Fluid models of plasma Alec Johnson Centre for mathematical Plasma - PowerPoint PPT Presentation

Fluid models of plasma Alec Johnson Centre for mathematical Plasma Astrophysics Mathematics Department KU Leuven Nov 29, 2012 Presentation of plasma models 1 Derivation of plasma models 2 Kinetic Two-fluid MHD Conclusion Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 1 / 40

Outline 1 Presentation of plasma models 2 Derivation of plasma models Kinetic Two-fluid MHD Conclusion Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 2 / 40

Modeling parameters Physical constants that define an Collisionless time, velocity, and space scale parameters: ion-electron plasma: p , s := n 0 e 2 ω 2 e (charge of proton), 1 plasma frequencies : , ǫ 0 m s m i , m e (ion and electron mass), 2 ω g , s := eB 0 gyrofrequencies : , 3 c (speed of light), m s ǫ 0 (vacuum permittivity). 4 t , s := 2 p 0 v 2 thermal velocities : , ρ s Fundamental parameters that B 2 characterize the state of a plasma: A , s := 2 p B v 2 0 Alfv´ en speeds : = , n 0 (typical particle density), 1 ρ s µ 0 m s n 0 � T 0 (typical temperature), 2 λ D := v t , s ǫ 0 T 0 Debye length : = n 0 e 2 , B 0 (typical magnetic field). 3 ω p , s r g , s := v t , s = m s v t , s Derived quantities: gyroradii : , ω g , s eB 0 p 0 := n 0 T 0 (thermal pressure) � δ s := v A , s c m s B 2 skin depths : = = µ 0 n s e 2 . p B := 2 µ 0 (magnetic pressure) 0 ω g , s ω p , s ρ s := n 0 m s (typical density). � v t , s � 2 = � r g , s � 2 . plasma β := p 0 p B = Collision periods: δ s v A , s v A , s = r g , s λ D = ω p , s τ sp : expected time for 90-degree c ω g , s . non-MHD ratio: deflection of species s via p . Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 3 / 40

Plasma model hierarchy Particle Maxwell : discrete particles: ( x p ( t ) , v p ( t )) 1    � large number of particles (per “mesh cell”) Kinetic Maxwell : particle density functions: f s ( x , v ) 2    � fast collisions ( τ ss → 0). two-fluid Maxwell : one gas for each species: ρ s ( x ) , u s ( x ) , e s ( x ) 3    � fast light waves ( c → ∞ ), charge neutrality ( λ D → 0). extended MHD : gas that conducts electricity: ρ ( x ) , u ( x ) , e ( x ) , B ( x ) ; 4 J = µ − 1 0 ∇ × B , E = u × B + η J + · · · .    � small gyroradius ( r g → 0) and gyroperiod ( ω g → ∞ ). Ideal MHD : a perfectly conducting gas: E = u × B . 5 Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 4 / 40

Fundamental model: particle-Maxwell (relativistic) Maxwell’s equations: Changing SI to Gaussian units: replace B with B / c . ∂ t B + ∇ × E = 0 , choose ǫ − 1 = 4 π . 0 ∂ t E − c 2 ∇ × B = − J /ǫ 0 , ∇ · B = 0 , ∇ · E = σ/ǫ 0 . Problem: model based on particles is not a computationally accessible standard of truth for Charge moments: most applications. σ := � p S p ( x p ) q p , Solution: replace particles with a particle density J := � function f s ( t , x , γ v ) for each species s . p S p ( x p ) q p v p , Particle equations: d t x p = v p , d t ( γ p v p ) = a p ( x p , v p ) , γ − 2 := 1 − ( v p / c ) 2 . p Lorentz acceleration: a p ( x , v ) = q p m p ( E ( x ) + v × B ( x )) Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 5 / 40

2-species kinetic-Maxwell (relativistic) Maxwell’s equations: “Collision” operator includes all microscale effects ∂ t B + ∇ × E = 0 , � ∂ t E − c 2 ∇ × B = − J /ǫ 0 , m ( C i + C e ) γ − 1 d ( γ v ) = 0 , conservation: where m = ( 1 , γ v , γ ) . ∇ · B = 0 , ∇ · E = σ/ǫ 0 . decomposed as: Charge moments: C ii + ← → σ := � � C i = � C ie , q s f s d ( γ v ) , s m s C ee + ← → C e = � J := � � C ei , q s v f s d ( γ v ) . s m s � m � C ss γ − 1 d ( γ v ) = 0 . where Kinetic equations: “collisionless”: ← → C sp ≈ 0 . ∂ t f i + v · ∇ x f i + a i · ∇ ( γ v ) f i = C i BGK collision operator ∂ t f e + v · ∇ x f e + a e · ∇ ( γ v ) f e = C e C ss = M s − f s � , τ ss Lorentz acceleration: where the entropy-maximizing distribution M shares a i = q i physically conserved moments with f : m i ( E + v × B ) , M = exp ( α · m ) , a e = q e m e ( E + v × B ) . � m ( M − f ) d ( γ v ) = 0 . Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 6 / 40

2-species kinetic-Maxwell (classical) Maxwell’s equations: “Collision” operator includes all microscale effects ∂ t B + ∇ × E = 0 , � ∂ t E − c 2 ∇ × B = − J /ǫ 0 , conservation: v m ( C i + C e ) = 0 , where m = ( 1 , v , � v � 2 ) . ∇ · B = 0 , ∇ · E = σ/ǫ 0 . decomposed as: Charge moments: C ii + ← → C i = � C ie , σ := � � q s f s d v , C ee + ← → s m s C e = � C ei , J := � � q s v f s d v . � � s m s v m � v m � C ii = 0 = C ee . where Kinetic equations: “collisionless”: ← → C sp ≈ 0 . ∂ t f i + v · ∇ x f i + a i · ∇ v f i = C i BGK collision operator ∂ t f e + v · ∇ x f e + a e · ∇ v f e = C e C ss = M s − f s � , τ ss Lorentz acceleration: where the Maxwellian distribution M shares physically conserved moments with f : a i = q i � � m i ( E + v × B ) , −| c | 2 ρ M = ( 2 πθ ) 3 / 2 exp , a e = q e 2 θ m e ( E + v × B ) . θ := �| c | 2 / 2 � . Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 7 / 40

2-fluid Maxwell Maxwell’s equations: Closures (neglect): Definitions:   R e d s ∂ t B + ∇ × E = 0 , t := ∂ t + u s · ∇ , en ≈ η · J + β e · q e ,   ∂ t E − c 2 ∇ × B = − J /ǫ 0 ,   c s := v − u s ,   R i = − R e ,   n s := ρ s / m s , ∇ · B = 0 , ∇ · E = σ/ǫ 0 .     Q s =: Q ex s + Q fr s ,   X ◦ := X + X T − I tr X   Charge moments: .   ≈ 3 2 K s n 2 ( T 0 − T s ) , Q ex 2 3   s   σ s := q s  Q fr := Q fr  σ := σ i + σ e , m s ρ s . i + Q fr Collisional sources:   e     J := J i + J e , J s := σ s u s . ≈ η : JJ + β e : q e J , � v ← →   R s := C s d v ,    Q fr i = Q fr  e m e / m i , � 1 2 � c s � 2 ← → Evolved moments:   Q s := C s d v .   � P ◦ s ≈ − 2 µ s : ∇ u ◦ s ,       1 ρ s Closing moments q s ≈ − k s · ∇ T s .   :=   f s d v ρ s u s v (intraspecies): 1 2 � c s � 2 ρ s e s � P s := c s c s f s d v , Evolution equations: p s := 1 3 tr P s , ∂ t ρ s + ∇ · ( u s ρ s ) = 0 , P ◦ s := P s − p s I , ρ s d s t u s + ∇ p s + ∇ · P ◦ s = σ s E + J s × B + R s � 1 2 c s � c s � 2 f s d v . q s := t e s + p s ∇ · u s + P ◦ ρ s d s s : ∇ u s + ∇ · q s = Q s Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 8 / 40

2-fluid MHD (extended) electromagnetism ( ∂ t E ≈ 0) Closures Definitions: (simplified): d t := ∂ t + u · ∇ , ∂ t B + ∇ × E = 0 , ∇ · B = 0 , Q := Q i + Q e J J = µ − 1 w = en , 0 ∇ × B ≈ η : JJ w i = m red m i w , Ohm’s law (evolution of J solved for E ) Q s = m red m s Q , w e = − m red w , E = η · J + B × u + m i − m e P ◦ s ≈ − 2 µ s : ∇ u ◦ m e J × B s , e ρ P d := m red n ww q s ≈ − k s · ∇ T s . e ρ ∇ · ( m e ( p i I + P ◦ 1 i ) − m i ( p e I + P ◦ + e )) m − 1 red := m − 1 + m − 1 . � � e i + m i m e ∂ t J + ∇ · ( uJ + Ju − m i − m e JJ ) e 2 ρ e ρ mass and momentum (total): ∂ t ρ + ∇ · ( u ρ ) = 0 ρ d t u + ∇ · ( P i + P e + P d ) = J × B energy evolution (per species): ρ i d t e i + p i ∇ · u i + P ◦ i : ∇ u i + ∇ · q i = Q i , ρ e d t e e + p e ∇ · u e + P ◦ e : ∇ u e + ∇ · q e = Q e ; Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 9 / 40

Resistive MHD MHD system: Fluid closure: P ◦ ≈ − 2 µ : ∇ u ◦ , ∂ t ρ + ∇ · ( ρ u ) = 0 (mass continuity) , ρ d t u + ∇ p + ∇ · P ◦ = J × B q ≈ − k · ∇ T . (momentum balance) , ∂ t E + ∇ · ( u ( E + p ) + u · P ◦ + q ) = J · E (energy balance) , Descriptions: ∂ t B + ∇ × E = 0 (magnetic field evolution) . ρ = total mass density The divergence constraint ∇ · B = 0 is maintained by exact ρ u = total momentum density solutions and must be maintained in numerical solutions. u = velocity of bulk fluid E = total gas-dynamic energy Electromagnetic closing relations: density p = total scalar pressure J := µ − 1 0 ∇ × B (Ampere’s law for current) , P ◦ = total deviatoric pressure ∇ u ◦ = deviatoric rate of “strain” E ≈ B × u + η · J (Ohm’s law for electric field) . (deformation) T = temperature q = total heat flux η = resistivity µ = viscosity k = heat conductivity Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 10 / 40

Outline 1 Presentation of plasma models 2 Derivation of plasma models Kinetic Two-fluid MHD Conclusion Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 11 / 40

Outline 1 Presentation of plasma models 2 Derivation of plasma models Kinetic Two-fluid MHD Conclusion Johnson (KU Leuven) Fluid models of plasma Nov 29, 2012 12 / 40