5 th EASW on Laboratory, Space and Astrophysical Plasmas. By C. Z. (Frank) Cheng

Transcription

1 5 th EASW on Laboratory, Space and Astrophysical Plasmas Alfven Waves and Instabilities in Tokamak By C. Z. (Frank) Cheng University of Tokyo, Japan National Cheng Kung University, Taiwan

2 Outline Alfven Waves in MHD & Two-fluid Models Alfven Waves in Tokamak - Toroidal Magnetic field configuration - Continuous spectrum & gaps - Existence of TAE (Toroidal Alfven Eigenmode) - RSAE (Reverse Shear Alfven Eigenmode) Alfven Instabilities in Tokamak - TAE, RSAE, Fishbone, etc. Fast Ion/alpha Transport due to TAEs

3 Mass density continuity equation: ρ t + ( ρv) = 0 Quasi-neutrality: J = 0 Momentum equation: ρ t + V V = J B P Adiabatic pressure law: ( ) )( γ ( ) t V Pρ s + = 5 / 3 is ratio of specific heat P t + V P+ γ sp V = 0 Ohm's law: E+ V B= 0 Parallel elecric field: EB= 0 Maxwell's Equations: B = Ideal MHD Model γ s = B J; E = ; B = 0 t 0

4 MHD Waves MHD waves are frequently observed in magnetosphere and laboratory plasmas MHD wave dispersion in uniform plasmas: - 3 branches of MHD waves: Shear Alfven waves Slow magnetosonic waves Fast magnetosonic waves

5 MHD Waves in Homogeneous Plasmas Plasma displacement : V ξ t Two wave equations: ω k V B k ξ = 0 ( )( ) A kv kc 1 k 0 ω s ( ) ( ) +β + ξ = A ω B γp γp where V A ; C s ; β ρ ρ B There are two possible solutions: (1) shear Alfven waves () magnetosonic waves

6 Shear Alfven Waves If B k ξ 0, then ω k V = 0, ( ) A ρω γ P+ B k γp but 0, and thus ( k ) 0. + ξ = kb B ρω Thus, the dispersion relation for shear Alfven waves is ω kv A = 0 Shear Alfven waves propagate along B with phase velocity V and its dispersion is independent of k A

7 Magnetosonic Waves ω kc s If ( k ξ) 0, then ( 1+β ) + 0, = kva ω but ω k V 0, and thus B k ξ = 0. A ( ) Dispersion relation for slow and fast magnetosonic waves: ω 1 = C s + VA ± Cs + VA 4CsVAcos θ ( ) ( ) k where k B = kbcosθ, + sign for fast magnetosonic waves, sign for slow magnetosonic waves. For β = C / V << 1, s A (1) slow magnetosonic waves: ω k C (sound waves); ω () fast magnetosonic waves: k s V A (compressional Alfven waves)

8 MHD Wave Characteristics Shear Alfven waves: ω = kv A kb ( B k) b = ik B( B k) ξ = δek 0 ω δ P= k ξ= B ξ= 0 ; b = kb = 0 Poynting flux : S = δe b = iωξ B b = iω b ξ B Magnetosonic waves: k ξ 0 ; δ P= γp k ξ 0 ; ( ) ( ) ( ) ω γ + kc + = 0 kv P B s A B ω C kc s ξ= ( s B kb k ξ) 0 ; b = i 1 B( k ξ) 0 ; ω ω ( B k) ξ= 0 ; ( B k) b = 0 ; kb = 0 Poynting flux : S = δe b = ( iωξ B) ( ξ ) ik B = ω( ξ B) k B kc s Bb=δP 1 γ P ω Fast magnetosonic mode: ω > kc s, Bb and δp are in phase. Slow magnetosonic mode: ω < kc, Bb and δp are out of phase. s

9 MHD Wave Equation in Nonuniform Plasmas for Slab Geometry The background magnetic field is chosen as ˆ ˆ Β Bx ( ) = Bz( xe ) z + By( xe ) y, ( Ρ+ ) = 0 The perturbed displacement vector is represented as iky ( y + kz z ωt) ξ ( xt, ) =ξ( xk, y, kz, ω ) e + Complex Conjugate The wave equation becomes d ρ (V + C )( ω k V )( ω k V ) dξ ( ) +ρ ( ω k V ) ξ = 0 dx ( )( ) dx A s A Sl x A x ω ω+ ω ω where slow mode velocity V CV s A Sl Cs + VA (k + k)(c s s + V) A ± (k + k)[k(c s s s + V) A + k(c s V)] A ω±

10 Wave Propagation outside Resonance Layer Away from the resonance location ( k ) ks 0 ( ) x A s A, we can consider i kx ( x) dx dξx WKB solution ξx ~ e and ~ ikx( x) ξx dx and the wave solution becomes If ω + ( ) k ( xv ) = ω k + k V A i kx ( x) dx and ξ ~ e is an oscillatory solution x If s A 0, x is imaginary kx ( x) dx and ξ ~ e is a damped solution x V ω k + k V < k x x > 0, k is real

11 Singular Solution near Resonance Layer Near field line resonance location x, where ω = k V =ω x d ω x ω x + ω x (x x ) +... ( ) ( ) ( ) A A 0 A 0 dx x= x d ω k V ω x (x x ) +... ( ) A A 0 dx x= x Then, the wave equation reduces to 0 0 ( ) 0 A A 0 d VA dξ x (x x [ 0) +...] + [(x x 0) +...] ξx 0 dx ( ω ω+ ) dx x= x0 Thus, the solutions near the resonance location x behave as [ ] [ ] ξ ln(x x ) c + c (x x ) + + d + d (x x ) + x Note that singular solutions can be resolved by plasma dissipation and kinetic effects. 0

12 Wave Propagation & Field-line Resonance At ω=ω A (x 0 ), compressional Alfven wave energy piles up and transfers energy to shear Alfven wave, which is absorbed by plasma via collisional or Landau damping. At resonance surface, mode conversion to kinetic Alfven waves via finite ion gyroradius effect (kρ i ). x 0 = Resonance surface

13 Two-Fluid Model (no electron inertia) Mass density continuity eqn. δρ + iρk ξ = 0 Isothermal plasma : P = nt ρt / mi δρt iρt n = ne = ni ; T = Te + Ti ; δp = = k ξ = ipk ξ mi mi MHD momentum eqn. ρω ξ = ik ( b) B ikδ P (no electron inertia) ( k b) B = k Bb ( b B) k Ion momentum eqn. ω minξ = ikδ Pi + en( E iωξ B) by assuming ui V = iωξ Maxwell's Eqns. ωb = k E ; j = ik b Parallel electric field: E = ik δ P / en e

14 Wave-Coupling Dispersion Relation ω ω kp ω ω kv kv B T + T P P β = = = P+ B kp 1 + = = k ρ s A A ρω ωci ρωci ωci i e where ρs ; miωci ρva B Coupling of shear Alfven wave and by (a) ion cyclotron frequency effect ( ω / ωci ) (b) ion gyroradius effect ( k ρs ) Kinetic Alfven Wave (KAW) : magnetosonic waves Consider ω / ω << 1 ; k << k ( ω ~ kv << kv ) ; β << 1 ci A A 4 ω Te δn k ρsva 1 ( ) 0 ; kv = A ω E i kv A e n - KAW: ω = [1 + k ρ ] kv s A - Ion FLR effect resolves the shear Alfven wave fieldline resonance singularity by mode conversion into kinetic Alfven wave.

15 Two-Fluid Model (with electron inertia) Electron dynamics : density continuity eqn. ωδ n + nk ue = 0 isothermal pressure δpe δnte ; T = Te + Ti momentum eqn. iωmenu e = ikδpe + en( E + ue B0 ) Ion dynamics : density continuity eqn. ωδ n + nk ui =0 isothermal pressure δpi δnti momentum eqn. iωm nu = ikδp en E + u B ( ) 0 i i i i Maxwell's Eqns. ωb = k E ; j = ne( u u ) = ik b i e

16 Wave-Coupling Dispersion Relation ( ) 1 1 k (Ti + Te S ) n Te + Ti ω ω ω k (Ti + TeS ) = mi B0 kv A kv ω A ωci ωcimi 1 ω m e ω n(te + T i) S kt e kva B0 For massless electron m = 0 and S=1 e n(t + T ) Consider ω ~k V ; ω / ω << 1 ; k << k ; β = << 1 e i A ci B0 ω m c V m T + T S 1 ; = ; = 1 e A e e i λ de ρ s kt e ωpe ωci mi ωcimi ω k V [1+ k A Te δn ρs k λ de ] ; E = i es n (1) Kinetic Alfven Wave (KAW) : () Inertial Alfven waves (IAW) : ω k V [1+ k ρ ] A s ω k V [1 k λ ] A de

17 Interaction between Alfven Waves and Fast Ions Zoo of Alfven Eigenmodes in Tokamak - Existence of TAE (Toroidal Alfven Eigenmode) - RSAE (Reverse Shear Alfven Eigenmode) - RTAE, BAE, GAE, HAE, KBM, Fishbone, CAE, HAE Alfven Instabilities in Tokamak - Destabilization of Alfven eigenmodes by fast ions - Strongly fast ion-driven Alfven instabilities Fishbone; Resonant TAE (R-TAE) Fast Ion/alpha Transport due to TAE - TAE, Fishbone

18 Tokamak Tokamak = toroidal magnetic chamber (Russian acronym) ITER

19 Tokamak Magnetic Field Tokamak magnetic field: B= B + B rdθ Rdϕ Field line equation: = Bp Bt Force Equilibrium: J B= P Field lines form closed surfaces. t p dϕ r Bt Safety factor q = ( number of turns magnetic field line goes around dθ RB p the toroidal direction when it goes 1 turn around the poloidal direction) q(r) is related to the stability of tokamak plasma.

20 Fast Ion Physics is Critical for Burning Plasmas Fast ions play essential roles in thermal plasma heating and current drive in burning plasma: Fast ions (100 kev - MeVs ) in NBI, N-NBI, ICRH provide auxiliary plasma heating 3.5 MeV α-particles produced in D-T fusion reaction provide dominant plasma heating in fusion reactors Fast ion driven AE instabilities (TAEs, RSAEs, RTAEs, EPMs, BAEs, fishbones, etc.) and significant fast ion loss caused by TAEs have been observed in all major magnetic confinement. In burning plasmas, α-particles are dominant heating source: P α (=W α /τ s ) > P aux (=W tot /τ E ), and control thermal plasma profiles, global plasma stability and confinement. In burning plasmas, significant α/fast ion loss can quench DT burning, degrades heating and current drive efficiency, and causes localized damage on first wall of fusion reactors.

21 Burning Plasmas P(r), n(r), q(r) Confinement, Disruption Control, MHD Stability Auxiliary Heating D-T fueling Current Drive Fusion Output α-heating α-cd Heating Power: P α > P aux Fast Ion Driven Instabilities Alpha Transport Need to understand α/fast ion physics and control α/fast ion confinement!

22 Alfven-Type Modes can interact with Fast Ions and be destabilized by Fast Ions Kinetic Ballooning Modes (KBM) (Cheng, 1981, 198; Cheng et al., 1995; Cheng & Gorelenkov, 004) Fishbones (resonant internal kink) (McGuire et al., 1983; Chen et al., 1984; Cheng, 1990). TAEs (Cheng et al, 1985; Cheng and Chance, 1986; Cheng et al.,1988; Fu and Van Dam, 1989; Wong et al., 1991, etc.) Resonant TAE (RTAE) (Cheng et al. 1995) Reverse Shear AE (RSAE) (Kusama et al., 1998; Breizman et al. 003; Takechi et al., 005) Beta Induced AE (BAE) (Chu et al., 199; Heidbrink et al., 199) Global AE (GAE) (Goedbloed, 1975; Appert et al., 198) Helicity Induced AE (HAE) (Nakajima et al., 199) Compressional AE (CAE) (Coppi et al., 1986; Gorelenkov and Cheng, 1995) TAEs are generic issue for all toroidal fusion devices!! TAEs are expected to be most serious in fast ion transport!

23 Zoo of Alfven Eigenmodes in Toroidal Devices Toroidal effects: B (, r θ) & B (, r θ) t coupling of poloidal harmonics p

24 Linearized MHD Equation for Alfven Eigenmodes in Toroidal Confiinement System MHD wave & stability equations in a magnetic field with nested flux surfaces: P1 P1 ξs ψ = C + D ξψ ξψ ξ ξs P1 E = F ξ ξψ where ξψ = ξ ψ ; ξs = ξ ( B ψ) / ψ ; P1 = δp+ δbb C, D, E, F are matrix operators involving only surface derivatives: B and ( B ψ ) If the E operator can be inverted, then P1 1 P1 ψ = ( C + DE F) ξψ ξψ A system of two coupled first-order derivative equations of P and ξ in ψ. NOVA code was developed to solve the linearized MHD equations Cheng and Chance, J. Compt. Phys. 71, 14 (1987) Cheng, Phys. Report ψ

25 MHD Wave Continuous Spectrum Cheng and Chance, Phys. Fluids 9, 3695 (1986) If det E = 0 for a given ω at ψ, MHD wave is singular at ψ ξs E = 0 determines continuous (field line resonance) spectrum. ξ Two branches of continuous spectrum : Shear Alfven mode : Slow mode : γsp ρω ψ ρω ψ B B ξ + = ( ) s ξ s γspk s ξ B B 1 γ P B B ( ξ) ( ξ) = K sξs B B s ξ ω ψ E = ξ Continuous spectrum equations show coupling s Eigenvalues ( ) of 0 form continuous spectrum between shear Alfven modes and slow modes through magnetic field geodesic curvature K and plasma pressure P. s of ψ.

26 Shear Alfven Continuous Spectrum = << Consider low- β large aspect ratio tokamak plasmas with β~ O( ε ), ε r/ R 1. Shear Alfven continuous spectrum equation ψ ρω ψ B B ξ + = s ξ s B B in( q ) Consider Y = ξ e ζ θ, Jacobian I= ( ψ θ ζ) s ρω ψ ψ Y Lagrangian functional L= I Y dθ = 0 B IB θ = = i( m nq) θ * Consider Y ae m, and we have al m m', mam 0 m m', m ρω ψ ψ i( m m') θ m',m = I )( )] where L [ ( m' nq m nq e dθ B IB The flux surfaces in cylindrical ( X, ϕ, Z) coordinate are given by X = R r r + O Z = r + O r cos θ ( ) ( ε ) ; sin θ ( ε ) ; ( ) = shift of flux surface ψ 1+ ( r/ R+ '( r))cos θ + O( ε ) ; I α( r)[1 + σ(r)cos θ + O( ε )] B L =Ω [ δ + ( r/ R+ + ' σ)( δ + δ )] m', m mm, ' mm, ' 1 mm, ' ( m nq)( m' nq)[ δmm, ' ( r / R ' σ)( δmm, ' 1 δmm, ' + 1)] O( ε ) where Ω = ( r) = / A( r) ; ρω α ω ω δ is Kronecker delta mm, ' 0 1

27 Shear Alfven Continuous Spectrum ω /4 A For a fixed n TAE exists with frequency inside continnum gap! 0 q=m/n q=(m+1)/n q=(m+)/n q=(m+0.5)/n q=(m+1+0.5)/n q(r) Consider only neighboring poloidal m and m+ 1 harmonics, det L = ( Ω Ω )( Ω Ω ) ( δω ) = 0 mm, + 1 m m+ 1 δ σ σ ε where Ω = ( r/ R+ + ' ) Ω ( r/ R+ ' ) ΩmΩm+ 1 ~ O( ), Ω = m nq( r), Ω m m+ 1 A = m + 1 nq( r). (1) Near q = ( m+ 1/ ) / n, Ω and Ω curves cross and Ω Ω 1/ 4 m m+ 1 m m+ 1 ω = ( ω /4)[1 ± ( r/ R+ ') + O( ε )] independent of m and n () ed ones. When Ωm Ωm+ 1 O(1), the solutions are uncoupl

28 Theoretical Discovery of TAE modes n = 1 TAE computed by NOVA code (Cheng & Chance, 1986) Existence of global TAE with frequency inside the continuum gap (n=1 fixed boundary mode with (ω/ω Α ) =0.5, ω Α = V A (0)/q(0)R) TAEs can exist for all toroidal n-modes (cavity-type modes) For each n, there can be many TAEs with different poloidal mode structures

29 Alfvén Continuum (n = 3) and TAEs in NSTX <β> = 10 % <β> = 33 % q 0 = 0.7, q 1 = 16 Large continuum gaps due to low aspect ratio even at high β. Many n TAEs with global structure are found.

30 High-n TAEs Analytical & numerical solutions of TAEs were obtained for both low-n modes (Cheng & Chance, 1986) and high-n modes (Cheng et al., Ann. Phys., 161, 1 (1985)) In a large aspect ratio tokamak with nonuniform q-profile and magnetic field intensity along B, high-n shear Alfven waves are described by d ω s + (1 cos ) 0 ε θ Φ= d θ ω A (1+ s θ ) θ k r, ε = r/r, s = rq /q, ω A = V A /qr For zero magnetic shear (s = 0), waves are described by the Mathiu equation: waves move in a periodic potential well, similar to electrons moving in a periodic lattice in solid state physics: continuous frequency bands (energy bands) and gaps There is an infinite number of frequency gaps centered at ω ~ jω Α /, where j = 1,,. The lowest continuum gap is bounded by ω = (1 ± ε ) / 4 ± ω A

31 High-n TAEs ω ω A TAE frequency s ω 1 ε For s << 1, ± 1 ( 1 s π / 8 ) ω A A ω 1 ε 4 For s > 1, ± 1+ ( 1 π / 7s ) ω For finite magnetic shear (s 0), periodicity in the wave potential is broken, similar to periodicity breaking by impurity atoms or other effects. TAEs are similar to discrete electron energy states in aperiodic lattice due to periodicity breaking in solid state physics. TAEs can exist for all toroidal n-mode numbers.

32 TAE Instability Fast ions resonate with TAEs if V h ~ V A. For B = 10T, n e = x cm -3, V A = 10 7 m/sec, V α = 1.3 x 10 7 m/sec. V h > 0.5 V A can be satisfied for α-particles, MeV protons in ICRH operation, and MeV N-NBI Deuterium ions. Necessary condition for fast ion instability drive: Free energy in fast ion pressure gradient overcomes velocity space damping: nq(v h /V A ) > (r/r)(l h /ρ h ). Sufficient condition for TAE instability: γ h (fast ion drive) > γ d (thermal plasma damping) Multiple TAEs are expected to be robustly unstable in burning plasmas (ITER)!! NOVA-K code was developed to study fast ion driven Alfven waves and instabilities Cheng, Phys. Reports 11, 1-51 (199)

33 First Experimental Observation of TAEs: TAE Excitation by NBI in TFTR NBI energy = 110 kev Wong et al. (1991) f TAE ~ V A /qr Observed frequency is consistent with theoretical TAE frequency

34 Large amplitude bursting TAEs cause fast ion loss in NSTX Fast neutron drops correlated with H α bursts; fast ions hitting wall? Small impact on soft x-ray emission. (Fredrickson et al., 003)

35 Bursting TAEs Cause Fast Ion Loss in NSTX (Fredrickson et al., 003) NSTX shot with B = 0.434T, R = 87 cm, a = 63cm, P NB = 3.MW. Single dominant mode being n= or 3, mode amplitude modulation represents "beating" of multiple modes. Bursting TAEs lead to neutron drop and cause 5 10% fast ion loss.

36 Bursting TAEs Observed in JT-60U (NNB) Slow FS mode (L-RSAE) lasts ~00ms Fast FS mode (H-RSAE) with bursting time of 1-5 ms Fast ion loss is associated with Bursting TAEs (ALE) with bursting time of µs and δb/b~10-3. (Shinohara et al., 00)

37 Bursting TAEs Cause Fast Ion Transport Loss After bursting TAE mode occurs neutron emission rate (S n ) drops and enhanced fast neutral fluxes (G) are observed. Bursting TAE modes cause enhanced transport of energetic ions via waveparticle resonant interaction. (Shinohara et al., 00)

38 Fast Ion Transport by Bursting TAEs neutron emission rate (10 14 m -3 s -1 ) r/a E3967 at 4.51sec before ALE aft er ALE fast ion density (a.u.) before ALE aft er ALE (ALE (Abruptly Large Event) is bursting TAE) (Ishikawa et al. 005) r/a

39 Fast Ion Loss due to TAE Modes Particles resonating with TAEs form particle drift orbit islands in phase space, e.g. (P ϕ, ϕ ωt/n); P ϕ = toroidal angular momentum, ϕ = toroidal angle. In general, numerical computations (e.g., Hamiltonian GC orbit Monte Carlo codes) must be performed due to complex magnetic field geometry and particle orbits Two basic fast ion loss mechanisms: - Near prompt loss boundary transient loss - Stochastic diffusion loss Alpha loss due to multiple TAEs in burning plasmas could be significant.

40 Fast Ion Loss due to TAE Modes If particle drift orbit islands overlap with the prompt loss domain, fast ions in drift orbit islands are lost to the first wall with Transient loss rate ~ (δb r /B) 1/. Stochastic diffusion loss occurs when multiple drift orbit islands overlap, and particles diffuse stochastically to the prompt loss domain. - Stochastic diffusion loss rate ~ (δb r /B) - Orbit stochastic thresholds: - δb r /B ~ 10-3 for single TAE mode - δb r /B ~ 10-4 for multiple TAEs.

41 Pitch Angle - Toroidal Angular Momentum Equilibrium magnetic field: B = ϕ ψ + RB ϕ in ( ψθϕ,, ) coordinate Energy: ε = mv / = µ B + mv / Magnetic moment: µ = mv /B Toroidal angular momentum: P = ZeA R/ c + mv RB / B ϕ ϕ ϕ = Zeψ / c + mv RB / B Pitch angle variable: Λ= µ B0 / ε Transit time: τtransit = πr0/ v0 Normalized quantities: Pˆ P / mω R ; Ω = ZeB / mc ϕ ϕ ϕ Poloidal flux: ψ ψ/ BR 0 0 Magnetic field: h( ψθ, ) B / B p ϕ 0 Λ (=µβ 0 /Ε) Prompt loss domain Counter passing confined domain Trapped confined domain Co-passing confined domain Pϕ / mω 0 R 0 Hsu and Sigmar, 199; Sigmar et al. 199

42 Alpha Particle Orbits Trapped particle orbits Mag. axis encircling co-passing orbit Z/a Z/a R/a R/a Hsu and Sigmar, 199; Sigmar et al. 199

43 n = 1 TAE Mode Structure δ B r B n = 1 Equilibrium parameters : a= 0.55 m, R = 1.75 m, B = 10 T 0 0 κ =, δ = 0.5 q(0) = 1.01, qa ( ) = 3.4 n = m T = kev 0 3 (0) 5 10, (0) 15 α slowing-down time: τ = 40 ms SD α transit time: τ = transit 6 s ψ p Cheng & Chance, 1987; Cheng, 199 ; Sigmar et al. 199

44 Resonant Alpha Particle Orbits Energy loss of resonant α δ B / B= r v / v ε/ ε 0 Z/a R/a t/ τ transit t/ τ transit Resonant α orbits are scattered by TAE α pitch angle changes from barely passing to trapped lost orbit at v = 0 αloses energy via resonance inteaction ( εε / is energy loss fraction) 0 Sigmar et al. 199

45 Alpha Particle Diffusion due to TAE n = 1, δ B / B= 10 4 r ˆ ˆ ˆ ϕ ϕ ϕ ( P ) P ( t) P (t) n = 1, δ B / B= 10 3 ( Pˆ ϕ ) r D Pˆ t pp = lim ( ϕ ) / t >> 1 D pp ( Pˆ ) ϕ t D pp ( δ B / B) r 3 t τ t τ / transit / transit Initially, 51 alpha particles with same ( ε = 3.5 MeV, P ), located far from prompt loss boundary, and randomly distributed in ( θϕ, ) and pitch angle v / v. Alpha resonant interacton with TAE gives ϕ dp / dt ~ ( v ω/ k )( δb / Bh) Diffusiveloss of resonant α's: ( Pˆ ) t ; D ( δb / B) ϕ ϕ pp r [ δ B /( 10 B)] r r Hsu and Sigmar, 199

46 Alpha Particle Loss Process (lost) N α Prompt loss Diffuive loss t/ τ transit N α (lost) [ δ B/( 10 B)] 3 α loss time: τ L N /( dn / dt) For δ B / B= 10 τ L r = 70 ms 3 Radial diffusion coefficient α = / 4τ L 1.1 / D a m s τ SD L = 18 ms SD α τ / τ 4 α Quasi-steady α loss is due to TAE-driven stochastic diffusion to prompt loss domain Lost alpha number Nα ( δ Br / B) Sigmar et al., 199

47 Alpha Particle Loss due to TAEs Alpha stochastic diffusion loss ( δ B / B) Using TFTR & DIIID geometries, simulated alpha loss could be serious due to small machine size relative to alpha banana orbit width Cheng et al., IAEA 1994 r

48 Alpha Particle Interactions with RF Waves Removal of helium ash (~ a few hundred kev) Convective bucket transport by externally driven RF waves with frequency chirping Alpha current drive Alpha channeling transfer alpha energy to thermal ions directly Plasma flow generation and control using fast ions Burning plasma devices such as ITER can explore these ideas!

49 Convective Bucket Transport for Helium Ash Removal EM waves with time-dependent frequency δbr = δbmn( r)sin[ mθ nς ω( t) t] Drift orbit island (or bucket) moves in the radial direction as wave frequency chirps (via wave-particle resonance): ω() t [ m/ qrt (()) n] ( v/ R) then, qrt ( ( ) m/( n+ω( trv ) / ) When ω(t) is varied slowly, particles in bucket move adiabatically with the bucket at radial convection velocity dr/ dt [ q R/ mv ( dq/ dr)]( dω/ dt) Bucket existence condition: - Frequency change rate < (particle bounce freq.) - (particle bounce freq.) ~ δb r /B How to excite frequency chirping waves with desirable δb r (r) in plasmas should be further studied. Hsu et al. (PRL, 1994)

50 Nonlinear Behavior of TAEs Saturation mechanism for weakly unstable TAEs is mainly due to quasi-linear local fast ion profile modification and nonlinear wave-particle trapping. Saturation will occur at δb r /B ~ (γ L /ω). If multiple TAEs are strongly unstable, it is possible to have a global flattening of fast ion profile resulting from an explosive avalanche-like domino effect. Quasi-linear ORBIT and nonlinear kinetic-mhd simulation codes have been developed to study TAE induced fast ion transport. Fully nonlinear studies of α-particle interaction with multiple strongly unstable TAEs should be performed with both selfconsistent simulations and burning plasma experiments. Cheng et al. IAEA, 1994

51 Summary A zoo of Alfven eigenmodes (TAE, RSAE, BAE, GAE, etc.) in toroidal plasmas have been discovered since mid 1980s Alfven eigenmodes exist because of breakup of continuous spectrum due to effects of periodicity of magnetic field, magnetic shear, plasma beta, etc. TAEs exist ubiquitously in toroidal plasmas. Unstable Alfven eigenmodes can interact with fast ions and cause serious fast ion loss in fusion plasmas Theories of Alfven eigenmodes (TAEs etc.) and fast ion transport are also applicable to space/astrophysical plasmas where there is flux rope or global nonuniform magnetic field.