Two-fluid 20-moment simulation of fast magnetic reconnection E. Alec Johnson Department of Mathematics KU Leuven These slides were presented in my behalf at CSE13 in Boston Massachusetts on February 25, 2013 Abstract. The 20-moment two-fluid-Maxwell model resolves diago- nal pressure tensor components near the X-point when compared with Vlasov simulations of fast magnetic reconnection, in contrast to the 10- moment model. This occurs because, unlike the hyperbolic 10-moment model, the 20-moment model admits heat flux, which is a modeling re- quirement to admit steady-state 2D symmetric (driven) magnetic recon- nection. Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 1 / 56
Outline 1 Hyperbolic plasma models 2 Magnetic reconnection: Vlasov vs. fluid simulations Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 2 / 56
Standard of truth: 2-species kinetic-Maxwell (classical) BGK collision operator Maxwell’s equations: C s = f θ − f s ∂ t B + ∇ × E = 0 , , ∂ t E − c 2 ∇ × B = − J /ǫ, τ s ∇ · B = 0 , ∇ · E = σ/ǫ. where Charge moments: � −| c | 2 ρ � � σ := � q s f θ = ( 2 πθ ) 3 / 2 exp , f s d v , s m s 2 θ � J := � q s v f s d v . s m s θ := �| c | 2 / 2 � . Kinetic equations: ∂ t f i + v · ∇ x f i + a i · ∇ v f i = C i ∂ t f e + v · ∇ x f e + a e · ∇ v f e = C e Lorentz acceleration: a i = q i m i ( E + v × B ) , a e = q e m e ( E + v × B ) . Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 3 / 56
Fluid approximation requires a numerical collision operator Fluid models are kinetic equation solvers. How to choose C s ? use a simple choice based on what you want to Characteristic speeds correspond to discrete resolve. velocities. BGK damps all components and moments Fluid models evolve moments. representing perturbation from Maxwellian at Moments define a representation F s selected the same rate τ − 1 . s from a finite-dimensional space F . Damping individual moments at tunable rates convergence F s → f s requires an infinite allows smooth transition to a model with more number of moments. moments. F is a good representation space when C is Faster damping for higher moments large but bad when C is small. corresponds to faster damping for finer-scale components of f s . How can we justify fluid models when physically C s = 0 ? by a multiscale framework: fluid models are a coarse-scale model used to accelerate convergence of the lowest moments of the kinetic equation. C s can be physically defined to incorporate all microscale effects not resolved by the coarse-scale model. in code, C s � = 0 is a numerical mechanism to regularize f s . collision period τ s selects the largest time scale for resolution of velocity-space detail. Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 4 / 56
Conserved Moment Evolution Convention: Products and powers of Substituting into equation (1) gives: vectors are defined by tensor products. General conserved moment evolution Conserved moments are moments of � � F n + ∇ · � F n + 1 ∨ F n − 1 + � F n × B monomials in v . Let χ = χ ( v ) . Take the ∂ t � = q E � + � C n m n χ th moment of the kinetic equation: � This is a hierarchy of moment evolution equations: � � χ ∂ t f + ∇ x · ( v f ) + ∇ v · ( a f ) = C F 0 + ∇ · � F 1 = � v ∂ t � C 0 , Integrate by parts to get F 1 + ∇ · � F 2 = q F 0 + � F 1 × B )+ � ∂ t � m ( E � C 1 , � � � � � � F 2 + ∇ · � F 1 + � F 2 × B F 3 ∨ ∂ t � = q E � + � C 2 , ∂ t v χ f + ∇ · v v χ f = v f a · ∇ v χ + v χ C m 2 � � (1) F 3 + ∇ · � F 4 ∨ F 2 + � F 3 × B ∂ t � = q E � + � C 3 , m 3 � � Choose χ = v n . Define F 4 + ∇ · � F 5 ∨ F 3 + � F 4 × B ∂ t � = q E � + � C 4 . m 4 � � F n := C n := � � v n f , v n C . Tensor notation: v v AB = A ⊗ B = tensor product, Since a = q m ( E + v × B ) , Sym A = symmetrization of tensor A , � � a · ∇ v ( v n ) ∨ Ev n − 1 + v n × B A ⊻ B = Sym ( A ⊗ B ) , and = q m n . A ∨ = B ⇐ ⇒ Sym A = Sym B . Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 5 / 56
Primitive moments Fluid closures should be Galilean invariant. The kinetic equation is Galilean-invariant, so we require fluid closures to be Galilean-invariant. Primitive moments are Galilean-invariant Definitions: � ρ = f v � v χ � χ � := ρ u := � v � c := v − u � F n := c n f v Specifying closure in terms of primitive moments F n ensures that closures are Galilean-invariant. Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 6 / 56
Mapping from conserved to primitive variables To express primitive moments in terms of conserved Multiplying by f and integrating over velocity, moments we observe that F 0 = ρ, n ( − 1 ) j � n � � c n ∨ =( v − u ) n ∨ u j v n − j . F 1 = 0 , = j j = 0 F 2 = � F 2 − ρ u 2 , Multiplying by C and integrating over velocity, F 3 − 3 u � F 2 + 2 ρ u 3 , F 3 ∨ = � C 0 = � C 0 , F 4 − 4 u � F 3 + 6 u 2 � F 2 − 3 ρ u 4 , F 4 ∨ = � C 1 − u � C 1 = � C 0 , F 5 − 5 u � F 4 + 10 u 2 � F 3 − 10 u 3 � F 2 + 4 ρ u 5 , F 5 ∨ = � C 2 = � C 1 + u 2 � C 2 − 2 u � C 0 , F 1 = ρ u . C 3 − 3 u � C 1 − u 3 � where we have used that � C 3 ∨ = � C 2 + 3 u 2 � C 0 , C 4 − 4 u � C 3 + 6 u 2 � C 1 + u 4 � C 4 ∨ = � C 2 − 4 u 3 � C 0 , In practice, when computing with conserved variables, we compute the primitive variables that we need for C 0 = 0 and in the where in the absence of production � the closing moment and then use one of the relations C 1 = 0. further absence of interspecies friction � = F 3 + 3 u � F 2 − 2 ρ u 3 , F 3 ∨ � = F 4 + 4 u � F 3 − 6 u 2 � F 2 + 3 ρ u 4 , F 4 ∨ � = F 5 + 5 u � F 4 − 10 u 2 � F 3 + 10 u 3 � F 2 − 4 ρ u 5 � F 5 ∨ to solve for the closing conserved moment. Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 7 / 56
Mapping from primitive to conserved variables To express conserved moments in terms of primitive Multiplying by f and integrating over velocity, moments we observe that F 0 = ρ, � n � n � � v n ∨ =( u + c ) n ∨ u j c n − j . F 1 = ρ u , = � j j = 0 F 2 = F 2 + ρ u 2 , � Multiplying by C and integrating over velocity, = F 3 + 3 uF 2 + ρ u 3 , F 3 ∨ � C 0 = C 0 , � = F 4 + 4 uF 3 + 6 u 2 F 2 + ρ u 4 , F 4 ∨ � C 1 = C 1 + uC 0 , � = F 5 + 5 uF 4 + 10 u 2 F 3 + 10 u 3 F 2 + ρ u 5 , F 5 ∨ � C 2 = C 2 + 2 uC 1 + u 2 C 0 , � where we have used that F 1 = 0. = C 3 + 3 uC 2 + 3 u 2 C 1 + u 3 C 0 , C 3 ∨ � = C 4 + 4 uC 3 + 6 u 2 C 2 + 4 u 3 C 1 + u 4 C 0 , C 4 ∨ � where in the absence of production C 0 = 0 and in the absence of interspecies friction C 1 = 0. Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 8 / 56
Collisional closures Conventional names for primitive moments: and where when computing � P := F 2 , Q := F 3 , R := F 4 , S := F 5 , c c n ( f θ − f ) C n = τ � δ t P := C 2 , δ t Q := C 3 , δ t R := C 4 . we have used that � Collisional moments for BGK collision operator: f θ = ρ, c � δ t P = − P ◦ τ , c f θ = 0 , c � δ t Q = − Q , cc f θ = P τ 3 c � = 3 PP /ρ − R ∨ δ t R , ccc f θ = 0 , and τ 4 c � where P ◦ := P − p I is deviatoric pressure and for BGK ∨ cccc f θ = 3 PP /ρ. τ = τ 3 = τ 4 . c Model coarsening: dial τ 4 ց 0 to smoothly transition from 35-moment to 20-moment model. dial τ 3 ց 0 to smoothly transition from 20-moment to 10-moment model. dial τ ց 0 to smoothly transition to the 5-moment model. Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 9 / 56
20-moment flux closure 20-moment Maxwellian-based closure [Grad49] Assumed distributions Entropy-maximizing closure : ∨ = 3 ( PP − P ◦ P ◦ ) /ρ. R f ( c ) = exp ( a · m ) , 20-moment Gaussian-based closure [GrothGRB03] where m = ( 1 , cc , ccc ) is the tuple of evolved moments and a is a tuple of coefficients. ∨ = 3 PP /ρ . R Maxwellian-based closure f = W M ( 1 + c ′ · m ) , Comparison of closures: where The Maxwellian-based closure assumes that the W M := exp ( c 0 · m 0 ) , velocity distribution is a Maxwellian times a m 0 = ( 1 , | c | 2 ) , polynomial. and c ′ · m is a polynomial orthogonal to 1 and | c | 2 in the The Gaussian-based closure assumes that the weight W M . velocity distribution is a Gaussian times a polynomial. Gaussian-based closure f = W G ( 1 + c ′ · m ) , Gaussian-based closure is a consistent generalization of the 10-moment model. where Gaussian-based closure is hyperbolic if heat flux is W G := exp ( c 0 · m 0 ) , small enough. m 0 = ( 1 , cc ) , Maxwellian-based closure is hyperbolic if heat flux and c ′ · m is a polynomial orthogonal to 1 and cc in the and deviatoric stress are small enough. weight W G . BGK would relax R to the Gaussian-based closure for R . Johnson (KU-Leuven) 20-moment two-fluid reconnection Feb. 25, 2013 10 / 56
Recommend
More recommend
