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!