J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch...

8/3/2019 J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution

1/23

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. [email protected] 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


2/23

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


3/23

Wire arrays cover a wide range of parameters

but exhibit the same physical processes


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


4/23

MAGPIE wire array experiments show intrinsically 3D phenomena

with scales ranging from a few m to several mm


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


5/23

Talk Outline


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


6/23

Plasma formation in wires depends on complex EOS and transport coefficients


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]


7/23

1D cold-start MHD simulations show formation of core-corona structure


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


8/23

In 2D(r,z) cold-start simulations of a single wire,

pinching of the corona excites the m=0 instability


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


9/23

Comparison to single wire data provides benchmark tests for

2D MHD code plus EOS and transport models therein


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)


10/23

3D behaviour of wires in arrays limits the application of

2D single wire calculations to scaling arguments


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


15/23

The precursor plasma is an apparently stable, uniform and long-lived, 1D plasma.


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.


16/23

There are at least 3 different theoretical modes of nested wire array dynamics


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


17/23

2D(x,y) simulations reproduce collapse dynamics of nested arrays on MAGPIE


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


18/23

Model inner and outer arrays on Z separately, first calculate radial plasma

flux from outer array, then use this to bombard the inner array.


240x7.5m 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.


19/23

2D(x,y) simulations predict the implosion modes of nested arrays on Z


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.


20/23

Even with 30% amplitude perturbation on (z) with 0.5mm wavelength,apparent modulation is much less than in experiment


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


21/23

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]


22/23

Modulating core resistivity versus z, gives results similar to experiment


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.


23/23

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 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.

J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch...

Documents

Transcript of J.P. Chittenden- Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch...