Energetic particle and Alfvén wave physics in fusion plasmas: a brief review

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 1 Energetic particle and Alfvén wave physics in fusion plasmas: a brief review Fulvio Zonca http://www.afs.enea.it/zonca Associazione Euratom-ENEA sulla Fusione, C.R. Frascati, C.P. 65-00044 - Frascati, Italy. Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, P.R.C. January 13.th, 2012 Symposium on Plasma Theory A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 12 13 January 2012, Beckman Center, University of California, Irvine

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 2 Why Energetic particles and Alfvén Waves (I) Possible detrimental effect of shear Alfvén instabilities on energetic ions in fusion plasmas was recognized theoretically before experimental evidence was clear SAW have group velocity v A B and of the same order of EP characteristic speed, v E v A Mikhailovskii 75 and Rosenbluth and Rutherford 75 conjecture the SAW excitation by resonant wave-particle interaction with MeV ions Main interest in the 60 s focused on the (electron) beam plasma system: O Neil, Malmberg, Mazitov, Shapiro, Ichimaru...

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 3 The beam-plasma system vs. EP-SAW interactions in tokamaks Similarities can be drawn but strong differences and peculiarities emerge depending on the strength of the drive: Advantages of using a simple 1-D system for complex dynamics studies Roles of mode structures, non-uniformity and geometry in determining nonlinear behaviors (Liu Chen s seminal contributions)

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 4 Why Energetic particles and Alfvén Waves (II) Experimental observation of fishbones in PDX [McGuire et al. 83] with macroscopic losses of injected fast ions...

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 5 Followed by numerical simulation of the mode-particle pumping (secular) loss mechanism [White et al 83]... ω ω 2 B... 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]

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 6 Conceptual breakthrough Roles of the continuous SAW spectrum and self-consistent interplay of energetic particle transport and mode nonlinear dynamics. Linear physics: in the years where the stabilizing effects of supra-thermal particles on MHD continuum modes was being investigated[berk 76, Coppi- Pegoraro 79, Hastie-Hesketh 81, Tsang-Sigmar 81] resonant excitation of modes belonging to the MHD continuous spectrum due to the interplay of wave-particle resonant interactions and the plasma non-uniformity Nonlinear physics: introduced the notion that strong (ballistic) transport may significantly alter plasma dielectric response and produce a completely new nonlinear dynamic behavior. This conceptual breakthrough still maintains its innovative strength and influences many of the ongoing research: complexity in burning plasmas

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 7 Alfvén waves in non-uniform systems Alfvén wave heating [Grossman and Tataronis 73] by phase mixing of the continuous spectrum [Chen and Hasegawa 74] and mode conversion to kinetic Alfvén waves [Hasegawa and Chen 75, 76] are good examples of complex behaviors and cross-scale couplings in magnetized plasmas. The shear Alfvén wave spectrum is continuous, due to spatial nonuniformities, and oscillations are characterized by a quasi-exponential behavior because of phase mixing [Barston 64, Grad 69]. δφ(x,t) t 1 exp{ iω A (x)t} ω 2 A(x) = k 2 (x)v 2 A(x) The radial( ψ) component of the plasma displacement(and δb) is strongly phase-mixed while the bi-normal (b ψ) component does not fade away (the compressional component is small for SAW).

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 8 Evidence of shear Alfvén continuous spectrum and phase-mixing from AMPTE CCE magnetic field data in the Earth s magnetosphere [Engebretson et al. 87]

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 9 Singular structures are formed at the resonant layer with the shear Alfvén continuous spectrum, intrinsically producing shorter and shorter scales, where kinetic and nonlinear physics become important The rate of resonant energy absorption, computed as ratio between Poynting flux into the resonance and wave energy density, is proportional to the wave intensity and independent of the detailed dissipation mechanism for transferring energy to the plasma as heat [Chen and Hasegawa 74]. The counterpart on the Alfvén wave is continuum damping. At the resonance layer, mode conversion to kinetic Alfvén waves [Hasegawa and Chen 76].

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 10 Parametric decays of kinetic Alfvén waves KAW parametric decay into daughter KAW and ion-sound wave, where cross-section is maximized when k 0 of pump and k 1 of daughter KAW are k 0 k 1 [Hasegawa and Chen 76]. This is contrary to ideal MHD case where the maximum decay rate is obtained for k 0 k 1 [Sagdeev 69]. Kinetic processes dominate for k 2 ρ2 i > ω/ω ci = O(10 2 ) and tend to make KAW k spectrum isotropic. For resonant excitation of Alfvén waves ( k r r k ), this has important consequences on radial transport. Similarly, KAW may decay into zonal flows and geodesic acoustic modes [Chen et al 10,11]. It is crucial to account for k ρ i 1 nonlinear gyrokinetic analyses [Frieman and Chen 82]. Energetic particles play peculiar role: have macro-scale profile variations and drive Alfvénic modes on meso-scales T E T e,t i : in nonuniform plasmas they mediate micro- to meso- to macro-scale dynamics [Chen et al. 07 11].

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 11 Alfvén waves in toroidal systems In two dimensional (doubly periodic) systems, the shear Alfvén continuous spectrum is modified[pogutse and Yurchenko 78, D Ippolito and Goedbloed 80] and broken by gaps [Kieras and Tataronis 82], due to lattice symmetry breaking for shear Alfvén wave moving along field lines. Inside the frequency gaps, at ω (1/2qR)lv A for equilibrium variations cos lθ, localized (in frequency and space) modes can exist: in the toroidicity induced gap (l = 1) one has Toroidal Alfvén Eigenmodes (TAE) [Cheng, Chen, Chance 85], essentially unaffected by continuum damping. This TAE paper [Cheng, Chen, Chance 85] has become the paradigm for all studies of various types of Alfvén Eigenmode excitations by energetic particles and related transports: first observation [Heidbrink 91, Wong 91]. It also discusses the continuos spectrum discretization by resistivity effects, later on extended to finite ion Larmor radius effects [Mett and Mahajan 82]: kinetic TAE as a counterpart of KAW.

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 12 The excitation of the shear Alfvén continuous spectrum by energetic particles at the characteristic frequencies of their motion was demonstrated by [Chen 94]: energetic particle modes (EPM) as a counterpart of fishbones. The Alfvén Wave Zoology [Heidbrink 02] } EPM Various non-uniformity effects allow different varieties of the same SAW species to exist [Chen and Zonca 07]. Unique and general theoretical framework for explanation of this variety and interpretation of observations: the general fishbone-like dispersion relation [Zonca and Chen 06,07].

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 13 Generalized fishbone-like dispersion relation In general, demonstrate that the mode dispersion relation can be always written in the form of a fishbone-like dispersion relation iλ+δw f +δw k = 0, where δw f and δw k play the role of fluid (core plasma) and kinetic (fast ion) contribution to the potential energy, while Λ represents a generalized inertia term. The generalized fishbone-like dispersion relation can be derived by asymptotic matching the regular (ideal MHD) mode structure with the general (known) form of the SA wave field in the singular (inertial) region, as the spatial location of the shear Alfvén resonance, ω 2 = k 2 v2 A, is approached. Examples are : Λ 2 = ω(ω ω pi )/ω 2 A for k qr 0 1 and Λ 2 = (ω 2 l ω 2 )/(ω 2 u ω 2 ) for k qr 0 1/2, with ω l and ω u the lower and upper accumulation points of the shear Alfvén continuous spectrum toroidal gap [Chen 94].

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 14 δw f is generally real, whereas δw k is characterized by complex values, the real part accounting for non-resonant and the imaginary part for resonant wave particle interactions with energetic ions. 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; see later): a discrete gap mode, or Alfvén Eigenmode (AE), for IReΛ 2 < 0; an Energetic Particle continuum Mode (EPM) for IReΛ 2 > 0. For both AE and EPM, the SAW accumulation point is the natural gateway through which modes are born at marginal stability [Chen and Zonca 07] Generally, global mode structures reflect nonlocal coupling to the shear Alfvén continuous spectrum [Zonca and Chen 92; Rosenbluth et al 92] and importantly depend on plasma non-uniformity and toroidal geometry.

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 15 Micro- and meso- scale excitation of low-frequency AE/EPM Λ 2 = 1 ω 2 A Excitation of low-frequency AE by thermal ions is important since it shows the cross-scale coupling physics mentioned earlier, mediated by the fast ions: AITG excitation mechanism (simplified form of Λ 2 ) [ ω 2 ( 7 4 + T ) e q 2 ωti ]+i 2 πq 2 e ω2 /ω2ti ω2 T i ωa 2 ( ωti ω ω )( Ti ω 2 ω ti ωti 2 + T ) 2 e. T i When ωω Ti > ω 2 ti accumulation point becomes unstable. When ωω Ti > ω 2 ti and equilibrium effects localize the AE, the Alfvénic ITG mode is excited (AITG) [Zonca and Chen 98, 99]. Recent numerical results emphasize thermal ion kinetic effects for kinetic BAE excitation by energetic particles ([Wang et al 10,11; Zhang et al 11].

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 16 R. Nazikian, et al. 06, PRL 96, 105006 R. Nazikian, et al. 06, PRL 96, 105006

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 17 Nonlinear equilibria and long time scale behaviors Complex behaviors in burning plasmas are expected to create long time scale behaviors as consequence of many interacting degrees of freedom Among all possible interactions, special roles are played by those that maintain the original equilibrium symmetry; even further, those that alter only radial profiles: zonal structures Drift wave interactions (random) with zonal structures generated by turbulence are investigated by [Diamond et al 98 05] Focus on long time-scale behavior of coherent nonlinear dynamics that are mediated by zonal structures (envelope self-modulations in space, also reflecting on velocity space structures): cross-scale couplings [Chen et al 00] Due to intrinsic coherent nature they can yield spontaneous modulational instability γ NL A

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 18 Radial Spreading of Drift Wave-Zonal Flow Turbulence via Soliton Formation [Guo and Chen 08; Guo et al 09]

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 19 Radial modulations in the fast ion profiles Spontaneous generation of zonal flows and fields [Chen et al NF01, Guzdar et al PRL01] are inefficient for sufficiently strong fast ion drive[zonca, Chen et al 00] crucial roles of plasma nonuniformity and geometry. Recent numerical simulations [Bass and Waltz 10] are consistent with this. Radial modulations in the fast ion profiles become dominant and yield fragmentation of energetic particle mode structures and modulations: modulational instability and frequency splitting.

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 20 Spontaneous hole-clump pair creation in phasespace Ifthesystemisclosetomarginalstability, such that NL particle characteristics are small compared to kr 1, the system can be assumed uniform [Berk, Breizman et al 90 99] Classic beam-plasma system[o Neil, Malmberg, Mazitov, Shapiro, Ichimaru... 60s] with addition of sources and sinks Numerical simulation results of spatially average particle distribution and wave spectrum [Berk et al 99]: hole-clump dynamics in phase space.

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 21 Nonlinear initial value problem for EPM Use the theoretical framework of the general fishbone-like dispersion relation: focus on precession resonance for trapped particles with one monochromatic EPM [Zonca and Chen 05 10] iλ(ω(t)+i t )A(r,t) = (δw f,mhd +δw f,ep +δw k,ept )A(r,t) δw k,ept = I J 2 k θ ω dk ˆF 0 (ω)/ r ω c ω dk ω(t) ω e iωt dω δw f,ep = I k θ F 0 ω c r I 2π2 e 2 Mc 2 q s R 0B 0 v / v =±1 J 2 0(λ Lk )J 2 0(λ dk ) δφ c (θ,θ k ) 2 dθ 0 EdE J 2 L dλ ω dk k 2 θ τ b J2 This problem is already in the suitable nonlinear form via ˆF 0 J 2 L J 2 J 2 0(λ Lk ) δφ c (θ,θ k ) 2 dθ

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 22 Within this approach, it is possible to systematically generate standard NL equations in the form (expand wave-packet propagation about envelope ray trajectories): drive/damping }{{} { ω 1 t γ ω ξ potential well }{{} } λ nq r +i(λ+ξ)+i θ k (nq θ k ) 2 2 r A(r,t) = NLTERMS {}}{{}}{ group vel. (de)focusing θ k solution of D R (r,ω,θ k ) = 0 and λ = ( θ 2 k 2 ) 2 D R / θk 2 ω D R / ω ; ξ = θ k( D R / θ k ) θk 2( 2 D R / θk 2) ω D R / ω ; γ = D I D R / ω Equations admit the well-known local limit, which is readily recovered.

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 23 The renormalized ˆF 0 expression The F 0 expression is obtained from the solution of the nonlinear GKE [Frieman and Chen PF82] with a source term S (collisions are added trivially; note that only trapped particles are considered for simplicity, v = 0). F 0 t = S c B 0 i r r (δk kj 0k δφ k δk k J 0 k δφ k ) Assuming an implicit summation on k ˆF 0 (ω) = i ωŝ(ω)+ i 2πω F 0(0) ck θ ωb 0 e m r ˆF 0 (ω x x )δφ k (x ) δφ k (x) J0kJ 2 0dk 2 ω dk ω dk +x ω [ ω d k δφ k (x) ω d k +x ω Q k,x x Q k,x x ˆF0 (ω x x )δφ k (x ) ] dxdx

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 24 This solution is valid for strong distortions of the equilibrium distribution function and is the analogue of the Dyson equation (G = G 0 + G 0 ΣG, with G 0 /G the bare/dressed propagators), as noted, e.g., by [Al tshul and Karpman 65]. The description does not include power spectrum generation and spatial bunching and accurate treatment of phase mixing on scales much longer than the wave-particle trapping time. Not a limitation for EPM nonlinear dynamics convective amplification of unstable front.

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 25 Monochromatic EPM: reasonable assumption, since EPM spectrum is peaked for optimizing resonance condition. δφ(t) = δ φ(τ)exp( iω(τ)t), ω(t) γ(t)ω(t). δˆφ k (ω) = iδ φ k (τ) 2π(ω ω(τ)) ˆF 0 (ω) = i i ωŝ(ω)+ ω StˆF 0 (ω)+ i 2πω F 0(0) ck θ e ωb 0 m ] ω dk Q k,ω(τ) ω +ω (τ) ω dk ω(τ) δˆφ k (ω) = r J2 0kJ 2 0dk iδ φ k (τ) 2π(ω +ω (τ)) [ ω dk ω dk +ω ω(τ) ˆF 0 (ω 2iγ(τ)) δ φ k (τ) 2 Q k,ω(τ) ω(τ) For evanescent drive and flat envelope, this problem is reduced to the waveparticle trapping and to [Berk et al PLA97, POP99] for ω B t 1. For increasing drive, EPM mode structures play a crucial role: particles are convected out efficiently since - with frequency locking - wave-particle resonances are de-correlated only in velocity space. (nqs) 1 < (γ/ω)(ω /ω) < (L ph/nqr) 1/2.

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 26 Differentspatiotemporalscalesentertheequationfortherenormalized ˆF 0 (ω) expression (in addition to the linear scales) [Chen and Zonca 05 10] the finite interaction time, describing how long a resonant particle effectively exchanges energy with the wave; it can be defined as Γ 1, i.e. the inverse nonlinear de-correlation time the finite interaction length r L, corresponding to the particle radial displacement necessary for detuning the wave-particle resonance by spatial nonuniformities The optimal condition for EPM to extract energy from particle distribution is obtained when after a π rotation in angle, action change de-correlates the wave-particle interaction Corresponds to a displacement ( r L ) the local mode width Analogy with fundamental laser and gyrotron physics problems

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 27 Zonca et al. IAEA, (2002) Avalanches and NL EPM dynamics x 10-3 0.25 0.2 0.15 0.1 8, 4 9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4 φ m,n (r) t/τ A0 = 60.00 X1 t=60 x 10-2 0.25 0.2 0.15 0.1 8, 4 9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4 φ m,n (r) t/τ A0 = 75.00 X10 t=75.009.008.007.006.005.004.003 8, 4 9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4 φ m,n (r) t/τ A0 = 90.00 X30 t=90 0.05 0 0 δα H 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r/a 1 4 2 0.05 0 0 δα H 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r/a 1 4 2.002.001 0 0 δα H 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r/a 1 4 2 0-2 -4 NL distor tion of free energy SR C 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r/a 0-2 -4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r/a 0-2 -4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r/a Importance of toroidal geometry on wave-packet propagation and shape

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 28 Vlad et al. IAEA-TCM, (2003) Propagation of the unstable front r max 0.85 [d(rn H )/dr] 0.80 [d(rn H 0.060 )/dr] max 0.055 0.75 0.050 0.70 0.045 0.65 0.040 0.60 0.035 0.55 0.030 0.50 linear phase 0 50 100 150 200 250 300 t/τ A convective phase diffusive phase 0.025 linear phase 0 50 100 150 200 250 300 t/τ A convective phase diffusive phase Gradient steepening and relaxation: spreading... similar to turbulence

Symposium on Plasma Theory: A Celebration of Professor Liu Chen s 40 Years of Scientific Accomplishments 29 Conclusions Studying the energetic particle and Alfvén wave behaviors in fusion plasmas leads to unveiling subtle and fundamental physics issues and to dealing with complex burning plasma behaviors. These behaviors reflect the cross-scale couplings, non-uniformities and geometries of the plasmas under investigations, which in general cannot be reduced to simpler descriptions. The relevant spatiotemporal scales of present day experiments are significantly different with respect to those of burning plasmas of fusion interest: mutual positive feedbacks between experiment, numerical simulation and theory are needed for developing a reliable predictive capability. Liu Chen s own seminal contributions have formed a school of thought and shaped two generations of researchers with an articulated, profound and long term vision of these issues.