Wave-particle interactions and nonlinear dynamics in magnetized plasmas

Transcription

1 Mathematics Department Colloquium, Bologna Wave-particle interactions and nonlinear dynamics in magnetized plasmas 1,2 1 ENEA, C.R. Frascati, C.P Frascati, Italy. 2 Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou , P.R.C. June 16.th, 2015, Mathematics Department Colloquium, Bologna University Acknowledgment: L. Chen 2,S.Briguglio 1,G.Fogaccia 1,A.V.Milovanov 1,Z.Qiu 2,G.Vlad 1 Consortium EUROfusion ER Project ER15-ENEA-03

2 Mathematics Department Colloquium, Bologna Outline (I) General aspects: the importance of toroidal geometry Mode structures and nonlinear interactions in tokamaks Quasi-periodic fluctuations Connection with applied mathematics problems Nonlinear energetic particle (EP) dynamics and transport (II) EP-Shear Alfvén Wave (SAW) interactions: unified theoretical framework SAW fluctuation spectrum excited by EPs in tokamaks Non-perturbative EP-wave interactions (III) Nonlinear Physics: wave-wave and wave-particle nonlinear interactions Governing equations: NL Schrödinger equation (III.1) Wave-wave nonlinear interactions (III.2) Wave-partice nonlinear interactions and phase-space transport Non-perturbative EP dynamics and phase locking Solution of Schwinger-Dyson Equation with strong instability EP Modes: convectively amplified solitons in active media (IV) Conclusions and Discussion

3 Mathematics Department Colloquium, Bologna Nuclear fusion in a donut ITER: International Thermonuclear Experimental Reactor under construction in Cadarache, France.

4 Mathematics Department Colloquium, Bologna not only physics and science

5 Mathematics Department Colloquium, Bologna (I) General aspects: the importance of toroidal geometry Toroidal geometry must be accounted for, since it crucially modifies Particle motions and transport processes Linear and nonlinear mode structures Plasma response: effective inertia, dispersiveness, etc.

6 Mathematics Department Colloquium, Bologna (I) General aspects: the importance of toroidal geometry The tokamak configuration Plasma current Magnetic field lines are twisting on a given magnetic flux surface Changing B 0 pitch: coupling of and structures ψ =poloidalmagneticflux B 0 = F (ψ) φ + φ ψ ζ ψ ζ = φ qθ q B 0 ζ B 0 θ = q(ψ)

7 Mathematics Department Colloquium, Bologna Mode structures and nonlinear interactions in tokamaks Courtesy of Y. Xiao et al., POP 22, (2015) Filaments Quasi-particles [Zonca et al, PPCF15] Representation based on the Poisson Summation Formula [Z.X. Lu et al., POP12] m eimθ =2π m δ(θ 2πm).

8 Mathematics Department Colloquium, Bologna Generic fluctuation δφ(r, θ, ζ) = m,n exp(inζ imθ)δφ m,n(r) can be decomposed as δφ(r, θ, ζ) =2π e inζ inq(θ 2πl) δ ˆφ n (r, θ 2πl) = e inζ imθ l,n Z e i(m nq)ϑ δ ˆφ n (r, ϑ)dϑ = m,n Z e inζ imθ m,n Z e i(m nq)ϑ P Bn (r, ϑ)[δφ]dϑ. Radial envelope (varying on meso-scales) and parallel mode structures (quasi-particles) δ ˆφ n (r, ϑ) =A n (r)δ ˆφ 0n (r, ϑ) A n (r)δ ˆφ 0n (ϑ). Reduces to well-known ballooning formalism when separationofradialscalelength applies L L A nq 1 L p /n [Z.X. Lu et al., POP12] Eikonal Ansatz: A n (r) exp i r nq (x)θ k (x)dx. k r nq θ k

9 Mathematics Department Colloquium, Bologna Mode structures can be represented by three degrees of freedom: the toroidal mode number n, theradialenvelopea n (r) (withscalelengthl A ); and the parallel (to B 0 )modestructureδ ˆφ 0n (r, ϑ), with only a slow radial variation on the equilibrium scale length L. Correspondingly, nonlinear interactions can take the following three forms: mode coupling between two ns, modulation of the radial envelope; and distortion of the parallel mode structure [L. Chen et al., PPCF05].

10 Mathematics Department Colloquium, Bologna Typical space time scales of low frequency plasma waves in tokamaks Consider a magnetized plasma with a sheared magnetic field: 2D tokamak equilibrium Magnetic shear k = k (ψ p ); ψ p magnetic flux (SAW continuum). In order to minimize kinetic damping mechanisms, compression and field line bending effects λ L, withl the system size Perpendicular wavelength λ L p /n can be significantly shorter than the characteristic scale length of the equilibrium profile L p for sufficiently high mode number n. Using the ordering k /k 1 and k L p 1, the2dproblemofplasma wave propagation can be cast into the form of two nested 1D waveequations: parallel mode structure radial wave envelope.

11 Mathematics Department Colloquium, Bologna Quasi-periodic fluctuations: solution strategy Awideclassofinterestingandpracticallyrelevantfluctuations, including SAW, are characterized by quasi-periodic fluctuation spectrum γ ω : growth rate is much smaller than fluctuation frequency. Parallel mode structures form on rapid time scale ω 1 : essentially fixed! but not necessarily linear. Nonlinear time scale is ideally ordered as τ NL γ 1.Remainingnonlinear interactions are coupling between different n s and modulation of the radial envelope and can be understood as interactions between particles and quasiparticles (filaments). Explicitly isolate nonlinear interactions, tobecalculatedforwavepackets constituted of quasi-particles. Straightforward for the unstable spectrum. Butissuesarisewiththestable spectrum.

12 Mathematics Department Colloquium, Bologna D PDE L( t, r, θ ; r, θ)δφ n = n [NL] n,n {}}{ symmetric 1D ODE L( t, ϑ,θ k ; r, ϑ)a n (r, t)δ ˆφ 0n (ϑ, r, t) = n [NL] n,n {}}{ symmetric }{{} δ ˆφ 0n (ϑ, r, t) L( t, ϑ,θ k ; r, ϑ)a n (r, t)δ ˆφ 0n (ϑ, r, t)dϑ = n [NL] n,n D n ( t,θ k ; r)a n (r, t) = n [NL] n,n

13 Mathematics Department Colloquium, Bologna Connection with applied math problems and computational methods: Multi spatiotemporal scale analysis a la Maslov Micro-local analysis and pseudo-differential operators (extended to nonlinear problems) Viable and advantageous numerical approach for practical calculations Systematic (higher-order) approaches for long time-scale analyses On long time scales and/or addressing nonlinear physics: role of stable spectrum issues arising [Z.X. Lu et al., POP 2013].

14 Mathematics Department Colloquium, Bologna NL Dynamics and fluctuation induced transport Description of resonant wave-particle interaction as particles interacting with quasi-particles. Quasi-particles carry energy and momentum. But unlike particles, quasiparticles are not conserved in number: occupation number A n (r, t). Fluctuation induced transport due to emission and re-absorption of toroidal symmetry breaking perturbations [Zonca et al. PPCF 2015]. Characteristic δ ˆφ 0n (r, ϑ) width L p. radial However, characteristic radial width of filaments nq 1 L p /n due to magnetic shear. Transport may become non-local when r 2 r 1 > nq 1 L p /n.

15 Mathematics Department Colloquium, Bologna NL Dynamics and action at a distance Collisionless plasmas are characterized by non-local interactions which mediate transport processes [Zonca et al. PPCF 2015].

16 Mathematics Department Colloquium, Bologna The importance of zonal structures Zonal structures (ZS) coherent micro/meso-scale radial corrugations of equilibrium in toroidal device plasmas. Zonal structures spontaneously excited by micro/meso-scale turbulence due to plasma instabilities; includingshear Alfvén waves (SAW) this talk. Zonal structures scatter instability turbulence to shorter-radial wavelength stable domain nonlinearly damp the instability Self-regulation of plasma instabilities! Nonlinear interaction by modulation of the radial envelope A n (r, t). Generation of quasi-particle multiplets δ ˆΦ n. More generally: phase-space zonal structures.

17 Mathematics Department Colloquium, Bologna Dyson Equation: single-n coherent nonlinear interaction Dyson Equation describes fluctuation induced transport in the presence of asingle-n quasi-particle Instability in strongly driven system. Non-perturbative interplay of SAW with Energetic Particles (EP). Energetic Particle Modes (EPM). Example later.

18 Mathematics Department Colloquium, Bologna Fishbone Paradigm for SAW-EP nonlinear interplay Consider ω ω d ω b 2integralsofmotion: µ and J = v dl. The system behaves as non-autonomous, non-uniform system with one degree of freedom. Reminiscenceof3Dequilibriumsystem. Crucial difference with the beam plasma system: non-autonomous, uniform system with one degree of freedom. Fishbone and bump-on-tail paradigms: unified theoretical framework (based on Dyson Equation). [Zonca et al. NJP 2015]

19 Mathematics Department Colloquium, Bologna Observation of fishbone oscillations Experimental observation of fishbones in PDX [McGuire et al. 83] with macroscopic losses of injected fast ions...

20 Mathematics Department Colloquium, Bologna Followed by numerical simulation of the mode-particle pumping (secular) loss mechanism [White et al 83]... ω ω 2 B Non-adiabatic!... 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]

21 Mathematics Department Colloquium, Bologna Outline (I) General aspects: the importance of toroidal geometry Mode structures and nonlinear interactions in tokamaks Quasi-periodic fluctuations Connection with applied mathematics problems Nonlinear energetic particle (EP) dynamics and transport (II) EP-Shear Alfvén Wave (SAW) interactions: unified theoretical framework SAW fluctuation spectrum excited by EPs in tokamaks Non-perturbative EP-wave interactions (III) Nonlinear Physics: wave-wave and wave-particle nonlinear interactions Governing equations: NL Schrödinger equation (III.1) Wave-wave nonlinear interactions (III.2) Wave-partice nonlinear interactions and phase-space transport Non-perturbative EP dynamics and phase locking Solution of Schwinger-Dyson Equation with strong instability EP Modes: convectively amplified solitons in active media (IV) Conclusions and Discussion

22 Mathematics Department Colloquium, Bologna (II) EP-SAW interactions in tokamaks: unified theoretical framework In tokamaks, SAW propagate in a periodic potential (Mathieu/Floquet). Gaps are formed in the SAW continuous spectrum when Bragg reflection condition is met (k = ωqr 0 /v A ): 2L = lλ, λ (2π/k), l =1, 2, 3,... B R 0 /R 1 (r/r 0 )cosθ There exist a unified theoretical framework for the description of linear and nonlinear dynamics of SAW excited by EPs in tokamaks: the General Fishbone Like Dispersion Relation (GFLDR) [Z&C POP (Aug 2014); RMP (submitted 2014)].

23 Mathematics Department Colloquium, Bologna Zoology of Shear Alfvén Waves excited by EP [RMP submitted 2014] } EPM Courtesy of W.W. Heidbrink POP 9, 2113,(2002)

24 Mathematics Department Colloquium, Bologna General Fishbone Like Dispersion Relation Kinetic Energy Principle General Fishbone Like Dispersion Relation (GFLDR) iλ(ω) =δw f + δw k (ω) Λ(ω): inertia (kinetic energy) due to background plasma Structures of continuum, gaps,andresonantabsorption δw f : δw (potentialenergy)due to background plasmas existence of discrete AEs (different types; depending on equilibrium) δw k : δw (activepotentialenergy)duetoeps instability mechanisms & new unstable modes: EPMs [Chen 1994] Simple limit [Chen, White, Rosenbluth 1984] Λ(ω) =ω/ω A (ω A = V A /qr 0 ), δw f 0 E ωd F EP δw k F EP (r, E): EP distribution function ω d ω r v

25 Mathematics Department Colloquium, Bologna The fishbone-like dispersion relation demonstrates the existence of two types of modes (note: Λ 2 = k 2 q2 R 2 0 is SAW continuum): adiscretegapmode,oralfvén Eigenmode (AE), forireλ 2 < 0; an Energetic Particle continuum Mode (EPM) for IReΛ 2 > 0. For AE non-resonant fast ion response provides a real frequency shift, i.e. it removes the degeneracy with the continuum accumulation point (Λ = 0); resonant wave-particle interaction gives the mode drive. δw f +IReδW k > 0: AEfrequencyis above the accumulation point δw f +IReδW k < 0: AEfrequencyis below the accumulation point For EPM iλ term represents continuum damping [Chen et al 1984, Chen 1994] IReδW k (ω r )+δw f =0 EP determine ω r γ/ω r =( ω r ωr IReδW k ) 1 (IImδW k Λ) determines γ/ω r

26 Mathematics Department Colloquium, Bologna Non-perturbative EPs: EPM radial localization Expression for δ W f ( s = rq /q < 1, α = 8πR 0 q 2 P /B0 2 < 1limit; x nq (r r 0 )) δ W f = s π ) (1+2κ(s) ααc + s π 2 κ(s) 8 8 x, 2 α c = s2 ; and κ(s) 1 ( 1+ 1 ) e 1/ s. 1+ s 2 s Governing equations for EPM structure: Λ = (1/2)[(ω 2 ω 2 l )/(ω2 u ω 2 )] 1/2 ; ω l,u =lower/upperlimitsofthesawcontinuum Simple model for fusion α particles: isotropic slowing down distribution function; P 0E =radialpressureprofile;e F =birthenergy;e c =critical energy F 0 = 3P 0E H(E F /m E E) 4πE F (2E) 3/2 +(2E c /m E ) 3/2

27 Mathematics Department Colloquium, Bologna Solve the linearized dispersion relation for EPM (α E = 8πR 0 q 2 P 0E /B2 0) α E = α E0 exp [ x 2 /(s 2 kϑl 2 2 pe) ] ( ) α E0 1 x2 /s 2 kϑ 2L2 pe iλa = s π 8 (1+2κ(s) ααc ) + 3π(r/R 0) 1/2 8 2 s α E0 ( A + s π A 8 κ(s) 2 x 2 1 x2 /s 2 k 2 ϑ L2 pe ) { 1+ ω [ ( ωdf ) ln ω df ω 1 + iπ] } A, Linear EPM radial localization is determined by the EP resonant drive: Important consequences on NL dynamics (phase locking: examplelater).

28 Mathematics Department Colloquium, Bologna Outline (I) General aspects: the importance of toroidal geometry Mode structures and nonlinear interactions in tokamaks Quasi-periodic fluctuations Connection with applied mathematics problems Nonlinear energetic particle (EP) dynamics and transport (II) EP-Shear Alfvén Wave (SAW) interactions: unified theoretical framework SAW fluctuation spectrum excited by EPs in tokamaks Non-perturbative EP-wave interactions (III) Nonlinear Physics: wave-wave and wave-particle nonlinear interactions Governing equations: NL Schrödinger equation (III.1) Wave-wave nonlinear interactions (III.2) Wave-partice nonlinear interactions and phase-space transport Non-perturbative EP dynamics and phase locking Solution of Schwinger-Dyson Equation with strong instability EP Modes: convectively amplified solitons in active media (IV) Conclusions and Discussion

29 Mathematics Department Colloquium, Bologna (III) Nonlinear Physics SAW turbulence spectrum and spectral transfers Proper assessments of heating/acceleration and transports Broad scope of ongoing research activities Formal solutions of nonlinear GFLDR: D = iλ L (δw f + δw k ) L [Zonca NF05, C&Z RMP submitted 2014] D = iλ NL + (δw f + δw k ) NL (III.1) Nonlinear Wave-Wave Interactions and spectral transfers: Λ NL. Understand and describe nonlinear processes in terms of breaking of the Alfvénic state cancellation of Reynolds and Maxwell stresses (III.2) Nonlinear Wave-EP Interactions and transports: δw NL k. Nonlinear dynamics of structures in the EP phase space phasespace zonal structures secular nonadiabatic resonant particle transport (τ TRANSP τ NL )onmeso-andmacro-scales

30 Mathematics Department Colloquium, Bologna Governing Equations: NL Schrödinger Equation D L n ω 0 Adopt the theoretical framework of the general fishbone like dispersion relation (GFLDR) [C&Z 06,07,14] and (envelope) wave-packet propagation decomposition in toroidal geometry (reviewed by [Z.X. Lu et al. 12]). ( i t i θ k0 nq r ) A n0 (r, t)+ DL n θ k0 ] A n0 (r, t) = ( δ ( i ) nq r θ k0 W NL f + δ A n0 (r, t)+ 1 2 ) NL W k n iλnl n 2 D L n θ 2 k0 [ ( i nq r θ k0 ) 2 + S ext n (r, t) k r nq θ k0 NLSE with integro-differential NL terms Slow spatiotemporal evolution. According to the GFLDR: Dn L = iλ L n ( δ W f + δ W ) L k n Λ n :generalizedinertia response δ W fn :potentialenergyduetofluid plasma response δ W kn :potentialenergyduetokinetic plasma response Fastest time-scale determines parallel mode structure consistent with Dn L

31 Mathematics Department Colloquium, Bologna Outline (I) Multi-scale dynamics in fusion plasmas: the role of energetic particles (EPs) Mode structures and nonlinear interactions in tokamaks Nonlinear dynamics and fluctuation induced transport The importance of zonal structures The fishbone paradigm for NL wave-particle interplay (II) EP-Shear Alfvén Wave (SAW) interactions: unified theoretical framework SAW fluctuation spectrum excited by EPs in tokamaks (III) Nonlinear Physics: wave-wave and wave-particle nonlinear interactions Governing equations: NL Schrödinger equation (III.1) Wave-wave nonlinear interactions Transport by mode conversion and parametric decay Zonal structures by Toroidal Alfvén Eigenmodes (III.2) Wave-partice nonlinear interactions and phase-space transport Non-perturbative EP dynamics and phase locking EP Modes: convectively amplified solitons in active media (IV) Conclusions and Discussion

32 Mathematics Department Colloquium, Bologna (III.2) Phase-space dynamics and transport Focus on one isolated resonance: Universal description of a nonlinear resonance [Chirikov 79] Non-autonomous system with 1 degree of freedom Bump-on-tail (uniform 1D beam-plasma) paradigm Source&Coll. [Berk&Breizman 90 11; Review by Breizman&Sharapov PPCF11] Applies near marginal stability Crucial importance of equilibrium geometry and nonuniformity [Zonca et al. 05, PPCF15; C&Z RMP submitted 2014]: Non-autonomous system with 1 degree of freedom in 3D Crucial roles of mode structures when r NL < λ Non-perturbative EP dynamics and phase-locking r NL > λ Interplay of mode structure and EP transport Avalanches Time dependent non-local behaviors from meso- to macro-scales

33 Mathematics Department Colloquium, Bologna Non-perturbative EPs and phase locking For non-perturbative EPs, EP transport reflects mode structure change frequency shift, consistentwithchangeind n (GFLDR). For increasing EP source strength, effects are visible on both AEs and EPMs [C&Z 95,96: Briguglio et al. 95,98]; recent Exp/Sim comparison in DIII-D [Wang et al. 13] Above excitation threshold: EPM has characteristic frequency of EPs and maximizes wave-ep power exchange Phase locking: EPMfrequencyadaptstolocalresonanceconditionto maximize wave-ep power exchange non-adiabatic evol. ω ω 2 B Similar to autoresonance [Meerson&Friedland 90; Fajans&Friedland 01] but with new twists: non-perturbative non-adiabatic. Similar to superradiance in FEL [Bonifacio et al. 90,94]: EPs amplify EPM over a finite cooperation length as they are convected outward ( A 2 ( r) 2 ) These behaviors are deeply connected.

34 Mathematics Department Colloquium, Bologna Autoresonance Autoresonance [Meerson PRA90] occurs when a nonlinear pendulum, driven to large amplitude, evolves in time to instantaneously match the nonlinear pendulum frequency with that of an external drive with sufficiently slow downward frequency sweeping [Fajans AJP01]. Autoresonance is a common phenomenon: it was first observed in particle accelerators, and has since been noted in atomic physics, fluid dynamics, plasmas, nonlinear waves, and planetary dynamics. Adiabatic Slow please!

35 Mathematics Department Colloquium, Bologna Superradiance Superradiance: For want of a better term, a gas which is radiating strongly because of coherence will be called superradiant [Dicke PR54] Superradiance: whenagroupofn emitters, such as excited atoms, interact with a common light field and the wavelength of the light is much greater than the separation of the emitters, thentheemittersinteractwiththelight in a collective and coherent fashion. Single atom: Coherent state: high intensity pulse N 2 Important difference with stimulated emission. Dependence on system size and geometry. E.g.: system cannot be too large to avoid reabsorption.

36 Mathematics Department Colloquium, Bologna Nonlinear Dynamics of a single-n coherent EPM NL dynamics of a single-n coherent EPM: excited by precessional resonance in the small orbit width limit (ω k = ω k (τ) = ω dk )byfusion-α s born at E F. Summation of the Dyson series for nearly periodic fluctuations (integral) with ω γ ω and dominant spatial gradients... (ω k (τ) =ω 0 (τ)+iγ(τ))... yields the following expression for the Laplace-transformed F 0 (t) ˆF 0 (ω) = i ω St ˆF 0 (ω)+ i ω Ŝ(ω) + i {[ 2πω F nc nc 0 (0) ω(dψ/dr) r ωk (τ)(dψ/dr) r ˆF 0 (ω 2iγ(τ)) ω ω k (τ)+ ω dk + nc ω k (τ)(dψ/dr) r ˆF0 (ω 2iγ(τ)) ω + ω k (τ) ω dk ] ˆω dk δ φ k0 (r, τ) 2 }.

37 Mathematics Department Colloquium, Bologna The effect of phase locking Neglect sources and collisions, and move to t-representation ( ) [( t F 0(t) 2kϑv 2 Eρ 2 2 n ωdn + (γ iω)e iωt ˆF ) 0 (ω) Ān0(r, LE dω t) ] 2 r (n ω dn ω 0 ) 2 +(γ iω) 2 r ω 0 Not a diffusion equation: Integro-differential equation. The NL radial envelope Ā n0 (r, t) self-consistentlyevolveswithf 0 (E,r,t). Phase locking may change structure and features of the nonlinear equation. (dominant radial transport at E const.) Ā n0 (r, t) ω 0 ṙ r n ω dn Integral kernel (γ iω) 1

38 Mathematics Department Colloquium, Bologna δw nk + δw nk = 3π(r/R 0) 1/2 8 2 s ω + ω 0 (τ)+iγ(τ) n ω dn ω 0 (τ) iγ(τ) ω e iωt r ˆF 0 (ω)dω v [{ linear dispersiveness }}{ α E 1+ ω ( ωdf ) ln ω df ω 1 + iπ ω +iπ ω k ω θρ 2 2 T E 1 E df m E α E A 2 1 t A 2 2 r 1 t ω df ( αe A 2)] α E = 8πR 0 q 2 dp E /dr s = r(dq/dr) m.shear ω df = ω dk (E F ) The operator 1 t must be interpreted in symbolic sense. Note that the NL time scale, τ NL t 1 A 1 (mode particle pumping [White PF83; Chen PRL84]) secular transport. Similar to superradiance in FEL [Bonifacio et al. 90,94]: EPs amplify EPM over a finite cooperation length as they are convected outward ( A 2 ( r) 2 )

39 Mathematics Department Colloquium, Bologna EPM: conv. amplified solitons in active media Nonlinear dynamics of phase-space structures that evolve non-adiabatically ( ω ωb 2 )onthesame time scale of the corresponding fluctuations Phase-space zonal structures [Zonca et al. 13,14; NJP15] NLSE for EPM envelope: look for solutions that can be expressed as the convectively amplified propagating (self-similar) wave-packet A(ξ,t) =U(ξ)e t γ(t )dt W (ξ)e iϕ(ξ)+ t 0 γ(t )dt, ξ k n0 (r r 0 ) k n0 (r r 0 (0) t 0 v g (t )dt ) with k n0 denoting the NL wave-vector and v g the NL group velocity. Solution is obtained by balancing the nonlinear term with the linear dispersiveness.,

40 Mathematics Department Colloquium, Bologna This optimal balance fixes k n0 v g but not the nonlinear group velocity One-parameter family (λ g < 1) of EPM wave packets ṙ 2 0 = v 2 g = λ 2 gv 2 E B ; v E B =(E B) r velocity ω EPM =Ω(r 0,λ g ;...) λ g k 2 n0 = κ 2 k 2 ϑ/λ 2 g ; κ = O(1) const. : ImΩ/ λ g =0 (Max wave EP power exchange) while U(ξ) =W (ξ)e iϕ(ξ) satisfies the nonlinear Schrödinger equation 2 ξ U = λ 0 U 2iU U 2. The value λ i1.33 corresponds to the ground state of the corresponding complex nonlinear oscillator.

41 Mathematics Department Colloquium, Bologna (IV) Conclusions and Discussion Wave-particle interactions and nonlinear dynamics in magnetized plasmas: Collective behaviors and self-consistent interplay in wave-particle interactions Framework for posing applied mathematics problems with new twists Plasma instabilities and nonuniform equilibria in complex geometries: Reduced phase-space dynamics description: non-autonomous nonuniform system with one degree of freedom Novel aspects: Schwinger-Dyson Equation with self-consistent field structures (SDE+NLSE) Uniform plasma limit: Beam-plasma system and BGK modes (connection with nonlinear Landau-damping [C. Villani 2010])

42 Mathematics Department Colloquium, Bologna Low frequency fluctuations: in general non-autonomous non-uniform system with two degrees of freedom: What is Arnold diffusion? How can we compute/measure it? Extensions to broad band (strong) turbulence: γ/ω 1 Challenge: combine solid understanding of single processes to investigate long time-scales behaviors for predicting, e.g., ITERperformance Practical impact: importantrolesoftheoryandsimulationinadvancing fusion science Broader implication: cross-fertilization with, e.g., space physics and astrophysics, condensed matter and accelerator/laser physics, applied mathematics.

43 Mathematics Department Colloquium, Bologna References [1] Chen L and Zonca F 2014 Rev. Mod. Phys. submitted [2] Xiao Y, Holod I, Wang Z, Lin Z and Zhang T 2015 Phys. Plasmas [3] Zonca F, Chen L, Briguglio S, Fogaccia G, Milovanov A V, Qiu Z,VladG and Wang X 2015 Plasma Phys. Control. Fusion [4] Lu Z X, Zonca F and Cardinali A 2012 Phys. Plasmas [5] Chen L, Zonca F and Lin Z 2005 Plasma Phys. Control. Fusion 47 B71 [6] Lu Z X, Zonca F and Cardinali A 2013 Phys. Plasmas [7] Zonca F, Chen L, Briguglio S, Fogaccia G, Vlad G and Wang X 2015 New J. Phys [8] McGuire K, Goldston R and Bell et al M 1983 Phys. Rev. Lett

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