Fluid models from multi-fluid to resistive MHD Alec Johnson Centre - PowerPoint PPT Presentation

Fluid models from multi-fluid to resistive MHD Alec Johnson Centre for mathematical Plasma Astrophysics Mathematics Department KU Leuven Nov 28, 2013 Abstract: The fundamental plasma equations consist of Maxwell’s equations for the electromagnetic field coupled to the kinetic equations for particle mo- tion. The two-fluid model replaces the kinetic equations with fluid equations and is appropriate when intraspecies collisions are frequent enough to keep the distribution of particle velocities nearly symmetric. On time scales for which plasma oscillations are rapid, positive and negative charges must balance, and the plasma acts like a single, conducting fluid described by the equations of resistive magnetohydrodynamics (MHD). Johnson (KU Leuven) Fluid models Nov 28, 2013 1 / 30

Outline 1 Vlasov: fluid in phase space 2 Presentation of plasma models 3 Derivation of plasma models 4 MHD Johnson (KU Leuven) Fluid models Nov 28, 2013 2 / 30

Conservation law framework Quantities: Balance law: � � t = time ( ∀ Ω) Ω U ( t 1 ) − Ω U ( t 0 ) � � t 1 X = position = − t 0 F ∂ Ω d A · U ( t , X ) = balanced quantity � � F ( t , X ) = flux function (e.g. F ( U ) ). ⇐ ⇒ ( ∀ Ω) d t Ω U = − d A · F ∂ Ω � S ( t , X ) = 0 (no production of U ). ⇐ ⇒ ( ∀ Ω) ( ∂ t U + ∇ · F ) = 0 Definitions: Ω Ω = arbitrary region ⇐ ⇒ ∂ t U + ∇ · F = 0 . d Ω : volume element d t d Ω S : production in volume element � n = outward unit vector d A = � n d A : surface element d t d A · F ( t , X ) = flux of U out of surface element. To see that flux is linear in d A , consider that Ω can be approximated by a set of cells in a rectangular grid. d t dA 1 F 1 gives flux across face perpendicular to first axis; dA 1 is area of projection of surface element onto first axis. Note: F = T in picture. Johnson (KU Leuven) Fluid models Nov 28, 2013 3 / 30

Balance law framework Quantities: Balance law: � � t = time ( ∀ Ω) Ω U ( t 1 ) − Ω U ( t 0 ) � � t 1 � � t 1 X = position = − t 0 F + ∂ Ω d A · t 0 S Ω U ( t , X ) = balanced quantity � � � F ( t , X ) = flux function (e.g. F ( U ) ). ⇐ ⇒ ( ∀ Ω) d t Ω U = − d A · F + Ω S ∂ Ω � S ( t , X ) = production of U . ⇐ ⇒ ( ∀ Ω) ( ∂ t U + ∇ · F − S ) = 0 Definitions: Ω Ω = arbitrary region ⇐ ⇒ ∂ t U + ∇ · F = S . d Ω : volume element d t d Ω S : production in volume element � n = outward unit vector d A = � n d A : surface element d t d A · F ( t , X ) = flux of U out of surface element. To see that flux is linear in d A , consider that Ω can be approximated by a set of cells in a rectangular grid. d t dA 1 F 1 gives flux across face perpendicular to first axis; dA 1 is area of projection of surface element onto first axis. Note: F = T in picture. Johnson (KU Leuven) Fluid models Nov 28, 2013 4 / 30

Transport Derivatives Given: Conservation of transported material: ρ ( t , x ) is transported by V t = time ⇐ ⇒ F := V ρ is a flux for ρ X = position ⇐ ⇒ ∂ t ρ + ∇ · ( V ρ ) = 0 V ( t , X ) = velocity field ⇐ ⇒ δ t ρ = 0 α ( t , x ) = arbitrary function ⇐ ⇒ d t ρ + ρ ∇ · V = 0 ρ ( t , x ) = density convected by V ⇐ ⇒ d t ln ρ = −∇ · V . d t := d d t Incompressible flow: δ t α := ∂ t α + ∇ · ( V α ) V is incompressible = “transport derivative” of α . ⇐ ⇒ d t ρ = 0 d t α := ∂ t α + V · ∇ α ⇐ ⇒ d t ln ρ = 0 = material derivative of α . ⇐ ⇒ ∇ · V = 0 ⇐ ⇒ d t α = δ t α ( ∀ α ) . Properties: δ t α = d t α + α ∇ · V . δ t ( αβ ) = d t ( αβ ) + ( ∇ · V ) αβ = ( d t α ) β + α ( d t β ) + ( ∇ · V ) αβ = ( δ t α ) β + α ( d t β ) . δ t ( ρβ ) = ρ d t β . Johnson (KU Leuven) Fluid models Nov 28, 2013 5 / 30

Vlasov equation Given: Theorem: Lorentz acceleration implies incompressible flow in phase space. x : position Incompressible means ∇ X · V = 0. v = . x : velocity ∇ X · V = ∇ x · v + ∇ v · a a = . v : acceleration ∇ x · v = 0 because x and v are ˜ f s : number distribution of species s . independent variables. ˜ f s ( t , x , v ) d x d v : number of particles of ∇ v · E ( t , x ) = 0 for same reason. species s in a region of state space with So ∇ v · a = q ∂ ∂ v i ǫ ijk v j B k ( t , x ) = 0. m volume d x d v . m s : particle mass of species s Vlasov equation (conservation of particles): q s : particle charge of species s f ( t , X ) is transported by V f s = m s ˜ f s : mass distribution of species s . ⇐ ⇒ ∂ t f + ∇ X · ( V f ) = 0 a s = q s m s ( E + v × B ) : Lorentz acceleration. ⇐ ⇒ ∂ t f + ∇ x · ( v f ) + ∇ v · ( a f ) = 0 X := ( x , v ) : position in state space. V := . (conservation form) X = ( v , a s ) : velocity in state space. ( v × B ) i = � � k ǫ ijk v j B k (cross product) ∂ t f + v · ∇ x f + a · ∇ v · f = 0 ⇐ ⇒ j ǫ ijk : Levi-Civita symbol ⇐ ⇒ ∂ t f + V · ∇ X f = 0 We suppress the species index s when focusing Remark: conservation form is preferred for on one species. taking fluid moments. Johnson (KU Leuven) Fluid models Nov 28, 2013 6 / 30

Outline 1 Vlasov: fluid in phase space 2 Presentation of plasma models 3 Derivation of plasma models 4 MHD Johnson (KU Leuven) Fluid models Nov 28, 2013 7 / 30

kinetic-Maxwell and the fluid limit Kinetic-Maxwell: Fluid approximation: particle equations: ∂ t ρ + ∇ · ( ρ u ) = 0 (mass) , ρ d t u + ∇ p + ∇ · P ◦ = J × B + σ E + R (momentum) , d t x p = v p , d t p + γ p ∇ · u + P ◦ : ∇ u + ∇ · q = 0 q # (energy) , p d t v p = e m p ( v p × B ( x p ) + E ( x p )) + r where we have used the definitions electromagnetic field: σ := � qS , J := � v qS , � c 2 mS , p := 1 ∂ t B + ∇ × E = 0 , 3 ρ := � mS , ρ u := � v mS , P := � cc mS , − c − 2 ∂ t E + ∇ × B = µ 0 J , R := � r mS , P ◦ := P − p I , c := v − u , c − 2 ∇ · E = µ 0 σ. ∇ · B = 0 , with the abbreviations charge-weighted moments: σ ( x ) := e � m := m p , p S p ( x − x p ) q # S := S p ( x − x p ) , p , � := � q := eq # J ( x ) := e � p , p , p S p ( x − x p ) q # p v p . v p = q p and the chain rule ∂ t S = − v · ∇ S . Plugging ˙ m p ( v p × B + E ) into the time-derivative of mass ( � Assuming that particle velocities for each species have p S p m p ), momentum ( � a symmetric distribution implies P ◦ s = 0 and q s = 0, p v p S p m p ), and energy giving Euler gas dynamics for each species, hence the ( � 2 v 2 1 p S p m p ) density yields gas (i.e. ideal two-fluid Maxwell plasma model. p fluid) equations. Johnson (KU Leuven) Fluid models Nov 28, 2013 8 / 30

Modeling parameters Physical constants that define an Collisionless time, velocity, and space scale parameters: ion-electron plasma: p , s := µ 0 n 0 ( ce ) 2 ω 2 plasma frequencies : , e (charge of proton), 1 m s m i , m e (ion and electron mass), 2 ω g , s := eB 0 gyrofrequencies : , m s 3 c (speed of light), t , s := 2 p 0 µ 0 (vacuum permeability). 4 v 2 thermal velocities : , ρ s MHD parameters that characterize B 2 A , s := 2 p B the state of a plasma: v 2 0 Alfv´ en speeds : = , ρ s µ 0 m s n 0 n 0 (typical particle density), 1 � λ D := v t , s T 0 T 0 (typical temperature), 2 Debye length : = n 0 µ 0 ( ce ) 2 , ω p , s B 0 (typical magnetic field). 3 r g , s := v t , s = m s v t , s gyroradii : , Derived typical quantities: ω g , s eB 0 � p 0 := n 0 T 0 (thermal pressure) δ s := v A , s c m s skin depths : = = µ 0 n s e 2 . B 2 ω g , s ω p , s p B := 2 µ 0 (magnetic pressure) 0 � v t , s � r g , s ρ s := n 0 m s (mass density). � 2 = � 2 . plasma β := p 0 p B = v A , s δ s Collision periods: v A , s = r g , s λ D = ω p , s c ω g , s . τ s : period of relaxation of non-MHD ratio: species s toward Maxwellian Johnson (KU Leuven) Fluid models Nov 28, 2013 9 / 30

Plasma model hierarchy kinetic-Maxwell 1  � fast collisions ( τ s − 1 → ∞ )   ideal two-fluid Maxwell : Euler gas for each species: ρ s , u s , p s 2   � fast oscillations ( e → ∞ )  relativistic ideal MHD : perfectly conducting gas 3   � fast light waves ( c → ∞ )  classical ideal MHD : perfectly conducting gas: E = B × u . 4 Johnson (KU Leuven) Fluid models Nov 28, 2013 10 / 30

two-fluid Maxwell → MHD Two-fluid Maxwell: Quasi-relativistic MHD ( e → ∞ ): gas evolution: gas evolution: ∂ t ρ s + ∇ · ( ρ s u s ) = 0 , ∂ t ρ + ∇ · ( ρ u ) = 0 (mass) , ρ s d s t u s + ∇ p s = J s × B + σ s E + R s , ρ d t u + ∇ p = J × B + σ E (momentum) , m red d s t p s + γ p s ∇ · u s = 2 d t p + γ p ∇ · u = 2 m s Q 3 J · η · J (thermal energy) . 3 magnetic field: electromagnetic field: ∂ t B + ∇ × E = 0 (magnetic field) , ∂ t B + ∇ × E = 0 , E = B × u + η · J (Ohm’s law) , − c − 2 ∂ t E + ∇ × B = µ 0 J , ∇ · B = 0 (divergence constraint) , c − 2 ∇ · E = µ 0 σ, ∇ · B = 0 , µ 0 J := ∇ × B − c − 2 ∂ t E (Ampere’s law for current) , J := J i + J e , J s := σ s u s , µ 0 σ := c − 2 ∇ · E σ s := ± e (quasineutrality) . σ := σ i + σ e , m s ρ s . definitions: closure: γ := 5 3 , − R i = R e = e 2 n e n i η · ( u i − u e ) d s t := ∂ t + u s · ∇ , � m − 1 m − 1 red := . ≈ en η · J , d t := ∂ t + u · ∇ , s Q = − � s s R s · u s ≈ J · η · J . Johnson (KU Leuven) Fluid models Nov 28, 2013 11 / 30