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INTRODUCTION TO INERTIAL CONFINEMENT FUSION R. Betti Lecture 25 Hydrodynamic Scaling
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### Transcript of R. Betti Lecture 25 Hydrodynamic ScalingR. Betti . Lecture 25 . Hydrodynamic Scaling . 2...

• INTRODUCTION TO INERTIAL CONFINEMENT FUSION

R. Betti

Lecture 25

Hydrodynamic Scaling

• 2

Hydrodynamic scaling is used to extrapolate OMEGA results to NIF laser energies

• The Euler equations are scale invariant and depend on a single dimensionless parameter

0vtρ ρ∂ +∇ =∂

0/itV Rτ = ˆ / iv v V= 0ˆ /r r R= 0ˆ R∇ = ∇

0v v v pt

ρ ∂ + ∇ +∇ = ∂

/ Ap p P=0/ρ ρ ρ=

2 23 1 5 1 02 2 2 2

p v v p vt

ρ ρ∂ + +∇ + = ∂

• 2 2 2 23 1 5 1ˆˆ ˆˆ ˆ ˆ ˆ ˆ 02 2 2 2

p Mach v v p Mach vρ ρτ∂ + +∇ + = ∂

ˆ ˆ ˆ ˆ 0vρ ρτ∂

+∇ =∂

2 ˆ ˆ ˆˆ ˆ ˆ ˆ 0vMach v v pρτ∂ + ∇ +∇ = ∂

22 0 i

A

VMachPρ

=

• The single dimensionless parameter is the Mach number

• Dimensionless Euler equations:

• • Keeping the Mach number fixed results in similar implosions

3/52 2 22 0

3/5 2/5~i i iA

A A A

V V VPMachP P Pρ

α α = =

Hydroequivalent implosions have: • Same implosion velocity • Same ablation pressure • Same adiabat

• • The stagnation pressure and density is the same for hydroequivalent implosions

3 4stag AP P Mach

−=

1 20stag Machρ ρ

−=

3/5

0 ~ APρα

• • Since the hydrodynamics is the same, the classical growth of the instabilities is the same for hydroequivalent implosions

• The classical growth factor is the same

0

stag te Growth Factorγηη

= = −

• • To achieve the same ablation pressure, hydroequivalent implosions require the same laser intensity or radiation pressure

2/3~A LP I

• To achieve the same implosion velocity, hydroequivalent implosions require absorbed laser energy proportional to mass

21 ~2 shell i Hydro abs Hydro abs L

M V E Eη η η=

• If the absorption fraction is scale invariant then

~L shellE M

Same for hydroequivlent implosions

• • The ablative stabilization of the RT instability is the same for hydroequivalent implosions

• The specific ablation rate is the same for hydroequivalent implosions

1/3~A Lm I

1/31/15 3/5

3/50

~ ~ ~( / )

A LA L

A

m IV IP

αρ α

• • Time scale is proportional to radius

0/itV Rτ =0

i

R tV

τ = 0~t R

2 30 0~ ~ ~L L LE Power time I R t I R×

• The laser energy is proportional to volume

• The laser power is proportional to surface

2 20 0~ ~LPower I R R

• The adiabat depends on initial shock strength that depends on initial ablation pressure (i.e. intensity). Keep IL(τ) fixed

• • Summary of hydroscaling

1/3L LMass ~ Volume ~ E R~E

1/3Ltime ~ R ~ E

20

( )I ( ) ~LLaser Power fixed

Rττ − =

2 2/3LPower ~ R ~ E

0/itV Rτ =

η(0)= Initial Nonuniformities ~ R ~ EL1/3

• To keep the same relative size of the nonuniformities the same:

Vi and α are the same

• 12

Hydrodynamic scaling of the no-alpha fusion yield

• 13

Hydro scaling

For 2x amplification

• Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15