pavel sasorov- dynamics of plasma jets in multiwire arrays
TRANSCRIPT
Dynamics of Plasma Jets in Multiwire Arrays
Pavel Sasorov
Institute of Theoretical and Experimental Physics, Moscow, 117218, Russia
Acknowledgments
to the Angara-5-1 team,
and especially to
E. Grabovski, S. Nedoseev and G. Oleinik
Outline
1. Introduction.Role of prolonged plasma production in implosion of high quality
2. Theory of plasma ablation rate
2.1 1d model (of 2001)
2.2 Comparison with magnetic probe measurements
2.3. 2d model; very thin plasma jets
2.4 Comparison with experimental data
3. Interaction of plasma jets in nested arrays
3.1 Theory of steady state flow (two regimes)
3.2 Comparison with experiments
4. Conclusions This theory is updated significantly by Sasorov, Oliver, Yu and Mehlhorn. However the publication has been not finished yet. Hence I will review only “old” results in this field and be as brief as possible.
1. Introduction.Role of prolonged plasma production in implosion of high quality
2. Theory of plasma ablation rate
2.1 1d model (of 2001)
2.2 Comparison with magnetic probe measurements
2.3. 2d model; very thin plasma jets
2.4 Comparison with experimental data
3. Interaction of plasma jets in nested arrays
3.1 Theory of steady state flow (two regimes)
3.2 Comparison with experiments
4. Conclusions
Motivations
• The process of prolonged plasma production combined with the effect of plasma rainstorm at the final stage of implosion are responsible for very high parameters of x-ray pulse obtained with the multiwire arrays.
• High quality of plasma compression and hence short duration of the x-ray pulse occur when there is a matching between the time of plasma formationfrom relatively cold remnants of the initial explosion of the wires and the time of compression of the array with matched mass. This statement was considered theoretically (Alexandrov et al. Plasma Phys. Reps. (2001)) and is obtained experimentally (Cuneo et al. PRE (2005)).Plasma ablation rate depends on interwiregap (S. Lebedev et al. Nucl. Fusion (2004))
There is an optimal number of wires for good X-ray pulse(Mazarakis et al. (2003))There is an optimum ratio of
ablation time to time of stagnation.
Cuneo et al. (2005): 6.0
optimum
≈stag
abl
tt
Motivations
Hence it is important now to have theoretical estimation of plasma ablation rate dependent on all main parameters of multiwire array and its initiation (on I, R0, N, …)
The next problem: How do flows from outer and inner arrays of nested array interact with each other?
1. Introduction.Role of prolonged plasma production in implosion of high quality
2. Theory of plasma ablation rate
2.1 1d model (of 2001)
2.2 Comparison with magnetic probe measurements
2.3. 2d model; very thin plasma jets
2.4 Comparison with experimental data
3. Interaction of plasma jets in nested arrays
3.1 Theory of steady state flow (two regimes)
3.2 Comparison with experiments
4. Conclusions
After a few nanoseconds After a few tens of nanoseconds
to the axis
Prolonged Plasma Production is caused by by heterogeneous structure of individual dense Z-pinches
Plasma flow in the case, when the interwire gap is less than diameters of plasma coronas around the wires
V. V. Alexandrov, et al., Plasma Phys. Reps., 27, 89 (2001).
Estimation of plasma production rate and structure of the accelerating boundary layer
toward the generatortoward the axis
cold products of initial explosion of wires
plasma flow
Ampere force
boundary layer; region of considerable current density and Ohmic heating
Hea
tflu
x
Old Simple 1d Estimation of the Plasma Production Rate
nscmg18.0 2
8.1
cm
MA μ⎟⎟⎠
⎞⎜⎜⎝
⎛=
LRIm
While effects of depletion of the plasma source do not work:
(theoretical 1d estimation)
V. V. Alexandrov, et al., Plasma Phys. Reps., 27, 89 (2001).
nscmg13.007.0 2
8.1
cm
MA μ⎟⎟⎠
⎞⎜⎜⎝
⎛×÷=
LRIm
After comparison of 1d MHD simulations of plasma dynamic inside arrays and results of magnetic probe experiments at Angara-5-1 (TRINITI, Russia)
DZP5-Proceedings(2002)
1. Introduction.Role of prolonged plasma production in implosion of high quality
2. Theory of plasma ablation rate
2.1 1d model (of 2001)
2.2 Comparison with magnetic probe measurements
2.3. 2d model; very thin plasma jets
2.4 Comparison with experimental data
3. Interaction of plasma jets in nested arrays
3.1 Theory of steady state flow (two regimes)
3.2 Comparison with experiments
4. Conclusions
2d Problem
2d theory
The previous 1d theory does not contain any parameters of array,whereas present experiments show unambiguously dependence on interwire gap and condition of wire initiation.
Main assumptions:
1. Periodic structure of wire array
2. Slab geometry
3. All spatial scales of the problem are much smaller than interwire gap
4. Very thin plasma jets of ideally conducting plasma and vacuum around them
B
y
x
z
V
y
x
Main equations for accelerating of very thin plasma jets in the frame of ideal MHD.
EcB
BBdxd
m
y
yx
−=
π=μ
==μ
+
v
vv
v
21
const 1
( ) ( )( ) ( ) xdxxBxBeexB
BBBdxdEmc
yx
x
x
yxy
′
Δ′−π
−′−−
Δ=
=π
∫∞
∞Δ′π
Δπ−
+
+
02
2
31
sh111
2
P
0==×∇=⋅∇ zBBBoutside the jets
0 1 2
0 1 2
Structure of steady state plasma flow and magnetic field for the case of periodic wire array
mBπ
=≥12
;32 2
0critcritpuller vvv
Super alvenic flow
0 1 2
By ρ
0 1 2
Bx
0 1 2x / Δ
v
y / Δ = 0.5
y = 0
y = 0
2==∞
∞∞
AcM v
Distribution of of Electric Current along the Jet
0 0.5 1 1.5x / D
0
2
4
6Bx ∝ j
0 1 2
( )⎩⎨⎧
<>
⋅Δ
⋅→=→0201
3320,0
3
03
xx
xByxBy
Influence of 2d effects on estimation ofablation rate
1. Only small part of the total current per wire flows in relevant vicinity of wires:I1 << I/N
2. Increasing of local magnetic field in plasma source: B ∝ I1/rwire
3. Decreasing of area of ablation
Main 2d effects:
Interplay of these three effects determines estimation of ablation rate that takes into account azimuth periodic structure of plasma source
Estimation of Plasma Production Rate
dissipation region(Ohmic heating; heat conduction;plasma diffusion across magn. field)
Assumption: Δ<<<< dds
ds, dissipation region depth;d , diameter of cylindrical cloud
formed by the cold products of initial wire explosion;
D , interwire gap.
( ) 3/1010 dBBB Δ∝→
( ) ddBdBm 3011
μμμ Δ∝∝ ( ) 3101
μ−μ Δ∝Δ= dBmm
31
cm
MAμ−μ
⎟⎠⎞
⎜⎝⎛
Δ⎟⎟⎠
⎞⎜⎜⎝
⎛≅
dRIkmL
28.1 ÷≈μ
Comparison of the 2d correction forplasma production rate with the data from IC
0 1 2 3 4 5 6 70
0.5
1
1.5
The curve
is superimposed simply on the
data by S. Lebedev, et al. (2003)
0.4mm
-1skm93 Δ×⋅=ablv
The theory gives:
4.02.0220 ⎟
⎠⎞
⎜⎝⎛ Δ
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎠⎞
⎜⎝⎛ Δ
⎟⎟⎠
⎞⎜⎜⎝
⎛∝∝
−
dRI
dRI
mB
LLabl
νμ
v
Updated Estimation for the Plasma Production Rate
8.1(???);nscmg18.0; 1-2-31
cm
MA =μ≈⎟⎠⎞
⎜⎝⎛
Δ⎟⎟⎠
⎞⎜⎜⎝
⎛≅
μ−μ
μkdRIkmL
Example:
where Δ is the interwire gap; and d is diameter of cylindrical clouds of relatively cold products of initial wire explosions. 1-μ/3 ≅ 0.4
Typical experiment at Angara-5-1 with magnetic probe measurements, that were used earlier to measure the rate of plasma production:
RL = 1 cm; N = 40; tungsten; I = 3 MA; let d be equal to 200 μm.
Δ = 1.5 mm; ds = 50 μm; Δ<<<< ddsB1 ≈ 2 B0; ( ) 5.2121~31 ÷Δ μ−d
Conclusions
• The case, when the interwire gap is much larger than width of the jets, differs from the 1d case:
• Only small part of total current flows in the region, where Ohmic heating and transport of the heat toward cold products of initial explosion of wires play significant role. It leads to suppression of the plasma production rate, which becomes now dependent on d/Δ also.
• Dependence on d, that is diameter of “cold” products of initial explosion of wires, means that the ablation rate as well as global dynamics of multiwire arrays depend on subtle details of wire initiation.
1-2-31
cm
MA nscmg18.0; μkdRIkmL
≈⎟⎠⎞
⎜⎝⎛
Δ⎟⎟⎠
⎞⎜⎜⎝
⎛≅
μ−μ
1. Introduction.Role of prolonged plasma production in implosion of high quality
2. Theory of plasma ablation rate
2.1 1d model (of 2001)
2.2 Comparison with magnetic probe measurements
2.3. 2d model; very thin plasma jets
2.4 Comparison with experimental data
3. Interaction of plasma jets in nested arrays
3.1 Theory of steady state flow (two regimes)
3.2 Comparison with experiments
4. Conclusions
Interaction of plasma flows in nested arrays
1
2
Assume that there is no interaction between jets apart from interaction through global magnetic field
(conservation of momentum, mass and magnetic flux)
Steady state plasma flow in nested arrayEquations in the bulk
( )BrdrdB
rv
drdv
π−=ρ
41
( )⎩⎨⎧
<+<<
−=ρ22211
1211
forfor
rrrmrmrrrrm
vr
const== cEvB
equation of motion
mass conservation
conservation of magnetic flux
( )( ) ( ) ( ) ( )3
03
020
02
0
22BrBrM
BrBrM
rr
−+=
( )rBBvrv 11)( = ( ) ( )
( )rrvvr
r rρ
=ρ
( ) ( ) ( )( )rB
rvrrM
24πρ=
Steady state plasma flow in nested arrayConjugation conditions at inner array
π+ρ=
π+ρ +
++−
−− 44
222
22
222
22BvBv
( ) 12 ≥< rrM
22222 mvv −ρ=ρ ++−−
++−− = 2222 BvBv
conservation of momentum
mass conservation
conservation of magnetic flux
outflowing into vacuum
Two regimes of steady state plasma flow in nested array
2
3
5
2
3
5
0.1
1
10
((dm
2/dt
) / (
dm1/
dt))
cr
0 0.2 0.4 0.6 0.8 1r2 / r1
subalvenic flowbetween arrays
superalvenic flowbetween arrays
super-sub-alvenic flow + shock wave
cr1 cr2
0 0.2 0.4 0.6 0.8 1r2 / r1
0
2
4
6
8
10
((dm
2/dt
) / (
dm1/
dt))
cr
Two regimes of steady state plasma flow in nested array
0 0.5 1 1.5r/RL
0
0.5
1
1.5
B /
B0
0 0.5 1 1.5r/RL
0
0.5
1
1.5
B /
B0
0 0.5 1 1.5r/RL
0
0.5
1
1.5
2
v (1
2πm
1 do
t / B
02 )
0 0.5 1 1.5r/RL
0
0.5
1
1.5
2
v (1
2πm
1 do
t / B
02 )
0 0.5 1 1.5r/RL
0.1
1
10
100
ρ (B
0/ m
1 do
t)2 / 1
2π
0 0.5 1 1.5r/RL
0
1
2
3
4
ρ (B
0/ m
1 do
t)2/ 1
2π
superalvenic subalvenicdm2/dt = dm1/dt
r2= 0.5 r1
dm2/dt = 3 dm1/dt
r2 = 0.6 r1
Both types of interaction of plasma flows in nested arrays were observed in experiments at Angara-5-1
750 800 850 900 9500
0.5
1
1.5
2
2.5x 106
Time_ns
I1 (r=30 mm)
I2 r=9 mm
SXR-powerA
I3 r=5 mm
t, ns
I, MA4
3
2
1
0
Itotal
SXR(a.u.)
I(<0.85R0)
I(<0.65R0)
0 40 80 120 160
Nested array:Outer array of ∅20 mm, 40 W 6 μm wiresInner array of ∅12 mm, 120 W 6 μm wires
Typical single array of the same radius:
microprobes 1 2 3
Wires
Alexandrov et al. IEEE Trans. Plasma Science (2002)
Grabovsky, Zukakishvili, Mitrofanov et al.Plasma Phys. Reps. (2005)
Both types of interaction of plasma flows in nested arrays were observed in experiments at Angara-5-1
750 800 850 900 950 1000 1050 1100
0
0.5
1
1.5
2
2.5
3x 106 #4087
at r=30 mm
at r=9 mm
at r=5 mm
SXR-power>50 eV
ΔT(30-5)
ΔT(30-9)
Time_ns
A
750 800 850 900 9500
0.5
1
1.5
2
2.5x 106
Time_ns
I1 (r=30 mm)
I2 r=9 mm
SXR-powerA
I3 r=5 mm
Outer array of ∅20 mm, 40 W 6 μm wiresInner array of ∅12 mm, 120 W 6 μm wires
Outer array of ∅20 mm, 40 W 6 μm wiresInner array of ∅12 mm, 60 W 6 μm wires
Outer array of ∅20 mm, 40 W 6 μm wiresInner array of ∅6 mm, 40 W 8 μm wires
1. Introduction.Role of prolonged plasma production in implosion of high quality
2. Theory of plasma ablation rate
2.1 1d model (of 2001)
2.2 Comparison with magnetic probe measurements
2.3. 2d model; very thin plasma jets
2.4 Comparison with experimental data
3. Interaction of plasma jets in nested arrays
3.1 Theory of steady state flow (two regimes)
3.2 Comparison with experiments
4. Conclusions
Conclusions
• Two quite different types of plasma flows interactions may occur in nestedmultiwire array. This fact was established both theoretically and experimentally.
• The first type: the flow between the arrays is superalvenic, and existence ofinner array does not influence at all on dynamics of plasma between arrays.
• The second type:
The flow between the arrays is subalvenic
Dynamics of plasma originating from the outer array is determinedsignificantly by existence and parameters of the inner array
Main part of the total current flows around initial position of the innerarray
• What type of possible regimes will take place depends on ratio of radiuses ofthe arrays r2/r1 and on ratio of ablation rates from the arrays.
Conclusions
• The case, when the interwire gap is much larger than width of the jets, differs from the 1d case:
• Only small part of total current flows in the region, where Ohmic heating and transport of the heat toward cold products of initial explosion of wires play significant role. It leads to suppression of the plasma production rate, which becomes now dependent on d/Δ also.
• Dependence on d, that is diameter of “cold” products of initial explosion of wires, means that the ablation rate as well as global dynamics of multiwire arrays depend on subtle details of wire initiation.
1-2-31
cm
MA nscmg18.0; μkdRIkmL
≈⎟⎠⎞
⎜⎝⎛
Δ⎟⎟⎠
⎞⎜⎜⎝
⎛≅
μ−μ
Thank you for your attention!