Accurate simulation of fast magnetic reconnection calls for higher-moment fluid models. E. Alec Johnson

Transcription

1 Accurate simulation of fast magnetic reconnection calls for higher-moment fluid models. E. Alec Johnson Centre for mathematical Plasma Astrophysics Mathematics Department KU Leuven Oct 30, 2012 Abstract: This talk argues that the simplest fluid model of plasma that can resolve steady fast magnetic reconnection is a hyperbolic two-fluid plasma model that evolves at least 13 moments in each species (i.e. including tensor pressure and heat flux). Models with fewer moments that admit fast reconnection involve physical deficiencies (anomalous closures) or numerical difficulties (due to complex diffusive closures or dispersive terms). A key point is that higher-moment hyperbolic models are the simplest fluid models that behave analogously to kinetic models, facilitating asymptotic-preserving kinetic-fluid stitching. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

2 Outline 1 Models of plasma 2 Extended MHD models 3 Simulation of fast magnetic reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

3 Fundamental model: particle-maxwell (relativistic) Maxwell s equations: t B + E = 0, t E c 2 B = J/ɛ 0, B = 0, E = σ/ɛ 0. Charge moments: σ := p Sp(xp)qp, J := p Sp(xp)qpvp, Particle equations: d t x p = v p, d t (γ pv p) = a p(x p, v p), γ 2 p := 1 (v p/c) 2. Lorentz acceleration a p(x, v) = qp m p (E(x) + v B(x)) Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

4 Fundamental collisionless parameters Physical constants that define an ion-electron plasma: 1 e (charge of proton), 2 m i, m e (ion and electron mass), 3 c (speed of light). Fundamental parameters that characterize the state of a plasma: 1 n 0 (typical particle density), 2 T 0 (typical temperature), 3 B 0 (typical magnetic field). Derived quantities: p 0 := n 0 T 0 (thermal pressure) p B := B2 0 2µ 0 (magnetic pressure) ρ s := n 0 m s (typical density). Subsidiary time, velocity, and space scale parameters: plasma frequencies: ω 2 p,s := n 0e 2 ɛ 0 m s, gyrofrequencies: ω g,s := eb 0 m s, thermal velocities: v 2 t,s := 2p 0 ρ s, Alfvén speeds: Debye length: gyroradii: skin depths: plasma β := p 0 p B another ratio := va,s 2 := 2p B = B2 0, ρ s µ 0 m sn 0 λ D := v t,s ω p,s = ɛ 0 T 0 n 0 e 2, r g,s := v t,s ω g,s = msv t,s eb 0, δ s := v A,s = c = ω g,s ω p,s = ( v t,s ) 2 ( rg,s ) 2. v A,s = δ s c v A,s = rg,s λ D = ωp,s ω g,s. ms µ 0 n se 2. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

5 Nondimensionalization Choose values for: x 0 (space scale) (e.g. ion skin depth δ i ), m 0 (mass scale) (e.g. ion mass m i ), q 0 (charge scale) (e.g. proton charge e), T 0 (temperature), (e.g. ion temperature T i ), B 0 (magnetic field) (e.g. ω g,i m i /e), and n 0 (number density) (e.g. something 1/x0 3). This implies typical values for: v 0 = T 0 /m 0 (velocity scale), t 0 = x 0 /v 0 (time scale), E 0 = B 0 v 0 (electric field), ρ 0 = m 0 n 0 (mass density), σ 0 = q 0 n 0 (charge density), J 0 = q 0 n 0 v 0 (current density), and S 0 = n 0 (no. pcls. per unit number density). Scale parameters are thus: ω 2 p,0 := n 0q 2 0 ɛ 0 m 0, ω g := q 0B 0 m 0, v t := v 0, v 2 A := 2p B ρ 0, λ D := v t ω p,0, r g := v t ω g, δ 0 := v A ω g. Making the substitutions t = ˆtt 0, σ = σσ 0, x = xx 0, J =ĴJ 0, q = qq 0, S p(x p) =Ŝp( xp)n 0, m = mm 0, c =ĉv 0, n = nn 0, v = vv 0, B = BB 0, E = ÊB 0v 0, =x 1 0 =x 1 0 x, in the fundamental equations gives an almost identical-appearing nondimensionalized system with only three nondimensional parameters: 1 ĉ = c v 0 (light speed), r g =: 1 r g (gyrofrequency or gyroradius), 2 ω g := t 0 ω g,0 = x 0 = 1 ω p 3 λd := λ D x0 (Debye length or plasma frequency). Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

6 Fundamental model: particle-maxwell (nondimensionalized relativistic) Maxwell s equations: ˆt B + Ê = 0, ˆt Ê ĉ2 B = Ĵ/ ɛ, B = 0, Charge moments: σ := pŝp( xp) qp, Ê = σ/ ɛ. Ĵ := pŝp( xp) qp v p, Particle equations: dˆt xp = vp, dˆt (γp v p) = â p( x p, v p), γ 2 p := 1 ( v p/ĉ) 2. Lorentz acceleration â p( x, v) = ω g q p m p (Ê( x) + v B( x) ) Nondimensional parameters: 1 ĉ: light speed. 2 ω g = 1/ r g: gyrofrequency or gyroradius. ( ) ( ) 2 3 ɛ = λ λd λd D = r r g : permittivity or g r g Debye length.... (Note: λ D r g Matching with SI units: Acceleration is scaled by ω g. = ωg ω p = v A c.) ω g = 1 if the time scale t 0 is chosen to be the gyroperiod. ω g can be absorbed into the definition of ɛ or λ D along with charge or electromagnetic field. Henceforth drop hats. Problem: model based on particles is not a computationally accessible standard of truth (for space weather). Solution: replace particles with a particle density function f s(t, x, γv) for each species s. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

7 Accessible standard of truth: 2-species kinetic-maxwell (relativistic) Maxwell s equations: t B + E = 0, t E ĉ 2 B = J/ ɛ, B = 0, Charge moments: E = σ/ ɛ. σ := q s s fs d(γv), m s J := q s s vfs d(γv). m s Kinetic equations: t f i +v xf i +a i (γv) f i = C i t f e+v xf e+a e (γv) f e= C e Lorentz acceleration: a i = ω g q i m i (E + v B), a e = ω g q e m e (E + v B). Collision operator includes all microscale effects conservation: m(c i + C e) γ 1 d(γv) = 0, where m = (1, γv, γ). decomposed as: C i = C ii + C ie, C e = C ee + C ei, where m C ss γ 1 d(γv) = 0. collisionless : C sp 0. BGK collision operator C ss = M f τ ss, where the entropy-maximizing distribution M shares physically conserved moments with f : M = exp (α m), m(m f )d(γv) = 0. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

8 Accessible standard of truth: 2-species kinetic-maxwell (classical) Maxwell s equations: t B + E = 0, t E ĉ 2 B = J/ ɛ, B = 0, Charge moments: E = σ/ ɛ. σ := q s s fs dv, m s J := q s s vfs dv. m s Kinetic equations: t f i +v xf i +a i vf i = C i t f e+v xf e+a e vf e= C e Lorentz acceleration: a i = ω g q i m i (E + v B), a e = ω g q e m e (E + v B). Collision operator includes all microscale effects conservation: m(ci + Ce) = 0, v where m = (1, v, v 2 ). decomposed as: C i = C ii + C ie, C e = C ee + C ei, where v m C ii = 0 = v m C ee. collisionless : C sp 0. BGK collision operator C ss = M f τ ss, where the Maxwellian distribution M shares physically conserved moments with f : ( ) ρ c 2 M = exp, (2πθ) 3/2 2θ θ := c 2 /2. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

9 Simplifications Independent and dependent simplifications: type assumption simplification result 1. r g log B 1 gyro-average gyro-kinetic/gyro-fluid 2. τ ss is small take moments fluid 3a. λd 0 quasineutrality 1 XMHD 3b. ĉ is large fast light classical 3c. ĉ (implies λ D = 0) 2 Galileo Ampere s law 3d. r g 0 (implies λ D = 0) 3 frozen flux Ideal MHD 4... yield a commuting lattice of increasingly simplified models... 1 λ σ = λd D E 0 as λd 0 if λ D = v A rg rg c is bounded. 2 assuming ĉ λd = ĉ is bounded ωp 3 λ assuming D = v A rg c is bounded 4 Relativistic MHD assumes finite c and quasineutrality in the frame of reference of the plasma. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

10 Model hierarchy Kinetic two-species Maxwell Kinetic two-species XMHD Kinetic MHD τ ss small 13-moment two-fluid Maxwell 13-moment two-fluid XMHD 13mom MHD 10-moment two-fluid Maxwell 10-moment two-fluid XMHD 10mom MHD τ ss 0 5-moment two-fluid Maxwell ĉ 5-moment two-fluid XMHD r g 0 Ideal MHD Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

11 Model hierarchy (gyro-averaged) gyro-kinetic 2species Maxwell gyro-kinetic 2species XMHD gk MHD gyro-7-moment 2fluid Maxwell gyro-7-moment 2fluid XMHD g7m MHD gyro-6-moment 2fluid Maxwell gyro-6-moment 2fluid XMHD g6m MHD 5-moment 2fluid Maxwell 5-moment 2fluid XMHD Ideal MHD Exactly gyro-averaged models fail to admit steady magnetic reconnection (for 2D problems rotationally symmetric about the origin) 5. 5 and also fail to admit heat flux perpendicular to the magnetic field Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

12 Requirements for steady 2D symmetric magnetic reconnection Consider the simplest reconnection scenario: steady 2D reconnection symmetric under 180-degree rotation about the X-point. Theorem Reconnection is impossible without viscosity or resistivity. Argument: Rate of reconnection is the electric field strength at the X-point. Electric field strength at the X-point is resistive electric field plus viscous electric field. Theorem (EAJ) Reconnection is impossible for any conservative model for which heat flux is zero. Argument: Steady reconnection requires entropy production near the X-point (via resistivity or viscosity). The X-point is a stagnation point. Without heat flux, heat accumulates at the X-point without bound. Observation: in kinetic simulations, fast reconnection is supported by viscosity, not resistivity. Conclusion: we need heat flux and viscosity in a fluid model of fast magnetic reconnection. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

13 5-moment multi-fluid plasma Evolution equations δ t ρ s = 0 ρ sd t u s + p s + P s = qsns(e + us B) 3 2 δ t p s + p s u s + P s : us + qs = 0 Evolved moments ρ s ρ su s = 3 2 ps Definitions 1 v 1 2 cs 2 δ t (α) := t α + (u sα) c s := v u s f s dv Relaxation (diffusive) flux closures: P s = (c sc s c s 2 I/3) f s dv = 2µ : e, 1 q s = 2 cs cs 2 f s dv = k T. n s := ρ s/m s Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

14 10-moment multi-fluid plasma Evolution equations δ t ρ s = 0 ρ sd t u s + P s = q sn s(e + u s B) δ t P s + Sym2(P s u s) + q s = q sn s Sym2(T s B) + R s Evolved moments ρs ρ su s = 1 v f s dv P s c sc s Definitions δ t (α) := t α + (u sα) c s := v u s Sym2(A) := A + A T n s := ρ s/m s T := P/n Relaxation source term closures: R s = c sc s C s dv= P s /τ, Relaxation (diffusive) flux closures: q s = c sc sc s f s dc s ( = 2 5 Sym k s Ks Sym3 ( ) T s ) T s T s Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

15 13-moment multi-fluid plasma Evolution equations δ t ρ s = 0 ρ sd t u s + P s = q sn s(e + u s B) δ t P s + Sym2(P s u s) + q s = q sn s Sym2(T s B) + R s δ t q s + q s u s + q s : u s + P s : Θ + P s θ s + R s = Sym3(P s sp s)/ρ s + qs qs B + qss,t ms Evolved moments ρ s ρ su s P = s q s Definitions 1 v c sc s 1 2 cs cs 2 δ t (α) := t α + (u sα) fs dv c s := v u s, Sym2(A) := A + A T, n s := ρ s/m s, Θ := P/ρ, T := P/n, θ := tr Θ/2, Relaxation source term closures: [ ] [ ] Rs csc s = q 1 C ss dv = 1 ss,t 2 cs cs 2 τ s Hyperbolic flux closures: [ ] qs [ ] csc sc s = R s c sc s c s 2 f s(c s) dc s [ ] P s Pr q s Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

16 Relaxation closures for viscosity and heat flux 5-moment: P = 2µ µ : e q = k k T 10-moment: t P = P /τ... q = 2 5 (k Sym K ( T ) ) Sym3 T T 13-moment: t P = P /τ... t q = q/ τ... Relaxation periods: τ τ 0 mt 3 /n, τ := τ/ Pr, τ 0 := 12π3/2 ln Λ ( ) 2 ɛ0. e 2 Diffusion parameters: µ =τp 2 5 k = µ m Pr Diffusion coefficients: k =I ϖ 2 (I ϖi ), µ = 1 2 (3I2 +I2 )+ 2 1+ϖ 2 (I I ϖi I ) ϖ 2 ( 1 2 (I2 I2 ) 2ϖI I ), ) K = (I I (I 2 + I2 ) ) + (I 3 1+ ϖ 2 I 2 ϖi I2 ( ) ϖ 2 2 (I2 I2 )I 2 ϖi I I +(k 0 I 3 + k 1I I 2 + k 2I 2 I + k 3 I 3 ), k 3 := 6 ϖ ϖ ϖ 4, k 2 := 6 ϖ2 + 3 ϖ(1 + 3 ϖ 2 )k ϖ 2, k 1 := 3 ϖ + 2 ϖk ϖ 2, k 0 := 1 + ϖk 1. Definitions: Pr := Prandtl no. (e.g. 2 3 or 1), ϖ := τ q m B, ϖ := τ q m B, b := B/ B, I := identity, I := bb, I := I bb, I := b I. In this frame the species index s is suppressed. All products of even-order tensors are splice products satisfying (AB) j1 j 2 k 1 k 2 := A j1 k 1 B j2 k 2, (ABC) j1 j 2 j 3 k 1 k 2 k 3 := A j1 k 1 B j2 k 2 C j3 k 3. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

17 Conclusion Available closures with nonzero heat flux: 1 For 5-moment and 10-moment gas dynamics, physical heat flux closures are diffusive and dependent on magnetic field: diffusive closures are time-averaged and must incorporate the rotational effects of the magnetic field heat conductivity is highly anisotropic in the presence of a strong guiding magnetic field. tokamak: parallel heat conductivity can be a million times stronger than perpendicular heat conductivity. need to avoid anomalous cross-field diffusion. need high-order accuracy or field-aligned coordinates. 2 if heat flux and viscosity are evolved, a simple relaxation closure can be used: 13-moment system: can use Grad s explicit closure with hyperbolicity limiting. 14-moment entropy-maximizing system evolves c s 4 f s dc s. (Maturing; see work by McDonald, Torrilhon, Groth.) Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

18 Outline 1 Models of plasma 2 Extended MHD models 3 Simulation of fast magnetic reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

19 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, µ 0 J = B c 2 t E, µ 0 σ = 0 + c E 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 (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

20 MHD: charge neutrality The assumption of charge neutrality reduces the number of gas-dynamic equations 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 u i = u + m i + m e J ne, u e = u m i m i + m e J ne. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

21 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 sj s + (q s/m s)p s) = (q 2 s /ms)n(e + us B) + (qs/ms)rs. Summing over both species and using charge neutrality gives net current evolution: ( t J + uj + Ju m ) ( ) ( ( i m e Pi JJ + e Pe = e2 ρ E + u m ) ) i m e J B Re. eρ m i m e m i m e eρ en A closure for the collisional term is Re en = η J + βe qe. (Note: η m red e 2 nτ slow ). Ohm s law is current evolution solved for the electric field: E =B u (ideal term) + m i m e eρ J B (Hall term) + η J (resistive term) + β e q e (thermoelectric term) + 1 eρ (mep i m i P e) [ ( + m im e e 2 ρ t J + uj + Ju m i m e eρ )] JJ (pressure term) (inertial term). Since J = µ 1 0 B, 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 (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

22 Equations of 5-moment 2-fluid MHD mass and momentum: t ρ + (uρ) = 0 ρd t u + (P i + P e + P d ) = J B Electromagnetism t B + E = 0, B = 0, J = µ 1 0 B Ohm s law E = Re en + B u + m i m e J B eρ + 1 eρ (me(p ii + P i ) m i(p ei + P e )) [ ] t J + (uj + Ju m i m e JJ) eρ + m im e e 2 ρ Pressure evolution 3 2 nd t T i + p i u i + P i : u i + q i = Q i, 3 2 nd t T e + p e u e + P e : u e + q e = Q e; Definitions: d t := t + u s, P d := m red nww, w = J en, Closures: P s = 2µ : ( u) q s = k T R e = η J + βe qe en Q s =? m 1 red := me 1 + m i 1. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

23 Equations of 10-moment 2-fluid MHD mass and momentum: t ρ + (uρ) = 0 ρd t u + (P i + P e + P d ) = J B Electromagnetism t B + E = 0, B = 0, J = µ 1 0 B Ohm s law E = Re en + B u + m i m e J B eρ + 1 eρ (mep i m i P e) [ ( t J + + m im e e 2 ρ Pressure evolution uj + Ju m i m e eρ )] JJ Definitions: d t := t + u s, P d := m red nww, w = J en, Closures: m 1 red := me 1 + m i 1. R s = 1 τ P s ( ) q s = 2 5 Ks Sym3 Ts T s T s R e = η J + βe qe en Q s =? n i d t T i + Sym2(P i u i ) + q i = q i m i Sym2(P i B) + R i + Q i, n ed t T e + Sym2(P e u e) + q e = qe m e Sym2(P e B) + R e + Q e Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

24 Equations of 13-moment 2-fluid MHD mass and momentum: t ρ + (uρ) = 0 ρd t u + (P i + P e + P d ) = J B Electromagnetism t B + E = 0, B = 0, J = µ 1 B 0 Ohm s law E = Re en + B u + m i me eρ J B + eρ 1 (mep i m i P e) [ t J + + m i me e 2 ρ ( uj + Ju m i me eρ JJ )] Diffusive relaxation closures: R e = η J + βe qe en Relaxation source term closures: R s = P s /τs q ss,t = q s/ τ s Hyperbolic flux closures: [ ] qs [ ] cscsc s = R s c sc s c s 2 f s(c s) dc s Interspecies forcing closures: [ ] Rs Q s q s,t =? Pressure evolution n i d t T i + Sym2(P i u i ) + q i = q i m i Sym2(P i B) + R i + Q i, n ed t T e + Sym2(P e u e) + q e = qe Sym2(Pe B) + Re + Qe me Heat flux evolution δ t q s + q s u s + q s : u s + P s : Θ + 3 s 2 Ps θs + 1 qs Rs = ρ(3θθ + 2Θ Θ) + 2 ms qs B + q ss,t + q s,t Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

25 Asymptotic-preserving hierarchy For efficient multiscale simulation, discretizations need to be asymptotic-preserving (AP) with respect to simpler models: h 0 micro-scheme(ɛ, h) micro-physics(ɛ) ɛ 0 ɛ 0 macro-scheme(h) h 0 macro-physics Micro-scheme is AP with respect to macro-physics if macro-scheme exists and is stable. In macro-physics limit, micro-physics waves disappear either are damped (strong limit) or become infinitely fast/fine (weak limit). Keys to designing good AP schemes: Thoroughly understand the limit (fit the framework). Use an implicit scheme to skip over unresolved microphysics (e.g. to damp fast waves). Preserve invariants (mimic the physics with conforming discretizations: scheme h 0 physics h 0 [disc. op. A] [op. A] [disc. op. B] h 0 [op. B] Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

26 Outline 1 Models of plasma 2 Extended MHD models 3 Simulation of fast magnetic reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

27 Frozen-in magnetic field lines charged particles spiral around magnetic field lines. viewed from a distance, the particles are stuck to the field lines. so magnetic field lines approximately move with the plasma. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

28 Magnetic Reconnection Start: oppositely directed field lines are driven towards each other. Field lines reconnect at the X-point. Lower energy state: change topology of field lines Results in large energy release in the form of oppositely directed jets 2D separator reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

29 Magnetic Reconnection Start: oppositely directed field lines are driven towards each other. Field lines reconnect at the X-point. Lower energy state: change topology of field lines Results in large energy release in the form of oppositely directed jets 2D separator reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

30 Magnetic Reconnection Start: oppositely directed field lines are driven towards each other. Field lines reconnect at the X-point. Lower energy state: change topology of field lines Results in large energy release in the form of oppositely directed jets 2D separator reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

31 Magnetic Reconnection Start: oppositely directed field lines are driven towards each other. Field lines reconnect at the X-point. Lower energy state: change topology of field lines Results in large energy release in the form of oppositely directed jets 2D separator reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

32 Magnetic Reconnection Start: oppositely directed field lines are driven towards each other. Field lines reconnect at the X-point. Lower energy state: change topology of field lines Results in large energy release in the form of oppositely directed jets 2D separator reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

33 Magnetic Reconnection Start: oppositely directed field lines are driven towards each other. Field lines reconnect at the X-point. Lower energy state: change topology of field lines Results in large energy release in the form of oppositely directed jets 2D separator reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

34 Magnetic Reconnection Start: oppositely directed field lines are driven towards each other. Field lines reconnect at the X-point. Lower energy state: change topology of field lines Results in large energy release in the form of oppositely directed jets 2D separator reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

35 Magnetic Reconnection Start: oppositely directed field lines are driven towards each other. Field lines reconnect at the X-point. Lower energy state: change topology of field lines Results in large energy release in the form of oppositely directed jets 2D separator reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

36 Magnetic Reconnection Start: oppositely directed field lines are driven towards each other. Field lines reconnect at the X-point. Lower energy state: change topology of field lines Results in large energy release in the form of oppositely directed jets 2D separator reconnection Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

37 Simplest plasma model: Ideal/Hall MHD In Ideal/Hall MHD Faraday s law is of the form t B + (B u }{{} c ) = 0, ideal E which says that u c is a flux-transporting flow for B. So magnetic reconnection is not possible in Ideal/Hall MHD. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

38 X-point analysis At the X-point the momentum equation ( Ohm s law ) reduces to rate of reconnection = E 3 (0) = R i en i + P i en i + m i e tu i (resistive term) (pressure term) (inertial term) Consequences: 1 Collisionless reconnection is supported by the inertial or pressure term. 2 For the 5-moment model the inertial term must support the reconnection; i.e. each species velocity at the origin should track exactly with reconnected flux. 3 For steady-state reconnection without resistivity the pressure term must provide for the reconnection. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

39 resistive MHD Resistive Ohm s law allows reconnection: E = B u c + η J. Magnetic reconnection is slow with uniform resistivity. Anomalous resistivity allows desired rate of reconnection. But a formula for anomalous resistivity has been elusive. Fast reconnection is not supported by physical anomalous resistivity (supported instead by anomalous viscosity). Hall MHD plus small resistivity allows fast reconnection (but supported by the wrong term in Ohm s law). Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

40 Collisionless models Observation: Fast reconnection always occurs on collisionless or weakly collisional spatial scales. Why: Mean free path reconnection scale. Therefore in model ignore interspecies collision operator (so zero resistivity). Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

41 GEM: Resistive MHD (η = ) Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

42 GEM: 5-moment 2-fluid-Maxwell: Johnson (KU Leuven) mi me = 25 magnetic reconnection needs higher-moment models Oct 30, / 56

43 GEM: 5-moment 2-fluid Maxwell: Johnson (KU Leuven) mi me = 1 (no Hall term) coarse magnetic reconnection needs higher-moment models Oct 30, / 56

44 GEM: 5-moment 2-fluid Maxwell: m i m e = 1, τ = 0, fine resolution Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

45 GEM: 2-fluid 10-moment ( m i m e = 1, τ = 0.2) Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

46 GEM ( m i m e = 25): kinetic models vs. 5- and 10-moment Reconnection rates compare well with with published kinetic results: model 16% flux reconnected Vlasov [ScGr06] t = 17.7/Ω i : PIC [Pritchet01] t = 15.7/Ω i : 10-moment t = 18/Ω i : 5-moment t = 13.5/Ω i : Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

47 Magnetic field at 16% reconnected magnetic field of [Pritchett01] Magnetic field lines for PIC at Ωi t = 15.7 [Pritchett01] Phys. Plasmas 13, "2006# H. Schmitz and R. Grauer Magnetic field for Vlasov at Ωi t = 17.7 [ScGr06] Johnson (KU Leuven) FIG. 2.!Color online" The out-ofplane magnetic field Bz!upper panel", the electron out-of-plane current je,z!middle panel", and the ion out-ofplane current ji,z!lower panel" at time "it = magnetic reconnection needs higher-moment models Oct 30, / 56

48 served in the outflow region. The peak values are, however, less than half of the inductive electric field. A striking feature in this picture is the almost circular ring around the X line, where the ions become demagnetized. The sheets of enhanced value along the separatrix are narrower than those Off-diagonal components of electron pressure tensor Figures pressure ten and P yz pla other eleme of Fig. 5 sh Off-diagonal components of the electron pressure tensor for 10-moment simulation at Ω i t = 18 Off-diagonal Downloaded 06 Apr 2010 to components Redistribution of subject the to electron pressure tensor for Vlasov simu- AIP license o lation at Ω i t = 17.7 [ScGr06] Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

49 Diagonal components of electron pressure tensor H. Schmitz and R. Grauer Diagonal components of the electron pressure tensor for 10-moment simulation at Ω i t = 18 Diagonal components of the electron pressure tensor for Vlasov simulation v i B z. This term becomes nonzero when ions can move observed from across the magnetic field lines in a region of a few ion inertial lengths Ω i t around = 17.7 the X line. [ScGr06] Again two peaks can be ob- term. peaks near the at served in the outflow region. The peak values are, however, Figures 5 a less than half of the inductive electric field. A striking feature pressure tensor in this picture is the almost circular ring around the X line, and P yz play a where the ions become demagnetized. The sheets of enhanced value along the separatrix are narrower than those of Fig. 5 show other elements Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

50 Electron gas at t = 16 Problem: the code crashes! Why? Look at electron gas dynamics: Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

51 Electron gas at t = 20 Problem: the code crashes! Why? Look at electron gas dynamics: Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

52 Electron gas at t = 26 Problem: the code crashes! Why? Look at electron gas dynamics: Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

53 Electron gas at t = 28 (just before crashing) Problem: the code crashes! Why? Look at electron gas dynamics: Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

54 Heating singularity ( m i m e = 25) A heating singularity develops between 20% and 50% reconnected flux which crashes the code or produces a central magnetic island. (T e ) yy becomes large. (T e ) xx becomes small. ρ e becomes small. electron entropy becomes large. These difficulties prompted me to study whether nonsingular steady-state solutions exist for adiabatic plasma models. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

55 Steady magnetic reconnection requires heat flux Theorem. 2D rotationally symmetric steady magnetic reconnection must be singular in the vicinity of the X-point for an adiabatic model. Argument. In a steady state solution that is symmetric under 180-degree rotation about the X-point, momentum evolution at the X-point says: rate of reconnection = E 3 (0) = R i en i + P i en i. Assume a nonsingular steady solution. Then at the origin (0) no heat can be produced, so R i = 0 at 0. Differentiating entropy evolution twice shows that P i = 0 at 0. So there is no reconnection. Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56

56 References [Br11] J. U. Brackbill, A comparison of fluid and kinetic models for steady magnetic reconnection, Physics of Plasmas, 18 (2011). [BeBh07] Fast collisionless reconnection in electron-positron plasmas, Physics of Plasmas, 14 (2007). [Ha06] A. Hakim, Extended MHD modelling with the ten-moment equations, Journal of Fusion Energy, 27 (2008). [HeKuBi04] M. Hesse, M. Kuznetsova, and J. Birn, The role of electron heat flux in guide-field magnetic reconnection, Physics of Plasmas, 11 (2004). [Jo11] E.A. Johnson, Gaussian-Moment Relaxation Closures for Verifiable Numerical Simulation of Fast Magnetic Reconnection in Plasma, PhD thesis, UW Madison, 2011 [JoRo10] E. A. Johnson and J. A. Rossmanith, Ten-moment two-fluid plasma model agrees well with PIC/Vlasov in GEM problem, proceedings for HYP2010, November [ScGr06] H. Schmitz and R. Grauer, Darwin-Vlasov simulations of magnetised plasmas, J. Comp. Phys., 214 (2006). Johnson (KU Leuven) magnetic reconnection needs higher-moment models Oct 30, / 56