Numerical methods for inertial confinement fusion Xavier Blanc - PowerPoint PPT Presentation

Numerical methods for inertial confinement fusion Xavier Blanc blanc@ann.jussieu.fr CEA, DAM, DIF, F-91297 Arpajon, FRANCE CEMRACS 2010 – p. 1

Outline High power laser facilities Experimental setting Modelling: hydrodynamics Modelling: radiative transfer Diffusion approximation Boundary conditions Frequency dependent diffusion Marshak waves Flux limitation Discretization Frequency Time Space CEMRACS 2010 – p. 2

Outline (continued) Diffusion schemes VF4 scheme Other schemes LapIn scheme P1 model PN model M1 model Boundary condition (continued) CEMRACS 2010 – p. 3

High power laser facilities • Laser MegaJoule (Bordeaux) + HiPER (PetAL) project • National Ignition Facility (Livermore) • LFEX (Osaka) LMJ project CEMRACS 2010 – p. 4

High power laser facilities • Laser MegaJoule (Bordeaux) + HiPER (PetAL) project • National Ignition Facility (Livermore) • LFEX (Osaka) NIF project CEMRACS 2010 – p. 4

Inertial confinement fusion Principle : implode a capsule of fusion fuel by laser pulses. Objective : Reaching conditions under which fusion reactions start. Mainly two strategies: • Direct drive : the target is directly heated by lasers • Indirect drive : the lasers heat the inner walls of a cavity. The walls emit X-rays toward the target. 10 mm 5 mm . LMJ / NIF project : indirect drive. CEMRACS 2010 – p. 5

Inertial confinement fusion 10 mm 5 mm . Indirect drive Advantage: Heating is more uniform. Drawback: Energy loss (up to 80%) in heating walls. CEMRACS 2010 – p. 6

Experimental setting CEMRACS 2010 – p. 7

Experimental setting CEMRACS 2010 – p. 8

Experimental setting • Size of capsule: ∼ 1 mm • Size of Hohlraum: ∼ 10 mm CEMRACS 2010 – p. 9

Experimental setting LMJ: 300 meters LMJ chamber: 10 meters Hohlraum: 10 mm Capsule: 1 mm CEMRACS 2010 – p. 10

Typical sizes Starting fusion reactions: need T ≥ 5 × 10 7 K Lawson’s criterion for reaching fusion ( τ confinement time, n e electronic density): n e τ ≈ 10 14 s cm − 3 . ρ ≈ 10 3 g cm − 3 , T ≈ 5 × 10 7 K, τ ≈ 10 − 9 s. Typically Hot spot at the center of the capsule Density Temperature 1.7e7 18 1.5e7 16 1.3e7 14 1.1e7 12 9e6 10 7e6 8 5e6 6 3e6 4 1e6 2 6e−5 8e−5 1e−4 1.2e−4 1.6e−4 1.8e−4 2e−5 4e−5 Radius (m) Main Hot spot fuel CEMRACS 2010 – p. 11

Modelling issues Coupling between: Laser – plasma � Fusion reactions ↔ Hydrodynamics ↔ Neutronics � Radiative transfer • Laser-plasma interaction • Hydrodynamical instabilities • Suprathermic particles • Loss of thermodynamic equilibrium • Dealing with uncertainties • .... CEMRACS 2010 – p. 12

Hydrodynamics Laser plasmas: hot, dense. Bitemperature compressible Euler equations ( d dt = ∂ ∂t + u · ∇ ):  dρ dt + ρ div( u ) = 0 ,        ρdu   dt + ∇ ( p e + p i ) = F r ,   ρdE e   dt + p e div( u ) − div ( χ e ∇ T e ) + γ ei ( T e − T i ) = Q r + S,       ρdE i   dt + p i div( u ) − div ( χ i ∇ T i ) − γ ei ( T e − T i )  = 0 , p e,i + e.o.s ( E e,i = ( γ e,i − 1) ρ = C v { e,i } T e,i ). F r radiative flux, Q r radiative energy ⇐ radiative transfer equation S laser energy drop m e 1 τ ei ∝ T 3 / 2 χ e ∝ T 5 / 2 γ ei = ρC ve , , . e e m i τ ei CEMRACS 2010 – p. 13

Radiative transfer I = I ν ( x, t, Ω) : specific radiative intensity ( Jm − 2 ) ν frequency, Ω direction of propagation. � I ν � ρ d + Ω · ∇ I ν + σ a ( I ν − B ν ( T e )) c dt ρ � � � I ν (Ω ′ ) d Ω ′ 3 � 1 + (Ω · Ω ′ ) 2 � + κ Th I ν − = 0 , 4 4 π S 2 where σ a = σ a ( ν, T e , ρ ) , and 1.6 b(x) 1.4 2 hν 3 1.2 1 B ν ( T ) = 1 c 2 hν kT − 1 e 0.8 � hν � 3 0.6 T 3 kT ∝ . hν kT − 1 0.4 e 0.2 � �� � b ( hν kT ) 0 0 2 4 6 8 10 CEMRACS 2010 – p. 14

Coupled system dρ 8 dt + ρ div( u ) = 0 , > > > > > > > ρ du > > dt + ∇ ( p e + p i ) = F r , > > < ρ dE e > + p e div( u ) − div ( χ e ∇ T e ) + γ ei ( T e − T i ) = Q r + S, > > > dt > > > > ρ dE i > > > dt + p i div( u ) − div ( χ i ∇ T i ) − γ ei ( T e − T i ) = 0 , : (+ equation of state) „ I ν « ρ d + Ω · ∇ I ν + σ a ( I ν − B ν ( T e )) c dt ρ I ν (Ω ′ ) d Ω ′ „ 3 « Z “ Ω · Ω ′ ´ 2 ” ` + κ Th I ν − 1 + = 0 , 4 4 π S 2 Z ∞ Z Ω ( σ a + κ Th ) I ν ( x, t, Ω) dν d Ω F r = 4 π , S 2 0 Z ∞ σ a ( I ν ( x, t, Ω) − B ν ( T e )) dν d Ω Z Q r = 4 π . S 2 0 CEMRACS 2010 – p. 15

Radiative transfer: theory Without hydro 1 ∂I ν ∂t + Ω · ∇ I ν + σ a ( I ν − B ν ( T )) c � � � I ν (Ω ′ ) d Ω ′ � 1 + (Ω · Ω ′ ) 2 � 3 I ν − + κ Th = 0 , 4 4 π S 2 � ∞ � ∂T σ a ( ν ) ( I ν ( x, t, Ω) − B ν ( T )) dν d Ω C v ∂t = 4 π , S 2 0 Initial conditions: I ν ( x, 0 , Ω) = I 0 T ( x, 0) = T 0 ( x ) . ν ( x, Ω) , Boundary conditions: I ν ( x, t, Ω) = I ext ∀ x ∈ ∂ D , ∀ Ω / Ω · n ( x ) ≤ 0 , ( x, t, Ω) , ν Theorem: (Golse, Perthame, 1986) Under "suitable hypotheses", the radiative transfer system is well- posed. CEMRACS 2010 – p. 16

Radiative transfer: theory Remark on the hypotheses: one of them reads ∀ ν > 0 , T �→ σ a ( ν, T ) is nonincreasing, and T �→ σ a ( ν, T ) B ν ( T ) is nondecreasing. Physically, this is not relevent. But implies accretiveness of semi-group. More realistic results on simpler system (Bardos, Golse, Perthame, Sentis, 1988) Coupled system: local in time existence: Lin (2007), Zhong, Jiang (2007). CEMRACS 2010 – p. 17

Diffusion approximation radiation almost isotropic ⇒ P1 approximation in Ω radiation is not Planckian (M-band of gold) CEMRACS 2010 – p. 18

Diffusion approximation P1 approximation in Ω : I ν ≈ cE ν + 1 3 Ω F ν . � � E ν = 1 c − F ν = − S 2 I ν d Ω , S 2 Ω I ν d Ω .  dE ν dt + div( F ν ) + σ a ( cE ν − B ν ( T )) = 0 ,       � 1 dF ν dt + − ΩΩ · ∇ I ν d Ω + σ a F ν = 0 .  c   S 2   � �� �  ≈ c 3 ∇ E ν Stationary approximation for the second equation: c ≪ 1 . � c � dE ν dt − div ∇ E ν + σ a ( cE ν − B ν ( T )) = 0 . 3 σ ν Nonlinear diffusion equation. CEMRACS 2010 – p. 19

Diffusion approximation – boundary condition Boundary condition on I ν ( x, t, Ω) ⇒ boundary condition on E ν ( x, t ) ? I ν ( x, t, Ω) = I ext ∀ x ∈ ∂ D , ∀ Ω / Ω · n ( x ) ≤ 0 , ( x, t, Ω) , ν � � | Ω · n | I ext ( x, t, Ω) d Ω := F in | Ω · n | I ν ( x, t, Ω) d Ω = ν ( x, t ) , ν Ω · n ≤ 0 Ω · n ≤ 0 � � � � c | Ω · n | d Ω+ 3 c ∇ E ν | Ω · n | Ω d Ω = F in − · 4 π E ν ν . 4 π 3 ( σ a ( ν ) + κ Th ) Ω · n ≤ 0 Ω · n ≤ 0 2 ∂E ν ∂n = 4 c F in E ν + ν . 3 ( σ a ( ν ) + κ Th ) Marshak boundary condition CEMRACS 2010 – p. 20

Diffusion approximation Dimensional analysis: 1 c ( σ a + κ Th ) ≈ ε 2 . 1 mean free path σ a + κ Th ≈ ε , mean free time Asymptotic analysis: t → ε 2 t , x → εx (Larsen, Badham, Pomraning, 1983). ε ∂I ν ∂t + Ω · ∇ I ν + σ a ε ( I ν − B ν ( T )) c � � � I ν (Ω ′ ) d Ω ′ + κ Th 3 � 1 + (Ω · Ω ′ ) 2 � I ν − = 0 , ε 4 4 π S 2 � ∞ � ∂T σ a ε ( I ν ( x, t, Ω) − B ν ( T )) dν d Ω εC v ∂t = 4 π S 2 0 Hilbert expansion: I ν = I 0 + εI 1 + ε 2 I 2 + . . . , T = T 0 + εT 1 + ε 2 T 2 + . . . . CEMRACS 2010 – p. 21

Diffusion approximation ε ∂I ν ∂t + Ω · ∇ I ν + σ a ε ( I ν − B ν ( T )) + κ Th ( I ν − K ( I ν )) = 0 , c ε order ε − 1 : � � � � I 0 ν − B ν ( T 0 ) I 0 ν − K ( I 0 = 0 ⇒ I 0 ν = B ν ( T 0 ) . σ a + κ Th ν ) order ε 0 : � ν − c ν ( T 0 ) T 1 � � � Ω · ∇ I 0 I 1 4 π B ′ I 1 ν − K ( I 1 ν + σ a + κ Th ν ) = 0 . � � Ω · ∇ I 1 σ a + κ Th Ω · ∇ I 0 1 ν ( x , t, Ω) = Ω · ∇ + Ω · ∇ ( ... ) . ν � � � 1 S 2 Ω · ∇ I 1 3( σ a + κ Th ) ∇ I 0 ν ( x , t, Ω) d Ω = div . ν CEMRACS 2010 – p. 22

Diffusion approximation order ε 1 : � � � T 1 � 2 ∂I 0 1 ν − c ν ( T 0 ) T 2 − c ν 4 π B ′ 4 π B ′′ ∂t + Ω · ∇ I 1 I 2 ν ( T 0 ) ν + σ a c 2 � � I 2 ν − K ( I 2 + κ Th ν ) = 0 , � d Ω � ∞ � � ∂T 0 ν ( T 0 )( T 1 ) 2 ν − c ν ( T 0 ) T 2 − c 4 π B ′ 4 π B ′′ I 2 C v = σ a 4 π dν. ∂t 2 S 2 0 Integrate over Ω and ν , sum: � ∞ � 1 ∂ Ω · ∇ I 1 ( x , t, Ω) dν d Ω � C v T 0 + ac ( T 0 ) 4 � + 4 π = 0 , c ∂t S 2 0 � � Ω · ∇ I 1 , Using the expression of � 1 T 0 � 4 �� 1 ∂ � � C v T 0 + ac ( T 0 ) 4 � � − div ∇ ac = 0 . c ∂t 3 σ R CEMRACS 2010 – p. 23