A field theory approach to plasma self-organization

Transcription

1 WILLIAM E. BOEING DEPARTMENT OF AERONAUTICS & ASTRONAUTICS A field theory approach to plasma self-organization Setthivoine You Jens von der Linden, E. Sander Lavine, Alexander Card, Evan G. Carroll, Manuel Azuara-Rosales University of Washington This work was supported by the U.S. Dept. of Energy Early Career Award DE-SC US-Japan Compact Torus Workshop, Irvine, CA (Aug 2016)

2 CT configuration from self-organization Spontaneous formation of new configuration subj. to constraint, init. conditions, to reduce or maximize energy. e.g. Strength Magnetostatic single fluid [Taylor, Woltjer, etc.]; Flowing two-fluid [Steinhauer, Ishida, etc.] Flowing neutral fluid [Moffat, etc.] No need for detailed dynamics; Can predict final state from initial & boundary conditions. Q: Open configurations? pressure, density, temperature gradients? L-H transition? kinetic regime? magnetic dynamo? dynamic collimation and stability? relativity? 1 A: (partial) Can transform plasma physics to a generalized form of Maxwell s equations, valid in all regimes. Maybe simpler to solve full system in this frame of reference? In physics, often, a suitable transformation simplifies analysis.

3 Demonstrate The equations can be expressed as Generalized Maxwell Equations Wanted the simplest, most general expression for plasmas. Helicity evolution valid in kinetic regimes, relativistic regimes too Wanted to see if helicity conservation, plasma self-organization, relaxation valid in/across regimes beyond fluid regime. Criterion for helicity conservation vs energy conservation Wanted to have a general criterion for when/where helicity is conserved Helicity conversion from one form to another Conversion of magnetic helicity to flows and vice-versa 2

4 Ingredients for magnetic confinement fusion concepts 3 ITER Physics Basis, 2007, Ch. 2 p.s41

5 Define relative canonical helicity K σrel = P σ Ω σ+ dv e.g. Can. momentum Can. vorticity Relative ԦP σ = m σ u σ + q σ ԦA Ω σ = ԦP σ ԦX ± = ԦX ± ԦX ref eqs. dk erel dt = dk irel dt = 2 නR Ω dv + 2 න S sep h i h e Ω d ԦS + (1) Also becomes K σ = m σ 2 K kinσ + m σ q σ K crsσ + q σ 2 K mag නu σ ω σ dv 2 නu σ B dv න ԦA B dv kinetic helicity cross helicity magnetic helicity q σ φ m σu σ 2 + P σ n σ enthalpy B u i P i Ω i ԦJ u e P e Ω e magnetic flux tube canonical flux tube 4 S.You, Phys. Plasmas 19, (2012); S. You, Plasma Phys. Control. Fusion, 56, (2014)

6 Define relative canonical helicity K σrel = P σ Ω σ+ dv e.g. Can. momentum Can. vorticity Relative Note: Ω = 0 ԦP σ = m σ u σ + q σ ԦA Ω σ = ԦP σ ԦX ± = ԦX ± ԦX ref Also becomes K σ = m σ 2 K kinσ + m σ q σ K crsσ + q σ 2 K mag නu σ ω σ dv kinetic helicity eqs. 2 නu σ B dv cross helicity dk erel dt = dk irel dt න ԦA B dv magnetic helicity = 2 නR Ω dv + 2 න S sep h i h e Ω d ԦS + q σ φ m σu σ 2 + P σ n σ enthalpy (1) B u i P i Ω i ԦJ u e P e Ω e magnetic flux tube canonical flux tube 4 S.You, Phys. Plasmas 19, (2012); S. You, Plasma Phys. Control. Fusion, 56, (2014)

7 Reconnection of two flux tubes with three possible results Two magnetic flux tubes Half twist each One magnetic flux tube, half twist One vorticity flow, half twist Two flow vorticity flux tubes. Half twist each 5 S.You, Plasmas Phys. Contol. Fusion, 56, (2014)

8 The model explains why bifurcation in CT formation depends on 1/S*. Ψ = m i f + q i ψ නB d ԦS ሶ K i = m i Δh f + q i Δh ψ න u i d ԦS Spheromak Counterhelicity merging Relaxes to spheromak +v A v A Δh m i v A 2 FRC Relaxes to FRC S L = L δ c = i ωpi L v ρ A i vthi Ratio of canon. helicity injection into vortex tube or magnetic flux tube is K f i ψ 1 K S i 6 S.You, Phys. Plasmas 19, (2012); E. Kawamori et al, Nucl. Fusion 45, 843 (2005)

9 Maxwell s Eqs. A μ = μ 0 j μ Particle {q; m} Ԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt Kinetic Ԧx, Ԧv, t f σ Stat. Mech. df σ dt = C n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ Fluid eqs. x 2 MHD n Ԧx, t U Ԧx, t P Ԧx, t Single fluid eq. + Ohm s law E + U B = System configuration B Ԧx, u Ԧx (quasi static evolution) P σfluid ρ σ u σ + ρ cσ ԦA h σfluid ρc ρ σu σ 2 + ρ c φ + ρφ g + P σ R σfluid R + P σfluid u σ + L σfluid n n Canonical Helicity p.o.v K σ = K kin + K cross + K mag dk σ dt = න dv Eqs. (canonical form) P v Ω = h R t no-work conserv. non-cons. Σ + Ԧv Ω = R 7 Canonical electric field (force-field) Σ h P t

10 Maxwell s Eqs. A μ = μ 0 j μ Particle {q; m} Ԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt Kinetic Ԧx, Ԧv, t f σ Stat. Mech. df σ dt = C n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ Fluid eqs. x 2 MHD n Ԧx, t U Ԧx, t P Ԧx, t Single fluid eq. + Ohm s law E + U B = System configuration B Ԧx, u Ԧx (quasi static evolution) P σfluid ρ σ u σ + ρ cσ ԦA h σfluid ρc ρ σu σ 2 + ρ c φ + ρφ g + P σ R σfluid R + P σfluid u σ + L σfluid n n Canonical Helicity p.o.v K σ = K kin + K cross + K mag dk σ dt = Eqs. (canonical form) P v Ω = h R t no-work conserv. non-cons. Σ + Ԧv Ω = R Note: න dv Σ = Ω t 7 Canonical electric field (force-field) Σ h P t

11 Maxwell s Eqs. A μ = μ 0 j μ Particle {q; m} Ԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt Particle Kinetic Ԧx, Ԧv, t f σ Stat. Mech. df σ dt = C n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ Fluid eqs. x 2 MHD n Ԧx, t U Ԧx, t P Ԧx, t Single fluid eq. + Ohm s law E + U B = System configuration B Ԧx, u Ԧx (quasi static evolution) P part m Ԧv + q ԦA h part mc mv2 + qφ + mφ g R part 0 P σfluid ρ σ u σ + ρ σ ԦA h σfluid ρc ρ σu σ 2 + ρ c φ + ρφ g + P σ R σfluid R + P σfluid u σ + L σfluid n n Canonical Helicity p.o.v K σ = K kin + K cross + K mag dk σ dt = Eqs. (canonical form) P v Ω = h R t no-work conserv. non-cons. Σ + Ԧv Ω = R න dv 8 Canonical electric field (force-field) Σ h P t

12 Maxwell s Eqs. A μ = μ 0 j μ Particle {q; m} Ԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt Particle Kinetic Ԧx, Ԧv, t f σ Stat. Mech. df σ dt = C n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ Fluid eqs. x 2 MHD n Ԧx, t U Ԧx, t P Ԧx, t Single fluid eq. + Ohm s law E + U B = System configuration B Ԧx, u Ԧx (quasi static evolution) P part γm Ԧv + q ԦA h part γmc 2 + qφ R part 0 P σfluid ρ σ u σ + ρ σ ԦA h σfluid ρc ρ σu σ 2 + ρ c φ + ρφ g + P σ R σfluid R + P σfluid u σ + L σfluid n n Canonical Helicity p.o.v K σ = K kin + K cross + K mag dk σ dt = Eqs. (canonical form) P v Ω = h R t no-work conserv. non-cons. Σ + Ԧv Ω = R න dv 8 Canonical electric field (force-field) Σ h P t

13 Maxwell s Eqs. A μ = μ 0 j μ Particle {q; m} Ԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt Particle P part γm Ԧv + q ԦA h part γmc 2 + qφ R part 0 Kinetic Ԧx, Ԧv, t f σ Stat. Mech. df σ dt = C Kinetic P σkin fp part = fm σ Ԧv + fq σ ԦA h σkin fh part = R σkin P part C + v f ԦaP part + L part f n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ Fluid eqs. x 2 P σfluid ρ σ u σ + ρ σ ԦA MHD n Ԧx, t U Ԧx, t P Ԧx, t Single fluid eq. + Ohm s law E + U B = h σfluid ρc ρ σu σ 2 + ρ c φ + ρφ g + P σ R σfluid R + P σfluid u σ + L σfluid n n System configuration B Ԧx, u Ԧx (quasi static evolution) Canonical Helicity p.o.v K σ = K kin + K cross + K mag dk σ dt = න dv Eqs. (canonical form) P v Ω = h R t no-work conserv. non-cons. Σ + Ԧv Ω = R (2) 9 Canonical electric field (force-field) Σ h P t

14 Is there a more fundamental p.o.v.? Where does the canonical form of the equation come from? The first two canonical Maxwell s eqs give us a clue: Canonical Gauss law (no monopole) Canonical Faraday s law Canonical Gauss law Canonical Ampere s law Ω = 0 Σ = Ω t Σ =? Ω =? 10

15 Is there a more fundamental p.o.v.? Where does the canonical form of the equation come from? The first two canonical Maxwell s eqs give us a clue: Lagrangian L Canonical Gauss law (no monopole) Canonical Faraday s law Euler-Lagrange eq. Canonical Gauss law Canonical Ampere s law Ω = 0 Σ = Ω t Σ =? Ω =? 10

16 Define a new Lagrangian L σ J μ P μ 1 4μ σ F μν F μν + L EinsteinHilbert Coupling interaction between sources J μ and field P μ Canon. four-current Canon. four-field Coupling interaction between field P μ components (its derivatives) J μ q xμ t = qc Ԧj = q Ԧv P μ h c, P Particle P par γmv + qa h par γmc 2 + qφ Kinetic P kin f P par h kin f h par Canon. charge density (switch) Canon. field tensor q ε σ R σ Ԧv σ Ω σ F μν μ P ν ν P μ P σfluid නP σkin d Ԧv σ h σfluid නh σkin d Ԧv σ 11 S.You, Phys. Plasmas 23, (2016)

17 In flat space L q Ԧv P q h εσ2 Ω2 2μ Balance between generalized kinetic energy and potential energy (enthalpy). Balance between generalized electrical energy (canon. forces) and magnetic energy (canon. vorticity) 12

18 Particle P par γmv + qa h par γmc 2 + qφ Kinetic P kin f P par h kin f h par L q Ԧv P q h εσ2 Ω2 2μ P σfluid නP σkin d Ԧv σ h σfluid නh σkin d Ԧv σ e.g. particle in e.m. field L part = 1 γ mc2 + q Ԧv ԦA qφ 12

19 Σ = h P t = q σ φ q σ ԦA t + = q σ E + Particle Σ = q σ E + Σ Ω = q σ B + Ω Kinetic Σ = f σ q σ E + Σ Ω = f σ q σ B + Ω Fluid Σ = ρ σ E + Σ Ω = ρ σ B + Ω L q Ԧv P q h εσ2 Ω2 2μ Field 1 2 ε 0E 2 B2 + 2q 1 2μ 0 2 εe B Ω Σ μ εσ 2 Ω 2 2μ L Maxwell L e.m. m L m m 13

20 Particle Σ = q σ E + Σ Ω = q σ B + Ω Kinetic Σ = f σ q σ E + Σ Ω = f σ q σ B + Ω Fluid Σ = ρ σ E + Σ Ω = ρ σ B + Ω L q Ԧv P q h εσ2 Ω2 2μ Field 1 2 ε 0E 2 B2 + 2q 1 2μ 0 2 εe B Ω Σ μ εσ 2 Ω 2 2μ L Maxwell L e.m. m L m m 13 Just a reformulation of existing Lagrangians. New Lagrangians that represent the collective coupling between kinetic distributions, & e.m. fields, between kinetic distributions themselves, and relativistic distributions

21 Eq. Insert L into Euler-Lagrange equation for space coordinates Ԧx: q Σ + v Ω = 1 2 εσ2 Ω2 2μ R Dissipative forces: Incomplete conversion between canon. electrical /"kinetic energy and canon. magnetic /"potential energy (canon. vorticity). Single particle: R par 0 (classical) R par 0 (relativistic) Kinetic: R σ kin Ԧv σ P σ par h σ par f σ P σ par df σ dt Fluid: 14 R σ flu R σα + R σσ + R σc + R σn

22 Canon. Maxwell equations Insert L into Euler-Lagrange equation for four-field P μ (with Lorenz gauge): D μ F μν = μ σ J ν μ μ P ν = μ σ J ν Σ = q ε Ω = μ q Ԧv + με Σ t ቐ h = q ε P = μ q Ԧv (also gives a canonical Poynting flux equation for Σ Ω) d Alembert wave operator 15

23 Canon. Maxwell equations Insert L into Euler-Lagrange equation for four-field P μ (with Lorenz gauge): D μ F μν = μ σ J ν μ μ P ν = μ σ J ν eq.! Σ = q ε Ω = μ q Ԧv + με Σ t ቐ h = q ε P = μ q Ԧv (also gives a canonical Poynting flux equation for Σ Ω) d Alembert wave operator 16

24 Hamiltonian, energy evolution Perform Legendre transformation of L H = 1 2 εσ2 + Ω2 2μ + q h Sum of energy density of the plasma field and the source enthalpy (if any) Equivalent derivation of eq. q Σ σ + Ԧv σ Ω σ inserting H into Hamilton s third equation: = R σ, by H = pሶ q i P i t + Ԧv P = H Using the Poynting theorem for Σ, Ω in the Hamilton equation H = L t t gives energy evolution equation 17 dh σ dt = නq h σ dt dv නq Σ σ Ԧv σ dv න Σ σ Ω σ μ σ d ԦS + නH σ u σ d ԦS (3)

25 Therefore E.M. Field E, B φ, Particle ԦA A μ P par γmv Particle + qa {q; m} h par γmc 2 + qφԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt Maxwell s Eqs. A μ = μ 0 j μ Can express eq. Solve as generalized Maxwell equations. [integrate] Canonical helicity transport is valid at all regimes. E.M. field sources ρ c, Ԧj Helicity vs energy conservation depends on density scale j μ length. P kin f P par h kin f h par Kinetic Ԧx, Ԧv, t lots of particles Kinetic f σ Stat. Mech. df σ dt = C Solve [integrate] f Ԧx, t averaging Dynamics Evolution, waves, etc. n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ P σfluid නP σkin d Ԧv σ h σfluid නh σkin d Ԧv σ c.o.m. MHD n Ԧx, t U Ԧx, t P Ԧx, t System state stability, etc. dk σ Energy p.o.v dt = δw K K i + K e = constant නU dv System configuration B Ԧx, u Ԧx (quasi static evolution) Canonical Helicity p.o.v K σ = K kin + K cross + K mag න dv L σ J ν P ν 1 4μ σ F μν F μν Fluid eqs. x 2 Single fluid eq. + Eqs. (canonical form) Ohm s law E + U qb = Σ + Ԧv Ω = R D μ F μν = μ σ J ν 18 Choose Particle R part 0 Kinetic R σkin = Ԧv σ P σpar h σpar f P σpar df σ dt R σfluid = R σα + R σσ + R σc + R σn

26 Change of (species) helicity vs change in energy? Take the ratio of evolution equations (Eq. 1)/(Eq. 3). e.g. isolated, fixed, non-conservative forces ΔK σ ΤK σ0 = 2 R σ Ω σ dv H σ0 2 Ω σ H σ0 ρ σ H σ0 ΔH σ ΤH σ0 R σ Ԧv σ dv K σ0 v σ K σ0 L s K σ0 Density scale length: shallow species helicity changes little steep species helicity changes a lot MST experiment: H e0 W mag 50 kj Ω e ρ ce B 0.2 kg m -3 s -1 n e m -3 B tor 0.12 T T e 300 ev K e0 23 mwb 2 => magnetic energy is dissipated 33 times more than magnetic helicity in an isolated, purely dissipative, magnetically dominated system. 1ΤL s 2 1ΤL circ + 1Τr L 19

27 Helicity conversion at the edge of plasmas where density gradients are steep Density scale length: shallow species helicity changes little steep species helicity changes a lot L-H transition as helicity-constrained relaxation? use kinetic form of canon. Maxwell s eq.? in spheromak? 21

28 First, examine helicity transport in cyl. geom. Generalization of n B = 0 h σ Ω σ = 0 Dot steady-state Σ σ + Ԧv σ Ω σ 0 with Ω σ Generalized induction equation for Ω σ Take curl of Σ σ + Ԧv σ Ω σ 0 So plasma motion canonical flux tube motion in many regimes beyond magnetostatic. M 2-9 Butterfly nebula 22

29 23 Mochi experiment just became operational Designed to study interaction between flows and magnetic fields

30 End

31 Energy equation Insert L into Euler-Lagrange equation for time t: q Ԧv P t = t 1 2 εe2 Ω2 2μ If the plasma field has a time dependence, energy changes.

32 Maxwell s Eqs. A μ = μ 0 j μ Solve [integrate] E.M. field sources ρ c, Ԧj j μ E.M. Field E, B φ, ԦA A μ Solve [integrate] f Ԧx, t Dynamics Evolution, waves, etc. Particle {q; m} Ԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt lots of particles Kinetic Ԧx, Ԧv, t f σ Stat. Mech. df σ dt = C averaging n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ c.o.m. MHD n Ԧx, t U Ԧx, t P Ԧx, t Fluid eqs. x 2 Single fluid eq. + Ohm s law E + U B = Choose

33 Maxwell s Eqs. A μ = μ 0 j μ Solve [integrate] E.M. field sources ρ c, Ԧj j μ E.M. Field E, B φ, ԦA Solve [integrate] f Ԧx, t Dynamics Evolution, waves, etc. System state stability, etc. A μ Particle {q; m} Ԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt lots of particles Kinetic Ԧx, Ԧv, t f σ Stat. Mech. df σ dt = C averaging n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ c.o.m. MHD n Ԧx, t U Ԧx, t P Ԧx, t Energy p.o.v δw නU dv Fluid eqs. x 2 Single fluid eq. + Ohm s law E + U B = Choose

34 Maxwell s Eqs. A μ = μ 0 j μ E.M. Field E, B φ, ԦA Solve [integrate] Solve [integrate] f Ԧx, t Dynamics Evolution, waves, etc. E.M. field sources ρ c, Ԧj j μ System state stability, etc. System configuration B Ԧx (quasi static evolution) A μ Particle {q; m} Ԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt lots of particles Kinetic Ԧx, Ԧv, t f σ Stat. Mech. df σ dt = C averaging n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ c.o.m. MHD n Ԧx, t U Ԧx, t P Ԧx, t Energy p.o.v δw නU dv Helicity p.o.v K mag න dv dk mag dt Fluid eqs. x 2 Single fluid eq. + Ohm s law E + U B = Choose

35 Maxwell s Eqs. A μ = μ 0 j μ E.M. Field E, B φ, ԦA Solve [integrate] Solve [integrate] f Ԧx, t Dynamics Evolution, waves, etc. E.M. field sources ρ c, Ԧj j μ System state stability, etc. System configuration B Ԧx, u Ԧx (quasi static evolution) A μ Particle {q; m} Ԧx t Ԧv t Electrodynamics d γm Ԧv = ԦF dt lots of particles Kinetic Ԧx, Ԧv, t f σ Stat. Mech. df σ dt = C averaging n σ Ԧx, t u σ Ԧx, t Ԧx, t P σ c.o.m. MHD n Ԧx, t U Ԧx, t P Ԧx, t Energy p.o.v δw නU dv Canonical Helicity p.o.v K σ = K kin + K cross + K mag dk σ dt = Fluid eqs. x 2 Single fluid eq. + Ohm s law E + U B = න dv Choose