Structures of the low frequency Alfvén continuous spectrum and their consequences on MHD and micro-turbulence

ENEA Structures of the low frequency Alfvén continuous spectrum 1 Structures of the low frequency Alfvén continuous spectrum and their consequences on MHD and micro-turbulence Associazione Euratom-ENEA sulla Fusione, C.R. Frascati, C.P. 65-00044 - Frascati, Italy. January 17.th, 2008 Tore Supra Workshop 16-18/01/2008 CEA, Cadarache, France Acknowledgments: S. Briguglio, L. Chen, G. Fogaccia, T.S. Hahm, A.V. Milovanov, G. Vlad

ENEA Structures of the low frequency Alfvén continuous spectrum 2 Background A variety of experimental observations have recently renewed the interest in the detailed structures of the Alfvén continuum at low frequencies: finite frequency fishbone oscillations at the GAM frequency and low-frequency feature of Alfvén Cascades (JET) observation of a broad band discrete Alfvén spectrum (DIII-D with n 2 40) excited by both energetic ions (low-n) and thermal ions (high-n) excitations of BAE modes by finite amplitude magnetic islands (FTU, TEX- TOR) evidence of GAM structures (DIII-D, CHS, JFT-2M, HL-2A, AUG, T10, TEXT) All these observations well fit within the present theoretical understanding, which poses new challenging questions to be addressed by next step experiments and theories of burning plasmas. Due to the degeneracy of GAM and BAE accumulation points, these questions encompass issues that impact macroscopic MHD as well as plasma micro-turbulence in a subtle way.

ENEA Structures of the low frequency Alfvén continuous spectrum 3 Structures of the low-frequency Alfvén cont. spectrum Shear Alfvén continuous spectrum is damped by phase mixing (H. Grad, 1969, PhTo 22 34) ω 2 = k 2 v2 A. Geometry effects: symmetry breaking in torus Shear Alfvén continuum with gaps (C.E. Kieras, et al. 1982, JPP 28 395). Low-frequency Shear Alfvén Wave (SAW) gap: ω ω i ω ti ; Λ 2 (ω) = k 2 v2 A (ideal MHD) accumulation point (at ω = 0) shifted by thermal ion kinetic effects (F. Zonca, et al. 1996, PPCF 38 2011) new low-frequency gap! Diamagnetic drift: KBM (H. Biglari, et al. 1991, PRL 67 3681) Thermal ion compress.: BAE (W.W. Heidbrink, et al. 1993, PRL 71 855) T i and wave-part. resonances: AITG (F. Zonca, et al. 1999, POP 6 1917) unstable SAW accumulation point localization unstable discrete AITG mode For simmetry reasons (later): BAE GAM degeneracy (F. Zonca, et al. 2006, PPCF 48 B15); (L. Chen, et al. 2007, NF 47 S727).

ENEA Structures of the low frequency Alfvén continuous spectrum 4 Experimental observations: JET Observation of finite frequency fishbone oscillations at the GAM frequency (F. Nabais, et al. 2005, PoP 12 102509) and low-frequency feature of Alfvén Cascades (B.N. Breizman, et al. 2005, PoP 12 112506).

ENEA Structures of the low frequency Alfvén continuous spectrum 5 R. Nazikian, et al. 06, PRL 96, 105006 R. Nazikian, et al. 06, PRL 96, 105006

ENEA Structures of the low frequency Alfvén continuous spectrum 6 The same modes are excited by a large amplitude magnetic island on FTU (P. Buratti, et al. 2005, NF 45 1446; S. Annibaldi, et al. 2007, PPCF 49 475). n = -1 HF mode n=+1 n=-1, m=-2 tearing mode Locking & unlocking P. Smeulders, et al. 2002, ECA 26B, D5.016

ENEA Structures of the low frequency Alfvén continuous spectrum 7 Experimental observations of GAM: DIII-D Observation of finite frequency zonal flow oscillations (GAM) and corresponding radial structures on DIII-D by BES (G.R. McKee, et al. 2006, PPCF 48 S123).

ENEA Structures of the low frequency Alfvén continuous spectrum 8 Experimental observations of GAM: CHS Observation of finite frequency zonal flow oscillations (GAM) and corresponding radial structures on CHS by HIBP (A. Fujisawa, et al. 2007, PPCF 49 845).

ENEA Structures of the low frequency Alfvén continuous spectrum 9 Experimental observations of GAM: JFT-2M & HL-2A Observation of finite frequency zonal flow oscillations (GAM) and corresponding radial structures on JFT-2M by HIBP (T. Ido, et al. 2006, PPCF 48 S41) and HL-2A by 3-step Langmuir probes (J.Q. Dong, et al. 2007, APS Conf.).

ENEA Structures of the low frequency Alfvén continuous spectrum 10 Experimental observations of GAM: AUG, T10 & TEXT Observation of finite frequency zonal flow oscillations (GAM) in AUG by DR (G.D. Conway, et al. 2005, PPCF 47 1165), T10 (A.V. Melnikov, et al. 2006, PPCF 48 S87) and TEXT (P.M. Schoch, et al. 2003, RSI 74 1846) by HIBP.

Good agreement with f BAE/GAM prediction The frequency of the low frequency modes agrees with the GAM/BAE dispersion relation. shot Evaluation at max. mode amp. : ρ=0 Association Euratom-Cea R. Sabot MHD & Fast particles 16/01/08 TORE SUPRA

ENEA Structures of the low frequency Alfvén continuous spectrum 11 BAE GAM degeneracy Kinetic expression of the GAM dispersion relation is degenerate with that of the low frequency shear Alfvén accumulation point (BAE) in the long wavelength limit (no diamagnetic effects). This degeneracy is not accidental [Zonca&Chen PPCF 2006, IAEA 2006, NF 2007]and is due to the identical dynamics of GAM (n = m = 0) and s.a. wave near the mode rational surface (nq m) under the action of geodesic curvature, the difference between the two branches is in the mode polarization In reference to experimental observations of modes at the GAM frequency, besides measuring the mode frequency, it is necessary to measure polarization and toroidal mode number to clearly identify the mode. BAE excitation: n m 0 excitation by both energetic ions (at the longest wavelengths) as well as via the AITG mechanism (at the shortest wavelengths) [Zonca etal. POP 1999]. Confirmed by observations on DIII-D [Nazikian etal. PRL 2006]. GAM excitation: n = m = 0 no linear excitation mechanism by spatial nonuniformity. Only instability mechanism is via velocity space: e.g., intense high-speed drifting beam such that F b / v > 0 at v qv ti /p; p =positive integers.

ENEA Structures of the low frequency Alfvén continuous spectrum 12 Experimental observations: fast ion driven GAM in JET Observation of frequency chirping oscillations at the GAM frequency excited in the presence of fast ion tails due to HFS ICRH (H.L. Berk, et al. 2006, NF 46 S888) large orbits...

ENEA Structures of the low frequency Alfvén continuous spectrum 13 GAM continuous spectrum In realistic plasmas: T e (r), T i (r), q(r) ωgam 2 2T i (r)/(m i R0) 2 (7/4 + T e (r)/t i (r)) = ωgam(r) 2 ω GAM varies radially ωgam(r) 2 forms a continuous spectrum r δj r (r, t) = 0 BAE-GAM degeneracy r { N 0 (r) Singular solution at ω 2 = ω 2 GAM(r) [ ω 2 2 (γ it i + T e ) (r) m i R 2 0 ]δe r } = 0 Generally r ( N0 (r)λ 2 (ω)δe r ) = 0 [Zonca&Chen PPCF 1996] Similar to Alfvén resonance [Chen&Hasegawa POF 1974]

ENEA Structures of the low frequency Alfvén continuous spectrum 14 Kinetic GAM δe r singular at r 0 where ω 2 = ω 2 GAM k r finite ion Larmor radius effects! ion and electron Linear mode conversion to Kinetic GAM (KGAM) propagating radially outward Similar to, e.g., Kinetic Alfvén Wave (KAW) [Hasegawa&Chen POF 1976] Dispersion relation of KGAM Assuming 1 k 2 rρ 2 i 1/q 2 and including higher order k 2 rρ 2 i corrections in GAM δf i expansion up to order O[(ω d /ω) 4 ] terms ω 2 = ω 2 GAM(r) + Cb i C > 0, b i = k 2 rρ 2 i

ENEA Structures of the low frequency Alfvén continuous spectrum 15 C > 0, complicated expression, lengthy: can be obtained from [Zonca, Chen, Santoro, Dong PPCF 1998] as a limiting case, using the degeneracy of BAE and GAM spectra [Zonca&Chen 2006, 2007] in the long wavelength limit b i > 0 when ω 2 > ω 2 GAM: propagation b i < 0 when ω 2 < ω 2 GAM: cut-off Radial wave equation and mode conversion of GAM Evidence of outward propagating GAM in JFT-2M [Ido etal. PPCF 2006] r { In nonuniform plasma k r = i / r Radial wave equation [ } N 0 (r) ρ 2 i(r)c(r) 2 r + 2 ω2 ωgam(r) ]δe 2 r Same as that for mode conversion of shear Alfvén wave [Hasegawa&Chen POF 1976] = 0

ENEA Structures of the low frequency Alfvén continuous spectrum 16 GAM: eigenmode vs. initial value problem Generally: look at the initial value problem solution of a driven nonlinear system: mode structures are determined by competition between linear and nonlinear dispersion. Nonlinear excitation of GAM favors short wavelengths (later): KGAM is preferentially excited by plasma turbulence and singular MHD structures. When the GAM continuum has a local extremum an eigenmode solution (cavity mode) is also possible when mode drive is provided by, e.g. anisotropic fast ions (JET) GAM/KGAM damping at short wavelength: collisionless damping due to resonances with high transit harmonics (L. Chen et al. Sherwood, US-TTF and APS 2007; submitted to EPL) γ GAM /ω GAM ( 1/k 2 rρ 2 i ) exp ( ωgam /ω dti ), k r ρ i q 2 > 1 For small drift orbits, k r ρ i q 2 < 1 (known case), ω ti = v ti /(qr 0 ) ) γ GAM /ω GAM ( π 1/2 /2 ) q 2 ( ω 3 GAM/ω 3 ti) exp ( ω 2 GAM /ω 2 ti Damping expressions are applicable to drive/damping analyses for GAM driven by fast ions on JET

ENEA Structures of the low frequency Alfvén continuous spectrum 17 Nonlinear excitations of Kinetic GAM Coherent 3-wave interactions [Chen, Lin, White POP 2000] Linear parametric instability: Pump DW KGAM Lower sideband Pump DW (ITG) δφ 0 : (ω 0,k 0 ) [ δφ 0 = A 0 e in 0ζ e imθ iω0t Φ 0 (n 0 q m) + c.c. m Zonal Mode (KGAM) δφ ζ : (ω ζ,k ζ ) and Pump DW modulation ] δφ ζ = [ A ζ e ik ζr iω ζ t + c.c. ]

ENEA Structures of the low frequency Alfvén continuous spectrum 18 Nonlinear excitation favors short (zonal) radial wavelengths KGAM is excited (L. Chen et al. Sherwood, US-TTF and APS 2007; submitted to EPL) Nonlinear dynamics δφ, δφ ζ growth depletes the pump δφ 0 3-wave nonlinear system with prey-predator self-regulation (d/dt γ 0n ) δa 0n = c B k θnk g δa nδa ζ (d/dt + γ n )δa n = c B k θnk g δa 0nδA ζ (d/dt + γ g )δa ζ = c 2B α ik θn k g δa 0n δa n Driven-dissipative system: limit cycle, period doubling, route to chaos, strange attractor [Wersinger etal PRL 1980].

ENEA Structures of the low frequency Alfvén continuous spectrum 19 Collective modes and DW turbulence E.m. plasma turbulence: theory predicts excitation of Alfénic fluctuations in a wide range of mode numbers near the low frequency accumulation point of s.a. continuum, ω (7/4 + T e /T i ) 1/2 (2T i /m i ) 1/2 /R (F. Zonca, L. Chen, et al. 96, PPCF 38, 2011;... 99, PoP 6, 1917): by energetic ions at long wavelength: finite Beta AE (BAE)/EPM by thermal ions at short wavelength: Alfvén ITG Magnetic flutter: may be relevant for electron transport (B.D. Scott 2005,NJP 7, 92; V. Naulin, et al. 2005, PoP 12, 052515) Recent observations on DIII-D confirm these predictions (R. Nazikian, et al. 06, PRL 96, 105006)

ENEA Structures of the low frequency Alfvén continuous spectrum 5 R. Nazikian, et al. 06, PRL 96, 105006 R. Nazikian, et al. 06, PRL 96, 105006

ENEA Structures of the low frequency Alfvén continuous spectrum 20 Generalized fishbone-like dispersion relation In general, it was demonstrated [Chen, POP 94] 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 (KBM) 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 (TAE). 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.

ENEA Structures of the low frequency Alfvén continuous spectrum 21 The fishbone-like dispersion relation demonstrates the existence of two types of modes: a discrete gap mode, or Alfvén Eigenmode (AE), for IReΛ 2 < 0; an Energetic Particle continuum Mode (EPM) [Chen POP 94] for IReΛ 2 > 0. For AE, the non-resonant fast ion response provides a real frequency shift, i.e. it removes the degeneracy with the continuum accumulation point, while the resonant wave-particle interaction gives the mode drive. Causality condition imposes δw f + IReδW k > 0 when AE frequency is above the continuum accumulation point: inertia in excess w.r.t. field line bending δw f + IReδW k < 0 when AE frequency is below the continuum accumulation point: inertia in lower than field line bending For EPM, ω is set by the relevant energetic ion characteristic frequency and mode excitation requires the drive exceeding a threshold due to continuum damping. However, the non-resonant fast ion response is crucially important as well, since it provides the compression effect that is necessary for balancing the positive MHD potential energy of the wave.

ENEA Structures of the low frequency Alfvén continuous spectrum 22 Fishbone modes: a celebrated example of EPM A celebrated example of EPM is the fishbone instability [Chen etal PRL 84; Coppi etal PRL 86], where i s [( R 2 0/v 2 A) ω (ω ω pi ) (1 + ) ] 1/2 = δ Ŵ f + δŵk, ω pi is the core ion diamagnetic frequency and q 2 is the enhancement of plasma inertia due to geodesic curvature [Glasser etal PF 75; Graves etal PPCF 00]. q 2 inertia enhancement is not the classic GGJ factor obtained in MHD [Glasser etal PF 75] and is mainly determined by trapped particle dynamics [Graves etal PPCF 00]. Wave-particle resonances with trapped particles can be crucial for determining the kink/fishbone stability in burning plasmas [Hu etal POP 06] Using a generalized expression of the inertia term Λ(ω) one readily demonstrates the existence of a high-frequency kink-fishbone branch at the frequencies of the Geodesic Acoustic Mode (GAM) [Zonca etal PPCF 07]. (ω ω pi, ω ti ) Λ 2 = 1 ω 2 A [ ω 2 ( 7 4 + T e T i ) q 2 ω 2 ti ] + i πq 2 e ω2 /ω2ti ω2 ωa 2 ( ωti ω ω ) ( Ti ω 2 ω ti ωti 2 + T ) 2 e. T i

ENEA Structures of the low frequency Alfvén continuous spectrum 23 Zonal Flows and Zonal Structures Very disparate space-time scales of AE/EPM, MHD modes and plasma turbulence: complex self-organized behaviors of burning plasmas will be likely dominated by their nonlinear interplay via zonal flows and fields Crucial role of toroidal geometry for Alfvénic fluctuations: fundamental importance of magnetic curvature couplings in both linear and nonlinear dynamics (B.D. Scott 2005,NJP 7, 92; V. Naulin, et al. 2005, PoP 12, 052515) Long time scale behaviors of zonal structures are important for the overall burning plasma performance: generators of nonlinear equilibria

ENEA Structures of the low frequency Alfvén continuous spectrum 24 Long time scale behaviors Depending on proximity to marginal stability, AE and EPM nonlinear evolutions can be predominantly affected by spontaneous generation of zonal flows and fields (L. Chen, et al. 2001, NF 41, 747; P.N. Guzdar, et al. 2001, PRL 87, 015001) radial modulations in the fast ion profiles (F. Zonca, et al. 2000, Theory of Fusion Plasmas, 17) EPM NL dynamics (F. Zonca, et al. 2000, Theory of Fusion Plasmas, 17) AITG and strongly driven MHD modes behave similarly

ENEA Structures of the low frequency Alfvén continuous spectrum 25 Discussions Burning plasmas are complex self-organized systems, whose investigation requires a conceptual step in the analysis of magnetically confined plasmas. Integrated numerical simulations are crucial to investigate these new physics; while fundamental theories provide the conceptual framework and the necessary insights. Verification against experimental observations in present day machines is a necessary step for the validation of physical models and numerical codes for reliable extrapolations to burning plasmas. Lack of understanding of some complex burning plasma behaviors can be likely filled in by increasingly complicated and more realistic modeling of plasma conditions as computing performances improve. However, some other unexplained behaviors may be just indications of fundamental conceptual problems: mutual positive feedbacks between theory, simulation and experiment will be necessary. Burning plasma physics is an exciting and challenging field: many examples of fundamental problems with broader applications and implications.