Nonlinear Alfvén Wave Physics in Fusion Plasmas

Transcription

1 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 1 Nonlinear Alfvén Wave Physics in Fusion Plasmas Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou , P.R.C. In collaboration with Fulvio Zonca, ENEA, Frascati, Italy, and IFTS Also at Department of Physics and Astronomy, University of California, Irvine, USA Reference: and Fulvio Zonca, Reviews of Modern Physics 88, (2016).

2 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 2 Outlines (I) Introduction (II) Alfvén Wave Induced Transports: (II.A) Single Particle Picture (II.B) Quasilinear Analysis (III) Nonlinear Wave Wave Interactions: (III.A) The Pure Alfvénic State (III.B) Parametric Decay Instabilities: Ideal MHD vs. Kinetic (III.C) Modulational Instabilities: Convective Cells, Zonal Flows and Currents (IV) Nonlinear Wave Particle Interactions: (IV.A) Fishbone Paradigm (IV.B) Frequency Chirping and Phase Locking (IV.C) General Approach: Dyson Equation (V) Summary and Discussions

3 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 3 (I) Introduction In magnetically confined fusion plasmas: Energetic/alpha charged particles Wave-Particle interactions Alfvén wave instabilities Electromagnetic Alfvén fluctuations: Breaking toroidal symmetry enhanced EP/α losses detrimental to the goal of self-sustaining burning fusion plasmas Enhanced anomalous losses Nonlinear kinetic-theoretic analysis necessary NO trustworthy magic theoretical models/formulae so far! Complex issues Nonlinearity, non-uniformity, geometries coupled together This talk Share some insights and current understandings Lots remain unknown!!

4 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 4 (II) Alfvén Wave Induced Transports ** Key points: Toroidal symmetry breaking and wave-particle resonance ** (II.A) Single Particle Picture B = ψ(r) (φ qθ) δψ = δψ p +δψ f, δψ f : field line displacement d ( ) [ dt δψ p= c 1 φ δê +(v d 1 δ ˆB ] ) b e iθ (β 1 neglect δb effect) Θ = Θ(X, t): wave-particle phase B (1/R 0 )[1 cosθ(r/r 0 )] ψ: magnetic surface / φ = 0 no transport magnetic surface P φ (banana center) = const.

5 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 5 Symbolically ( ) [ δψ p = i Θ 1 c 1 φ δê +(v d 1 δ ˆB ] ) b e iθ transport maximizes when dθ/dt = Θ = 0 wave-particle resonance Θ const. X = X 0 +δx Θ(X 0,t) = 0 linear (primary) resonance e.g., ω n ω d +pω b = 0; for trapped particles ω n ω d +(m n q)ω t +pω t = 0; for circulating particles Θ(X 0,t) 0 but Θ(X 0 +δx,t) = 0 nonlinear (secondary) resonances DIII-D: EGAM AUG: RMP

6 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 6 Nonlinearities in Wave-Particle phase Similar to µ-breaking at sub-cyclotron harmonics [Chen et al., 2000] (1+l)ω = nω c ; l = positive integers For ideal MHD waves δe = 0 Finite transport due to the v d δ ˆB term Correct B 0 correct transport! (II.B) Quasilinear Analysis [Chen, JGR (1999)] Axisymmetric Tokamak [C&Z, Springer-Nature Book] Resonant particles transports t F 0 + (δẋδg res)+ v (δ v δg res ) = 0 (...) = transit/bounce averaging of (...)

7 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 7 ( t +v l +v d ) δg L g δg = iqf 0 δh ( 1 F 0 QF 0 = i t + F 0 b ) v v Ω δh = q δφ v m c δa : Gyroaveraging [ δx b = Ω +b ] δh v δ v = b δh Symbolically δg res = iπδ( il g )QF 0 δh Particle s transport t [N] S + 1 Γ ψ = V ψ ψ [ ψ [ V ψγ ψ ] δẋδg res = 0 ; S ] v S V ψ = dl B 0... v d 3 (...)v [N] S : flux surface averaging

8 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 8 Trapped Particles: δh = 1 2 m,n Γ ψ = Γ ψc +Γ ψd [ ] δĥm,ne i(nφ mθ ωm,nt) +c.c. Convective: Γ ψc = π 2 Diffusive: Γ ψd = π 2 M 2 c q [ δ(n ωd +pω b ω m,n ) λ m,n,p 2 n ω ] m,n )F 0 M E v S [ δ(n ωd +pω b ω m,n ) λ m,n,p 2 m,n p( δĥ m,n 2 M 2 c q m,n p ( δĥm,n 2 Convective & Diffusive intrinsically coexist! n 2c q ] )F 0 ψ v S

9 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 9 Formally: Γ ψc Γ ψd depends on detailed spectral information roughly Γ ψc / Γ ψd O( ω/ ω ) Similar analysis for circulating particles

10 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 10 (III) Nonlinear Wave Wave Interactions (III.A) The pure Alfvénic state Nonlinear self-consistent SAW solution Infinite, uniform, ideal magnetohydrodynamic (MHD) fluid m( t +u )u = P +J B/c u 0 = 0 = J 0, B 0 = B 0ˆb Shear Alfvén waves (SAW) Negligible magnetic compression δb δb Incompressible δu 0 δ m 0,δP 0.

11 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 11 Ponderomotive force m0 t δu = F (2) p +δj B 0 /c F (2) p = δb 2 /8π Mx Re Mx (δb )δb /4π : Maxwell stress Re m0 (δu )δu Nonlinear SAW equation Ideal MHD approximation: δe 0 c 2[ (b 0 ) 2 V 2 A 2 t : Reynolds stress δj (2) = (c/b 0)b 0 [Re+Mx] ] 2 δφ+4π t ( δj (2) ) = 0 SAW NL Vorticity Eq.

12 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 12 Pure Alfvénic state: Walén relation δu W /V A = ±δb W /B 0 [ t ±V A b 0 ]δφ W = 0 : Re+Mx = 0 δj (2) = 0 [ (b 0 ) 2 V 2 A 2 t] δφw = 0 Co- or Counterpropagating SAW δφ W : solution to nonlinear SAW equations Nonlinear wave-wave interactions Breaking pure Alfvénic states [C&Z, PoP, 2013]: Finite ion compressibility: ion sound perturbations along B Microscopic-scales (ρ i ) Kinetic Alfvén Waves [a/ρ i > O(10 3 )] Enhancedelectron-iondecoupling EnhancedδE ( δe /δe O(k 2 ρ2 i)) Geometries: continuous and discrete SAW spectra [e.g., Toroidal Alfvén Eigenmodes (TAE)].

13 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 13 (III.B) Parametric Decay Instabilities: finite ion compressibility Coupling to the ion-sound wave via parallel ponderomotive force Ideal MHD macro-scale theories [S&G 1969] Resonant decay Ω 0 = (ω 0,k 0 ) = Ω S +Ω A Backscattering: Counter-propagating SAWs

14 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 14 Parametric Dispersion Relation ǫ S ǫ A = C I eδφ 0 /T e 2 ǫ S : ISW ǫ A : SAW C I O(k ρ 2 2 i)cos 2 θ, Coupling maximizes around θ = 0,π; k 0 k TAE: Hahm & Chen, PRL 74, 1995

15 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 15 Gyrokinetic micro/meso-scale theory [C&Z, EPL, 2011; H&C 1976] k ρ i O(1) enhanced electron ion decoupling Parametric Dispersion Relation ǫ SK ǫ A K = C K eδφ 0 /T e 2 ǫ SK : Kinetic ISW ; ǫ A K : KAW C K O [ (Ωi ) ] 2 (k ρ i ) 6 sin 2 θ, ω 0 Maximizes around θ ±π/2 (k 0 k ) and k ρ i O(1) Simulation by Y. Lin et al. [PRL 2012]

16 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 16 Quantitative & Qualitative differences C K > C I for 1 > k ρ i 2 > ω 0 /Ω i < O(10 2 ) Implications to transports Ω 0 : Mode converted KAW k 0 k 0rˆr MHD regime: k k rˆr no P θ breaking little transport! Kinetic regime: k k θˆθ large P θ breaking significant transport! Dayside Earth s Magnetopause Applications to fusion plasmas??

17 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 17 (III.C) Modulational Instabilities convective cells [C&Z, PRL2012] zonal flows & currents Zonal structures Coherent micro/meso-scale radial corrugations of equilibrium in toroidal device plasmas. Examples: Zonal Flow Zonal Current [More generally: phase-space zonal structures (Zonca et al., NJP 2015)]

18 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 18 Zonal structures spontaneously excited by micro/meso-scale turbulence due to plasma instabilities. Zonal structures scatter turbulence to shorter-radial wavelength stable domain nonlinearly damp the instability. Self-regulation of plasma instabilities! In toroidal plasmas continuous and discrete spectra Continuous spectrum ω 2 = k 2 (r)v 2 A(r) Re+Mx 0 negligible nonlinear contributions Discrete spectrum AEs finite nonlinear contribution Spontaneous excitation of zonal structures via modulational instability of a finite-amplitude TAE wave. Other discrete AEs can also break the Alfvénic State and stimulate interesting nonlinear wave-wave interactions.

19 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 19 FILR: Kinetic Alfvén Waves Electrostatic convective cells (zonal flow) Magnetostatic convective cells (zonal current) [Zonca et al., EPL 2015] for theory and simulation

20 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 20 (IV) Nonlinear Wave Particle Interactions (IV.A) Fishbone Paradigm Excitations via wave-particle interactions tapping EP s finite P EP expansion free energy. Magnetically trapped charged particles precess in φ Precessional frequency ω d E = v 2 /2 ω = ω d resonant particles secularly move in the radial (R) direction

21 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 21 µ = v 2/2B v2 R = adiabatic invariant moving outward particle looses energy gh P EP / r < 0 Net loss of charged particle kinetic energy Fishbone instability (Analogues to Rayleigh-Taylor instability)

22 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 22 (IV.B) Frequency Chirping and Phase Locking Θ = ω ω d (ψ,µ,e) 0 Frequency chirping ω < 0, ( ω d < 0) Θ dω dt ω d ψ δ ψ 0 Resonance maintained nonlinearly (phase-locking) Maximal wave-particle power exchange EPs secularly move outward Radial decoupling from the wave due to finite radial mode width Non-adiabatic: ω > ω2 B Nonlinear time scale < Wave trapping period

23 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 23 Fishbone simulations [Fu et al., 2006] Frequency chirping P φ (radial) redistribution of beam ions P φ = P φ [R]

24 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 24 Electron fishbone [Vlad et al., 2012] Reviewed [Vlad et al., NJP 2016] T = 300 (linear) T = 900 (saturation)

25 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 25 Frequency chirping + EP radial redistribution Secular radial motion Θ = nφ ωt Θ = n ω d ω 0 Phase locking

26 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 26 Nonlinear saturation occurs when δû n /γ L r s r s mode structure width Wave-EP interaction domain [Vlad et al., 2012] simulation results

27 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 27 (IV.C) General Approach: Dyson Equation Focus on EP nonlinear physics Conservation of (µ,j) 1D problem in P φ or r [ t + φ ] φ +δṙ n r f EP (r,φ,t µ,j) = 0; δṙ n = Decompose into n = 1 and n = 0 components ) ( ωdn ω n δu n, δu n = c δe θ B f EP = F 0EP (r,t µ,j)+δf n (r,t µ,j)exp(inφ)+c.c. Emission and absorption of toroidal symmetry breaking fluctuations evolution of F 0EP in r (P φ ) over t EP redistribution F 0EP evolution equation analogues to the Dyson Equation (cf. Al tshul & Karpman, 1966)

28 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 28 Simplifying arguments (C&Z RMP2016 for detailed analysis) Focusing on the resonant particles Frequency chirping and phase-locking rapid redistribution ω dn ω 0 < γ iω ω τ 1 nl [ ] t F 0EP = t 1 1 δû n 2 H(r) r r H(r)F 0EP + Source + Dissipation Resonant EPs convect outward with radial speed δû n

29 ASIPP 40 th Anniversary Nonlinear Alfvén Wave Physics in Fusion Plasmas 29 (V) Summary and Discussions In burning plasmas EP/α s Alfvén instabilities anomalous EP/α losses: Crucial issues! Wave - induced transports Kinetic processes: Symmetry breaking & linear and nonlinear Wave - Particle resonances NL Wave - Wave and Wave - Particle interactions on an equal footing Plasma Physics uniqueness Nonlinear kinetic effects and realistic mode structures Crucial roles Nonlinearities, Nonuniformities, geometries of B intellectually challenging and practically important No simplistic magic shortcut Serious analytical, simulation, experimental works and collaborations