Lecture 11. Instabilities and transport in burning plasmas. Fulvio Zonca

Transcription

1 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 1 Lecture 11 Instabilities and transport in burning plasmas Associazione Euratom-ENEA sulla Fusione, C.R. Frascati, C.P Frascati, Italy. March 28.th, 2013 Tor Vergata University of Rome, MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico March 2013

2 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 2 Lecture Plan Part I 1. Drift and drift Alfvén waves (fluid and kinetic instabilities) 2. Charged fusion products as source of instability in thermonuclear plasmas 3. Kinetic energy principle applied to shear Alfvén and MHD waves: unified description of low frequency waves Part II 4. Qualitative estimate of fluctuation induced transport: random walk and mixing length and their intrinsic limits 5. Role of zonal structures: new paradigm for turbulent transport 6. Resonant particle transport: the fishbone mode case 7. Nonlinear coherent interaction and avalanche transport: nonlinear Schrödinger equation models

3 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 3 Teaching Material Sources: Large portions of these lecture notes are based on Lectures I gave for the IFTS Intensive Courses on Advanced Plasma Physics-Spring 2009 and 2010 at the Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou, China. These Lecture Notes are all available online by clicking on the given hyperlinks. Similar material, also available online, can be obtained from the Lecture Series-Winter 2013 on Kinetic theory of meso- and micro-scale Alfvénic fluctuations in fusion plasmas, at the Max-Planck-Institut für Plasmaphysik, Garching, February 19-22, (2013). Lecture Notes: Available at electronic form on my personal webpage (follow hyperlinks) and on the 2013 ENEA MASTER webpage, hosted by the Tor Vergata University of Rome. At the end of the lecture, a list of specific reading material is given explicitly. Should you have difficulty in finding literatures and papers, please contact me at fulvio.zonca@enea.it.

4 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 4 Review of ideal MHD (MASTER Lectures L2,L4,L5,L6) Continuity equation: d dt ρ+ρ v = 0 Momentum equation: ρ d dt v = J B c Equation of state: d dt p+γp v = 0 Ohm s law: E = E+ v B c Faraday s law: E = 1 c = 0 B t p d dt = t +v Ampère s law: B = 4π c J

5 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 5 Waves in an infinite homogeneous magnetized plasma Uniform B = B 0 z (no eq. current), v 0 = 0, but finite ρ 0 and p 0. Three purely oscillation modes are found: (v 2 A = B2 0/(4πρ 0 ), v 2 S = γp 0/ρ 0 ) Shear Alfvén Wave (SAW): ω 2 = k 2 v2 A (ρ 1 = p 1 = v 1 = 0) Fast and Slow Magnetoacoustic Waves (FMW/SMW): ω 2 = (k 2 /2)(v 2 A +v2 S )[ 1±(1 α 2 ) 1/2] (α 2 = 4(k 2 /k2 )v 2 A v2 S /(v2 A +v2 S )2 ) E: Derive the three branches above. Show that, for β = vs 2/v2 A 1, the FMW becomes the Compressional Alfvén Wave (CAW) with ω 2 = k 2 va 2 and 4πp 1 /(B 0 B 1z ) = O(β); meanwhile the SMW becomes the Sound Wave (SW) with ω 2 = k 2 v2 S and v 1y/v 1z = O(β). E: Ideal MHD implies E = 0; yet it gives the SW! Solve the apparent paradox.

6 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Drift and drift Alfvén waves Roles of plasma nonuniformity and equilibrium magnetic geometry: Uniform plasmas: purely oscillating MHD waves Nonuniform plasmas: density, temperature, etc. gradients are free energy sources which can drive unstable fluctuations In toroidal systems, the situation is further complicated by equilibrium magnetic geometry: magnetic drifts and new particle orbits impact instabilities (MASTER L6) and transport (MASTER L7) Drift and drift Alfvén waves: micro-instabilities that dominate fluctuation induced transport Micro-instabilities: dominated by wavelengths of the order of (electron/ion) Larmor radii; much smaller than macroscopic system size Modification at short scale of SW (nearly e.s.) and SAW (δe 0)

7 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 7 The importance of toroidal geometry Toroidal geometry must be accounted for, since it crucially modifies Particle motions and transport processes Linear and nonlinear mode structures Plasma response: effective inertia, dispersiveness, etc.

8 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 8 Tokamak: 2D axisymmetric equilibrium (MAST.L2) Jacobian J = ( ψ θ φ) 1 B ψ = 0 B = B 1 ψ θ+b 2 φ ψ B = 0 θ B 2 = 0 B 2 = B 2 (ψ) Choose ψ = Φ p /2π B 2 = 1; Φ p = JB θdψdφ = 2π B 2 (ψ)dψ B = F(ψ,θ) φ+ φ ψ F = F(ψ) only if there are no toroidal forces on the system!

9 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 9 (4π/c)J = F φ+r 2 φ (R 2 ψ) (4π/c)J B = ψ (R 2 ψ) R 2 F F +J 1 φ θ F F = F(ψ) only if there are no toroidal forces on the system! E: What is the effect of rotation? And of anisotropic pressure tensor (still diagonal)? Straight Field line flux coordinates 2D with Generic ζ = φ ν(θ,ψ) 2π θ 0 0 ζ 2π Choose ν(θ,ψ) such that q = B ζ/b θ = q(ψ) Clebsch representation for B B = α(ψ,θ,ζ) ψ = (ζ qθ) ψ. E: Show that magnetic field lines are intersections of surfaces α(ψ, θ, ζ) = α 0 and ψ = ψ 0. Further freedom of choosing θ is used to select J = ( ψ θ ζ) 1 Examples are: J = J H (ψ): Hamada coordinates J = Ĵ B (ψ)/b 2 : Boozer coordinates

10 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Particle drifts in nonuniform systems Strongly magnetized plasmas(master L7): for(equilibrium) scale lengths much larger than ion Larmor radius and for (fluctuation) time scales much longer than the cyclotron period, particle motions are essentially free streaming along B 0 and gyromotion v = v b+v (e 1 cosα+e 2 sinα) v 2 /2B 0 = µ = const e 1 e 2 = b = B 0 /B 0 α = Ω = eb 0 /(mc) Particle drifts: at next order (in the drift parameter ρ L /R, MASTER L7) particle drifts are E B drift: v E = (c/b 2 0)E B 0 Magnetic drift (curvature and B): v d = Ω 1 b (µ B 0 +v 2 κ) κ b b (curvature) E: Show that particle drifts enter at the first order in the drift parameter.

11 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture B - drift

12 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Fluid drifts in nonuniform systems From fluid equations (MASTER L4), mn du dt = F +ne c u B d dt t +u it is generally possible to show that, under a force density F (not including Lorentz), the fluid plasma responds with a drift v F = c neb 0 F b In the case of inertia, F = mn u, the fluid drift is called polarization drift v p = 1 du b Ω dt E: Show that particle drifts enter at the first order in the drift parameter.

13 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture T - diamagnatic drift and N - diamagnatic drift In the following: intuitive derivation of ion temperature gradient (ITG) drift-wave dispersion relation, based on particle and fluid drifts and on the electrostatic approximation (MASTER L10). δe δφ E: Demonstrate that electrostatic approximation does not imply δb = 0.

14 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Ion Temperature Gradient (ITG) driven modes ITG are one of the dominant drift wave instabilities that are responsible of turbulent transport, together with Trapped Electron Modes (TEM) and Electron Temperature Gradient (ETG) driven modes They are essentially electrostatic in nature and are characterized by microscales (see later) that range from electron to ion Larmor radii.

15 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture ITG disp. relation in the local e.s. approximation Electrostatic approximation: δe δφ: need only one scalar field equation in order to derive the wave dispersion relation. Micro-instabilities in the long wavelength approximation: assume that λ < ρ i and λ λ D (Debye length; MASTER L4) Adiabatic electron response: n e = n e0 exp ( ) e δφ T e n e n e0 e T e δφ Poisson s equation (unit charge ions): ( 2 e 2 δφ = 4πn e0 1 T ) e δn i T e eδφn e0 δφ LHS RHS k2 λ 2 D

16 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Quasi-neutrality condition (unit charge ions) yields the relevant field equation and mode dispersion relation: δn i n e0 = eδφ T e Need only to solve for ion dynamics (adiabatic electrons). From linearized (in the fluctuation strength) ion continuity equation, assuming δφ exp( iωt) (stability problem): ( ) ni iωδn i + δ(n i v i )+B 0 δ v i = 0 B 0 E: Derive the (quasi-neutrality) equation above step by step, noting that and are defined with respect to the equilibrium magnetic field B 0 and = b.

17 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Perpendicular ion dynamics: the equilibrium particle drifts are diamagnetic and magnetic (κ, B 0 ). Perturbed (fluctuation induced) particle drifts are δe B 0, diamagnetic and polarization drifts δe B 0 -drift: δ(n i v ie ) = (n i0 c/b 0 )b δφ n i0 δv E diamagnetic drift: δ(n i v i ) = (c/eb 0 )b δp i n i0 δv pi polarization drift: δ(n i v ip ) = (n i0 /Ω i )b t (δv E +δv pi ) E: Justify the perturbed particle drifts derived above, filling in the details step by step. Can you give a simple justification of why polarization drift is higher order with respect to δe B 0 and diamagnetic drifts? The perturbed particle drifts are incompressible to the lowest order: for this reason polarization drift plays an important role even if it enters at higher order.

18 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Using the formal presence of b in the δe B 0 and diamagnetic drifts: ( [δ(n i v ie )+δ(n i v i )] = n i0 c δφ+ 1 ) ) n i0 e δp i ( bb0 }{{} + c b δφ n magnetic curvature i0 B } 0 {{} convective effect Ion pressure perturbation is derived from the equation of state, assuming u 0, i.e., nearly incompressible response. In fact, this is shown above and is intrinsically connected with the minimization of potential energy for MHD modes (MASTER L5 and L6; see also later). t δp i +δv E p 0i = 0 δp i ω pi = ω pi +ω pi = ct i eb 0 b = ω pi e δφ E : Derivethis! p i0 ω T ) i ( i ) (operator) ( ni0 n i0 + T i T i

19 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture E: Show that in a low-pressure toroidal plasma, given the magnetic curvature κ = b b ) ( bb0 1 ( b κ+ B ) 0 2 b κ B 0 B 0 B 0 Introducing the definition of magnetic drift frequency ω di = ct ) i ( bb0 ( i ) 2cT i b κ ( i ) (operator) e eb 0 the divergence of perpendicular ion particle flow becomes (ρ 2 i Ω 2 i (T i /m i ) δ(n i v i ) = i n [( i0e 1 ω ) ( pi ω di ω ni +ω 1 ω pi )ρ 2i ] 2 δφ T i ω ω E: Using the results given above for the polarization drift and other particle drifts, fill in the details leading the above expression and, in particular, the (last) polarization contribution. Hint: remember that λ < ρ i R 0.

20 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Parallel ion dynamics: from the parallel force balance equation (MASTER L4), noting v i0 = 0 and letting v 2 ti T i /m i B 0 δ in i0 m i ωδv i = δp i en i0 δφ ( ) ni v i B 0 = i n i0evti 2 T i ω ( 1 ω ) ( ) pi 1 B 0 δφ ω B 0 T i e δn i n i0 = Note: In general λ < ρ i R 0 but λ R 0 ρ i : two scale length structures of plasma turbulence. In summary: [ ] ( 1 ω ) pi ωdi ω ( ni + 1 ω ) ( pi ρ 2 }{{ ω ω}}{{} ω ω i 2 δφ v2 ti 1 ω ) ( ) pi 1 B }{{} ω 2 0 δφ ω B }{{ 0 } curvature convection FLR compression (SW)

21 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture [ ( 1 ω pi ω This equation is substituted back into the quasineutrality condition with adiabatic electrons, δn i /n e0 = eδφ/t e (p.16), and with equilibrium quasineutrality n e0 = n i0 yields the e.s. ITG wave equation ) ωdi ω ( ω ni ω + T e T i ) + ( 1 ω pi )ρ 2i ] ( 2 δφ v2 ti 1 ω ) ( ) pi 1 B ω ω 2 0 δφ ω B 0 = 0 The role of plasma nonuniformity and (toroidal) equilibrium geometry is crucial. Difficult problem to solve even for linear stability analyses. Nonlinear theory and simulation becomes very challenging. In the local limit, ik local ITG dispersion relation (k 2 k2 ) [ ( 1 ω pi ω ) ωdi ω ( ω ni ω + T ) i T e ( 1 ω ) ] pi k 2 ω ρ 2 i + k2 v2 ( ti 1 ω ) pi ω 2 ω = 0 Recovers the sound wave (SW) in the uniform plasma limit. E: Show this!

22 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Assume cold ions, T i = 0, no curvature and finite density gradient, with ω ne = (T e /T i )ω ni, c 2 s = T e /m i and ρ 2 s = c 2 s/ω 2 i electron DW ω ne ω 1 k2 ρ 2 s + k2 c2 s ω 2 = 0 ω = ω ne ± ω 2 ne +4(1+k 2 ρ2 s)k 2 c2 s 2(1+k 2 ρ2 s)

23 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Assumeω ni = k ρ i = ω di = 0andstrongpressuregradient ω pi /ω 1 ITG mode 1+ k2 c2 s ω 2 ω pi ω = 0 3 roots ω = ( ) k c 2 2 1/3 sω pi neutrally stable! ω = ( ( ) k c 2 2 1/3 1±i ) 3 sω pi one unstable root! 2 More generally, in toroidal geometry, the situation is more complex: beyond the scope of the present lecture. Modes with higher ω pi grow faster increased transport for short wavelength modes (λ < ρ i ) micro-instabilities. Growth rates of the order of the mode frequency broad band turbulence spectrum yielding turbulent transport.

24 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Fluid and kinetic instabilities Drift waves (DW) are essentially e.s. and result from the effect of diamagnetic drifts and magnetic field geometry in nonuniform toroidal plasmas of fusion interest. DW are characterized by short wavelength (λ < ρ i ) Need of kinetic descriptions. Kinetic descriptions are also needed for proper treatment of Landau damping (MASTER L10). DW, as modification of the SW branch, are affected by Landau damping Additional motivation for DW to prefer short wavelengths. In addition to ITG, other DW are important for determining turbulent transport in tokamaks: TEM and ETG (p.14; beyond the scope of this lecture). For these DW turbulences, all considerations above apply in general.

25 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture From MHD, we know of another branch, the shear Alfvén wave (SAW; p.5), which is nearly incompressible (unlike the fast magneto-acoustic wave; MASTER L5 and L6). SAW are less affected by Landau damping than SW, since δe 0 and frequency is generally (but not always) higher SAW can produce both drift Alfvén turbulence, as well as longer wavelength fluctuations,which can interact with and be destabilized by energetic particles that are present in burning plasmas: as charged fusion products and/or from additional heating/current-drive (MASTER L1, L10).

26 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Energetic particles and Shear Alfvén Waves Possible detrimental effect of shear Alfvén instabilities on energetic ions in fusion plasmas was recognized theoretically before experimental evidence was clear SAW have group velocity v A B and of the same order of EP characteristic speed, v E v A Mikhailovskii 75 and Rosenbluth and Rutherford 75 conjecture the SAW excitation by resonant wave-particle interaction with MeV ions Topic of Lecture 2, Lecture 3 and Lecture 4 of the Lecture Series-Winter 2013 Main interest in the 60 s focused on the (electron) beam plasma system: O Neil, Malmberg, Mazitov, Shapiro, Ichimaru...

27 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture The beam-plasma system vs. EP-SAW interactions in tokamaks Similarities can be drawn but strong differences and peculiarities emerge depending on the strength of the drive: Advantages of using a simple 1-D system for complex dynamics studies Roles of mode structures, non-uniformity and geometry in determining nonlinear behaviors(lecture 5 and Lecture 6 of L.S.-Winter 2013)

28 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Conceptual breakthrough Fromp.25: SAWarelessaffectedbyLandaudampingthanSW,sinceδE 0 and frequency is generally (but not always) higher: ω 2 = k 2 v2 A k2 v2 ti for β i = 2vti/v 2 A 2 1 SAW can produce both drift Alfvén turbulence, as well as longer wavelength fluctuations,which can interact with and be destabilized by energetic particles that are present in burning plasmas: as charged fusion products and/or from additional heating/current-drive (MASTER L1, L10). Drift Alfvén waves at short wavelength (λ < ρ i ), as DW (p. 23), may have growth rates of the order of the mode frequency broad band turbulence spectrum yielding turbulent transport. However, longer wavelengths (λ > ρ E ; energetic particle Larmor radius) are also easily excited by energetic particles (see above).

29 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Energetic particle driven SAW have longer wavelength and typically smaller growth rate, γ L /ω 1, than drift Alfvén turbulence Important role of resonant particle transport (MASTER L8). Roles of the continuous SAW spectrum and self-consistent interplay of energetic particle transport and mode nonlinear dynamics. Part II.

30 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture SAW equation in low-β nonuniform plasmas Express perturbed magnetic field and velocity in terms of a vector and scalar potential ( (ω/c)δa δφ ): δb = δa δv = c B 2B δφ Derive SAW equation from: i) quasineutrality condition δj = 0 and ii) parallel Ampère s law δ J + B ( δj /B ) = 0 For SAW δa δa ˆb (E: show it!) k 2 k 2 (see later) δj (c/4π) 2 δa

31 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture The perpendicular current is obtained from the perpendicular momentum equation δj = c B 2 B d dt δv + c B 2 B δp + + J B δb (δb B) B 2 J Neglect the coupling with FMW/CAW 4πδp+BδB = 0 (E: show it!) and consider parallel Ohm s law, b δφ (1/c) δa / t = 0. Then substitute into δj = 0 (E: fill in missing algebra) and obtain (κ = b b) ( ω 2 v 2 A ) δφ + ( 1 ( )] B 0 )[ 2 B0 2 B 0 δφ 4πγ P [( 0 B B κ ) ] 2δφ+4π [( B 0 2κ ][( ) B 0 B0 2 B 0 P ] 0 ) B0 2 δφ = 0

32 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Take the limit P 0 /B ( ω 2 v 2 A ) δφ + ( 1 ( )] B 0 )[ 2 B0 2 B 0 δφ = 0 In a 1D equilibrium (cylinder or slab), take the Fourier decomposition δφ = exp(ik z z imθ)δφ m (r). For ξ m (r) = mc/(rωb 0 )δφ m (r), this equation takes the form of the Hain-Lüst equation (K Hain and R Lüst 1958, Z. Naturforsch. 13a 956) d ( α(r r 0 ) d ) ξ m (r)+βξ m (r)+γ(r r 0 )ξ m (r) = 0 dr dr For r r 0, ω 2 ω 2 A(r) α(r r 0 ) 0, ω 2 A(r) = k 2 (r)v2 A(r). Where the 2nd order ODE sees the highest order derivative vanish, we expect boundary layer with singular structures in the fluctuating field. Sign of SAW continuous spectrum. E: Show that boundary layer appears when r r 0.

33 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Canonical (Fourier) form of the Hain-Lüst equation; ω 2 A (r) = k2 (r)v2 A (r) 1 d ( r 3 (ω 2 r 3 dr A(r) ω 2 ) d ) ξ m (r) m2 1 (ω dr r A(r) ω 2 2 )ξ 2 m (r) = 0 Solution is singular near r 0 where ǫ A (r 0 ) = ω 2 A (r 0) ω 2 = 0. So ξ m (r) (radial displacement) has two scale: fast (singular) x = (r r 0 ) and slow (equilibrium) r, with x r formally Near r 0, ǫ A = ǫ A(r)x and lowest order solution is (ξ m = ξ (0) m +ξ (1) m +...) At lowest order ξ (0) m = 0; thus, ξ (0) m = C 0 (r)/ǫ A(r)lnx ξ (0) mθ = i r m xξ (0) m = i rc 0(r) mǫ A (r)x

34 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Initial value form of Hain-Lüst equation; localized solutions r 2 /m 2 2 x 1 ) (ω 2A(r)+ 2 r t 2 r ξ(r,t) = 0 Solution is found as r ξ(r,t) = C 0 (r)exp(±iω A (r)t). Time asymptotically, x ±iω A (r)t. Thus, ξ(r,t) = i C 0(r) ω A (r)t exp(±iω A(r)t) Poloidal displacement oscillates indefinitely. From ξ = 0 ξ θ (r,t) = i k θ r ξ(r,t) = i C 0(r) k θ exp(±iω A (r)t)

35 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Satellite measurements of SAW Engebretson et al. 1987

36 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Torus: Shear Alfvén continuous spectrum with gaps 4 ω 2 2 ω o ω ω nq-m gap 1.5 SAW continuum damping r(d/dr)ω A (ChenandHasegawa 1974), with r the typical mode width SAW are most easily exited near (d/dr)ω A = 0 (SAW continuum accumulation points) Equilibrium non-uniformity create the local potential well for bound states to exist Frequencygap is dueto toformation of standing waves by two degenerate counter-propagating SAW: k,m+1 = k,m, i.e. nq (m + 1) = nq + m q = (2m + 1)/(2n). (Kieras and Tataronis 1982)

37 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture The Alfvén Wave Zoology (Heidbrink 2002) Various non-uniformity effects allow different varieties of the same SAW species to exist (Zoology; Heidbrink 2002). } EPM Unique and general theoretical framework for explanation of this variety and interpretation of observations: the general fishbone-like dispersion relation (Zonca and Chen 2006) Lecture 3 in the Lecture Series-Winter 2013

38 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Typical space time scales of low frequency plasma waves Consider a magnetized plasma with a sheared magnetic field: 2D equilibrium Magnetic shear k = k (ψ p ); ψ p magnetic flux. In order to minimize kinetic damping mechanisms, compression and field line bending effects λ L, with L the system size Perpendicular wavelength λ L p /n can be significantly shorter than the characteristic scale length of the equilibrium profile L p for sufficiently high mode number n. Using the ordering k /k 1 and k L p 1, the 2D problem of plasma wavepropagationcanbecastintotheformoftwonested1dwaveequations: parallel mode structure radial wave envelope.

39 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Fourier harmonics δφ m (r,t) have two scale structures: (nq ) 1 due to 1 < k qr = (nq m) < 1: mode-structure L A L p due to equilibrium variation: radial envelope

40 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Eikonal Ansatz for the radial envelope make it possible to solve the 2D problem of plasma wave propagation in the form of two nested 1D wave equations: provided (k r nq θ k ) nq θ k (nq θ k ) 2 1 2DPDE L( t, r, θ ;r,θ)δφ = 0 {}}{ symmetric 1DODE L( t, η,θ k ;r,η)a(r,t)δφ(η,r,t) = 0 {}}{ symmetric }{{} δφ(η,r,t)l( t, η,θ k ;r,η)a(r,t)δφ(η,r,t)dη = 0 D = δw f +δw k iλ 1DΨDE D( t,θ k ;r)a(r,t) = 0

41 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture General (unified) kinetic energy principle In general, demonstrate that the mode dispersion relation can be always written in the form of a fishbone-like dispersion relation (Chen et al 1984) 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 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 (Chen 94).

42 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 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. The fishbone-like dispersion relation demonstrates the existence of two types of modes (note: Λ 2 = k 2 q2 R 2 0 is SAW continuum; see later): a discrete gap mode, or Alfvén Eigenmode (AE), for IReΛ 2 < 0; an Energetic Particle continuum Mode (EPM) for IReΛ 2 > 0. For EPM, the iλ term represents continuum damping. Near marginal stability (Chen et al 84, Chen 94) IReδW k (ω r )+δw f = 0 determines ω r γ/ω r = ( ω r ωr IReδW k ) 1 (IImδW k Λ) determines γ/ω r

43 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 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 Λ 2 = λ 2 0(ω l ω) ; ω > ω l Λ i Λ 2 δw f +IReδW k < 0 when AE frequency is below the continuum accumulation point: inertia in lower than field line bending Λ 2 = λ 2 0(ω ω u ) ; ω < ω u Λ i Λ 2

44 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture For AE, iλ represents the shift of mode frequency from the accumulation point For both AE and EPM, the SAW accumulation point is the natural gateway through which modes are born at marginal stability (note, however, that unstable continuum may exist; see Lecture 4 in the Lecture Series-Winter 2013). 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.

45 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Qualitative estimate of fluctuation induced transport Assuming a turbulent bath of fluctuations δφ(k) (here k stands for both k and ω k ), the growth of δφ(k) due to the available free energy source (in the absence of fluctuation induced transport), may be written as d δφ(k) = γ(k)δφ(k) dt In the presence of fluctuations, turbulent transport competes with the fluctuation growth. Assuming that transport is diffusive, with effective diffusion coefficient D, one may estimate the reduction to the fluctuation growth as Dk 2. Thus, d dt δφ(k) = ( ) γ(k) Dk 2 δφ(k)

46 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture In stationary conditions, the previous equation allows to estimate the fluctuation induced transport coefficient as ( ) γ(k) D max mixing length estimate k k 2 From MASTER L7, this result could be obtained from a random walk transport process characterized by a typical step-size k 1 and a timestep γ(k) 1. This picture is by far too simplistic: Mixing length estimate predicts transport only in regions that are unstable to fluctuations: this is in contrast with experimental observations and numerical simulation results Transport in stable regions may be due to turbulence spreading: long time scale behaviors in burning plasmas and complex systems Special role of zonal structures: linearly stable structures with poloidal and toroidal symmetric nature nonlinear equilibria

47 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture The effect of zonal structures on turbulent transport Zonal structures are toroidally and poloidally symmetric structures: notably zonal flows, zonal currents, corrugation of radial profiles. Zonal Flow Zonal Current [More generally: phase-space zonal structures] Zonal structures scatter turbulence to shorter wavelength (stable) spectrum Mechanism for regulation of turbulence fluctuation level and turbulent transport.

48 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Zonal Flows (ZF) are common in plasmas Zonal Flows on Jupiter Drift Waves Drift waves + Zonal flows Paradigm Change P.H. Diamond, et al PPCF 47, R35 ZFs peculiarities No direct radial transport No linear instability Turbulence driven

49 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture ZFs regulate turbulence: effect on transport Z. Lin, et al. 1998, Science 281, 1835 Transport: local process predicted to scale as I: confirmed by numerical simulations (Z. Lin, etal. 1999, PRL83, 3645; 2004, PoP11, 1099) Drift wave intensity is determined by global equilibrium properties: turbulence spreading (X. Garbet, et al. 1994, NF 34, 963; P.H. Diamond, et al. 1995, PoP 2, 3685) Any size scaling of turbulent transport can be reduced to dependence of I on L.

50 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Resonant particle transport: the fishbone case Experimental observation of fishbones in PDX [McGuire et al. 83] with macroscopic losses of injected fast ions...

51 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Followed by numerical simulation of the mode-particle pumping (secular) loss mechanism [White et al 83] and the theoretical explanation of the resonant internal kink excitation by energetic particles and the (model) dynamic description of the fishbone cycle [Chen, White, Rosenbluth 84]

52 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Coherent self-consistent nonlinear interactions Within the approach of p.40, it is possible to systematically generate standard NL equations in the form (expand wave-packet propagation about envelope ray trajectories) of nonlinear Schrödinger equations: drive/damping }{{} { ω 1 t γ ω ξ D(ω+i t,r, r )A(r,t) = NL TERMS potential well }{{} } λ nq r +i(λ+ξ)+i θ k (nq θ k ) 2 2 r A(r,t) = NLTERMS {}}{{}}{ group vel. (de)focusing θ k solution of D R (r,ω,θ k ) = 0 and ( ) θ 2 λ = k 2 D R / θk 2 2 ω D R / ω ; ξ = θ k( D R / θ k ) θk 2( 2 D R / θk 2) ω D R / ω ; γ = D I D R / ω

53 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Zonca et al. IAEA, (2002) Avalanches and NL EPM dynamics x , 4 9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4 φ m,n (r) t/τ A0 = X1 t=60 x , 4 9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4 φ m,n (r) t/τ A0 = X10 t= , 4 9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4 φ m,n (r) t/τ A0 = X30 t= δα H r/a δα H r/a δα H r/a NL distor tion of free energy SR C r/a r/a r/a Importance of toroidal geometry on wave-packet propagation and shape

54 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture Vlad et al. IAEA-TCM, (2003) Propagation of the unstable front r max 0.85 [d(rn H )/dr] 0.80 [d(rn H )/dr] max linear phase t/τ A convective phase diffusive phase linear phase t/τ A convective phase diffusive phase Gradient steepening and relaxation: spreading... similar to turbulence

55 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture EPM solitons and convective amplification Detailed analyses are given in Lecture 6 in the Lecture Series-Winter Look for solutions that can be expressed as the convectively amplified propagating (self-similar) wave-packet A(ξ,t) = U(ξ)e t γ(t )dt W(ξ)e iϕ(ξ)+ t 0 γ(t )dt, ξ ξ 0 k n0 sk ϑ (x x 0) k n0 sk ϑ ( t ) x sk ϑ v g (t )dt 0 with k n0 denoting the nonlinear wave-vector and v g the nonlinear group velocity., Adopt the general procedure for isolating the behavior of the wave-packet soliton; i.e., balance the nonlinear term with the linear dispersiveness.

56 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture This optimal balance fixes k n0 v g, while the nonlinear group velocity is set by maximum wave-particle power transfer. k 2 n0v 2 g = 3π(r/R 0) 1/2 α H 2 2κ(s) ( ω t ) 0 k 4 ω ϑveρ 2 2 LEexp 2 γ(t )dt df 0, The shape U(ξ) of the self-similar wave-packet satisfies the NL equation 2 ξu = λ 0 U 2iU U 2. The value λ i1.32 corresponds to the ground state of the corresponding complex nonlinear oscillator.

57 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture On the nonlinear Schrödinger equation The EPM soliton formation and convective wave-packet amplification yield 2 ξu = λ 0 U 2iU U 2. This is similar but not the same as the equation a nonlinear oscillator in the so-called Sagdeev potential V = ( U 2 +U 4 )/2, which generates the equation of motion 2 ξu = U 2U 3, and gives U = sech(ξ). This is the solution that appears in the problem of ITG turbulence spreading via soliton formation [Z. Guo et al PRL 09].

58 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture More generally, this form appears in soliton-like solutions of the nonlinear Schrödinger equations; e.g. the Gross-Pitaevsky equation describing the ground state of a quantum system of identical bosons [Gross 61; Pitaevsky 61] the envelope of modulated water wave groups [Zakharov 68] thepropagationoftheshortopticalpulseofafelinthesuperradiant regime [Bonifacio 90] The complex nature of EPM equation is novel and connected with the peculiar role of wave-particle resonances, which dominate the nonlinear dynamics of EPMs via resonant wave-particle power exchange, whose maximization yields two effects: the mode radial localization, similar to the analogous mechanism discussed for the linear EPM mode structure the strengthening of mode drive Imλ 0 > 0, connected with the steepening of pressure gradient, convectively propagating with the EPM wave-packet

59 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture References and reading material J.P Freidberg, Ideal Magneto Hydrodynamics, Plenum Press, 1987 L. Chen, Waves and Instabilities in Plasmas World Scientific Publication Co., 1987 R. B. White, Theory of Tokamak Plasmas, North Holland, 1989 T.M. O Neil, Phys. Fluids 8, 2255 (1965). T.M. O Neil and J.H. Malmberg, Phys. Fluids 11, 1754 (1968). T.M. O Neil, J.H. Winfrey and J.H. Malmberg, Phys. Fluids 14, 1204 (1971). M.J. Engebretson et al., J. Geophys. Res. 92, (1987). L. Chen and A. Hasegawa, Phys. Fluids 17, 1399 (1974). C. E. Kieras and J. A. Tataronis, J. Plasma Phys. 28, 395 (1982). W. W. Heidbrink, Phys. Plasmas 9, 2113 (2002).

60 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture F. Zonca and L. Chen, Plasma Phys. Control. Fusion 48, 537 (2006). L. Chen, Phys. Plasmas 1, 1519 (1994). R.B. White, R.J. Goldston, K. McGuire K. et al., Phys. Fluids 26, 2958, (1983) L. Chen, R.B. White and M.N. Rosenbluth, Phys. Rev. Lett. 52, 1122, (1984) S. T. Tsai and L. Chen, Phys. Fluids B 5, 3284 (1993) F. Zonca and L. Chen, Phys. Plasmas 3, 323, (1996) F. Zonca, S. Briguglio, L. Chen, G. Fogaccia and G. Vlad, Nucl. Fusion 45, 477, (2005) F. Zonca, S. Briguglio, L. Chen, G. Fogaccia and G. Vlad, Theoretical Aspects of Collective Mode Excitations by Energetic Ions in Tokamaks ; in Theory of Fusion Plasmas, pp , J.W. Connor, O. Sauter and E. Sindoni (Eds.), SIF, Bologna, (2000). Z. Guo, L. Chen and F. Zonca, Phys. Rev. Lett. 103, (2009).

61 MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture F. Zonca and L. Chen, Phys. Plasmas 7, 4600 (2000). E. P. Gross, Nuovo Cimento 20, 454 (1961). L. P. Pitaevsky, Sov. Phys. JETP 13, 451 (1961). V. E. Zakharov, J. Appl. Mech. Tech. Phys. 9, 190 (1968). R. Bonifacio, L. De Salvo, P. Pierini and N. Piovella, Nucl. Instrum. Methods Phys. Res., Sect. A 296, 358 (1990).