Geodesic Acoustic and related modes A. Smolyakov* Acknowledgements X. Garbet, C. Nguyen (CEA Cadarache) V.I. Ilgisonis, V.P. Lakhin, A.Melnikov (Kurchatov Institute) * University of Saskatchewan, Canada 5th IAEA Technical Meeting on Theory of Plasmas Instabilities 05 07 Sep 011, Austin, Texas USA
Motivation Geodesic Acoustic Modes (GAM) are relatively high frequency eigen-modes supported by plasma compressibility in toroidal geometry. A Zonal Flow branch: Coupling to the drift-wave turbulence, regulation and control of turbulence? A finite frequency mode of neoclassical plasma rotation. Relation to neoclassical transport? Relation to Alfven modes/cascades/bae? Energetical particles drive? Effects on transport regulation/modulation?
5 khz mode ; CHS, JFT-M, T-10, TEXT-U Fujisawa, NF 007
McKee, PPCF 006 DIII-D, Asdex-U
GAM: Plasma theory success story? Theoretically predicted well before the experimental verification Fujisawa, NF 007
GEODESIC ACOUSTIC WAVES IN HYDROMAGNETIC SYSTEMS WINSOR N ; JOHNSON JL ; DAWSON JM PHYSICS OF FLUIDS 11 448, 1968 1968-1996 3 citations before 1971 (self) 1996/1997 Lebedev et al, Novakovskii, et al. Phys Plasmas- papers 1997-1999: Rewoldt, Hinton/Rosenbluth, Z. Lin,... ~30 citations
A short history of GAM 1968 : Geodesic acoustic modes predicted: Winsor, Johnson, Dawson Fully forgotten between 1968 and 1996 011: Every large or small tokamak has seen one or several variety of GAM; many sightings in numerical simulations, Dozens of theory papers. 1973: Mikhailovskii, NF: Electromagnetic drift wave instabilities (in current terminology finite m,n GAM/BAE) 1977: Mazur, Mikhailovskii, NF, Beam driven Alfven waves: 7/4 coefficient surfaces 1996: Levedev, Yushmanov, Diamond, Smolyakov, PoP, Relaxation of poloidal rotation, 7/4 surfaces again from kinetic calculations 1999: Mikhailovskii, Sharapov, : Electromagnetic drift wave instabilities, Plasma Phys Reports, GAM+BAE+ drift effects
Relation to BAE 1993, Heidbrink et al, What is the beta-induced Alfven eigenmode? oscillations with ω v Ti / R and ω v A / qr 199: Chu, Green et al., Coupling of Alfven and sound continuum via geodesic curvature creates low frequency gap 1995 Huysmans, Goedbloed: Global Beta Induced modes 1996-008: Zonca, Chen, et al., Unstable Alfven modes in the continuous spectrum: GAM dispersion relation with 7/4, electromagnetic (Alfven modes), Alfven Ion Temperature Gradient modes (AITG) 001--008: Berk, Sharapov, Breizman, Gorelenkov, Kolesnichenko, Fu, Nazikian, and others: Alfven cascades, Chirping modes: BAE/Alfven waves zoology Beta Alfven Acoustic modes BAAE (PhysLett A 007, Gorelenkov), Energetical particles GAM- EGAM (Fu, PRL 008)
Terminology: GAM-BAE degeneracy Geodesic Acoustic Modes: GAM Φ =Φ ( ) 0 0 r principal component with m = n = 0, but m=-1,+1, +,-, side-bands Beta-induced Alfven Eigen modes: BAE finite m and n (low numbers),, Amn exp imθ inς ( Φ ) ( ) mn k = m nq qr = ( ) / 0 ω = k v + A k = m nq qr c R s ( ) / 0
Lost link in GAM genealogy GAM: Winsor, Johnson, Dawson 1968 Green, Winsor 1973; Green, Johnson, Winsor 1971 Stringer spin-up, Hassam, Drake 1993 (Stringer 1969) Stringer spin-up is GAM driven by poloidal modulations of turbulent transport Linear GAM dispersion equation is explicitly given in Hassam-Drake paper
First experimental observations of GAMs GAM : H1- heliac, Shats et al., PRL 00; DIII- D, McKee et al, PoP 003 Edge quasicoherent mode in TEXT (0 khz): Tsui, PRL 1993, PoP 1993 Low m, narrow peak, edge localized (around q=3), nonlinearly correlated with shortwavelength drift wave turbulence Coherent potential oscillations in JIPPT-IIU tokamak, Hamada, 1995 Explodes from 005 and on...
GAM and BAE again JET n=0 chirping modes, Berk et al., NF 006 Geodesic frequency (compressibility) Correlation with square root of electron temperature ω = k v + c / R A s effects in finite m,n Alfven Cascade modes, Breizman, et al., PoP 005 Geodesic frequency
Development of GAMs theory GAM frequency, dispersion relation, radial propagation Zonca, Hinton, Rosenbluth, Hallatschek, Watari, GAM damping: Sugama, Zonca, Gao, Zhou,.. Radial structure, radial propagation, transport modulations, GAM spectroscopy Itoh, Diamond, Hallatschek, Gao, Sasaki, Nonlinear theory Guzdar, Chakrabarti, Itoh, GAM/BAE coupling, drift effects, Gorelenkov, Kolesnichenko, Elfimov,.
GAM is a rotational mode in toroidal plasma supported by plasma compressibility (geodesic) Involves perturbations of the electrostatic potential and pressure side-bands pˆ ± 1 Φ ( r 0 ) Dispersion relation: Geodesic: / R / R C s Ion sound: C s q
Adiabatic index: Discrepancy between MHD and kinetic theory in fluid limit Ideal MHD theory, Winsor et al. Kinetic calculations but in the fluid limit, Lebedev, Diamond et al 1996 Zonca, Chen 1996 1/q terms are dropped here Adiabatic indices for ions and electrons?
Extended MHD for GAMs: Anisotropic pressure perturbations are important Perturbed parallel viscosity (pressure anisotropy) is of the same order as pressure side-bands
Extended MHD (Grad equation) for parallel viscosity Mikhailovskii 1984, Smolyakov 1998 Different compressibility indices for pressure and parallel viscosity Restores agreement between fluid and kinetic calculations (Smolyakov et al, 005)
Coupling of Alfven and acoustic continua due to the geodesic curvature creates several modes These modes differ by their polarization and have different adiabatic (compressibility) indices
Mode polarization and coupling One can define principal and toroidal side -bands Principal component Parallel wave-vectors: GAM: m=n=0 BAE: m and n are finite (but low) Polarization is defined by the combinations:
High frequency electromagnetic side-band modes induced by geodesic coupling GAM/BAE parity GAM/BAE frq: Two Alfvenic side band modes v A / qr v A / qr Compressibility induced Alfven modes, Zheng, Chen, 1998 r -1 r 0 r 1 Smolyakov 008
Electromagnetic components in GAMS Compressibility of the diamagnetic current induces the parallel current at the second harmonic: Φ + Φ cos q Φ ( ) θ β 0 Φ ± ±, A Electromagnetic, second side-bands harmonics can be substantial, especially in high beta plasmas Second harmonics play crucial role in formation of eigenmode structure
GAM eigen-mode structure Huysman, 005, ω eig =0.0763, below the maximum CASTOR, Berk et al, NF 006 Local maximum is required in the frequency profile Global features are present in m=1 and m= harmonics No eigen mode is found without m= component
Ion sound cont. Analytical eigen-mode Second harmonics Alfven cont., nd Compessibility of the diamagnetic current
Eigen-mode dispersion equation Non-monotonic sound frequency profile Eigen-mode frequency, Lakhin, et al., 010 nd harmonic, vector-potential β q
GAMs in high beta plasma Local dispersion relation taking into account second harmonics = β q ε << 1 GAM in high beta plasms ε >> 1
GAM dispersion in kinetic theory Finite ion Larmor radius: Ion parallel motion Finite average geodesic curvature: -Ion Zonca, Nguyen, Sugama, (008) -Electron dd Electromagnetic effects: Smolyakov et al PPCF 008 NF 010
Ordering in two-fluid/kinetic models Two temperatures: T i and T e Different parallel wave-vectors put principal and side-band components into different regimes Ion fluid ordering: ω >> vti / qr >> k0v Ti Electron parallel motion ordering: Electrons are adiabatic in the side-bands: ω << v Te / qr but the principal component is -hydrodynamic ω >> k 0 v Te Can be in the -adiabatic regime (finite m, n ; BAE?) ω << k 0 v Te k ( m nq) / 0 0 = qr =
Electron Electromagnetic Effects Electromagnetic effects Zhou, PoP 007; Wang et al, PoP 011: The second side-band harmonics are electromagnetic Smolyakov et al., PPCF 008, NF 011 There are exist two regimes depending on the radial mode localization Electron side-bands are electrostatic for K >> 1 Electron side-bands are electromagnetic for K 1
Electron response 1. Electron drift kinetic equation. Electron density and current are found as moments of the distribution function 3. Amperes law is used to find magnetic potential 4. Electron density (principal and side-bands) are written in terms of the electrostatic potential
First side band is electromagnetic and finite axisymmetric density perturbation for large scale modes
Electron density K >>1 Electromagnetic modification factors: Electron side-bands are electrostatic for Electron side-bands are electromagnetic for K >>1 K 1
The parameter that controls the electron response: ω v Ti / R Electrostatic approximation K >>1 Electromagnetic approximation c ω pi cm for n0 10 K <<1 14 3 = cm Basic frequency is not affected by dispersion
Collisionless (Landau) γ GAM damping Watari, PoP 006, Sugama, JPP 006 4 4 4 π vti q R ω q R ω = q exp.. 4 qr + vti vti 6 6 6 1 qrω qrω + ( krρiq) exp.. + 6 4 4vTi 64vTi Qiu, Zonca. PPCF 009, higher order transit terms in the large orbit width limit Trapped particles can enhance GAM damping, Chavdarovskii, Zonca PPCF 010; Zhang, Lin PoP 010 Landau damping is large for small q (no GAMs in the core) Collisional damping is large for large ν ιι 4ν Novakovskii PoP, 1997 γ 7 ii q
Radial eigen-mode structure and radial propagation? JFT-M, Ido, NF 006 Frequency approximately constant in radius Outward radial propagation
Radial profile of observed GAM oscillations (constant) T-10, Melnikov Local GAM frequency
GAM frequency is approximately constant in radial direction Two frequency peaks observed at all locations, Δf=4-5 khz, T-10 Melnikov 6 4 0 18 16 freq, khz 14 1 10 8 6 4 n=0.8 *4/3 = 1.07 0 0 4 6 8 10 1 14 16 18 0 4 6 8 30 r, cm
Frequency peak splitting (two peaks)? Conway, PPCF 008
GAM dispersion and radial propagationn Local theory ω = ω ( r) ω ω 0 + -> Continuum spectrum = ω v = 0 -> No radial propagation rg k ( = 1 αk ) -> Mode dispersion = ω v rg 0 k
Radial group velocity Hinton, Rosenbluth 1997, Zonca EPL 009, Hager, Hallatschek, PoP 009, Itoh,... Hallatschek, PoP 009
GAM dispersion Small scale modes: K >>1 GAM dispersion in the electrostatic limit: Hinton, Rosenbluth, Diamond 1996; Zonca, PPCF 1996, EPL, 008; Sugama PoP 006, Nguyen, PoP 008 Large scale (electromagnetic) modes: K <<1 Dispersion scale length:
GAM dispersion and eigen-mode structures ( ) 0 1 k α ω ω + = -> Mode dispersion Discrete eigen-mode structure () 0 1 = φ ω ω φ α r r ( ) 0 0 0 1 ) ( r r r r + = ω ω ω 4 1/ 1 Δ r r ω α Localization: Global eigenmode requires a non-monotonic profile (RS)
Radial eigen-modes for monotonic profiles Itoh, 006, Zonca 008, Sasaki 006, Gao, 009, ω ω 0 + ( 1 αk ) = -> Mode dispersion φ ω α r ω () r 1 φ = 0 ω = ω + ω r ( r ) 0 ( r) 0 r0 Airy type eigen-function (radial propagation) ω = ω 0 No eigen-frequency (is defined by the source ), EPM
GAM excitation mechanisms? Energetical particles: NBI, RF heating Mode localization (eigen-mode formation) can be facilitated by energetical particles, EPM modes (Fu, Berk) How GAMs are excited in Ohmic plasmas -instability source? -localization (eigen-mode)? Nonlinear (Reynolds stress) drive? Nonlinear eigen mode formation? Chakrabarti, Guzdar, Zonca,..., Itoh, Transport modulations (Stringer spin-up) Itoh, Hallatschek, 005
Are GAMs driven by turbulent fluctuations? Signatures of predator-prey type behavior Conway, PRL 011
Drift effects on GAM/BAE modes with a finite m or Dispersion and instability of drift waves due to the average geodesic curvature (Drift effects in BAE by Gorelenkov, Kolesnichenko)
GAM frequency forms radial stair-like sequence Large amplitude in high density gradient region ASDEX, Conway, PPCF 008
Compressibility effects on dispersion and instability of drift waves (finite m) Electron drift waves Ion sound and FLR dispersion Average geodesic curvature dispersion Generalized inertia
Ion drift mode destabilized by averaged geodesic curvature/geodesic ITG ω /ω i *i ω /ω r *i η i
GAMs and neoclassical theory (of plasma rotation and transport) Hierarchy of relaxation times for neoclassical rotation Poloidal flow damping at ion-ion collision time and transit Landau frequency (GAM damping) neoclassical ambipolarity is established at this time scale Slow diffusion of toroidal momentum due to perpendicular viscosity Damped (driven) GAM oscillations induce non-automatically ambipolar neoclassical transport 1/ mi me Γ neo
In summary In general, higher order (dispersive) corrections to GAM/BAE are different Depend on mode localization Have several different contribution (electron and ion magnetic drift, parallel motion, electromagnetic corrections Large scale GAMs/BAE have substantial electromagnetic component k c / ω pi < 1 It is not clear what are the mechanisms for GAM excitation and localization (Ohmic plasmas) Relation of GAM and neoclassical transport is to be further studied (Stringer spin-up and transport modulations)