Physics of RF Heating

1. Physics of RF Heating J.-M. Noterdaeme with the support of M. Brambilla, R. Bilato, D. Hartmann, H. Laqua, F. Leuterer, M. Mantsinen, F. Volpe, R. Wilhelm Max Planck Institute for Plasmaphysics, Garching Avanced Course for the European Fusion Doctorate October 2008

2. Neutral Beam Injection Beam duct Ion source Accellerator Neutraliser Neutral beam Magnetic filter Ionisation Thermalisation Electricity -> other form (kinetic energy of particles) Transport to plasma (outside part) (inside part)

3. R Resonance zone Wave heating Antenna Wave to particles Electricity -> other form (electromagnetic oscillations) Transport to plasma (outside part) transmission lines antenna (inside part) waves Thermalisation

4. Wave propagation and absorption sets the frequency range that can be used. Wave heating: very tight combination of physics and technology

5. What will be addressed here? • plasma • (coupling) • waves • absorption how, why those frequencies, mechanisms, practical applications • with side glances at • technology, to show that we can do it • experiments, to show that it works • emphasize • concepts • physical understanding • goal • working knowledge for experiments

6. Waves • waves in neutral gas • pressure waves • EM waves in vaccuum • waves in medium with free charged particles, magnetised successive approximations/simplifications • electrons, several ion species, arbitrary distribution function plasma kinetic theory • electrons, ions (one or more types), fluid equations cold plasma -> two fluid -> CMA diagram • one electrically conducting fluid MHD

7. In general: approximations in plasmas Kinetic equation (full distribution function) how important are collisions? not important important induced fields wrt applied fields? how dominant are collisions? not important important not dominant  >  dominant  <  partice orbit theory warm plasma wave theory cold plasma wave theory MHD theory

8. Approximations elctrons + (multiple) ions cold -> “fluid” equations, no temperature effect warm -> finite larmor radius plays a role • coupling • cold plasma • propagation • cold plasma • (warm plasma) • power absorption • warm plasma EC IC (EC) (IC) EC IC

9. Power Absorption • energy in the wave can be affected by energy transfer between wave and particles(wave-particle resonance) or other wave(wave resonance) • absorption mechanism always particles • note : • wave-particle resonance : only few particles fulfill resonance conditions • wave resonance : collective effect : all particles fulfill conditions

10. Interaction between Absorption and Waves • wave fields • absorption • change in distribution function Wave fields affected by change in distribution function Absorption affected by change in distribution function

11. R Resonance zone Transport of energy in plasma: waves Antenna Wave to particles Electricity -> other form (electromagnetic oscillations) Transport to plasma (outside part) transmission lines antenna (inside part) waves Thermalisation

12. Wave Equation Maxwell‘s Equations generalized Ohm‘s law

13. Dispersion relation • set of homogenous, linear equations for Ex, Ey, and Ez, • has non trivial (different from 0) solutions provided the determinant vanishes • det = 0 is known as the dispersion relation

14. k k|| Plane waves fronts of constant phase Plane waves k External magnetic field B0 • fixed by „generator“ k response of plasma equivalent k=k(w) dispersion relation D(w,k)=0

15. Wave Equation Maxwell‘s Equations generalized Ohm‘s law Plane waves

16. Wave Equation generalized Ohm‘s law Maxwell‘s Equations Plane waves

17. Wave Equation Dispersion relation • the equation is exact • will become approximate only in terms of the model used for

18. Propagation in cold unmagnetized Plasmasimply depends on wp k || E : k (current carried only by electrons) (equ. of motion of electrons) Plasma frequency: EM waves Langmuir oscillations

19. Wave cutoff and resonance reflexion tunnelling vph>c! „cutoff“ 1. N  0 „resonance“ 2. N  vph -> 0 wave „gets stuck“ wave energy dissipation

20. k2perp k2inf x Wave resonance thermal effects become important • wave resonance • Energy density ~ A2 • Energy flux vA2 => • when v -> 0 then A must increase -> damping mechanisms amplified • v -> 0 also means l-> 0 < r • simple example

21. Ohm‘s law 0 Goal: determine j=j(E) Small perturbation cold plasma equation of motion of particles solve for v1 as function of E1 cold current

22. Generalisation Fokker - Planck Vlasov linearized Vlasov equation warm plasma • 6 waves: • 2 cold • pressure driven • electron and ion Bernstein • accoustic branch

23. Continuing the cold plasma case

24. Characteristic frequencies Plasmafrequencies Cyclotron frequencies Upper Hybrid frequency Lower Hybrid frequency

25. Ordering in tokamaks for the electrons further

26. - ~ E B0 + Wave propagation and absorption • ECRH • electron cyclotron resonance heating • LH • lower hybrid frequency • ICRF • Ion cyclotron range of frequencies Transport from outside plasma to inside: wave propagation (wave cut-off and resonance) Transfer of energy from wave to particles: resonance condition (wave-particle) ion unmagnetized, oscillate with E1 electrons oscillate with E1 x B0 drift

27. Electron / Ion Cyclotron Range • electron cyclotron • only electron dynamics (ions fixed background) • v >> vthe • relativistic effects • ion cyclotron • ions (multiple) and electron dynamics • characteristic propagation velocity is the Alfven speed • comparison with the electron thermal velocity

28. Approximations made • plane wave: ignored initial conditions • Te= Ti = 0 • infinite plasma: ignored boundary conditions • homogeneous in space: equilibrium values are constant • B = B0, uniform, static • no free streaming • no dissipative effects, no collisions, no forces quadratic in v • small amplitude B0 >> B1 Consequences • no finite temperature effects • no streaming effects such as • sound waves • particle bunching • Landau damping • shock waves

29. Dispersion relation, cold plasma case with 2 solutions for N2 form of solution depends on S, P, R, L, 

30. k2perp k2inf x • 2 solutions for (k/k0)2, function of  • if > 0 -> propagating • if < 0 -> non-propagating • propagating solution k/k0 = +/- • waves travelling in opposite directions • if two propagating solutions • smaller v  = w /k -> slow wave • larger -> fast wave • (k/k0)2 can change sign • by going through 0 -> cut-off • reflection • evanescent wave • by going through infinity -> resonance • absorption • reflection and transmission • cut-off -> independent of angle • resonance -> depending on angle Cut-off Resonance

31. Classification of waves • phase velocity • fast • slow • direction of propagation • k parallel to B0: according to polarisation (with respect to B0, in other wrt propagation direction) • right -> direction of rotation of electrons • left -> direction of rotation of ions • k perpendicular to B0 • ordinary: E1 // B0 • extraordinary: E1 perp to B0

32. k  B w X-Mode 3 Frequency ranges for Plasmaheating O-Mode wUH UHR wce UH-Wave wpe ECRH LH ICRH LH-Wave c wLH LHR wci Alfvèn-Wave k 0 Parallel and perpendicular propagation k || B w R-Wave L-Wave wce ECR Electron- Cyclotron-Wave c Whistler-Wave wci ICR Ion- Cyclotron-Wave Alfvèn-Wave k 0 cut-off Resonance

33. k || B w R-Wave L-Wave wce ECR Electron- Cyclotron-Wave c Whistler-Wave wci ICR Ion-Cyclotron-Wave Alfvèn-Wave k 0 k  B X-Mode w wUH O-Mode wce UHR UH-Wave wpe wLH c LH-Wave LHR wci Alfvèn-Wave k 0

34. R • 2 solutions for (k/k0)2, function of , n, B • if > 0 -> propagating • if < 0 -> non-propagating • propagating solution k/k0 = +/- • waves travelling in opposite directions • if two propagating solutions • smaller v  = w /k -> slow wave • larger -> fast wave • (k/k0)2 can change sign • by going through 0 -> cut-off • reflection • evanescent wave • by going through infinity -> resonance • absorption • reflection and transmission • cut-off -> independent of angle • resonance -> depending on angle L B O X L R B X

35. wce2 wci2 w2 w2 wp2 w2 cut-off Resonanz =>Magnetic field CMA diagram L= 1 • 2 solutions for (k/k0)2, function of , n, B • if > 0 -> propagating • if < 0 -> non-propagating • propagating solution k/k0 = +/- • waves travelling in opposite directions • if two propagating solutions • smaller v  = w /k -> slow wave • larger -> fast wave • (k/k0)2 can change sign • by going through 0 -> cut-off • reflection • evanescent wave • by going through infinity -> resonance • absorption • reflection and transmission • cut-off -> independent of angle • resonance -> depending on angle P=0 S=0 1 R= => Plasma density

36. Easier to show with mi / me = 2.5

37. CMA detail

38. Use of the CMA diagram B(R) Wave wce/w (=Magnetic field) wce/w (=Magnetic field) O-/X-Mode with kB High field- upper Hybrid - Resonanz = "ordinary wave" (O-Mode) mit E || B „ECR“ 1 O-Mode cut-off Low field- coupling X- cut off 0 = “extra-ordinary wave” (X-Mode) mit E  B 1 0 2 (=Density) path through the Plasma!

39. wce/w (=Magnetic field) B(R) Wave wpe /w 2 2 ECR O-/X-Mode with kB High field- upper Hybrid - Resonanz „ECR“ 1 O-Mode cut-off Low field- coupling X- cut off 0 1 0 2 (=Density) path through the Plasma! X1-Mode O-Mode EC-“Resonance“ EC-“Resonance“ “HF-cut-off“ O-Mode cut-off ..if ne >ncrit B UH-Resonanz B E  B E || B B(R) B(R) ne ne R R

40. R Resonance zone Wave heating Antenna Wave to particles Electricity -> other form (electromagnetic oscillations) Transport to plasma (outside part) transmission lines antenna (inside part) waves Thermalisation

41. Wave propagation • approximations • wave equations • dispersion relation • characteristic frequencies • classification of waves • parallel and perpendicular propagation • cut-off and resonances • CMA diagram • application to ECRH • Absorption • interaction between wave and particles

42. Approximations elctrons + (multiple) ions cold -> “fluid” equations, no temperature effect warm -> finite larmor radius plays a role • coupling • cold plasma • propagation • cold plasma • (warm plasma) • power absorption • warm plasma EC IC (EC) (IC) EC IC

43. Interaction between a wave and a charged particle Force on an electron Integration along an unperturbed orbit gives for the momentum increase Energy transfer only if is satisfied With relativistic effects we have => Interaction only with resonant particles in velocity space The same is valid for ions.

44. Landau damping: Increase of parallel momentum Collisionless Damping Energy transfer only if Resonance condition: k Condition for damping The deformation of the distribution function increases the energy of the electron system.

45. Cyclotron Damping: increase of perpendicular momentum Cyclotron Damping (Doppler shifted) Energy transfer only if Resonance condition:

46. Interaction between a wave and a charged particle Force on an electron Integration along an unperturbed orbit gives for the momentum increase Energy transfer only if is satisfied With relativistic effects we have => Interaction only with resonant particles in velocity space The same is valid for ions.

48. 3 frequency regions for plasma heating: w X-Mode ECRH LH ICRH wR O-Mode w=wUH wce wpe Upper Hybrid Wave Lower Hybrid Wave w=ck wL w=wLH wci Alfvèn-Wave k 0 Wave propagation and absorption B0 ne r/a 1 X-mode Cutoff UHR O-wave X1-wave X1-wave X2-wave Upper hybrid resonance EC-resonance

49. Mode Conversion OXB-Heating EC-resonance Mode conversion process under certain launch angles and for minimum density. UH-resonance O-mode O mode converts into X mode at O-mode cutoff. X-mode converts into electrostatic electron (Bernstein) wave. Bernstein wave absorbed by electron cyclotron damping. No upper density limit. B-mode X-mode O-mode cutoff X-mode cutoff

50. O-X-B-Mode O2-Mode X2-Mode 1019 1020 1021 Plasma density ECRH: Operation Scenarios for W7-X Plasma density range Determines the microwave frequency: (2.5 T, n=2, 140 GHz for W7-X) Determines the density range Cyclotronfrequency: Plasmafrequency: (m-3)