A case for 13-moment two-fluid MHD E. Alec Johnson Department of Mathematics University of Wisconsin–Madison April 5, 2012 Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 1 / 29
Outline The purpose of this talk is to build a case that a 13-moment 2-fluid model is the simplest fluid model of plasma that can resolve steady fast magnetic re- connection and avoid anomalous cross-field transport in a highly magnetized plasma. The argument: A study of the terms of the XMHD Ohm’s law and entropy evolution at the 1 X-point of steady 2D reconnection invariant under 180-degree rotation reveals that nonzero heat flux and viscosity are model requirements. In a highly magnitized plasma, anomalous cross-field transport is difficult 2 to avoid unless spatially higher-order-accurate methods are used. Higher-order-accurate positivity-preserving methods are available for hyperbolic models but not yet for diffusive models. Relaxation closures are non-diffusive and trivial for the 13-moment model and are independent of the magnetic field. In contrast, relaxation closures for quadratic-moment models are diffusive, become complicated in the presence of a magnetic field, and become ill-conditioned when the magnetic field becomes strong. As a consequence, implicit methods are necessary and it is difficult to design appropriate preconditioners, particularly for positivity-preserving methods. Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 2 / 29
Model hierarchy (Two-species kinetic-Maxwell) � (13-moment two-fluid Maxwell) − − − − → (13-moment two-fluid MHD) � � (10-moment two-fluid Maxwell) − − − − → (10-moment two-fluid MHD) � � (5-moment two-fluid Maxwell) − − − − → (5-moment two-fluid MHD) � (5-moment MHD) Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 3 / 29
2-species kinetic-Maxwell with Gaussian-BGK collision operator Kinetic equations: Collision operator moments with f and has pseudo-temperature � Θ equal to an Conservation dictates: � � affine (not necessarily convex!) C ii + ← → ∂ t f i + v · ∇ x f i + a i · ∇ v f i = � C ie := C i combination of the m � m � C ii = 0 = C ee , pseudo-temperature Θ and its C ee + ← → v v ∂ t f e + v · ∇ x f e + a e · ∇ v f e = � � isotropization: C ei := C e � Θ − 1 · c / 2 � m ( C i + C e ) = 0 − c · � ρ e v f � Θ = � , Maxwell’s equations: where m = ( 1 , v , � v � 2 ) . det ( 2 π � Θ) � � ∂ t B + ∇ × E = 0 , C ss + ← → The source term C s = � C sp is Θ := � cc � = cc f d v / f d v , ∂ t E − c 2 ∇ × B = − J /ǫ 0 , specified by physics, but there is some freedom in how to allocate C s among � C ss � ∇ · B = 0 , ∇ · E = σ/ǫ 0 , and ← → Θ := νθ I + ν Θ , ( ν + ν = 1 ) , C sp . For weakly collisional plasma � � � v m C s ≈ 0, and � ν := 1 / Pr = τ/ � q s τ. C ss can be chosen to σ := f s d v , dominate � m s C sp . Here � τ is the heat flux relaxation period, s � τ is the relaxation period of deviatoric � Gaussian-BGK collision q s Θ respects entropy if � pressure, and C � Θ J := v f s d v operator m s For C ss we obtain relaxation is positive definite (i.e. 0 < ν ≤ 3 / 2). s In the limit ν ց 0 heat flux goes to zero closures with a Gaussian-BGK and the solution approximates hyperbolic collision operator which relaxes Lorentz force law Gaussian-moment (10-moment) gas toward a Gaussian distribution: dynamics. qi f � Θ − f a i = mi ( E + v × B ) , � C ss = C � Θ = , Use of a Gaussian-BKG collision � τ qe operator allows one to tune the viscosity a e = me ( E + v × B ) where the Gaussian distribution f � µ = p τ and the thermal conductivity Θ k = 5 µ shares physically conserved m Pr . 2 Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 4 / 29
5-moment multi-fluid plasma Evolution equations δ t ρ s = 0 ρ s d t u s + ∇ p s + ∇ · P ◦ s = q s n s ( E + u s × B ) + R s 3 2 δ t p s + p s ∇ · u s + P ◦ s : ∇ u s + ∇ · q s = Q s Evolved moments Relaxation (diffusive) flux closures: � � ρ s 1 P ◦ ( c s c s − � c s � 2 I / 3 ) f s d v , s = = f s d v ρ s u s v 3 1 2 � c s � 2 � 2 p s 2 c s � c s � 2 f s d v 1 q s = Definitions Interspecies forcing closures: δ t ( α ) := ∂ t α + ∇ · ( u s α ) � � c s := v − u s � R s � � ← → v = C s d v ≈ 0 n s := ρ s / m s 1 2 � c s � 2 Q s Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 5 / 29
10-moment multi-fluid plasma Evolution equations δ t ρ s = 0 ρ s d t u s + ∇ · P s = q s n s ( E + u s × B ) + R s δ t P s + Sym2 ( P s · ∇ u s ) + ∇ · q s = q s n s Sym2 ( T s × B ) + R s + Q s Evolved moments Relaxation (diffusive) flux closures: � ρ s 1 � = f s d v ρ s u s v q s = c s c s c s f s d c s P s c s c s Definitions Relaxation source term closures: δ t ( α ) := ∂ t α + ∇ · ( u s α ) � c s c s C s d v = − P ◦ c s := v − u s R s = s /τ, Sym2 ( A ) := A + A T Interspecies forcing closures: n s := ρ s / m s � � v T := P / n � ← � R s � → = C s d v ≈ 0 c s c s Q s Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 6 / 29
13-moment multi-fluid plasma Evolution equations δ t ρ s = 0 ρ s d t u s + ∇ · P s = q s n s ( E + u s × B ) + R s δ t P s + Sym2 ( P s · ∇ u s ) + ∇ · q s = q s n s Sym2 ( T s × B ) + R s + Q s q ss , t + ← → δ t q s + q s · ∇ u s + q s : ∇ u s + P s : ∇ Θ s + P s · ∇ θ s + ∇ · R s = Sym3 ( P s P s ) /ρ s + q s m s q s × B + � q s , t Evolved moments Relaxation source term closures: � � � R s � � � � � C ss d v = − 1 P ◦ 1 c s c s ρ s s = 2 c s � c s � 2 v � 1 Pr q s ρ s u s q ss , t τ s = f s d v c s c s P s 2 c s � c s � 2 1 Hyperbolic flux closures: q s � � q s � � � Definitions c s c s c s = f s ( c s ) d c s c s c s � c s � 2 R s δ t ( α ) := ∂ t α + ∇ · ( u s α ) Interspecies forcing closures: Sym2 ( A ) := A + A T , � c s := v − u s , R s v n s := ρ s / m s , T := P / n , ← → = c s c s Q s C s d v ≈ 0 Θ := P /ρ, θ := tr Θ / 2 , ← → 2 c s � c s � 2 1 q s , t Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 7 / 29
MHD: Maxwell’s equations Models which evolve Maxwell’s equations and classical gas dynamics fail to satisfy a relativity principle. Magnetohydrodyamics (MHD) remedies this problem by assuming that the light speed is infinite. Then Maxwell’s equations simplify to ∂ t B + ∇ × E = 0 , ∇ · B = 0 , c − 2 ∂ t E , ❳❳❳❳ c − 2 ∇ · E ✘ µ 0 σ = 0 + ✘✘✘✘ ❳❳❳ ✘ µ 0 J = ∇ × B − ✘✘✘ ❳ ❳ This system is Galilean-invariant, but its relationship to gas-dynamics is fundamentally different: variable MHD 2-fluid-Maxwell E supplied by Ohm’s law evolved (from gas dynamics) (from B and J ) J J = ∇ × B /µ 0 J = e ( n i u i − n e u e ) (comes from B ) (from gas dynamics) σ σ = 0 (quasineutrality) σ = e ( n i − n e ) (gas-dynamic constraint) (electric field constraint) Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 8 / 29
MHD: charge neutrality The assumption of charge neutrality reduces the number of gas-dynamic equa- tions that must be solved: net density evolution The density of each species is the same: n i = n e = n net velocity evolution The species fluid velocities can be inferred from the net current, net velocity, and density: m e J m i J u i = u + ne , u e = u − ne . m i + m e m i + m e Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 9 / 29
MHD: Ohm’s law For each species s ∈ { i , e } , rescaling momentum evolution by q s / m s gives the current evolution equation ∂ t J s + ∇ · ( u s J s + ( q s / m s ) P s ) = ( q 2 s / m s ) n ( E + u s × B ) + ( q s / m s ) R s . Summing over both species and using charge neutrality gives net current evolution: � � � P i � � � � � e 2 ρ m i − m e P e m i − m e R e ∂ t J + ∇ · uJ + Ju − JJ + e ∇ · − = E + u − J × B − . e ρ m i m e m i m e e ρ en A closure for the collisional term is R e en = η · J + β e · q e . Ohm’s law is current evolution solved for the electric field: E = B × u (ideal term) + m i − m e J × B (Hall term) e ρ + η · J (resistive term) + β e · q e (thermoelectric term) 1 + e ρ ∇ · ( m e P i − m i P e ) (pressure term) � � �� uJ + Ju − m i − m e + m i m e ∂ t J + ∇ · JJ (inertial term). e 2 ρ e ρ Ohm’s law gives an implicit closure to the induction equation, ∂ t B + ∇ × E = 0 (so retaining the inertial term entails an implicit numerical method). Johnson (UW-Madison) 13-moment two-fluid MHD Apr. 5, 2012 10 / 29
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