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

3.5~A radP T

• 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

3~A radm T

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

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