Lecture 4. Relationship of low frequency shear Alfvén spectrum to MHD and. microturbulence. Fulvio Zonca

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 1 Lecture 4 Relationship of low frequency shear Alfvén spectrum to MHD and microturbulence 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. February 21.st, 2013 Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013, Kinetic theory of meso- and micro-scale Alfvénic fluctuations in fusion plasmas 19 22 February 2013, IPP, Garching

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 2 The roles of low frequencies Lecture 2 assessed the important role of SAW for energetic particle transport in burning plasmas magnetically confined in toroidal geometry Lecture 3 constructed the general theoretical framework for investigating SAW fluctuations based on the general fishbone like dispersion relation This Lecture: look at specific applications, favoring low frequencies, since theyarethenaturalonesonwhichbothenergeticaswellasthermalparticles can resonantly excite collective modes characterized by the respective scale lengths: fast ions mesoscales, thermal particles microscales. Similar temporal scales of disparate phenomena facilitates their interplay and dictates long time-scale nonlinear dynamic response (importance for the fusion burn)

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 3 21 Reduced kinetic SAW model equations. Assume ω ds ˆω ds =(m s c/e s )(µ + v 2 /B)Ω κ, Ω κ =(k b) κ Vorticity Quasineutrality are: ( ) k 2 B B δψ kϑ 2 B2 + ω(ω ω pi) v 2 A k 2 δφ kϑ 2 + 4π k k b ( ) P ϑ 2 + P Ωκ δψ = B2 e 2 s F 0s (δφ δψ) + m s E s=i s E ( e [ω tr θ i(ω ω d )] s δk s =i m) s s=i 4πe s k 2 ϑ c2 ωˆω dsδk s 4πeE k 2 ϑ c2 J 0(k ρ E )ωˆω de δk E e s δk s =0, QF 0s [J 0 (k ρ s )(δφ δψ)+ ( ˆωd ω ) s, ] J 0 (k ρ s )δψ.

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 4 BAE dispersion relation and high-frequency fishbones Assume massless electrons and decompose drift-kinetic ion response as ( [ e F0 δf i = m) i E δφ QF ] 0i ω δψ + δk i Definitions: F 0i the equilibrium distribution function, E = v 2 /2 the energy per unit mass, QF 0i = (ω E + ˆω ) i F 0i, ˆω i F 0i = (m i c/eb)(k b) F 0i. Scalar fields are δφ and δψ, related with the vector potential fluctuation δa by δa i(c/ω)b δψ (δe = b (δφ δψ)). Fields are obtained from quasi-neutrality and vorticity equations. Simplify ballooning space notations; δφ stands for δˆφ etc (Lecture 1).

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 5 The particle distribution function δk i is derived from the drift-kineticequation ( e ( ωd ) ] [ω tr θ i (ω ω d )] i δk i = i QF 0i [(δφ δψ) + δψ m) i ω i Definitions: ω tr = v /qr is the transit frequency, k 2 = k2 ϑ [1+(sθ αsinθ)2 ] and ω di is the magnetic driftfrequencyω di (θ) = g(θ)k ϑ m i c(v 2 /2+v2 )/ebr, g(θ) = cosθ+[sθ αsinθ]sinθ. Quasi-neutrality and vorticity equations are [ 1 Bb B ( 1+ 1 ) τ ] k 2 b δψ kϑ 2 + ω2 v 2 A (δφ δψ) = T i ne δk i ( 1 ω ) pi k 2 δφ+ α ω kϑ 2 q 2 R 2g(θ)δψ = 4πe kϑ 2c2ωω diδk i

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 6 BAE are important since they can be excited by both fast ions (long wavelengths) as well by thermal ions (short wavelengths; AITG). This was predicted theoretically (PPCF96 2011) and then observed experimentally (Nazikian etal PRL 2006). For BAE one typically has ω ω ti ω pi k v A. Important effects are expected from kinetic interaction in the radial local region (kinetic layer) k qr 0 β 1/2. Since (nq m) and ballooning angle θ are conjugate variables w.r.t. the Fourier integral transform (Lecture 1), relevant kinetic physics is expected at large θ. Typical paradigmatic case for application of BF to drift Alfvén turbulence as well as MHD (nospecificationof mode numberso far; justk qr 0 β 1/2 ). Derivation of layer equation via asymptotic expansion in β 1/2. Lowest order solution yields δk (0) i = (e/m) i (QF 0i /ω)(δφ (0) δψ (0) ), which yields δψ (0) = δφ (0) when substituted back into the quasi-neutrality condition.

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 7 At the next order, δφ (1) = e iθ δφ (1+) + e iθ δφ (1 ) with a corresponding δk (1) i = e iθ δk (1+) i +e iθ δk (1 ) i. Let ω (±) tr = [1±(nq m)](v /qr 0 ). Then i [ ] ±ω (±) tr ω δk (1±) i ( e = i m) i ( e QF ( ) 0i δφ (1±) i m) i QF 0i v 2 /2+v2 R 0 ω ci ω k r 2i δφ(0) Note that the dominant effect comes from the geodesic curvature (large k r θ), causing radial magnetic drifts. Sideband generation is evident in the wave-particle interaction as well as in the ω effect in QF ( ) 0i, which is computed at m 1 [Zonca NF 09, Lauber PPCF 09]. Substituting back into the quasi-neutrality and letting ω (±) ti = (2T i /m i ) 1/2 [1±(nq m)]/qr 0 δφ (1±) = i ct i N m (ω/ω (±) ti ) eb 0 D m 1 (ω/ω (±) ti ) k r ωr 0 δφ (0)

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 8 N(x) = D(x) = Definitions: Z(x) = π 1/2 e y2 /(y x)dy and subscripts m and m 1 indicate poloidal mode numbers to compute ( 1 ω ) ni [x+ ( ) 1/2+x 2 Z(x) ] ω Ti [ ( ) x 1/2+x 2 + ( 1/4+x 4) Z(x) ], ω ω ( )( 1 1+ 1 ) ( + 1 ω ) ni Z(x) ω Ti [ ( x+ x 2 1/2 ) Z(x) ], x τ ω ω Corresponding solutions of the drift-kinetic equation are δk (1±) i = i ct i eb 0 e/m i ω ω (±) tr [ m i 2T i ( ] v 2 2 +v2 )QF 0i N m(ω/ω (±) ti ) D m 1 (ω/ω (±) ti ) QF( ) 0i k r ωr 0 δφ (0)

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 9 The large θ vorticity equation becomes finally (canonical form yielding the general fishbone like dispersion relation) ( ) 2 θ 2 +Λ2 δψ (0) = 0 Definitions: δψ (0) = [1+(sθ αsinθ) 2 ] 1/2 δψ (0) ( Λ 2 = ω2 1 ω ) [( pi +q 2 ω2 ti 1 ω ni ωa 2 ω 2ωA 2 ω ω Ti ω ( ( (ω/ω (+) ti )G(ω/ω (+) ti )+(ω/ω ( ) ti )G(ω/ω ( ) ti ) )( ) (ω/ω (+) ti )F(ω/ω (+) ti )+(ω/ω ( ) ti )F(ω/ω ( ) ti ) (ω/ω (+) ti )N m (ω/ω (+) ti ) N m 1(ω/ω (+) ti ) D m 1 (ω/ω (+) ti ) +(ω/ω( ) ti )N m (ω/ω ( ) ti ) N m+1(ω/ω ( ) D m+1 (ω/ω ( ) F(x) = x ( x 2 +3/2 ) + ( x 4 +x 2 +1/2 ) Z(x), G(x) = x ( x 4 +x 2 +2 ) + ( x 6 +x 4 /2+x 2 +3/4 ) Z(x), ) ti ) ti ) )]

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 10 Details of sideband generation (ω ) are unnecessary for drift Alfvén waves; they are needed for moderate wavelength MHD-type modes BAE dispersion relation is obtained by asymptotic matching of the large θ solution with internal region 2 θδψ (0) (s αcosθ) 2 [ 1+(sθ αsinθ) 2 ] 2 δψ(0) + αcosθ [ 1+(sθ αsinθ) 2 ]δψ(0) = 0. Definition: δw f as MHD-fluid potential energy δw f = 1 [ ( ) ] θ dθ δψ (0) ID 2 (s αcosθ) 2 + [ 2 1+(sθ αsinθ) 2 ] 2 αcosθ δψ [ 1+(sθ αsinθ) 2 ] (0) ID 2. General fishbone-like dispersion relation (δψ (0) e iλ θ for θ 1)) iλ = δw f +δw k (fastionsincluded)

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 11 Λ 2 = 1 ω 2 A Reminder from Lecture 3: Λ 2 = k 2 q2 R 2 0 represents the SAW continuum; i.e., frequency gap is at Λ 2 < 0 To demonstrate that Λ 2 expression contains the physics of low-frequency (BAE) gap formation in the SAW continuum, take the fluid limit in which MHD is valid, ω ω ti Large argument expansion of the plasma Z function yields: [ ( 7 ω 2 4 + T ) ( e q 2 ω 2 ωti ti T i ω ω )( Ti ω 2 ω ti ]+i πq 2 e ω2 /ω2ti ω2 ωa 2 ω 2 ti + T ) 2 e. T i From solution of Lecture 2 Exercise, the low frequency gap in MHD should occur at ω 2 < γ(t i +T e )/(m i R 2 0). Comparing results with kinetic theory: γ e = 2 electrons behave as adiabatic massless fluid in 2D γ i = 7/2???

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 12 Result is not surprising since MHD has many simplifications; including isotropic pressure response 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 and neglecting damping) i ω ω A [1 ( 7 4 + T ) ] 1/2 e q 2ω2 ti = δw T i ω 2 f +δw k When compression effects are important [Zonca et al NF 09], it is generally important to take into account the deviation of the mode structure from the usual rigid plasma displacement [Kolesnichenko et al NF 10]. Anticipating BAE/GAM degeneracy...

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 13 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). Λ 2 = k 2 0 v2 A

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 14 Micro- and meso- scale excitation of low-frequency AE/EPM With the general expression of Λ (k = 0) Λ 2 = ω2 ω 2 A ( 1 ω ) pi ω N m(ω/ω ti ) 2 +q 2ωω ti ω 2 A [( 1 ω ) ni F(ω/ω ti ) ω Ti ω )] ( Nm+1 (ω/ω ti ) D m+1 (ω/ω ti ) + N m 1(ω/ω ti ) D m 1 (ω/ω ti ) ω G(ω/ω ti) low frequency AE/EPM can be excited by both thermal ions (micro-scale) and energetic ions (meso-scales) iλ = δw f +δw k (fastionsincluded)

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 15 Λ 2 = 1 ω 2 A Excitation of low-frequency AE by thermal ions is most easily seen using the simplified expression of Λ derived earlier: AITG excitation mechanism [ 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! The unstable continuum is not a concern (E: compute how much the mode can grow before mode converting to KAW. Hint: compute how long takes for a wavepacket born at small θ-ballooning to reach large θ where KAW physics is important). When ωω Ti > ω 2 ti and equilibrium effects localize the AE, the Alfvénic ITG mode is excited (AITG)

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 16 R. Nazikian, et al. 06, PRL 96, 105006 R. Nazikian, et al. 06, PRL 96, 105006

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 17 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: GAM, e.s. with small magnetic component; BAE, e.m. with Alfvénic structure. 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.

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 18 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 non-uniformity. Only instability mechanism is via velocity space: e.g., intense high-speed drifting beam such that F b / v > 0. EGAM excitation by a radially narrow [Fu PRL 2008] (weak coupling to GAM continuum) or broad [Qiu et al POP 2012] (weak coupling to GAM continuum) energetic particle beam. Mode conversion to kinetic GAM [Zonca and Chen EPL 2008].

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 19 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...

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 20 GAM continuous spectrum In realistic plasmas: T e (r), T i (r), q(r) ω 2 GAM 2T i (r)/(m i R 2 0)(7/4+T e (r)/t i (r)) = ω 2 GAM(r) ω GAM varies radially ωgam 2 (r) forms a continuous spectrum r δj r (r,t) = 0 BAE-GAM degeneracy { [ N 0 (r) ω 2 (7T } i/2+2t e )(r) ]δe r m i R0 2 r = 0 Singular solution at ω 2 = ω 2 GAM (r) Generally r ( N0 (r)λ 2 (ω)δe r ) = 0 [Zonca&Chen PPCF 1996] Similar to Alfvén resonance [Chen&Hasegawa POF 1974]

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 21 Kinetic GAM δe r singular at r 0 where ω 2 = ω 2 GAM ion and electron k r finite ion Larmor radius effects! 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 ω 2 = ω 2 GAM(r)+Cb i C > 0,b i = k 2 rρ 2 i

r Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 22 C > 0, complicated expression, lengthy: can be obtained using the degeneracy of BAE and GAM spectra [Zonca&Chen 2006, 2007, 2008] 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 In nonuniform plasma k r = i / r Radial wave equation [ρ { N 0 (r) 2i(r)C(r) 2 r 2 +ω2 ω 2 GAM(r) Same as that for mode conversion of shear Alfvén wave [Hasegawa&Chen POF 1976] Evidence of outward propagating GAM in JFT-2M [Ido etal. PPCF 2006] ]δe r } = 0

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 23 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 = δŵf +δŵ k, 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]. 1.6q 2 (R 0 /r) 1/2 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]

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 24 The 1.6q 2 (R 0 /r) 1/2 is identical to Zonal Flow polarizability (Rosenbluth and Hinton 1998): not accidental! Similar argument to BAE/GAM degeneracy. Analytical expression of Λ 2 recently obtained, smoothly connecting MHD to BAE/GAM frequencies (Chavdarovski 2009). Trapped particle compressions generally lower the BAE accumulation point frequency [Chavdarovski and Zonca PPCF 09] and are needed for a correct description of experimental observations [Lauber et al PPCF 09]. RP: Repeat the calculation of Λ 2 including the degeneracy removal between poloidal sidebands due to both finite m and finite k qr 0. Do the same for computing the terms involving Finite Larmor Radius and Finite drift Orbit Width RP: Use the above results to derive a very general dispersion relation for SAW at low frequency, including discretization of the SAW continuum by FLR/FOW in the frequency range of the so called finite Beta induced Alfvén Acoustic Eigenmodes ω < ω ti

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 25 Acoustic wave couplings The notion of Beta induced Alfvén Acoustic Eigenmode (BAAE) was originally formulated on the basis of a fluid approach, which cannot be applied to collisionless plasmas of fusion interest. This approach was later on extended to kinetic analyses of circulating thermal particles [Gorelenkov etal POP 2009]. Gorelenkov N N, Van Zeeland M A, Berk H L et al. 2009 Phys. Plasmas 16 056107 cylindrical Ω 2 uncoupled continuum m 1 n m m+1 n n A a q torus BAAE gap β 4/3 TAE gap ~ RSAE BAE gap ~ β m n ε GAM/ BAE BAAEs q FIG. 1. Color Schematic of the low-frequency Alfvénic and acoustic continuum in a cylinder a and in a torus b.

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 26 Simple derivation of kinetic BAAE. The derivation closely follows that for BAE/KBM with circulating particle response only. For Alfvénic polarization, the dominant component is flute-like and satisfies δψ (0) = δφ (0) (ideal Ohm s law). For appreciable acoustic polarization, we mustconsiderano(1)sidebandδφ s = e iθ δφ (+) +e iθ δφ ( ) ; i.e., thesideband does not enter as first order modulation as for BAE/KBM. Note that the δψ s sideband component is negligible at low-β, due to the vorticity equation constraints. At the lowest order of the β 1/2 asymptotic expansion, solution of the quasineutrality condition implies δψ (0) = δφ (0) (as for the BAE/KBM problem) and D m 1 (ω/ω (±) ti )δφ (±) = 0 So, no connection is provided at the lowest order between δφ (±) and δφ (0). This shows that, at the lowest order, the true acoustic mode (no SAW coupling) is merely the usual (sideband) e.s. drift-wave, given by D m 1 (ω/ω (±) ti ) = 0.

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 27 At the next O(β 1/2 ) order, the solution of the quasi-neutrality condition gives δφ (±) = i ct i N m (ω/ω (±) ti ) k r δφ (0) eb 0 D m 1 (ω/ω (±) ωr 0 i.e. the same link between sidebands and flute-like component that was written for the BAE/KBM case. The only difference is that, in the present case, we may generally have δφ (±) δφ (0) for D m 1 (ω/ω (±) ti ) β 1/2, or even δφ (±) δφ (0) for a purely electrostatic polarization [Zonca and Chen PPCF 96]. The final consistency condition is obtained from the solution of the vorticity equation up to O(β), which yields the same expression for Λ 2, obtained above. ti )

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 28 Summarizing: The BAAE dispersion relation reduces trivially to the GFLDR in the limit where trapped particle dynamics are neglected; however, they play crucial roles for ω < ω ti [Chavdarovski PPCF 09, Lauber et al PPCF 09]. Both SAW and acoustic polarizations are considered on the same footing in the GFLDR framework. The notion of the low frequency Kinetic Thermal Ion (KTI) gap [Chen and Zonca NF 07] is the most general and appropriate for interpretation of experimental data. The acoustic polarization has stronger Landau damping due to the typically lower fluctuation frequency and the stronger a.c. electric field component

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 29 Numerical solutions for the kinetic spectrum Note the stronger damping of the acoustic/mixed polarization branch (logscale), consistent with theoretical predictions. This is the main reason why the Alfvénic branch is the one of practical interest for interpreting experimental observations for meso-scale fluctuations, driven by energetic particles. Micro-scales typical of micro-turbulence are not treated here. Fixed parameters are v 2 Ti /v2 A = 0.01, ω ni/ω Ti = 0.1, ω Ti /ω Ti = 0.2, q = 2 and τ = 2.

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 30 From theory to experiment: one single view... A variety of experimental observations have renewed the interest in the detailed structures of the Alfvén continuum at low frequencies: finite frequency fishbone oscillations at the GAM frequency and lowfrequency 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, TEXTOR) evidence of GAM structures (DIII-D, CHS, JFT-2M, HL-2A, AUG, T10, TEXT)

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 31 New interest was attracted by observations of BAAE [Gorelenkov etal 07] at frequencies below the BAE accumulation point and more recent evidence of Sierpes modes in ASDEX Upgrade [Garcia-Munoz PRL 08] interpreted as BAE excited by energetic ions generated by ICRH (ICRH) [Ph. Lauber NF 08], and of ICRH driven BAE in Tore Supra [R. Sabot et al. NF 09]. For interpretation of these experimental data, it is necessary to use kinetic theories for the proper treatment of wave-particle resonant interactions with thermal particles. 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 e.g. 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.

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 32 Structures of the low-frequency SAW spectrum Low-frequencyShearAlfvén 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 (, et al. 1996, PPCF 38 2011) new low-freq. gap! Kinetic Thermal Ion (KTI) gap (L. Chen 2007, NF 47 S727) 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 (, et al. 1999, POP 6 1917) unstable SAW accumulation point localization unstable discrete AITG mode For physics analogy: BAE GAM degeneracy (, et al. 2006, PPCF 48 B15); (L. Chen, et al. 2007, NF 47 S727).

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 33 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 (, 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)

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 16 R. Nazikian, et al. 06, PRL 96, 105006 R. Nazikian, et al. 06, PRL 96, 105006

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 28 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 (, 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) Theory describes well the nonlinear excitation of BAE modes in FTU by magnetic islands (S.V. Annibaldi et al. 07, PPCF 49, 475), when FLR/FOW effects are included (, et al. 98, PPCF 40, 2009).

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 29 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). Recent theoretical descriptions by [A. Biancalani et al. 10, 11]. n = -1 HF mode n=+1 n=-1, m=-2 tearing mode Locking & unlocking P. Smeulders, et al. 2002, ECA 26B, D5.016

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 30 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

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 31 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 (, et al. 2000, Theory of Fusion Plasmas, 17) EPM NL dynamics (Lecture 6) AITG and strongly driven MHD modes behave similarly

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 32 Crucial points and Summary There is unified theoretical framework that allows explaining a variety of experimental observations of shear Alfvén waves with one single fishbonelike dispersion relation. The various phenomenologies are different only apparently. The existence of continuous spectra made of radial singular structures, both for GAM and SAW, plays a crucial role in experimental observations that depend on the source spatial profile and temporal coherence There is a relationship of MHD and SAW in the kinetic thermal ion frequency gap with microturbulence, Zonal Flows and Geodesic Acoustic Modes, which has importance in determining long time scale dynamic behaviors in burning plasmas. (Un)expected behaviors: ITG turbulence enhanced by EGAM [D. Zarzoso, submitted to PRL].

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 33 References and reading material and L. Chen, Structures of the low frequency Alfvén continuous spectrum and their consequences on MHD and micro-turbulence, CP1069, Theory of Fusion plasmas: Joint Varenna-Lausanne International Workshop edited by O. Sauter, X. Garbet and E. Sindoni (AIP, 2008) p. 355-60. L. Chen and A. Hasegawa, Phys. Fluids 17, 1399, (1974). A. Hasegawa and L. Chen, Phys. Fluids 19, 1924, (1976). L. Chen, R.B. White and M.N. Rosenbluth, Phys. Rev. Lett. 52, 1122, (1984) et al, Plasma Phys. Control. Fusion 38, 2011, (1996) et al, Phys. Plasmas 6, 1917, (1999), S. Briguglio, L. Chen, G. Fogaccia and G. Vlad, Theoretical Aspects of Collective Mode Excitations by Energetic Ions in Tokamaks, Theory of Fusion Plasmas, pp. 17-30, J.W. Connor, O. Sauter and E. Sindoni(Eds.), SIF, Bologna,

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 34 (2000). L. Chen and, Nucl. Fusion 47, S727, (2007) G. Fu, Phys. Rev. Lett. 101, 185002 (2008). and L. Chen, Europhys. Lett. 83, 35001 (2008). Z. Qiu, and L. Chen, Phys. Plasmas 19, 082507 (2012). I. Chavdarovski and, Plasma Phys. Controlled Fusion 51, 115001 (2009). Ph. Lauber et al, Plasma Phys. Controlled Fusion 51, 124009 (2009). Ya. I. Kolesnichenko, V. V. Lutsenko and R. B. White, Nucl. Fusion 50, 084017 (2010). N. N. Gorelenkov et al., Phys. Plasmas 16, 056107 (2009). N. N. Gorelenkov, H. L. Berk, E. Fredrickson and S. E. Sharapov, Phys. Lett. A 370, 70 (2007)

Max-Planck-Institut für Plasmaphysik Lecture Series-Winter 2013 Lecture 4 35 M. Garcia-Munoz et al., Phys. Rev. Lett. 100, 055005 (2008). Ph. Lauber and S. Günter, Nucl. Fusion 48 084002 (2008). R. Sabot et al, Nucl. Fusion 49 085033 (2009).