s.v. lebedev- implosion dynamics of wire array z-pinches
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Implosion dynamics of wire array Z-pinches
S.V. LebedevMini-course on Z-pinches
Monterey, CA 18-19 June 2005
Imperial College London
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 20052
Acknowledgements
D.J. AmplefordS.N. BlandD. Bliss S.C. BottJ.P. ChittendenM. CuneoJ. DavisC. DeeneyJ. GoyerC. Jennings M.G. Haines G.N. Hall D.A. Hammer J.B.A. Palmer S.A. Pikuz
D.D. RyutovT. SanfordT.A. ShelkovenkoD. SinarsA. VelikovichE. Waisman
………………
Experiments at Imperial College are supported by Sandia National Laboratories and by NNSA DOE
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 20053
Outline
• Introduction: why wire arrays?
• Overview of the implosion dynamics
• Early stages of plasma formation
• Implosion phase in wire arrays and the X-ray pulse
• Trailing mass and trailing current
• Different implosion modes of nested wire arrays
• Possible non-MHD effects
• Other configurations/applications of wire array Z-pinches
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 20054
0-D implosion
RIBRRm
πμ
μπ
422
20
0
2
0 −=⋅−=&&
22
2
)(ττ
fd
rdr ⋅Π−=⋅
Thin conducting shell with axial current
Equation of motion of a shell driven by the pressure of magnetic field (m0 is mass per unit length) :
200
2max
2max0
4 RmtI
πμ
≡Π
In dimensionless variables
r = R / R0, τ = t / tmax f(τ) = I(τ) / Imax
Π is a dimensionless scaling parameter:identical implosions for identical values of Πand the same current pulse shape (f(τ))
0.0 0.5 1.00.0
0.5
1.0
0.0
0.5
1.0
Time
Π=20
Π=5
Rad
ius
Cur
rent
I ~ sin2(t)
Implosion trajectory
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 20055
X-ray power from Z-pinch implosion or why wire arrays?
rayXWmVCU−⇒⇒
22
22
VRWP X δτ
τ~∝
Imploding plasma shell Energy Power
•Small mass to maximize implosion velocity
•Wire array is equivalent to ~50nm thick foil and wires should rapidly merge into a shellWire array Z-pinch
Wire array Shell
This transition does not happen!
3-D effects are important throughout the implosion!
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 20056
Wire array Z-pinches: two stage implosion dynamics
Two-stage implosion dynamics
• “Slow” ablation of wires and radial redistribution of mass
• Snowplough-like final implosion phase, stabilised by the peaked on axis density profile
Wires survive for ~3/4 of the implosion time!
0.5 1.00.0
0.5
1.0
Rad
ius
time
coronal
plasma
Trailing mass
StagnationPrecursor pinch
Snowplow-likefinal implosion
0-D
Ablation of wire cores
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 20057
Wires in wire array (magnetic field configuration)
wNRgap δπ
>>= 02
0 2 4 6 8 10x, mm
0
2
4
6
Inter-wire separation >> initial wire diameter
“Global” and “private” magnetic flux
∑= −⋅−+
−⋅−⋅=
N
CosrrrrCosrr
NIB
122
00
)(2)(
2 β βββ
ββθ θθ
θθπ
μ
∑= −⋅−+
−⋅⋅=
N
r CosrrrrSinr
NIB
122
00
)(2)(
2 β βββ
ββ
θθθθ
πμ
0.6 0.8 1.0 1.2 1.4 1.6-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
B/B
shel
l
radius
shell
array:between wires
array:through wire
Magnetic field lines (N=8)
Magnetic field distribution (N=16)
( )π
δπδ
μπ
μ2
@2
/2
00 gapNIBRIB wireshell =≡=≡
( >200μm) ( ~5-20 μm)
Magnetic field distribution[ Felber & Rostoker, Phys. Fluids 1981]
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 20058
Wires in a wire array (R and L)
HNr
RNR
RhLwire
oretout
10
0
0 108.6)ln1(ln2
−⋅≈⋅
+=π
μ
Typical Ohmic heating time to ~Tmelt:
τ ~9ns
Typical “cold” array resistance: R~25 mΩ/cm for 6mg, N=300 tungsten array
Typical inductance:
Typical array resistance at T=Tmelt:
R~8.7 Ω/cm 0 2 4 6 8 100.0
0.1
0.2
0.3
0.4
0.5
0
1000
2000
3000
4000
R (O
hm)
time (ns)
T
R
I (kA)
T (K
), I (
kA)
Z arrayW 6mg, 1cm longN=300
After plasma formation current is in the coronal plasma: Rplasma =?
L/Rcold ~ 30ns
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 20059
Experimental set-up: MAGPIE (1MA) and Z (20MA)
Wire arrays:
X-pinch in current return Z-Beamlet and spherical crystals
MAGPIE Z
1ns, 10μm
hν ~ 3-5keV
MAGPIE Z
R (mm) 4-8mm 10-20mm
N ~32 ~300
timplosion ~250ns ~100nsCurrent per wire
30 kA 60 kADiagnostics: X-ray/optical imaging, laser probing
Radiography:
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200510
Core-corona structure of plasma in wire arrays (1MA)
Wires remain at initial positions until ~80% of implosion
Non-uniformity of coronal plasma formation imprints on the cores
Sharp outward and a shallow inward edges of wire cores
Radiography Laser probing Radiography
Same λ
Dense, stationary wire cores surrounded by low density coronal plasma
0.0 0.5 1.0 1.5 2.0
16
18
20Al
250μm
arrayedge
Film
den
sity
(a.u
.)
Radial position (mm)0.5 1.0 1.5
W100μm
twowires
Shape of wire cores is not cylindrical
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200511
Core-corona structure of plasma in wire arrays (1MA)
Wires remain at initial positions until ~80% of implosion
Non-uniformity of coronal plasma formation imprints on the cores
Maximum ablation and maximum plasma flow are at different axial positions!
Radiography Laser probing Radiography
Same λ
Radiography Laser probing
Dense, stationary wire cores surrounded by low density coronal plasma
flow
gap
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200512
Precursor plasma flow in wire arrays (1MA)
Inward streaming of the coronal plasmaEnd-on laser probing End-on XUV Radial optical streak
X-ray image of precursor
• Plasma on axis at t ~50% timpl
(V~15cm/μs from end-on measurements)
• “Inertially confined” precursor column on axis
• Implosion starts at t ~80% timpl
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200513
Core-corona structure of plasma in wire arrays (Z)
Radiography shows wire cores until ~60% of implosion
Non-uniformity of coronal plasma formation
Maximum ablation and maximum plasma flow are in different axial positions!
Precursor column on axis
Radiography Visible imagingD. Sinars et al, PoP 2005. D. Bliss et al, ICOPS 2004.
End-on X-ray imageM. Cuneo et al., PRE 2005
precursor
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200514
“Delayed” implosion trajectories
MAGPIE Z
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
0-D Al
N=16 N=32
W N=32 N=64
R /
R0
t / t imp
shell-likeimplosion
N=32, 8mm
• During the first 80% - 60% of time the JxBforce is not applied to the cores, accelerating instead the coronal plasma
• Fast acceleration – not all mass participates
• Rate of plasma formation is the most important parameter during the first phase
M. Cuneo et al., PRE 2005
0-D
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200515
Ablation rate of wires in a wire array
Ablation in conical array Ablation time depends on array radius
The JxB force of the “global” magnetic field determines the ablation rate of wires in an array
32x15μm Al arrays driven by the same current pulse have different ablation times
Direct application of single wire results is not possible !
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200516
“Rocket” model of ablation and mass redistribution
0
20
4 RI
dtdmV
πμ
−=
∫=t
abl
dtIRV
tm0
2
0
0
4)(
πμδ
Ablation of stationary wire cores
(Rocket model)
rI
dtrdm
πμ4
20
2
2
0 −=
Ablated mass:
Momentum balance:
JxB force is only acting on the coronal plasma
(0-D model)
ablVRI
dtdm
0
20
4πμ
−=
202
02
0 )]([8
),(ablabl V
rRtIVrR
tr −−⋅=
πμρ
Ablation rate:
Radial profile of the ablated material:
Concept of “ablation velocity” (Vabl) highlights the main dependence of the ablation rate on current and array radius (Vabliis a weak function)
Lebedev et al., PoP 2001
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200517
Ablation rate of wires in a wire array
Ablation in conical array Ablation time depends on array radius
Starting time of the implosion is consistent with ablation of about half of array mass:
32x15μm Al arrays driven by the same current pulse have different ablation times
Direct application of single wire results is not possible !
∫=t
abl
dtImRVm
tm
0
2
00
0
0 4)(
πμδ
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200518
Early time XUV radiation as indicator of ablation rate
Linear wire array
abl
global
VBI
dtdm
⋅
×−=
2
32x15μm Al arrays driven by the same current pulse show different level of XUV emission
)(/4~),(/300~/
~ WionkeVAlioneVdtdm
PE rad
The same current, but larger Bgl (x6.6)
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200519
Variation of ablation rate with inter-wire gap
Ablation time of an array “Ablation velocity” versus inter-wire gap
0 5 10 15 20 25 30 350.0
0.5
1.0
1.5
2.0N=8R=8mm
N=16R=8mm
N=32R=8mm
N=64R=8mm
N=16, R=4mm
N=32, R=4mm
N=64, R=4mm
Vab
l (10
7 cm
/s)
gap / core size
Al Wπ (critical ratio)
Rapid increase of ablation rate for the gaps below “critical”:
change in the magnetic field topology at δcr ~ 3 x (core size)
Could be an additional dependence of Vabl on wire diameter [D. Sinars et al., PoP 2005]
0 2 4 6 8 10x, mm
0
2
4
6
7 8 9 10x, mm
0.0
0.5
1.0
1.5
2.0
Magnetic field
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−−⋅=
4.3exp15.1)( xxf
∫=t
abl
dtImRVm
tm
0
2
00
0
0 4)(
πμδ
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200520
Implosion similarity of wire arrays
max0max /)()ˆ(/ˆ/ IItfRrrtt ττ ===
22
2
)(ττ
fd
rdr ⋅Π−=⋅)
)
200
2max
2max0
4 RmtI
πμ
≡Π
0-D similarityIn dimensionless variables:
0-D dimensionless scaling:identical implosions for identical Πand the same current pulse shape f(τ)
How the similarity criteria should change to account for the redistribution of mass by the precursor plasma?
∫⋅=
t
abl
dtIRVmm
tm
0
2
00
0
0 4)(
πμδ
ablVRI
dtdm
0
20
4πμ
−=
Ablation rate:
Ablated mass:
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200521
Implosion similarity of “ablating” wire arrays
)//()ˆ,ˆ(ˆ/ˆ/ 2000 RmtrRrrtt m ρρτ ===
[ ]22
))ˆ1((ˆˆ2
),ˆ(ˆ rKIr
Kr −⋅−⋅⋅
⋅Π= τπ
τρ
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0
0.5
1.0 Π ~ 6
Cur
rent
time
Abl
ated
mas
s fra
ctio
n
K=0.2(Magpie)
K=0.7(Z ?)
Ablated mass fraction:
∫∫ ⋅⋅Π=⋅⋅
⋅⋅
=ττ
ττπμτδ
0
2
0
20200
220
0
)~()~(4
)( dIKdItV
RRm
tImm
mabl
m
Πis fixed ⇒ the same 0-D implosion time and the same trajectory+K is fixed ⇒ the same degree of mass redistribution
Dimensionless variables:
Density profile:Π K
0.0 0.2 0.4 0.6 0.8 1.00.0
0.1
0.2
0.3
0.4
Π ~ 6
K=0.7, τ =0.7(Z ?)
K=0.2, τ = 0.97(Magpie)
Den
sity
Radius
KMagpie ~0.2
KAngara ~0.2
KZ ~ 0.4 - 0.7
KSphinx ~0.7
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200522
Distribution of mass at the start of implosion phase
Implosion phase:Snowplough-like implosion of the distributed mass
Stabilisation by density profile
Does all mass participate in the implosion?
~50% of mass is inside the array
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Π ~ 6
K=0.2, τ = 0.97(Magpie)
K=0.7, τ =0.7(Z ?)
Mas
s fra
ctio
n
Radius
~50% of mass in wire cores at R0
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200523
Axial non-uniformity of the ablation rate
Axial modulation of the ablation rate is responsible for existence of trailing mass in wire array Z-pinches
Process of wire ablation is the same for 1MA and 20MA currents
1MA
MAGPIE
20MA
Z (SNL)
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200524
Transition to the implosion phase
Axial modulation of ablation rate formation of “breaks” in the wires
Imploding current sheath, formed by a number of “magnetic bubbles”
Laser probing (Al, N=16) XUV images (Cu, N=8)
precursor
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200525
Snowplough-like implosion in W arrays
Laser probing of 32 x 4 µm tungsten wire array
Implosion of current sheath through the plasma pre-fill
-8 -6 -4 -2 0 2 4 6 850
100
150
200
250 precursor
initial array diameterOpt
ical
den
sity
(a.u
.)
Radius (mm)-8 -6 -4 -2 0 2 4 6 8
0
50
100
150
200
initial array diameter
implodingplasmapiston
precursor
Opt
ical
den
sity
(a.u
.)
Radius (mm)
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200526
Snowplough implosion phase
implosion trajectory
10%
60%mass fractionin the piston:
0% 10% 60%
Rad
ius
(mm
)
time (ns)100 150 200 250 300 3500
2
4
6
80-D
experiment streak
x-ray: outer inner
Two possible implosion scenarios:No current through the gaps Current re-strike
Trailing mass All mass implodes
ρ(r) from “rocket” model
Initial piston mass is adjusted to fit implosion trajectory
~40% of array mass is left behind
32 x 15µm Al array on MAGPIE
Imploding sheath on end-on X-ray images
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200527
Snowplough implosion phase on Z facility
202
02
0 )]([8
),(ablabl V
rRtIVrR
tr −−⋅=
πμρ 32 )(),()(
21)( ablp VVtr
dtdr
dtdmtP −⋅∝⋅= ρ
0 2 4 6 8 100
2
4
6
dens
ity (m
g/cc
)
Radius (mm)
Density profile along implosion trajectory
Vabl = 1.45x107 cm/s
Vpiston = Vabl
Density profile from the ablation model:
Radiation power from the inelasticallyaccreted plasma:
0
5
10
60 80 100 120 1400.01
0.1
1
10
100
60 80 100 120 140
time (ns)
P_exp
Rad
ius
(mm
)
Shot674nf rpeak
Vabl = 1.45x107 cm/s
t0 = 71.4ns
Pow
er (T
W)
P_model
time (ns)
R0D rad
The “foot” of the X-ray pulse is produced by the snowplow radiation
~35% of array mass is left behind
M. Cuneo et al., PRE 2005
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200528
Expansion of precursor during the implosion phase
dtdm
RVP
p
ablkin ⋅=
π2 lRETnZP
pith 23
2)1(π
⋅=+=
Equilibrium radius of precursor during wire ablation phase:
M. Cuneo et al., PRE 2005
dtdmlV
ERabl ⋅⋅
⋅=34
radQFdtdE
dtdR
⋅−⋅=∝ )1( α
Kinetic pressure of plasma flow is equal to the precursor thermal pressure
TlnRZE ip2)1(
23 π+=
Equilibrium precursor radius:
Heating of precursor by the snowplow radiation leads to increase of equilibrium radius
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200529
Implosion and stagnation phase
Laser probingXUV images (32x10μm Al)X-ray peak
Transition from small λ modulation on wires to the global m=0 mode with λ ~2mm (Al)
Fraction of mass is left behind, and it is gradually reducing with time
Significant fraction of mass is left behind even at the time of the X-ray peak
Development of m=1 mode after the peak of X-ray pulse
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200530
Implosion and stagnation phase
X-ray signals
XUV images (32x10μm Al)X-ray peak
Expansion of precursor during the implosion phase
Start of the main X-ray pulse at ~time of the current sheath collision with precursor
Fast electrons (~100keV e-beam) during the X-ray pulse phaseX-ray>10keV
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200531
Spatial structure of stagnated pinch (MAGPIE)
Axially resolved X-ray spectrum (Al)
Correlation between positions of “hot spots”and the most pronounced m=0 structures in the trailing plasma
Al K-shell spectral lines
Continuum radiation from localised “hot spots”
Similar “hot spots” emitting continuum radiation were observed on Z [Sinars et al., JQSRT, in press]
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200532
Spatial structure of stagnated pinch (Z)
X-ray image of W array on Z Radiography of W wire array on Z[Deeney et al., PRL 1998] [Sinars et al., PRL 2004]
The spatial structure of the stagnated pinch could be related to the axial distribution of the trailing mass?
Pinch diameter ~1.5mm
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200533
Trailing mass at the time of the radiation pulse
Laser probing X-ray image
~10 -30% of mass is trailing at time of X-ray peakgradual clearing of trailing mass throughout the X-ray pulse
Trailing mass on Z (Cuneo et al. PRE 05)
Trailing mass could prevent efficient delivery of current (magnetic energy) to the radiating pinch
X-ray peakX-ray peak
Diameter of X-ray emitting region is a small fraction of the plasma diameter
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200534
Trailing mass at the time of the radiation pulse
Laser probing X-ray image
~10 -30% of mass is trailing at time of X-ray peakgradual clearing of trailing mass throughout the X-ray pulse
Secondary implosions could act as a mechanism transporting magnetic energy to stagnated pinch
Trailing mass could prevent efficient delivery of current (magnetic energy) to the radiating pinch
X-ray peakX-ray peak
Diameter of X-ray emitting region is a small fraction of the plasma diameter
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200535
Dynamic modes of nested wire arrays
Model of two plasma shells:
• Mitigation of R-T instability by the inner shell
• X-ray pulse at the strike
3-D reality:
• No current in the inner array
• Initial array transparency > 99%
• Interpenetration of the arrays and implosion due to fast transfer of current to the inner array
Plasma shell
Wires Davis et al., APL 1997, Deeney et al., PRL 1998, Terry et al. PRL 1999,
Lebedev et al., PRL 2000, Deeney et al., PRL 2004, Cuneo et al PRL 2005
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200536
Current division in nested wire arrays
)ln1(ln2
0
NrR
NRRhL
wire
arr
arr
retarr ⋅
+=π
μ)(ln
20
arr
ret
RRhM
πμ
=
Inductive current division
For large wire number arrays (e.g. on Z) only a small fraction of total current (~2%) should be in the inner array
For arrays with N~16 (e.g. on MAGPIE) current fraction is ~ 20%
Resistivity could play a role in the current division
)ln()2(
)(Nr
RMLL
MLII
wire
o
inout
out
total
inner
⋅∝
−+−
=
Array inductance Mutual inductance
Current fraction in the inner array
A. Velikovich et al., PoP 2002
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200537
Nested wire arrays operating in a current switching mode
Current pulse through the inner array is suppressed
0 10 20 30 40 500
2
4
6 6% of totalcurrentcurrent in
inner array
curr
ent (
kA)
time (ns)
Current pulse through the inner array is controlled by the phasetransitions in the wires of outer and then inner arrays
Small core size of the inner array wires
Inner array retain high transparency (~98%)
Lebedev et al., PRL 2000, Bland et al, PoP 2003
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200538
Nested wire arrays operating in a current switching mode
No momentum transfer at “strike”
nested
sinlge arrayradi
us (m
m)
0
4
8
150 200 250 3000
5
10single array
PC
D (a
.u.)
time (ns)
nestedarray
Trajectory and the X-ray pulse
Current from the sheath switches into the inner array at “strike”
Decay of snowplough emission, plasma piston coasts to the axisAblation phase of the inner array after current pulse [Cuneo et al., PRL 2005]
Interaction radiation pulse in talk by M. Cuneo
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200539
Nested wire arrays offer control of the radiation pulse shape
150 200 250 3000
1nestedRin=2mm
pcd3s0802 pcd3s0830
PCD
(a.u
.)
time (ns)
0
5
10
nestedRin=4mmP
CD
(a.u
.)
pcd1s0726 pcd1s0802
02468
singlearray
nested
radi
us (m
m)
0.0
0.5
1.0
curr
ent (
MA
)
time (ns)0
1
2
3
PC
D (a
.u.)
stagnationstrike
precursor
b
current
X-ray PCD
0 50 100 150 200 250 3000
2
a
singlearray
nestedarray
PC
D (a
.u.)
0.0
0.2
0.4
0.6
0.8 experiment: nested array
0-D, 100% currenttransfer
I = 100%(single array)
Rad
ius
(cm
)
c
experiment: single array
100 150 200 250 3000.0
0.2
0.4
0.6
0.8
Different current division between the outer and the inner array affects the X-ray pulse shape
Effect of inner array radius on X-ray pulse shape
Implosion trajectories X- ray pulses
Iout = 65%
Iout = 78%
Rad
ius
(cm
)
time (ns)
d
Iin = 22%
Iin = 35%
I = 100%(single array)
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200540
Some “hot” topics (very subjective)
“Optimal” wire number
Effect of electric field polarity on plasma formation
Non-MHD effects?
Effects of turbulence on plasma resistivity?
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200541
Effect of wire number on X-ray power
0 5 10 15 20 25 30 350.0
0.5
1.0
1.5
2.0
2.5
3.0
Vab
l (10
7 cm
/s)
gap / core size
N=300
N=120N=50
f(x) = 2.2*(1-exp(-x/3.4))
N=600
Rise-time of the X-ray pulse for Al and W arrays
Mazarakis et al., Pl. Dev. Oper. 2005
0 2 4 6 8 100
1
2
3
4
5
6
dens
ity (m
g/cc
)
radius (mm)
V=0.8, t_0=61.6ns V=1.0, t_0=65.1ns V=1.2, t_0=68.1ns V=1.45, t_0=71.5ns V=2.2, t_0=80ns 0.8
11.21.45
2.2
Change in the density profile along the implosion trajectory
Expected scaling of Vabl withwire number for Z conditions
Optimal wire number in wire array implosions:• at large gaps – improvement with wire number due to statistics of uncorrelated perturbations
• small gaps – degradation due to less stabilisation of the R-T by the density profile
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200542
Non-MHD or “initial conditions” effects?
The wires in the array should have the same current.
However, the top and the bottom halves show very different dynamics!
The reason for this appears to be in the sign of the radial electric field on the wires
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200543
Effect of polarity of the radial electric field
-1 -0.5 0 0.5 10
20
40
60
80 Distance (mm)
Electric Field (AU)
“Standard” array:
Er <0
“Long” array:
Er <0 at the top
Er >0 at the bottom half
Long single array Electric field in“long” array
The polarity of the radial electric field changes in the middle of the “long” array
Difference in:•time of plasma formation •core size•ablation rate (Vabl)•implosion time
The half with Er>0 behaves as a standard array (in which Er<0) !?
Holes in electrodes
8mm
MITL
Anode
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200544
Effect of polarity of the radial electric field
The polarity of the radial electric field changes in the middle of the “long” array
Laser probing High resolution XUV image
Size of the wire core shadows:
Standard array “Long” array
300-350μm bottom half ~350μm
top half ~100μm
Later “breakage” of the wires and later start of the implosion for the top half of the array
Different ablation rate:Standard /bottom half top halfVabl ~15cm/μs Vabl ~40cm/μs
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200545
Axial plasma flow: two-fluid MHD?
Plasma has an axial velocity component (from cathode to anode), especially on the outward side of the wires
Is this related to the mechanism responsible for the axial modulation of ablation rate?
MAGPIE Z
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200546
X-ray pulse: kinetic energy or current convergence?
0.0
0.1
0.2
0.3
0.0
0.5
1.0
curr
ent o
n ax
is (M
A)
Iin0117
Iin0121
Iin0128
tota
l cur
rent
(MA
)
Id0117
Id0121
Id0128
0 100 200 3000
1
2
3
4
time (ns)
pcd
(a.u
.) pcd1s0117
pcd1s0121
pcd1s0128
Configuration with post-convolute transfers a fraction of current from the wires into precursor column, leading to delay in implosion
m=1 instability in the precursor
Total current
Precursor current
X-ray pulse
220 240 260 280 300
0
1
2
3
4
5
6
0
5
10
15
20
PC
D (a
.u.)
time (ns)
pcd5s0921 pcd5s1013
pcd2s0921 pcd2s1013
Different current left in the wires different array mass to keep the same implosion time
different kinetic energy ( x2)
The same current after stagnation and the same X-ray pulse
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200547
Trailing mass at the time of the radiation pulse
Laser probing X-ray image
~10 -30% of mass is trailing at time of X-ray peakgradual clearing of trailing mass throughout the X-ray pulse
Trailing mass on Z (Cuneo et al. PRE 05)
Trailing mass could prevent efficient delivery of current (magnetic energy) to the radiating pinch
X-ray peakX-ray peak
Diameter of X-ray emitting region is a small fraction of the plasma diameter
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200548
What limits the current flowing through the trailing mass?
)21~( −⋅=> ξξ scrit Cuu
For Spitzer resistivity all current should remain in the trailing mass• High non-uniformity of the trailing mass
• Anomalous resitivity – Ion Acoustic Turbulence:
The following conditions should be satisfied(see e.g. Ryutov et al., RMP, 72, 167 (2000)):
1. Current velocity should exceed critical velocity:
i
i
i
ies m
Tm
TZTC
22 ⋅>
+=
ZTT i
e 7>
2. Ion sound speed should exceed ion thermal speed by, e.g., factor of 2:
For typical Z array(m0 =6mg, tungsten)
If mtr / m0 = 10%
u > ucrit for Itr > 3 MA
For high Z plasma satisfied even for Te=Ti
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200549
Is I.A.T. responsible for sub-quadratic power scaling?
secrite Cenuenj ⋅=⋅= ξ
∫ ⋅== trp
str m
AmZeCrdrrjI ξπ2)(
Current density in the trailing mass is saturated at the threshold of ion acoustic turbulence:
Total trailing current is proportional to the trailing mass:
Ohm’s law with Ion Acoustic Turbulence
Only the current flowing through the radiating pinch is useful!
0 5 10 15 20 25 300.0
0.5
1.0
1.5
2.0
2.5
0
200
400
Ene
rgy
(MJ)
current (MA)
0-D R=10mm R=20mm Exp R=10mm Exp R=20mm
Pow
er (T
W)
(τ=5
ns)
Z ZR
R=2cm
R=1cm
A =183, Z =8, T =30eV, α = 0.1, ξ = 1.1
Trailing plasma parameters:
)ln()(4
020
0
ptr R
RIIhW −⋅=π
μ
Magnetic energy delivered to the array axis:
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200550
Some other configurations of wire arrays
Radiatively cooled supersonic plasma jets:“hydrodynamic” “magnetic”
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200551
The “dynamic” life story of wire array Z-pinches
0.5 1.00.0
0.5
1.0
Rad
ius
timecoronal
plasma
Trailing mass
StagnationPrecursor pinch
Snowplow-likefinal implosion
0-D
Ablation of wire cores
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200552
References used in this talk1. D. D. Ryutov, M. S. Derzon, M. K. Matzen, “The physics of fast Z pinches”, Rev. Mod. Physics 72, 167 (2000)
2. M. K. Matzen, Phys. Plasmas 4, 1519 (1997)
3. T. W. L. Sanford et al, - “Improved Symmetry Greatly Increases X-Ray Power from Wire-Array Z-Pinches”, Phys. Rev. Lett. 77 5063 (1996)
4. M. K. Matzen et al., - “Pulsed-power-driven high energy density physics and inertial confinement fusion research”, Phys. Plasmas 12, 055503 (2005)
5. F.S. Felber, N. Rostoker, “Kink and displacement instabilities in imploding wire arrays”, Phys. Fluids 24 1049 (1981)
6. M.E. Cuneo et al., - “Characteristics and Scaling of Tungsten-Wire-Array Z-Pinch Implosion Dynamics at 20 MA”, Phys. Rev., E71, 046406 (2005).
7. S.V. Lebedev et al., - “Physics of Wire Array Z-Pinch implosions: Experiments at Imperial College”, Plasma Physics and Controlled Fusion, 47, A91 (2005)
8. S.V Lebedev. et al., – “Snowplow-like behaviour in the implosion phase of wire array Z pinches” , Phys. Plasmas, 9, 2293 (2002)
9. S.V. Lebedev et al., - “Effect of discrete wires on the implosion dynamics of wire array Z pinches” , Phys. Plasmas, 8, 3734 (2001)
10. C. Deeney et al., - ” Enhancement of X-Ray Power from a Z Pinch Using Nested-Wire Arrays”, Phys. Rev. Letters, 81, 4883 (1998)
11. S.V., Lebedev et al.,– “Plasma formation and the implosion phase of wire array Z-pinch experiments”, Laser and Particle Beams, 19, 355 (2001)
12. D. B. Sinars et al., - ” Mass-Profile and Instability-Growth Measurements for 300-Wire Z-Pinch Implosions Driven by 14–18 MA”, Phys. Rev. Letters, 93, 145002-1 (2004)
13. D. B. Sinars et al., “Measurements of the mass distribution and instability growth for wire-array Z-pinch implosions driven by 14–20 MA”, Phys. Plasmas, 12, 056303 (2005)
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200553
References used in this talk14. A. L. Velikovich, I. V. Sokolov, A. A. Esaulov, - ” Perfectly conducting incompressible fluid model of a wire array
implosion”, Phys. Plasmas, 9, 1366 (2002)
15. D.B. Sinars et al., - ” Measurements of K-shell Ar spectra from z -pinch dynamic hohlraum experiments made using a focusing spectrometer with spatial resolution”, JQSRT – (in press)
16. E. M. Waisman et al., - “Wire array implosion characteristics from determination of load inductance on the Z pulsed-power accelerator”, Phys. Plasmas, 11, 2009 (2004)
17. J. Davis, N. A. Gondarenko, A. L. Velikovich, - “Fast commutation of high current in double wire array Z-pinch loads”, Appl. Phys. Letters, 70, 170 (1997)
18. R. E. Terry et al., - “Current Switching and Mass Interpenetration Offer Enhanced Power from Nested-Array Z Pinches”, Phys. Rev. Letters, 83, 4305 (1999)
19. S.N. Bland et al., – “Nested wire array z-pinch experiments operating in the current transfer mode”, Phys. Plasmas, 10, 1100 (2003)
20. C. Deeney et al., - “Spectroscopic Diagnosis of Nested-Wire-Array Dynamics and Interpenetration at 7 MA”, Phys. Rev. Letters, 93, 155001-1 (2004)
21. M. E. Cuneo et al., - “Direct Experimental Evidence for Current-Transfer Mode Operation of Nested Tungsten Wire Arrays at 16–19 MA”, Phys. Rev. Letters, 94, 225003-1 (2005)
22. S.V. Lebedev et al., - “Effect of core-corona plasma structure on seeding of instabilities in wire array z-pinches”, Phys. Rev. Lett. 85, 98 (2000)
23. C. A. Coverdale et al., - “Optimal Wire-Number Range for High X-Ray Power in Long-Implosion-Time Aluminum Z Pinches”, Phys. Rev. Letters, 88, 065001-1 (2002)
24. M. G. Mazarakis et al., - “Tungsten wire number dependence of the implosion dynamics at the Z-accelerator”, Plasma Devices and Operations, 13, 157 (2005)
S.V. Lebedev, Implosion dynamics of wire array Z-pinches. Z-pinch mini-course, Monterey, CA, June 19, 200554
References used in this talk25. C. Deeney, C.A. Coverdale, M.R. Douglas, - “A review of long-implosion-time z pinches as efficient and high-power
radiation sources”, Laser and Particle Beams 19, 497 (2001)
26. S.V. Lebedev et al., - “Two different modes of nested wire array Z pinch implosions” - Phys. Rev. Lett., 84 1708 (2000)
27. S. N. Bland et al., – “Use of linear wire array Z-pinches to examine plasma dynamics in high magnetic fields”, Phys. Plasmas, 11, 4911 (2004)
28. S.V. Lebedev et al., - “The dynamics of wire array z-pinch implosions”, Phys.Plasmas, 6, 2016 (1999)
29. W. A. Stygar et al., - “X-ray emission from z pinches at 107 A: Current scaling, gap closure, and shot-to-shot fluctuations”, Phys. Rev. E 69, 046403 (2004)
30. V. V. Aleksandrov et al., - “Dynamics of Heterogeneous Liners with Prolonged Plasma Creation”, Plasma Physics Reports, 27, 89 (2001)
31. I. K. Aivazov, V. D. Vikharev, G. S. Volkov, L. B. Nikandrov, V. P. Smirnov, and V. Ya Tsarfin, JETP Lett. 45, 28 (1987)
32. Bekhtev M B et al, Sov. Phys.—JETP 68 955 (1989)
33. C. Deeney et al., - “Power enhancement by increasing the initial array radius and wire number of tungsten Z pinches”, Phys. Rev. E 56, 5945 (1997)
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