The Trilateral International Workshop on Energetic Particle Physics(Nov.10-12,2017)-会议-浙江大学聚变理论与模拟中心.pdf

The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 1 The fishbone burst cycle∗ Fulvio Zonca1,2 and Liu Chen2,3 http://www.afs.enea.it/zonca 1 2 ENEA C.R. Frascati, C.P. 65 - 00044 - Frascati, Italy. Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, P.R.C. 3 Dept. of Physics and Astronomy, University of California, Irvine CA 92697-4575, USA November 12.th, 2017 The Trilateral International Workshop on Energetic Particle Physics, November 10–12 2017, IFTS – ZJU, Hangzhou, P.R.C. ∗ [Reviews of Modern Physics 88, 015008 (2016)] [Unpublished work (ArXiV) (2007)] Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics Observation of fishbone oscillations ✷ Experimental observation of fishbones in PDX [McGuire et al. 83] with macroscopic losses of ⊥ injected fast ions ... Fulvio Zonca and Liu Chen The fishbone burst cycle – 2 The Trilateral International Workshop on Energetic Particle Physics ✷ The fishbone burst cycle – 3 Followed by numerical simulation of the mode-particle pumping (secular) loss mechanism [White et al 83] ... ω̇ ≈ ωB2 ✷ ... and the theoretical explanation of the resonant internal kink excitation by energetic particles and the (model) dynamic description of the fishbone cycle [Chen, White, Rosenbluth 84] Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 4 The fishbone dispersion relation ✷ The problem was solved by [Chen, White, Rosenbluth 84]: first example of the General Fishbone Like Dispersion Relation (GFLDR; n = 1) i|s|Λn = δ Ŵnf + δ Ŵnk . ✷ Definitions: drop subscript n for simplicity, ∆q0 = 1 − q0 , Λ 2 = (ω/ωA2 )(ω − ω∗pi ) 2 2 1/2 1 + 0.5q + 1.6q (R0 /r) [Graves al. 2000] Z et rs 2 δ Ŵf = 3π∆q0 13/144 − βps rs2 /R02 ; r2 (dβ/dr)dr βps = −(R0 /rs2 )2 0 [Bussac et al. 1975] 2 δ Ŵk = 2 rs Z rs 0 3 r dr Z EdEdλ X π 2 R0 e2 τb ω̄ 2 Z +∞ vk /|vk |=± E = v 2 /2 , λ ≡ µB0 /E d c2 q m ω0 (τ ) −∞ ω + ω0 (τ ) e−iωt Qk,ω0 (τ ) F̂0 (ω)dω ω̄d − ω0 (τ ) − ω Qk,ω0 (τ ) F̂0 ≡ (ω0 (τ )∂E + Ω−1 k × b · ∇) F̂0 Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics ✷ The fishbone burst cycle – General Fishbone Like Dispersion Relation (GFLDR) ⇒ Kinetic Energy Principle [Z&C POP14] i|s|Λ(ω) = δ Ŵf + δ Ŵk (ω) s = magnetic shear ◦ Λ(ω): “inertia” (kinetic energy) due to background plasma ⇒ Structures of continuous spectrum, gaps, and resonant absorption ◦ δ Ŵf : “δW ” (potential energy) due to background plasmas ⇒ existence of discrete AEs (different types; depending on equilibrium) ◦ δ Ŵk : “δW ” (active potential energy) due to EPs ⇒ instability mechanisms & new unstable modes: Energetic Particle continuum Modes (EPMs)/Fishbones [Chen 1994] ✷ For EPM/fishbone ⇒ iΛ represents continuum damping [Chen et al 84] IReδ Ŵk (ωr ) + δ Ŵf = 0 ⇒ determines ωr , γ/ωr = (−ωr ∂ωr IReδ Ŵk )−1 (IImδ Ŵk − |s|Λ) Fulvio Zonca and Liu Chen (non − perturbative) ⇒ determines γ/ωr 5 The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – Outstanding questions • Why does it chirp? ⇒ connected with nature of wave-EP interaction, and mode dispersion relation (non-perturbative) • What is the frequency sweeping rate? ⇒ connected with wave-EP power exchange, and self-consistent (non-perturbative) interplay of mode structure and EP transport • What is the mechanism of EP loss? ⇒ connected with non-adiabatic frequency sweeping and phase-locking Fulvio Zonca and Liu Chen 6 The Trilateral International Workshop on Energetic Particle Physics Why does it chirp? The fishbone burst cycle – 7 ⇒ Generation of the distribution δfk due to the interaction of f0 with δφk , corresponding to the solution of the GFLDR. The diagram of the process is defined in the top frame, while the solution of the “Dyson” equation corresponds to the summation of all terms in the Dyson series (bottom). Nonlinear distortion of f0 due to emission and absorption of the field δφk . Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 8 ✷ Nonlinear distortion of f0 is accompanied by self-consistent nonlinear evolution of mode structure (frequency), due to the non-perturbative wave-EP interaction ⇒ frequency chirping. ✷ Not all modes can chirp: fishbone real frequency (dispersion relation) is set by the characteristic EP precession rate, which depends of radial position ⇒ continuum of possible fishbone frequencies controlled by EP radial profile; dominated by the strongest growing mode. ✷ Non-perturbative EP response (dispersion relation), via the selection of the strongest growing mode (maximization of wave-EP power transfer) also dictates the optimal chirping rate for preserving the resonance condition throughout the nonlinear phase ⇒ phase-locking [C&Z RMP16]: ω ≃ ω̄d nq c ∂ ∂ |ω̇| ≃ δvr ω̄d ≃ δφ ω̄d ∝ ωB2 ∂r r B0 ∂r non − adiabatic frequency sweeping Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 9 Nonlinear wave-EP interactions are accounted for via (n = 1) Dyson equation ∗ Qk0 ,ω0 (τ ) i i nc ∂ i e F̂0 (ω) = StF̂0 (ω) + Ŝ(ω) + F̄0 (0) + ω ω 2πω m ω(dψ/dr) ∂r ω0∗ (τ ) # ) Qk0 ,ω0 (τ ) F̂0 (ω − 2iγ0 (τ )) F̂0 (ω − 2iγ0 (τ )) 2 × ω̂dk0 δ φ̄k0 (r, τ ) + . ∗ ω − ω0 (τ ) + nω̄dk0 ω0 (τ ) ω + ω0 (τ ) − nω̄dk0 ✷ ✷ Solution strategy of the nonlinear problem (small fluctuation amplitude expansion): • At the lowest order, the problem is solved satisfying the linear dispersion relation • At next order the nonlinear dynamics describes the mode amplitude and frequency evolution; and EP transport Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics ✷ The fishbone burst cycle – 10 Introduce the radial displacement δ ξ¯rk0 (r, τ ) = nqc kϑ c δ φ̄k0 (r, τ ) = − δ φ̄k0 (r, τ ) . ω0 B0 ω0 rB0 ✷ From the Dyson Equation, assume |ω∗EP | ≫ |ω0 | ⇒ ω0 (τ )−1 Qk0 ,ω0 (τ ) F̄0 ≃ (ω0 Ω)−1 (−nq/r)∂r F̄0 . Furthermore, dψ/dr = B0 (r/q).Then, by direct substitution, at the two lowest orders in γ0 /ω0 i i i nω̄dk0 ∂ i F̄0 (0) − 2 F̂0 (ω) = StF̂0 (ω) + Ŝ(ω) + ω ω 2πω ω ω0r ∂r (γ0 + iω) + (nω̄dk0 − ω0r )(γ0 /ω0r ) ¯ 2 ∂ × δ ξrk0 (r, τ ) F̂0 (ω − 2iγ0 (τ )) . 2 2 (nω̄dk0 − ω0r ) + (γ0 + iω) ∂r ✷ This expression has to be employed within the δ Ŵk expression. Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics ✷ The fishbone burst cycle – 11 Solution of Dyson equation reveals [C&Z NJP15; RMP16]: • phase locking: resonance condition preserved throughout the nonlinear phase • phase bunching: periodic modulation due to EP phase synchronization caused by resonance tuning & detuning (non-adiabatic chirping) ✷ Numerical solution is required in general. Analytical progress is possible assuming: • considering deeply trapped EP with radially localized interaction with internal kink-like mode structure • describing the average evolution of the phase space zonal structure (PSZS) [C&Z NJP15; RMP16] Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 12 Nonlinear fishbone dynamics (I) ✷ Thanks to these assumptions we canZdefine resonant particle βE as X r π2 τb ω̄d2 βEr (r; ω0 (τ )) = 2 2 m |Ω| 2 EdEdλ B0 q vk /|vk |=±1 Z ∞ (γ0 − iω) −iωt e F̂0 (ω)dω . × 2 2 −∞ (ω̄d − ω0r ) + (γ0 − iω) ✷ With this expression, " # 1/2 Z rs 1/2 R0 r ∂ R0 2 r Imδ Ŵk = − q βEr (r; ω0 (τ )) dr rs 0 rs r ∂r R0 Z rs R0 2 ∂βEr 2 βEr dr −rq −q , = rs 0 ∂r 2 rs ✷ Resonant particle βE describes the PSZS βE that controls wave-particle power exchange. Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics ✷ The fishbone burst cycle – 13 In the same way, we can define non-resonant particle βE as Z 2 X r π τb ω̄d2 β̂E (r; ω0 (τ )) = 2 2 m |Ω| 2 EdEdλ B0 q vk /|vk |=±1 Z ∞ (ω̄d − ω0r ) −iωt e F̂0 (ω)dω . × 2 2 −∞ (ω̄d − ω0r ) + (γ0 − iω) ✷ With this expression, R0 L Reδ Ŵk ≃ Reδ Ŵk = − rs ✷ Z rs 0 q 2 r rs R0 r 1/2 ∂ ∂r " r R0 1/2 β̂E (r; ω0 (τ )) dr , Non-resonant particle βE describes the PSZS βE that controls mode frequency. Fulvio Zonca and Liu Chen # The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 14 Nonlinear fishbone dynamics (II) ✷ As reference case, consider a simple isotropic slowing down distribution, H(EF /mE − E) 3P0E , F̄0 = 3/2 3/2 4πEF (2E) + (2Ec /mE ) where H denotes the Heaviside step function and the normalization condition is chosen such that the EP energy density is (3/2)P0E for EF ≫ Ec , and EP energy is predominantly transferred to thermal electrons by collisional friction as it occurs for α-particles in fusion plasmas. ✷ Reconsider the expressions of Reδ Ŵk and β̂E (r; ω0 (τ )); and δ Ŵf + Reδ ŴkL ≃ 0 . ✷ For ReΛ2 > 0, this equation (real part of GFLDR) determines the real frequency of the fishbone mode for |γ0 /ω0 | ≪ 1 [Chen 1984]. Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 15 ✷ The fishbone frequency is set by the condition ω0 /ω̄dF ≃ const, to be computed at the position of the radial shell where the most significant EP contribution is localized. ✷ The GFLDR (imaginary part; for |γ0 /ω0 | ≪ 1), gives the evolution of the fishbone amplitude γ0 −∂Reδ ŴkL /∂ω0r = Imδ Ŵk −|s|Λ(ω0r ) . ✷ This equation expresses the competition between EP drive and continuum damping; and can be cast as ( Z 1/2 rs 2(R0 /rs ) R0 ∂ 2 2 r − ln |δξr0 | = q ∂t rs r 0 −∂Reδ ŴkL /∂ω0r ) " # 1/2 r ∂ rs × . |s|Λ(ω0r ) βEr (r; ω0 (τ )) dr − ∂r R0 R0 Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 16 ✷ The simplest way to close the (GFLDR) evolution for the fishbone amplitude is to obtain the evolution equation for βEr (r; ω0 (τ )) directly from the Dyson equation for F̂0 (ω). ✷ Calculation of δ Ŵk and βEr (r; ω0 (τ )) involves a vel. space and freq. integral (γ0 + iω) + (nω̄dk0 − ω0r )(γ0 /ω0r ) ¯ 2 ∂ ∝ δ ξ (r, τ ) F̂0 (ω − 2iγ0 (τ )) rk0 2 2 (nω̄dk0 − ω0r ) + (γ0 + iω) ∂r ie−iωt [ω0r + i(γ0 − iω)] [(nω̄dk0 − ω0r ) + i(γ0 − iω)] × ω (nω̄dk0 − ω0r )2 + (γ0 − iω)2 ✷ 2 2 Recall that δ ξ¯rk0 (r, t) = exp(2γ0 t) δ ξ¯rk0 (r, τ ) . For strongly growing modes, γ0 ∼ |ω| ⇒ γ0 ∼ τN−1L , as for SAW excited by EP in tokamaks. ✷ Calculations can be done noting, for |a|, |b| ≪ 1, Rea > 0 and Reb > 0 π x2 → δ(x) 2 2 2 2 (x + a )(x + b ) a+b 1 π → δ(x) . 2 2 2 2 (x + a )(x + b ) ab(a + b) Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 17 Nonlinear fishbone equations With the previous result, the nonlinear equation for βEr (r; ω0 (τ )) is [C&Z RMP16] " !#) 1/2 1/2 ( ∂ r q ∂ r R0 |ω0 |2 |δξr0 |2 βEr . ∂t βEr = β̇ErS − νext βEr +∂t−1 r r ∂r q ∂r R0 ✷ ✷ From external source and collision operator, plus the definition of βEr (r; ω0 (τ )), we have Z 2 X γ0 r π 2 τb ω̄d S(t) , β̇ErS ≡ 2 2 m |Ω| 2 EdEdλ 2 2 B0 q (ω̄d − ωr0 ) + γ0 vk /|vk |=±1 2 π r νext βEr ≡ −2 2 m |Ω| 2 B0 q Z EdEdλ X vk /|vk |=±1 Fulvio Zonca and Liu Chen τb ω̄d2 γ0 StF0 (t) , 2 2 (ω̄d − ωr0 ) + γ0 The Trilateral International Workshop on Energetic Particle Physics ✷ Resonant EPs convect outward with radial speed |δun | ⇒ Nonlinear saturation occurs when |δun |/γL ∼ rs The fishbone burst cycle – 18 ✷ [Vlad et al., 2012] simulation results [rs ∼ mode structure width → Wave-EP interaction domain] ✷ Consistent with numerical simulation results by [GY Fu et al POP 2006]. ✷ Near marginal stability regime explored by [M. Idouakass et al POP16; 2017 tbs] analytically and numerically ✷ Electron-fishbone simulation results with HMGC code [Vlad et al NF13]; [Vlad et al NJP16]. Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 19 What is the frequency sweeping rate? ✷ Connected with wave-EP power exchange, and self-consistent (nonperturbative) interplay of mode structure and EP transport Determined by the phase locking condition, ω ≃ ω̄d , and by the PSZS evolution equation for βEr !#) " 1/2 ( 1/2 R0 q ∂ r r −1 2 2 ∂ βEr . ∂t βEr = β̇ErS − νext βEr +∂t |ω0 | |δξr0 | r r ∂r q ∂r R0 ✷ ✷ By balancing linear and nonlinear terms, the PSZS radial propagation speed is the instantaneous E × B velocity. ✷ Corresponding frequency chirping is non-adiabatic: ω̇0 . ωB2 ⇒ EP transport due to resonance tuning/detuning ⊕ radial decoupling. Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 20 What is the mechanism of EP loss? ✷ Mode particle pumping [White et al 83, Chen et al 84] ⇒ Connected with non-adiabatic frequency sweeping and phase-locking ✷ Non-adiabatic autoresonance [C&Z NJP15; RMP16] • EPs amplify fishbone over a finite interaction time, τI , as they are convected outward Z τI (ω0r (τ ) − ω̄d ) dτ ≃ π 0 • PSZS slips over EP population and amplification continues (resonance tuning/detuning) until particles are radially de-coupled • Strongly driven fishbone saturation [Fu et al 06; Vlad et al 13,16] corresponds to radial decoupling within τI [Near marginal stability studied by M. Idouakass et al 2017 tbs] • Similar to amplification of a short FEL pulse [Bonifacio et al. 90,94] (finite interaction time fishbone ⇔ finite interaction length FEL pulse) Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 21 FEL amplification and fishbones [Giannessi et al, 2005] Profile of the radiation pulse vs. the longitudinal electron beam coordinate Longitudinal phase space ✷ Steady state FEL theory: radiation slips forward over the electron beam as required by resonance condition ✷ Slippage effect ⇒ finite cooperation length over which electrons can emit with longitudinal coherence Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 22 Reduced nonlinear equations: predator-prey model ✷ Original work on fishbone proposed a qualitative model for fishbone cycle ⇒ secular EP loss mechanism. ✷ This understanding implies |δξr0 | ∼ rs |γL /ω0 | ✷ From [Chen et al. 1984], fishbone cycle can be described as dβ/dτ = S − Aβc , dA/dτ = γ0 (β/βc − 1) A . ✷ This system of equations can be obtained from our NL fishbone equations. • the first one from βEr (r; ω0 (τ )) → β equation • the second one from the fishbone amplitude evolution equation Here, τ is a normalized time, A = |δξr0 |/rs is the normalized fishbone amplitude and γ0 is a measure of the linear growth rate. Furthermore, ∂t−1 ∼ τN L ∼ rs /(|ω0 ||δξr0 |) in the βEr (r; ω0 (τ )) evolution equation; and ∂r2 ∼ −1/rs2 . [C&Z RMP16] Fulvio Zonca and Liu Chen The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 23 ✷ The solution of the predator-prey model is cyclic; i.e., it can be generally written as F (A, β) = const, where F (A, β) has a maximum at the fixed point position β = βc , A = S/βc . ✷ A crucial feature of the model is the linear dependence on A of the loss term in the β evolution equation. ⇒ Manifestation of secular resonant EP losses by mode particle pumping [White et al 1983]. ✷ It illuminates the role of sources and collisions as well as their link to the fluctuation strength: S ↔ β̇ErS − νext βEr Z 2 X γ0 r π 2 τb ω̄d S(t) , β̇ErS ≡ 2 2 m |Ω| 2 EdEdλ 2 2 B0 q (ω̄d − ωr0 ) + γ0 vk /|vk |=±1 2 π r νext βEr ≡ −2 2 m |Ω| 2 B0 q Z EdEdλ X vk /|vk |=±1 Fulvio Zonca and Liu Chen τb ω̄d2 γ0 StF0 (t) . 2 2 (ω̄d − ωr0 ) + γ0 The Trilateral International Workshop on Energetic Particle Physics The fishbone burst cycle – 24 Conclusions ✷ Nonlinear theory of fishbone excitation by EP is at hand [C&Z RMP16] and supports the qualitative model for fishbone cycle proposed originally [Chen et al 1984]. ✷ Present nonlinear theory can be extended to include generation of zonal structures; i.e., wave-wave and wave-particle interactions on the same footing. Work is in progress [L. Chen, Z. Qiu and F. Zonca]. ✷ Interesting mutual and positive feedbacks of theory development with numerical simulation results. Fulvio Zonca and Liu Chen