2.0x10
100
200
300
Radius in µm
400
1.5x10 1.0x10 5.0x10 0.0
300
*
Eqn. of State Perfect Gas
7
7
6
0
100
200
300
Radius in µm
400
1 0.1 0.01 1E-3
1E-4 0 12 1.2x10
400
Temperature (eV)
0.01
10
1.0x10
12
8.0x10
11
6.0x10
11
4.0x10
11
2.0x10
11
100
200
300
0.0 0
400
Total Pressure (Pa)
0.1
Density (kg/m )
1
10
2
10
1000
100
100
2
Current Density (A/m )
Temperature (eV)
3
Density (kg/m )
100
1E-4 0 10 2.5x10
1000
1000
Current Density (A/m )
1000
100
200
300
Radius in µm
400
100 10
Te Ti
1 0.1
0.01 0 9 2.5x10
Z 100
200
300
*
400
Eqn. of State Perfect Gas
9
2.0x10
9
1.5x10
9
1.0x10
8
5.0x10 0.0
0
100
200
300
Radius in µm
10ns 25ns Once vaporised core expands at roughly it’s sound speed. Magnetic field becomes sufficient to pinch corona back onto core, triggering increased current flow and Ohmic heating core. Surface regions drop to low density and are readily ionised. Core continues to expand at roughly constant speed. Current gradually transfers from core to corona, which heats and expands. Density is now low enough for pressure to be close to perfect gas value. Core pressure 1. 0.3 - 1. 0.1 - 0.3 0.03 - 0.1 0.01 - 0.03 0.003 - 0.01 0.001 - 0.003 0.0003 - 0.001 0.0001 - 0.0003
4
0
1
2
0
2
1
0
1
2
0
0
1
2
R axis in mm
R axis in mm
R axis in mm
R axis in mm
R axis in mm
25ns
30ns
35ns
40ns
45ns
Short wavelengths at early times give way to longer wavelengths as plasma expands so that λ / 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” APS-DPP, Quebec 2000
Comparison to single wire data provides benchmark tests for 2D MHD code plus EOS and transport models therein 1.8 Corona Min. Corona Max. Core Min. Core Max. Corona Exp. Core Exp.
1.6
Radius (mm)
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
Experiment at 51ns
Simulation at 51ns
Experiment at 85ns
0
20
40
Simulation at 85ns
60
80
100
120
Time (ns)
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
140
2
For example comparison to laser probing and X-pinch radiography of 100µm 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.
Areal Density ( µg/cm )
160 120 100 80 60 40 20 0
-0.4
-0.2
0.0
0.2
0.4
Radius (mm)
APS-DPP, Quebec 2000
3D behaviour of wires in arrays limits the application of 2D single wire calculations to scaling arguments
Necks penetrate cores forming X-ray bright-spots Growth dependent on current per wire
4
4
3
3
Z axis in mm
Amplitude and wavelength increase as corona expands
Z axis in mm
M=0 instability in single wires
2
1
0
1
0
1
2
R axis in mm
24ns
36ns
48ns
24 x 25µm Al on SATURN 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
148ns
Growth is a weak function of current per wire
2
0
0
1
2
R axis in mm
64 x 15µm Al on SATURN
~90% of mass remains in core For low current per wire, instability doesn’t penetrate core, perturbation amplitude remains small For higher current per wire (N < 40), core is penetrated before array implodes, perturbation amplitude in each wire reaches 100%, perturbation for whole array ~ 1/√N APS-DPP, Quebec 2000
2D(x,y) plane simulations show how core-corona structure radically alters implosion dynamics 8x15µm Aluminium on MAGPIE 8
8
25 ns
60 ns
0
2
4
6
4 2 0
8
0
X Axis in mm
2
4
6
8
X Axis in mm
Y Axis in mm
2
6
6
Y Axis in mm
4
125 ns
90 ns
6
Y Axis in mm
Y Axis in mm
6
0
8
8
4 2 0
4 2 0
0
2
4
6
8
X Axis in mm
0
2
4
6
8
X Axis in mm
Use 1D cold-start to initialise 8
8
180 ns 6
Y Axis in mm
Y Axis in mm
6 4 2 0
Low density corona accelerated by jxB and swept around wire cores forming radial plasma streams
210 ns
0
2
4
6
X Axis in mm
8
Dynamical balance between rate of material ejection from core and injection into interior of array
4 2
Streams reach axis at ½ implosion time forming precursor 0
0
2
4
6
8
Cores remain stationary until 80% of implosion time
X Axis in mm
Majority of array mass on axis prior to implosion Dynamics and structures accurately reproduce experiments APS-DPP, Quebec 2000
The same phenomena persist for higher wire numbers and larger, faster rising currents 48x17.5µm Aluminium on SATURN
Low density parts of plasma streams merge early on Cores remain stationary and intact until 80% of implosion time During flight, cores merge to form azimuthally symmetric shell. Significant fraction of array mass on axis prior to implosion Radial mass profile agrees well with initial conditions assumed by Whitney and Thornhill [IEEE Transactions on Plasma Science 26 p1168 (1998)]. Absence of axial dimension means short, sharp radiation pulse obtained. APS-DPP, Quebec 2000
Use reduced zone around one wire for high resolution e.g. One of 32 15µm Al wires on MAGPIE In this case, jxB force redirects ablating material towards array axis without applying force directly to the core
0.2
0.2
0.1
0.1
0.1
0.0
Y Axis in mm
0.2
Y Axis in mm
Y Axis in mm
Close-up of core region
0.0
0.0
-0.1
-0.1
-0.1
-0.2
-0.2
-0.2
7.9
8.0
8.1
X Axis in mm
Log ρ(x,y) and log v(x,y)
7.9
8.0
X Axis in mm
jz(x,y)
8.1
7.9
8.0
8.1
X Axis in mm
Log ρ(x,y) and jxB(x,y) APS-DPP, Quebec 2000
Ohmic heating of core material determines implosion trajectory Dynamical balance between core ablation and mass injection
Cool cores
R
Residual cores remain stationary until 80 % of implosion time. Mass averaged radius versus time
jz similar to thin shell model Warm cores
Once Te > 1eV current transferred to cores,
T
R
some jxB force applied directly cores rapidly heat and expand
jz
Trajectory similar to thin shell model
Close cores
T R
Reduces injection of material and current between cores. Trajectory similar to thin shell model
jz
T APS-DPP, Quebec 2000
The precursor plasma is an apparently stable, uniform and long-lived, 1D plasma. Precursor lifetime > 100ns
Gated soft X-ray images of precursor indicate that equilibrium radius is a strong function of material carbon
aluminium
Initial formation phase is collisionless.
Once collisonal, converges to a two component equilibrium of high density Can be modelled in high resolution stationary precursor and lower density convergent radial plasma stream. with 1D MHD
tungsten
60
Pressure balanced by ρv2 of bombarding stream. Little or no current.
Temperature in eV
Density in Kg/m
3
1.2 stationary precursor 1.0 0.8 flux through boundary
0.6 0.4 0.2 0.0 0.0
radially convergent stream 0.5
1.0
1.5
2.0
2.5
50 40
Kinetic energy delivered (ρv3A) is the roughly balanced by radiation losses.
30 20 10 0 0.0
3.0
0.5
Radius in mm
1.0
1.5
2.0
2.5
3.0
Vr in ms
-1
Radius in mm
12 10
Z star
8
Density ratio between precursor and stream ~[(γ+1)/(γ-1)]2. Data suggests for Al γ≈5/3. For W precursor density much higher ⇒ γ≈1.1
0.0
4
-5.0x10
6 5
-1.0x10
4 2 0 0.0
5
-1.5x10 0.5
1.0
1.5
2.0
Radius in mm
2.5
3.0
0.0
Similar to a test developed to evaluate different artificial viscosity formulations in 1D hydrodynamics (W.F. Noh, J. Comp. Phys. 72, p78 (1987).
0.5
1.0
1.5
2.0
Radius in mm
2.5
3.0
Ideal test-bed for opacity measurements, X-ray laser experiments and benchmarking radiation hydrodynamics codes. APS-DPP, Quebec 2000
There are at least 3 different theoretical modes of nested wire array dynamics Hydrodynamic Collision (or Shell on Shell) Mode 20 18 16 14 12 10 8 6 4 2 0 60
70
80
90 100 110 120
Time in ns
Transparent Inner (or Current Transfer) mode 20 18 16 14 12 10 8 6 4 2 0 60 70 80 90 100 110 120
Time in ns
Flux Compression (or Magnetic Buffer) mode 20 18 16 14 12 10 8 6 4 2 0 60
Outer Inner
70
80
90 100 110 120
Time in ns
APS-DPP, Quebec 2000
2D(x,y) simulations reproduce collapse dynamics of nested arrays on MAGPIE 1.0
1.0
0.2
0.6 0.4
0.4
0.6
0.8
0.0 0.0
1.0
0.2
0.4
0.6
0.8
1.0
0.4 0.2
0.6 0.4 0.2
0.6
0.8
0.6
0.8
1.0
X Axis in cm
0.0 0.0
0.4
0.6
0.8
1.0
0.8
0.0 0.0
1.0
0.6
0.8
0.4
0.6
240 ns 0.8
0.4
0.6 0.4 0.2
0.2
0.4
0.6
X Axis in cm
X Axis in cm
0.8
1.0
0.0 0.0
0.2
0.4
0.6
0.8
X Axis in cm
Outer Sim. Inner Sim. Outer Expt. Inner Expt.
8
Radius in mm
6 5 4 3 2 1 0 160
180
200
220
240
1.0
1.0
0.6
0.0 0.0
1.0
0.8
X Axis in cm
0.2
0.4
0.2
230 ns
0.4
7
Compression of this flux by implosion of outer produces sufficient current to drive inner ahead of outer.
0.6
9
16+16 x 15µm Aluminium (equal length arrays)
Small fraction of current flowing through inner array produces Bθ between arrays
0.4
0.8
0.6
X Axis in cm
Inner wires heated by bombarding plasma streams from outer
0.2
1.0
0.2
0.4 0.2
200 ns
0.0 0.0
0.6
X Axis in cm
0.2
0.2
0.4
0.0 0.0
1.0
0.8
Y Axis in cm
Y Axis in cm
Y Axis in cm
0.6
0.4
0.4
150 ns 0.8
0.2
0.2
1.0
90 ns
0.6
X Axis in cm
1.0
0.8
0.0 0.0
0.0 0.0
1.0
0.8
0.2
X Axis in cm
X Axis in cm
Inner
0.4
Y Axis in cm
0.2
0.6
0.2
0.2
240 ns
0.8
Y Axis in cm
0.4
Y Axis in cm
0.6
1.0
230 ns
0.8
0.8
Y Axis in cm
Y Axis in cm
0.8
0.0 0.0
1.0
200 ns
150 ns
Y Axis in cm
90 ns
Y Axis in cm
1.0
Outer
260
Time in ns
APS-DPP, Quebec 2000
1.0
) Axis in mm
Model inner and outer arrays on ‘Z’ separately, first calculate radial plasma flux from outer array, then use this to bombard the inner array. 240x7.5µm tungsten wires on a 40mm diameter 0.2
20 ns
0.0
Similar features to lower wire number cases,
Y (or ) Axis in mm Y (or ) Axis in mm ) Axis mm in mm θ) inAxis
-0.2 18.5
19.0
19.5
Dense wire cores retain most of the mass until implosions commences.
20.0
0.2
40 ns
Low density corona swept around cores forming radial plasma streams.
0.0 -0.2 18.5
19.0
19.5
20.0
0.2
60 ns
0.0
At 75ns precursor stream extends down to 6mm and contains 20% of mass. In 2D, the remaining 80% is in a 1mm wide shell
-0.2 18.5
19.0
19.5
20.0
0.2
70 ns
0.0
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 conditions for simulation of an inner array wire.
-0.2 18.5
19.0
19.5
20.0
X (or R) Axis in mm APS-DPP, Quebec 2000
2D(x,y) simulations predict the implosion modes of nested arrays on ‘Z’
107ns
-0.50 8.0 0.50
8.5
9.0
9.5
10.0
10.5
11.0
X (or R) Axis in mm
0.25 0.00 -0.25 -0.50 8.0 0.50
8.5
9.0
9.5
10.0
10.5
11.0
X (or R) Axis in mm
0.25 0.00 -0.25 -0.50 8.0 0.50
8.5
9.0
9.5
10.0
10.5
11.0
X (or R) Axis in mm
0.25 0.00 -0.25 -0.50 8.0 0.50
8.5
9.0
9.5
10.0
10.5
11.0
X (or R) Axis in mm
0.25 0.00 -0.25 -0.50 8.0
8.5
9.0
9.5
10.0
X (or R) Axis in mm
10.5
11.0
Y (or θ) Axis in mm
-0.25
Y (or θ) Axis in mm
0.00
Y (or θ) Axis in mm
0.25
Y (or θ) Axis in mm
Y (or θ) Axis in mm
Plasma stream from outer of 240x7.5µm W
0.50
Y (or θ) Axis in mm
104ns
Y (or θ) Axis in mm
102ns
Y (or θ) Axis in mm
98 ns
Y (or θ) Axis in mm
83 ns
Y (or θ) Axis in mm
Inner array of 60x10.5 µm W wires
0.50
Inner array wires see 20ns of low density coronal bombardment followed by main mass in 1mm wide shell.
0.25 0.00 -0.25 -0.50 8.0 0.50
8.5
9.0
9.5
10.0
10.5
11.0
X (or R) Axis in mm
At first little expansion of inner wires ⇒ outer material streams through, setting up bow–shock.
0.25 0.00 -0.25 -0.50 8.0 0.50
8.5
9.0
9.5
10.0
10.5
11.0
X (or R) Axis in mm
0.25
Later bombardment by denser main mass heats each wire with ~100GW of kinetic flux.
0.00 -0.25 -0.50 8.0 0.50
8.5
9.0
9.5
10.0
10.5
11.0
10.5
11.0
X (or R) Axis in mm
0.25 0.00 -0.25 -0.50 8.0 0.50
8.5
9.0
9.5
10.0
X (or R) Axis in mm
0.25
Inner wires expand rapidly allowing effective momentum transfer. Compression of magnetic flux carried by plasma stream effectively increases momentum transferred. Trajectory similar to hydrodynamic collision mode with reduced radiation at collision.
0.00 -0.25 -0.50 8.0
8.5
9.0
9.5
10.0
X (or R) Axis in mm
10.5
11.0
Trajectories consistent with transparent inner mode require ≤ 30 wires in inner. APS-DPP, Quebec 2000
Even with 30% amplitude perturbation on ρ(z) with 0.5mm wavelength, apparent modulation is much less than in experiment 50 ns
z r
80ns
Side-on laser schlieren of Al arrays on MAGPIE show: modulation in corona from ~60ns roughly constant amplitude (r+ - r-) and wavelength 100ns
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 different axial positions Maybe this isn’t an MHD instability at all ? APS-DPP, Quebec 2000
3D simulation of m=0 instability in ideal MHD equilibrium pinch: growth rate agrees well with analytic theory
Similar results have been obtained for m=1 instabilities [S.G. Lucek, private communication] APS-DPP, Quebec 2000
Modulating core resistivity versus z, gives results similar to experiment
220ns
240ns
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2 0.0 -0.2 -0.4 -0.6 -0.8 7.0
7.5
8.0
R Axis in cm
8.5
Z Axis in mm
0.8
Z Axis in mm
Z Axis in mm
150ns
0.2 0.0 -0.2 -0.4 -0.6
0.2 0.0 -0.2 -0.4 -0.6
-0.8 7.0
7.5
8.0
R Axis in cm
8.5
-0.8 7.0
7.5
8.0
R Axis in cm
8.5
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. APS-DPP, Quebec 2000
Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution Conclusions 2D “cold-start” models illustrate important processes involved in plasma formation phase and 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 j∧B 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’ ? More experimental data is needed. APS-DPP, Quebec 2000
