Waves in Plasmas. Francesco Volpe Columbia University. Mirai Summer School, 9-10 August 2012

Transcription

1 Mirai Summer School, 9-10 August 01 Waves in Plasmas Francesco Volpe Columbia University Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 1

2 Itinerary to New York Collective Thomson Scattering measurement of T i 1998 Liquid Metal Flow Simulations Univ. Trieste, Italy Electron Bernstein Wave (EBW) Emission and Current Drive (CD) W7-AS Stellarator (Garching, Germany) EBW Emission and Heating MAST Spherical Tokamak (Culham, UK) Electron Cycl.CD Stabilization of Neoclassical Tearing Modes Neoclassical Tearing Mode and Locked Mode control ITER Tokamak DIII-D Tokamak (San Diego, USA) 008 Electron Cyclotron Resonant Heating AUG Tokamak (Garching, Germany) 009 EBWs MST RFP (UW-Madison) Collaboration on ECE and MHD FTU Tokamak (Frascati, Italy) DIII-D Tokamak Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas

3 3 rd Part nd Part 1 st Part Outline Need for Heating Ohmic Heating and Need for Auxiliary Heating Neutral Beam Injection Waves Propagation Absorption Heating by Waves Motivated student + interesting topic Ion Cyclotron Lower Hybrid Electron Cyclotron a-particle Heating Current Drive Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 3

4 Motivation for Heating Plasma forms at few ev, but D-T nuclei fuse at 0-300keV Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 4

5 Ohmic Heating Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 5

6 Plasma Current confines Tokamak Plasma and heats it by Joule Effect Plasma = secondary of a transformer I plasma B q Bj a R F Ohmic heating: Plasma stability requires: q r a a B R B Therefore: j p oh B 0 R j I oh Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 6

7 The hotter the plasma, the more difficult to Ohmically heat it. Dominant loss mechanism: bremsstrahlung loss power p b. p b Z n e T e [Wm -3,m 3,keV] Ohmic heating power: p oh j Plasma resisitivity: T e 3 T e 10 n e 0 1 Z B R [kev,m -3,T,m] Ohmic heating alone: T e only a few kev Need for additional heating power! Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 7

8 Introduction to Neutral Beam Injection Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 8

9 Energetic Neutral Particles penetrate Plasma and release Energy 1. Injection of neutral fuel atoms (H, D, T) at high energies (E b > 50 kev). Ionization in the plasma H,D,T B-field 3. Beam particles confined 4. Energy Release Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 9

10 Energy is released by Collisions and Charge Exchange Ion collision: Electron collision: Charge exchange: H H fa st H fast fast H e H H H H fa st fast fast H H e e Example: beam intensity: (x) I exp x / I 0 E b0 = 70 kev tot n m 3 m 1 n tot 0.4m In large reactor plasmas: beam cannot reach core! Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 10

11 NB Injector = Accelerator + Neutralizer Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 11

12 Wave Propagation Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 1

13 Maxwell + Ohm Wave Equation Maxwell s Equations for Plane Waves ik E ib ik H i 0 0 ik E ik B 0 E j and Generalized Ohm s law k E k j, k,, where = conductivity tensor, combine in the wave equation k k E, k K E, 0 k K 1 K : dielectric tensor c i 0 Dispersion relation det N 1 N K(, N) 0 N k c Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 13

14 j (thus, thus Disp.Rel.) derived from Lorenz force dv q s (E vb) dt ms 0 Linearized (small perturbations, v v 0 v 1 ) E E 0 E 1 B B 0 B 1 and Fourier-transformed: q iv j 1 s q s n s v s 1 E1 v1b0 ms Solve for v 1 and plug in 1,s where s= e, i, Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 14

15 Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 15 Propagation in cold unmagnetized Plasma simply depends on p (HF, static ions) j env m e E e v i E m n e i j e e p e e m i n ie i K 0 1 E c E k k E k p e 0 e e, p m n e Plasma frequency: Langmuir oscillations p E : k EM waves ( sol.) 1 p k c N k x,k y p E : k

16 Propagation in cold magnetized Plasma also depends on c K K K 0 xx yx K K xy yy 0 K 0 0 zz y x ps Kxx Kyy 1 s cs cs ps Kxy K yx i s ps K zz 1 s Cyclotron frequency: cs ps qsb m s 0 z B Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 16

17 Propagation in cold plasma magnetized along z Wave equation: xx yx NN where N yy N only admits solution(s) if det( ij -N d ij +N i N j )=0 xy 0 N N ps xx yy 1 s cs cs ps xy yx i s ps ps zz 1 s zz 0 N N E E E Dispersion Relation Eigen-polarizations=(Eigen)modes = solutions of Eigen-value Problem for null eigen-value, det( ij -N d ij +N i N j )=0. x y z 0 notation: N =N x N =N z Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 17

18 extraordina ry (X) mode Special case 1: perpendicular to B (N z =0) X-Mode 3 frequency regions for plasma heating: y N x z B R ce pe L O-Mode UH Upper Hybrid Wave Lower Hybrid Wave LH ECRH LH ICRH ci 0 Alfvèn-Wave k Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 18

19 Resonances and Cutoffs N 0 cutoff reflection tunnelling v ph = / k >c! N resonance v gr = / k 0 wave gets stuck wave energy Dissipation, Mode conversion Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 19

20 Special case : propagation along B (N x =0) y L R R-Wave x N B ce L-Wave ECR Electron Cyclotron Wave z Whistler-Wave solutions: Nz 1, xx i xy ci 0 ICR Ion- Cyclotron-Wave Alfvèn-Wave k of polarization: E E x y i Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 0

21 Band gaps depend on direction of propagation and polarization Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 1

22 Interferometry Because, in unmagnetized plasma, or B but O-mode, In conclusion, Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas

23 Polarimetry Note: polarization rotation is Df/ Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 3

24 Case 3: arbitrary oblique propagation: Appleton-Hartree dispersion relation After algebra, B where and Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 4

25 Upper hybrid oscillations In unmagnetized plasma or along B, B, where restoring Lorentz force is maximum, Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 5

26 Waves refracted, reflected, scattered, etc. by the Plasma extract info on n e, B, T e,i etc. pe n e c B transmitter B 0 receiver v th,i,e T e,i Transmission n e n e =n e, crit Cutoff of ordinary wave Reflectometry n e, B Interferometry n e E d z Scattering T e, T i N=1 N < 0 Absorption n e Emission T e, n e Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 6

27 In warm plasma, Lorenz force replaced by Vlasov Equation or Cold plasma theory breaks down where kv c th 1 n c 1 k v th Linearized Vlasov equation: q tf1 vxf1 m for perpendicular wavelength close to wave-particle resonance (within few Doppler widths) Lorenz force (single particle): iv s E v q m Larmor radius v B0 vf1 E1 vb0 vf0 Complicated form of K ij involving Bessel Functions New set of (electrostatic) waves q B0 ms Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 7

28 Summary for the 1 st Part Importance of heating In general, plasma heating is species-selective and anisotropic Ohmic heating suffers from unfavourable T e -dependence of resistivity and from disruption limit on max current density j Accelerator + Neutralizer = Neutral Beam Injector Maxwell, Ohm, Lorenz (and, in warm plasma, Vlasov) Dispersion relation Wave propagation, plethora of modes, resonances and cutoffs Derived Dispersion Relation for cold unmagnetized plasma; discussed results in presence of B Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 8

29 3 rd Part nd Part 1 st Part Outline Need for Heating Ohmic Heating and Need for Auxiliary Heating Neutral Beam Injection Waves Propagation Absorption Heating by Waves Ion Cyclotron Lower Hybrid Electron Cyclotron a-particle Heating Current Drive Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 9

30 Wave Absorption Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 30

31 At t=0 At t=t/4 Landau Damping (Electrostatic) waves, k E E E v ph t vt Group of resonant particles: On average: v k coll slower particles are accelerated faster particles are decelerated z z Strong wave-particle interaction if: v ph t=vt f(v) k v vk f v v ph 0 Net loss of wave energy No collisions necessary! Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 31 v ph

32 Cyclotron Damping Electromagnetic wave E k E x left-handed polarized wave z tip of electric field vector E y v ph = / k Resonance condition: k v ls where l 1,, (3,...) g Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 3

33 nd and higher Harmonics damped if wave-field non-uniform on length-scale of Larmor radius gradient of electric field l E r l v v 0 E B B B v t=0 t=t/ t=t s T=period of the wave Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 33

34 EM wave heating: waves are damped where resonance is fulfilled localized heating! iso-b lines Resonant Layer Excitation: external or at plasma edge Wave: propagating / evanescent Antenna Propagation can be complicated codes: qb c m c R Ray tracing Beam tracing Full wave Resonant particles acquire energy at expense of wave, then thermalize with the others Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 34

35 Frequencies with good absorption Cyclotron Resonance generally i th k v s 1 Then good absorption where l s Electrons: 8 GHz / B[T] Electron Cyclotron Resonance Heating ECRH Hydrogen: 15 MHz / B [T] Ion Cyclotron Resonance Heating* ICRH Landau Resonance e v th k.3ghz T kev cm Lower Hybrid Heating LH 1 e * Landau Resonance and Magnetic Pumping also contribute to ICRH Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 35

36 Ion Cyclotron Resonance Heating Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 36

37 Ion Cyclotron Resonance Heating Dispersion relation has two solutions: fast wave E B 0 n > x m -3 slow wave E B 0 n < 1 x m -3, Problem: near = ci wave is right-handed, but ions resonate with (absorb from) left-handed polarization! Solution: Inject a minority species. Wave right-handed at majority resonance, = cm but damped at minority resonance, = cm Majority heating also possible, by Doppler broadening: E + =0 at = cm, but finite at = cm kv th,i Preferrable, but needs tunnel of cutoff region cm 10cm Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 37

38 ICRH - Wave Propagation ASDEX Upgrade F. Meo, P. Bonoli Alcator C-Mod Re(E y ) Multiple current straps Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 38

39 ICRF Technology - Generator 4 amplifier chain final stage: tetrode with MW / 10 sec tunable between 30 and 110 MHz efficiency 60% final stage tetrode Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 39

40 ICRH - Wave Power Transport Transmission line Matching network ASDEX-Upgrade generator antenna 50W Coaxial transmission lines 0cm diam., low loss Matching network antenna resistance 50W dependent on plasma Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 40

41 ICRH - Wave excitation Fast wave I ant W7-AS Antenna Strap antenna E hf Faraday screen Slow wave Bant B 0 B hf k B hf E hf k B 0 Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 41

42 Lower Hybrid Heating & Current Drive Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 4

43 Lower Hybrid Heating solutions of dispersion relation: slow wave (exhibits lower hybrid res.) fast wave i LH e 10cm, 1cm n e >10 17 m -3 at antenna, to enter plasma k > k c to reach center. Lower k k Hybrid resonance sw fw k too low, power stays near plasma edge radius radius k sufficiently high, slow wave travels into plasma, absorption at LH or before Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 43

44 LH - Wave Propagation Depends on n e and B. For k > k crit v gr, v ph independent of k. all launched power flows into same direction. antenna structure Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 44

45 Klystron and Grill Beam dump 3.7 GHz 500 kw 3 sec klystron waveguide grill -wave output -wave input anode cathode Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 45

46 LH - Wave Excitation Slow wave ASDEX Multiple wave guides E wg E hf B hf k B 0 Fast wave E hf B hf k B 0 Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 46

47 Summary for the nd Part Electrons can surf electrostatic waves (Landau Damping) Or gyrate in phase with circular e.m.waves (Cyclotron Damping), and so gain energy Processes are resonant well-localized Two examples: Ion cyclotron, strap antenna Lower hybrid, microwaves, grill Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 47

48 3 rd Part nd Part 1 st Part Outline Need for Heating Ohmic Heating and Need for Auxiliary Heating Neutral Beam Injection Waves Propagation Absorption Heating by Waves Ion Cyclotron Lower Hybrid Electron Cyclotron a-particle Heating Current Drive Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 48

49 Electron Cyclotron Resonance Heating Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 49

50 Electron Cyclotron Resonance Heating Dispersion relation has two solutions for perpendicular propagation: ordinary (O)-mode E B 0 extraordinary (X)-mode EB 0 mm No low density cut-off, but high density cutoff. Ions can be assumed stationary, but relativistic electron mass has to be included. Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 50

51 ECRH - O1 mode heating Reflection at cut-off region Dispersion k c p B Density gradient leads to diffraction away from plasma center. ne O-mode cutoff Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 51

52 ECRH - X1 mode heating Resonance inaccessible from low field side because no propagation between cutoff and upper hybrid resonance. X accessible from LFS i.e. second harmonic heating ne Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 5

53 ECRH Sources: Gyrotrons Window Diamond collector B field superconducting coils resonator annular electron beam Up to 1 MW cw (>30min) Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 53

54 Mirrors correct wavefront and polarization and match sources to Transmission Lines Matching Optics Unit to HE11 line polarizer 1 phase correcting mirrors gyrotron Design IPF Stuttgart to long pulse load spherical short load pulse 1 MW, load 1 s (CNR Milano) polarizer Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 54

55 Power is transmitted by mirrors or waveguides and launched by steerable mirrors ECRH launching mirrors in sector 5 launcher mirrors: Cr / Cu / Au - coated graphite Localized, adjustable Heating & CD suppresses MHD instabilities such as Neoclassical Tearing Modes (NTMs) Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 55

56 a-particle Heating Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 56

57 a-particle Heating D-T fusion reaction: D T n (14.1MeV) leaves plasma 4 He (3.5 MeV) if heats plasma sufficient ly confined long Heating power density: 0. nd nt v E where v Ti peaked heating profile a-particles need to be well confined through large plasma currents in tokamaks optimized stellarator fields Loss mechanisms: field ripples MHD events Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 57

58 Evidence for a-particle heating D, T experiments only done on JET and TFTR JET 30 MW a-particle heating NBI heating 16 MW (max) 30 kev T i0 T e0 0 sec Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 58

59 Analogy: coal oven - fusion oven Energy Activation energy Heat Energy gain Reaction time n Sustain reaction Ignition Heating n p p n Sustain reaction Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 59

60 Current Drive Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 60

61 Non-inductive current drive Asymmetric velocity distribution can be a side effect of plasma heating. ions j q n v fv dv electrons s s s Needed for : Efficiency: Theory: Steady-state tokamak current profile control in tokamaks MHD-mode stabilisation. (bootstrap current compensation in stellarators) th j p n m v v. coll ell e n e e ll v coll 1 Experiment: 0 3 n 10 m Rm IA ex Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 61 e P W th

62 v Current drive with EM waves Parallel momentum injection: required total electron momentum 10 th v k ve + change of electron momentum / wave energy + many electrons - large fraction of trapped electrons th k ve - 3 / coll Te 3 v + coll - small change of electron momentum 4 kgm s OH 1 v v LH IC (EC) Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 6

63 Fisch-Boozer: Heating electrons of v >0 or <0 makes them less resistive net current Electron velocity distribution Resonant electrons: - n ce / g - k v =0 k 0 (oblique launch) Trapped cone Preferential heating of electrons with v >0 become less collisional less resistive net current Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 63

64 Current drive efficiencies Efficiency LHCD ICCD ECCD NICD 0.1 x Te [ 10 kev] <0.1 x Te [ 10 kev]. x Te [ 10 kev] Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 64

65 Outlook to ITER Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 65

66 ECRH in ITER will use 4 gyrotrons, connected to 3-4 Upper Launchers and 1 Equatorial Launchers Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 66

67 ECRH/ECCD in ITER will serve several purposes Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 67

68 Summary Heating Heating scheme Advantages Disadvantages Ohmic efficient Cannot reach ignition Not in stellarators NBI reliable close to torus negative ions necessary LH Efficient current drive Antenna close to plasma Off-axis ECRH ICRH Reliable flexible Ion-heating Central heating Electron heating (density limit) Antenna close to plasma Antenna coupling Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 68

69 Auto-evaluation Quiz 1. What makes auxiliary Heating necessary?. What s the only frequency relevant to propagation in the simplest plasma you can think of (cold, unmagnetized, unbounded, no impurities, etc.etc.)? 3. What s, in your opinion, the main advantage of wave heating? 4. What s the common principle shared by all CD methods? 5. Can you imagine a distribution function f(v) that delivers energy to the wave by the Landau mechanism? Answers in next slide Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 69

70 Answers to the Quiz 1. Because T e -3/. pe 3. Localized and adjustable 4. Asymmetry: asymmetric resistivity (Fisch-Boozer), asymmetric trapping (Ohkawa), uncompensated ion and electron flows (NBCD) 5. Bump-on-tail Mirai Summer School, Japan, August 9, 01 F. Volpe Waves in Plasmas 70