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=0
ˆ 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 LM 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 1/3
Ltime ~ 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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