Alfvén Eigenmodes in Spherical Tokamaks S.E.Sharapov, M.P.Gryaznevich, and the MAST Team Euratom/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxfordshire, UK H.L.Berk Institute for Fusion Studies, University of Texas at Austin, Austin, Texas, USA S.D.Pinches Max-Plank Institute for Plasmaphysics, Euratom Association, Garching, Germany S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 1
INTRODUCTION • The primary motivation for the spherical tokamak (ST) concept is its predicted high- β β β β limit [1]. Record value of volume-averaged β β β β ≅ ≅ 40% was achieved in START NBI-heated plasmas [2]. The concept of ≅ ≅ high- β β burning plasma STs is considered [3]. β β • Alfvén instabilities are of major concern for magnetic fusion as they can lead to losses/redistribution of fast ions including alpha-particles. • Lots of Alfvén instabilities excited by NBI-produced energetic ions have been observed on START and MAST: - fixed-frequency modes in TAE and EAE frequency range; - frequency-sweeping “chirping” modes; - fishbones; - modes at frequencies above the AE frequency range. These instabilities in ST experiments: - provide a test-bed for testing theoretical models on Alfvén instabilities in ITER; - stimulate experimental studies of energetic-ion-driven instabilities over broad range of plasma beta, up to β β β (0) ≥ β ≥ ≥ 1 proposed for burning STs [3] ≥ [1] Y-K M Peng and D J Strickler, Nuclear Fusion 26 (1986) 769 [2] M P Gryaznevich et al., Phys. Rev. Lett. 80 (1998) 3972 [3] H R Wilson et al., Proc. 19 th IAEA Fusion Energy Conf. (2002) IAEA-CN-94/FT/1-5 S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 2
WHY ALFVÉN INSTABILITIES ARE COMMON IN STs? • Tight aspect ratio ( R 0 / a ∼ ∼ ∼ ∼ 1.2 ÷ ÷ ÷ ÷ 1.8) limits the value of magnetic field at level B T ∼ ∼ ∼ ∼ 0.15 ÷ ÷ ÷ ÷ 0.6 in present-day STs ⇒ Alfvén velocity in ST is very low V A = B T / (4 π π π π n i m i ) 1/2 ≅ ≅ 10 6 ms -1 (START) ≅ ≅ (compare, e.g. to Joint European Torus (JET), where V A ≅ ≅ ≅ 7 × ≅ × × 10 6 ms -1 ) × • Even a relatively low-energy NBI, e.g. 30 keV hydrogen NBI on START had speed V NBI ≅ ≅ ≅ 2.4 × ≅ × × 10 6 ms -1 > V A , × • The super-Alfvénic NBI can excite Alfvén waves via the fundamental resonance V NBI = V A . Free energy source for the Alfvén instability: radial gradient of beam ions, ( γ γ γ γ / ω ω ω ) AE ∝ ω ∝ ∝ - q 2 r AE ( d β ∝ β β beam / dr ) β S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 3
WIDE RANGE OF PLASMA / BEAM PARAMETERS ON STs Ratio β β β β fast / β β β β thermal in STs can be higher than what is obtained in other tokamaks 8246 0.8 8221 5 8493 0.7 8498 fast fraction 9166 4 0.6 7051, low density 8438 β fast , % 0.5 3 0.4 2 0.3 0.2 1 0.1 0 0.0 0 2 4 6 8 10 12 0.0 0.5 1.0 1.5 2.0 β t , % 3/2 /n e ~ τ slowdown T e Typical values of β β fast and β β thermal in MAST discharges β β β β Ratio β β fast / β β β β thermal vs. slowing-down time in MAST β β (TRANSP analysis by M.Gryaznevich) discharges. The spread is caused by difference in NBI power and plasma density. ⇓ both ‘perturbative’ AEs (TAEs) and ‘non-perturbative’ Energetic Particle Modes can exist S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 4
WIDE RANGE OF PLASMA / BEAM PARAMETERS ON STs Thermal plasma β β β β thermal can be as high as β β β β thermal (0) ∼ ∼ 1. High beta can affect Alfvén ∼ ∼ instabilities in two ways (at least). 1) High plasma pressure suppresses TAEs; 2) Thermal ion Landau damping plays a stronger role. Indeed, since β β i ≡ ≡ 8 π π n i T i /B T 2 =(2T i /m i ) × × (4 π π n i m i / B T β β ≡ ≡ π π × × π π 2 )=(V Ti /V A ) 2 Alfvén waves interact stronger with thermal ions as β β β β thermal increases. Limiting cases: low- β β β β discharges: V Ti << V A ≤ ≤ ≤ V beam <<V Te . Instability is determined by fast ion profile, while thermal ≤ ions play a stabilising role (via V | | i = V A /3 resonance); || | | | | discharges with β β i ∼ ∼ 1: β β ∼ ∼ V Ti ∼ ∼ V A << V beam <<V Te . Stability/instability is determined by thermal ions ∼ ∼ S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 5
OBSERVATIONS ON START (LOW- β β β β DISCHARGES) • START: R 0 ≈ 0.3 ÷ 0.37 m; a ≈ 0.23 ÷ 0.3 m; I P ≈ 300 kA; B 0 ≈ 0.15 ÷ 0.6 T • Hydrogen beam co-injected into D plasmas: E NBI ≅ 30 keV, P NBI ≤ 0.8 MW • Modes with fixed frequencies f AE ≅ 200-250 kHz (#35305), lasting for 1-5 ms, were observed in pulses with P NBI ≤ 0.5 MW and in early phase of some pulses with P NBI ≤ 0.8 MW, when β β T ≤ β β ≤ ≤ ≤ 3-5% • Mode frequency ∼ TAE frequency f TAE ≡ V A / 4 π qR 0 ∼ 200 kHz • Poloidal mode numbers of the excited modes, m = 1-4, are in agreement with the strongest drive estimate for TAE, ∆ ∆ ∆ ∆ orbit ∼ ∼ ∼ r TAE / m ∼ • Both Toroidal and Elliptical AEs (frequency range f EAE ≈ 2 f TAE ) were observed S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 6
1.0 1.0 (b) (a) Power, a.u. 0.5 0.5 0.0 0.0 100 200 300 400 500 0 200 400 600 800 1000 f, kHz f, kHz Mirnov coil signal Fourier power spectra of: (a) fixed-frequency TAE at t ~ 26ms, START, shot #35305, β β < 3%; β β (b) fixed-frequency EAEs in the EAE gap, t ~ 26.7ms, START #36484, β β ~ 4%. β β S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 7
OBSERVATIONS ON MAST (LOW- β β β β DISCHARGES) • MAST: R 0 ≈ 0.9 m; a ≈ 0.7 m; I P ≈ 1.35 MA (achieved in 2003); B 0 ≈ 0.4 ÷ 0.7 T; • D beam co-injected into D plasmas: E NBI ≅ 45 keV, P NBI ≤ 3.2 MW • Both TAE and EAE observed on MAST, but the modes are longer lasting (>20 ms), more numerous, with a broader range of unstable n ’s. Fine “pitchfork” splitting of the spectrum is often observed (as shown in the Figure (b) for MAST discharge #2884). f, kHz 1.0 (a) (b) “pitchfork” splitting 200 Power, a.u. 0.5 100 0 0.0 40 80 120 t,ms 100 200 300 400 500 S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan f, kHz 8
NONLINEAR EVOLUTION OF TAE INSTABILITY � Strong source ν eff > γ L - γ d F HOT Weak source ν eff < γ L - γ d 0 r r TAE JG04.458-1c Non-linear TAE behaviour depends on competition between the field of the mode that tends to flatten distribution function near the resonance (effect proportional to the net growth rate γ γ≡ γ γ ≡ ≡ ≡γ γ γ γ L - γ γ γ d ) and the γ collision-like processes that constantly replenish it (proportional to ν ν eff ) ν ν S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 9
NONLINEAR EVOLUTION OF TAE INSTABILITY Nonlinear equation for TAE amplitude 30 t t − τ [ ] dA / 2 2 ( ) (a) (b) = − φ τ − ν τ τ + τ A i 2 3 2 exp( ) exp 2 / 3 ∫ ∫ dt 1 20 20 0 0 ( ) × − τ − τ − τ − τ − τ τ τ A t A t A t d d * ( ) ( ) 2 10 10 1 1 1 A(t) derived in [4] describes four different regimes of TAE: - 10 0 20 30 100 10 0 50 a) Steady-state (observed); 20 100 (c) (d) b) Periodically modulated (observed as ‘pitchfork- 10 50 splitting’ effect); 0 0 c) Chaotic; JG04.458 - 3c - 10 - 50 d) Explosive regimes of TAE-behaviour as functions of ν≡ν eff / γ - 20 - 100 0 50 100 150 0 5 10 • Explosive regime in a more complete non-linear model [5] leads to frequency-sweeping ‘holes’ and ‘clumps’ on the perturbed distribution function. [4] H.L.Berk, B.N.Breizman, and M.S.Pekker, Plasma Phys. Reports 23 (1997) 778 [5] H.L.Berk, B.N.Breizman, and N.V.Petviashvili, Phys. Lett. A234 (1997) 213 S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 10
ON THE HOLES AND CLUMPS THEORY • Beyond the ‘explosive’ regime, theoretical prediction shows two long-living thermal fluctuations on the perturbed distribution function. • These long-living Bernstein-Greene-Kruskal (BGK) nonlinear waves sweep in frequency away from the starting frequency, with frequency sweep related to the particle trapping frequency in the TAE field: δω ∝ ω ω ∝ δ 1 / 2 t t B 3 / 2 1 / 2 ; ( ) b b TAE S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 11
MAST: FREQUENCY-SWEEPING MODES ARISING FROM TAEs f, 140 kHz 120 100 80 64 66 68 70 72 t, ms Primary suspect: hole-clump frequency-sweeping pairs S.E.Sharapov, M.P.Gryaznevich et al, 10 th ST Workshop, 29 September - 1 October 2004, Kyoto, Japan 12
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