Mod elisation et m ethodes num eriques pour l etude du transport - PowerPoint PPT Presentation

Mod´ elisation et m´ ethodes num´ eriques pour l’´ etude du transport de particules dans un plasma S´ ebastien Guisset Soutenance de th` ese de l’Universit´ e de Bordeaux Talence, le 23 Septembre 2016 Directeur de th` ese : St´ ephane Brull , Institut Math´ ematiques de Bordeaux Co-Directeur de th` ese : Emmanuel d’Humi` eres , Laboratoire Celia S´ ebastien GUISSET Study of particle transport in plasmas September 2016 1 / 46

Physical context ֒ → Contribution to the modelling and numerical methods for the transport of charged particles in plasmas ֒ → Hot plasmas created by lasers General context: Understanding of the processes leading to ignition of the fusion reactions by inertial confinement Multiphysics processes: Related research areas: ◮ Laser-plasma absorption ◮ Hypersonic flows ◮ Neutron production ◮ Radiotherapy ◮ Radiative transfer ◮ Magnetic confinement fusion ◮ Transport of particles ◮ Astrophysics ֒ → long time regimes studies (hydrodynamics scales) S´ ebastien GUISSET Study of particle transport in plasmas September 2016 2 / 46

Outline 1. Modelling in plasma physics: the angular moments models 2. Numerical methods for the study of the particle transport on large scales 3. First step towards multi-species modelling: the angular M 1 model in a moving frame 4. Conclusion / Perspectives S´ ebastien GUISSET Study of particle transport in plasmas September 2016 3 / 46

Modelling Plasma: set of totally ionised atoms. Electronic transport, fixed ions Kinetic description: electron distribution function f ( t , x , v ), ֒ → Resolution of the Vlasov or Fokker-Planck-Landau equation ∂ f + q α ∂ t + v . ∇ x f m α ( E + v × B ) . ∇ v f = C ee ( f , f ) + C ei ( f ) , � �� � � �� � � �� � collisional terms advection term force term Accurate but numerically expensive (usually limited to short scales) Hydrodynamic description: cheap but less accurate for far equilibrium regimes ֒ → describe kinetic effects on fluid time scales is challenging! ֒ → Intermediate description, angular moment models. S´ ebastien GUISSET Study of particle transport in plasmas September 2016 4 / 46

Angular moments models ֒ → Angular moments extraction: v = ζ Ω with ζ = | v | . � � � f 0 ( ζ ) = ζ 2 f 1 ( ζ ) = ζ 2 f 2 ( ζ ) = ζ 2 f ( v ) d Ω , f ( v )Ω d Ω , f ( v )Ω ⊗ Ω d Ω . S 2 S 2 S 2 Set of admissible states 1 � � ( f 0 , f 1 ) ∈ R × R 3 , f 0 ≥ 0 , A = | f 1 | ≤ f 0 . Angular moments model  ∂ t f 0 + ∇ x . ( ζ f 1 ) + q m ∂ ζ ( f 1 . E ) = 0 ,   ∂ t f 1 + ∇ x . ( ζ f 2 ) + q m ∂ ζ ( f 2 E ) − q m ζ ( f 0 E − f 2 E ) − q m ( f 1 ∧ B ) = 0 .   ֒ → Closure relation? 1 D. Kershaw, Tech. Report (1976). S´ ebastien GUISSET Study of particle transport in plasmas September 2016 5 / 46

The P N closure Spherical Harmonic expansion 2 + ∞ n f ( t , x , ζ, Ω) = 1 � � A m n f m n ( t , x , ζ ) Y m n (Ω) , 4 π n =0 m = − n with n = (2 n + 1)( n − | m | )! Y m n (Ω) = P | m | (cos θ ) e im ϕ , A m , n ( n + | m | )! n ( z ) are the associated Legendre functions 3 . and P m ֒ → Positivity of the distribution function is required → Positive P N closure 4 ֒ → We prefer a closure based on a entropy minimisation criterion 5 ֒ 2 Pomraning, Pergamon Press (1973). 3 Abramowitz and Stegun. Dover Publications (1964). 4 Hauck and McLarreen. Siam J. Sci. Comput. (2010). 5 G.N. Minerbo, J. Quant. Spectrosc. Ra. (1978). S´ ebastien GUISSET Study of particle transport in plasmas September 2016 6 / 46

The M 1 closure Determination of f 2 as a function of f 0 and f 1 : Entropy minimisation problem 6 , 7 . � � � � H ( f ) / ∀ ζ ∈ R + , ζ 2 ζ 2 min S 2 f (Ω , ζ ) d Ω = f 0 ( ζ ) , S 2 f (Ω , ζ )Ω d Ω = f 1 ( ζ ) , f ≥ 0 � with H ( f ) = ζ 2 S 2 ( f ln f − f ) d Ω . Entropy minimisation principle 8 : ◮ positivity f (Ω , ζ ) = exp( a 0 ( ζ ) + a 1 ( ζ ) . Ω) ≥ 0 , ◮ hyperbolicity ◮ entropy dissipation Expression of f 2 : � 1 − χ ( α ) Id + 3 χ ( α ) − 1 | f 1 | ⊗ f 1 f 1 � f 2 = f 0 , 2 2 | f 1 | χ ( α ) = 1 + | α | 2 + | α | 4 with , α = f 1 / f 0 . 3 6 G.N. Minerbo, J. Quant. Spectrosc. Ra. (1978). 7 D. Levermore, J. Stat. Phys. (1996). 8 B. Dubroca and J.L. Feugeas. C. R. Acad. Sci. Paris Ser. I (1999). S´ ebastien GUISSET Study of particle transport in plasmas September 2016 7 / 46

Advantages and limitations of the M 1 model Advantages ◮ Intermediate models (compromise) ◮ Application to radiative transfer and radiotherapy ◮ Accurate for isotropic configurations or configurations with one dominant direction 9 Limitations ◮ Validity of angular moments models for kinetic plasma studies? ◮ Complex configurations in collisionless regimes 10 (not presented here, see chapter 2) ֒ → Adapted for collisional plasma applications 9 Dubroca, Feugeas and Frank. Eur. Phys. J. (2010). 10 Guisset, Moreau, Nuter, Brull, d’Humi` eres, Dubroca, Tikhonchuk. J. Phys. A Math. Theor. (2015). S´ ebastien GUISSET Study of particle transport in plasmas September 2016 8 / 46

Collisional operators Electronic Fokker-Planck-Landau equation ∂ f ∂ t + v . ∇ x f + q m ( E + v × B ) . ∇ v f = C ee ( f , f ) + C ei ( f ) , � � v ′ ∈ R 3 S ( v − v ′ )[ ∇ v f ( v ) f ( v ′ ) − f ( v ) ∇ v f ( v ′ )] dv ′ � C ee ( f , f ) = α ee div v , � � 1 | u | 3 ( | u | 2 Id − u ⊗ u ) . C ei ( f ) = α ei div v S ( v ) ∇ v f ( v ) , S ( u ) = ֒ → C ee non-linear: complex angular moments extraction Simplification � F 0 = f 0 C ee ( f , f ) ≈ Q ee ( F 0 ) = C ee ( F 0 , F 0 ) 11 , 12 ζ 2 = S 2 fd Ω . ֒ → Angular moments extraction 11 Berezin, Khudick and Pekker J. Comput. Phys. (1987). 12 Buet and Cordier J. Comput. Phys. (1998). S´ ebastien GUISSET Study of particle transport in plasmas September 2016 9 / 46

Collisional operators Electronic M 1 model:  � qE �  ∂ t f 0 + ∇ x . ( ζ f 1 ) + ∂ ζ m f 1 = Q 0 ( f 0 ) ,   � qE � − qE  ∂ t f 1 + ∇ x . ( ζ f 2 ) + ∂ ζ m f 2 m ζ ( f 0 − f 2 ) = Q 1 ( f 1 ) .   Collision operators � � ζ 2 A ( ζ ) ∂ ζ ( f 0 Q 1 ( f 1 ) = − α ei 2 f 1 Q 0 ( f 0 ) = α ee ∂ ζ ζ 2 ) − ζ B ( ζ ) f 0 , ζ 3 , � ∞ � ∞ min( 1 ζ 3 , 1 min( 1 ζ 3 , 1 µ 3 ) µ 3 ∂ µ ( f 0 ( µ ) µ 3 ) µ 2 f 0 ( µ ) d µ, A ( ζ ) = B ( ζ ) = µ 2 ) d µ. 0 0 ֒ → Admissibility requirement Modification: admissible M 1 model 13  � qE �  ∂ t f 0 + ∇ x . ( ζ f 1 ) + ∂ ζ m f 1 = Q 0 ( f 0 ) ,   � qE � − qE  ∂ t f 1 + ∇ x . ( ζ f 2 ) + ∂ ζ m f 2 m ζ ( f 0 − f 2 ) = Q 1 ( f 1 ) + Q 0 ( f 1 ) .   13 J. Mallet, S. Brull and B. Dubroca. KRM (2015). S´ ebastien GUISSET Study of particle transport in plasmas September 2016 10 / 46

Collisional operators Fundamental properties of the M 1 collisional operators 14 , 15 : ◮ admissibility ◮ H-theorem (entropy dissipation) ◮ conservation properties ◮ caracterisation of the equilibrium states ֒ → Long time behavior: derivation of the plasma transport coefficients Boltzmann → Chapman-Enskog expansion: Navier-Stokes Fokker-Planck-Landau → Spitzer-H¨ arm approximation: Electron collisional hydrodynamics Electronic M 1 model → Spitzer-H¨ arm approximation: Electron collisional hydrodynamics ֒ → different plasma transport coefficients 14 Mallet, Brull, Dubroca. KRM (2015) 15 Guisset, Brull, Dubroca, d’Humi` eres, Tikhonchuk. Physica A (2016). S´ ebastien GUISSET Study of particle transport in plasmas September 2016 11 / 46

Electron collisional hydrodynamics Strongly collisional fully ionised hot plasma: f ( t , x , ζ, Ω) = M f ( ζ, T e ( t , x ) , n e ( t , x )) + ε F ( t , x , ζ, Ω) , where ε = λ ei / L , m e ζ 2 � m e � 3 / 2 � � M f ( ζ, T e ( t , x ) , n e ( t , x )) = n e ( t , x ) exp − , 2 π T e ( t , x ) 2 T e ( t , x ) F ( t , x , ζ, Ω) = F 0 ( t , x , ζ ) + F 1 ( t , x , ζ ) . Ω . Density and energy conservation laws:  ∂ n e ∂ t + ∇ x . ( n e u e ) = 0 ,   ∂ T e ∂ t + u e . ∇ x ( T e ) + 2 2 2  3 T e ∇ x . ( u e ) + 3 n e ∇ x . ( q ) = 3 n e j . E ,  where � + ∞ � + ∞ j = − en e u e = − 4 π e q = 2 π F 1 ( m e ζ 2 − 5 T e ) ζ 3 d ζ. F 1 ζ 3 d ζ, 3 3 0 0 ֒ → Closure: derivation of F 1 S´ ebastien GUISSET Study of particle transport in plasmas September 2016 12 / 46

Plasma transport coefficients Long time behavior � eE ∗ 2 T e ∇ x ( T e )( m e ζ 2 � 1 = − 2 α ei ζ 3 F 1 + 1 ζ 2 Q 0 ( ζ 2 F 1 ) , M f ζ T e + − 5) T e with E ∗ = E + (1 / en e ) ∇ x ( n e T e ) . → Solve an integro-differential equation 16 ֒ → Expansion 17 , 18 of F 1 on the generalised Laguerre polynomials ֒ Closure j = σ E ∗ + α ∇ x T e , q = − α T e E ∗ − χ ∇ x T e . 16 L. Spitzer and R. H¨ arm. Phys. Rev. (1953). 17 S.I. Braginskii. Rev. Plasma Phys. (1965). 18 S. Chapman. Phil. Trans. Roy. Soc. (1916). S´ ebastien GUISSET Study of particle transport in plasmas September 2016 13 / 46