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 310027, 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, 015008 (2016).
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
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!!
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.
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
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 (...)
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
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
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
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.
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.
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)].
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
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
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]
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??
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)]
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.
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
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
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)
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
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]
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)
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
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
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)
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
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