jmn 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

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

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

jmn Wave propagation and absorption sets the frequency range that can be used. Wave heating: very tight combination of physics and technology PhysicsTechnology         Electromagnetic energy Transmission lines Antenna Coupling Waves Waves -> Particles Transfer to bulk

jmn 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

jmn 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

jmn 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 partice orbit theory cold plasma wave theory not dominant  > dominant  < warm plasma wave theory MHD theory

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

jmn 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

jmn Interaction between Absorption and Waves wave fields absorption change in distribution function Absorption affected by change in distribution function Wave fields affected by change in distribution function

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

jmn Wave Equation generalized Ohm‘s law Maxwell‘s Equations

jmn Dispersion relation set of homogenous, linear equations for E x, E y, and E z, has non trivial (different from 0) solutions provided the determinant vanishes det = 0 is known as the dispersion relation

jmn Plane waves k External magnetic field B 0 k || kk fronts of constant phase  fixed by „generator“ k response of plasma equivalent k=k(  ) dispersion relation D( ,k)=0

jmn Wave Equation generalized Ohm‘s law Plane waves Maxwell‘s Equations

jmn Wave Equation generalized Ohm‘s law Plane waves Maxwell‘s Equations

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

jmn Propagation in cold unmagnetized Plasma simply depends on  p (current carried only by electrons) Plasma frequency: Langmuir oscillations EM waves k :E || k (equ. of motion of electrons)

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

jmn Wave resonance thermal effects become important wave resonance –Energy density ~ A 2 –Energy flux vA 2 => –when v -> 0 then A must increase -> damping mechanisms amplified –v -> 0 also means  -> 0 <  simple example x k 2 perp k 2 inf

jmn Ohm‘s law Goal: determine j=j(E)Small perturbation 0 cold plasma equation of motion of particles current cold solve for v 1 as function of E 1

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

jmn Continuing the cold plasma case

jmn Characteristic frequencies Plasmafrequencies Cyclotron frequenciesUpper Hybrid frequency Lower Hybrid frequency

jmn Ordering in tokamaks for the electrons further

jmn 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) Wave propagation and absorption - E ~ B0B0 + ion unmagnetized, oscillate with E 1 electrons oscillate with E 1 x B 0 drift

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

jmn Approximations made plane wave: ignored initial conditions T e = T i = 0 infinite plasma: ignored boundary conditions homogeneous in space: equilibrium values are constant B = B 0, uniform, static no free streaming no dissipative effects, no collisions, no forces quadratic in v small amplitude B 0 >> B 1 Consequences no finite temperature effects no streaming effects such as –sound waves –particle bunching –Landau damping –shock waves

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

jmn Resonance 2 solutions for (k/k 0 ) 2, function of  –if > 0 -> propagating –if non-propagating propagating solution k/k 0 = +/- –waves travelling in opposite directions if two propagating solutions –smaller v  =  /k -> slow wave –larger -> fast wave (k/k 0 ) 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 x k 2 perp k 2 inf Cut-off

jmn Classification of waves phase velocity –fast –slow direction of propagation –k parallel to B 0 : according to polarisation (with respect to B 0, in other wrt propagation direction) right -> direction of rotation of electrons left -> direction of rotation of ions –k perpendicular to B 0 ordinary: E 1 // B 0 extraordinary: E 1 perp to B 0

jmn Electron- Cyclotron-Wave ECR ICR Ion- Cyclotron-Wave Whistler-Wave Alfvèn-Wave L-Wave R-Wave k   ce  ci c 0 k || B cut-off Resonance Parallel and perpendicular propagation  UH  LHR UHR k  LH Alfvèn-Wave 0 O-Mode X-Mode LH-Wave UH-Wave c  pe  ci  ce ECRH LH ICRH k  B 3 Frequency ranges for Plasmaheating

jmn Electron- Cyclotron-Wave ECR ICR Ion-Cyclotron-Wave Whistler-Wave Alfvèn-Wave L-Wave R-Wave k   ce  ci c 0 k || B  ce  UH  LHR UHR k  LH Alfvèn-Wave 0 O-Mode X-Mode LH-Wave UH-Wave c  pe  ci k  B

jmn solutions for (k/k 0 ) 2, function of , n, B –if > 0 -> propagating –if non-propagating propagating solution k/k 0 = +/- –waves travelling in opposite directions if two propagating solutions –smaller v  =  /k -> slow wave –larger -> fast wave (k/k 0 ) 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 R L B OX L R B X

jmn CMA diagram p2p2 22 => Plasma density  ce 2 22 =>Magnetic field  ci 2 22 1 1 cut-off Resonanz P=0 L=  S=0 R=  2 solutions for (k/k 0 ) 2, function of , n, B –if > 0 -> propagating –if non-propagating propagating solution k/k 0 = +/- –waves travelling in opposite directions if two propagating solutions –smaller v  =  /k -> slow wave –larger -> fast wave (k/k 0 ) 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

jmn Easier to show with m i / m e = 2.5

jmn CMA detail

jmn Use of the CMA diagram O-Mode cut-off „ECR“ O-/X-Mode with k  B  ce   Magnetic field  (=Density) X- cut off upper Hybrid - Resonanz High field- Low field- coupling path through the Plasma! B (R) Wave = "ordinary wave" (O-Mode) mit E || B = “ extra-ordinary wave ” (X-Mode) mit E  B  ce   Magnetic field 

jmn ECR R B(R) nene “HF-cut-off“ UH-Resonanz EC-“Resonance“ X 1 -Mode EC-“Resonance“ R B(R) nene B O-Mode O-Mode cut-off..if n e >n crit B (R) Wave O-Mode cut-off „ECR“ O-/X-Mode with k  B  ce   Magnetic field   pe /  2 2 (=Density) X- cut off upper Hybrid - Resonanz High field- Low field- coupling path through the Plasma! E || B E  B B

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

jmn 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

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

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

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

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

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

jmn

jmn Wave propagation and absorption UHR O-wave X1-wave 1r/a nene B0B0 EC-resonance Upper hybrid resonance X-mode Cutoff X1-wave X2-wave  UH  k LL Alfvèn-Wave 0 O-Mode X-Mode Lower Hybrid Wave Upper Hybrid Wave  =ck  pe RR  LH  ci ECRH LH ICRH 3 frequency regions for plasma heating:  ce

jmn Mode Conversion OXB-Heating O-mode X-mode Mode conversion process under certain launch angles and for minimum density. 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. X-mode cutoff O-mode cutoff B-mode UH-resonance EC-resonance

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

jmn ECRH System Collector Resonator Electron- beam typ. 30A, 80kV mm-Wave quasi-optical or waveguide transmission EC-Resonance Gyrotron bis 1MW 80····170GHz Superconducting Magnet Window

jmn ECRH - Gyrotrons superconducting coils diamond window annular electron beam resonator conversion to Gaussian beam collector Presently development of 1 MW cw gyrotrons

jmn Needs: -Local current drive -Synchronized injection -Fast detection of NTM by ECE -Extremely fast mirror control and power modulation NTM Stabilisation EC-resonance NTM-island ECRH-Beam

jmn Removal of the magnetic Island by ECCD Current drive (P ECRH / P total = %) results in removal method has the potential for reactor applications

jmn Lower Hybrid system Klystron kW 2,5  3,7 GHz “Grill” = “phased array”- Antenna Stack of Waveguides Wave  0,5  vacuum N ||  2 Front view: N || NN N = conserved - quantity! E ~ evaneschent layer n e increases “Grill-Antenna“

jmn solutions of dispersion relation: slow wave (exhibits lower hybrid res.) fast wave n e > m -3 at antenna, to enter plasma k || > k c to reach center. Lower Hybrid Heating k || too low, power stays near plasma edge sw fw kk radius k || sufficiently high, slow wave travels into plasma, absorption at LH or before kk Lower Hybrid resonance radius

jmn LH - Wave Propagation Depends on n e and B. Antenna structure

jmn Klystron and Grill Beam dump cathode anode  -wave input  -wave output 3.7 GHz 500 kW 3 sec  klystron waveguide grill

jmn LH - Wave Excitation Fast wave k B0B0 E hf B hf Slow wave k B0B0 E hf B hf Multiple wave guides E wg ASDEX

jmn Current drive on ASDEX Example: Lower hybrid current drive, 1.3 GHz / 2.4 MW / 3 sec Leuterer F., Eckhardt D., Soeldner F.X., et al., Phys. Rev. Lett. 55 (1985) 75 The plasma current can be ramped up or the OH-transformer can be recharged with loop voltage = 0with loop voltage < 0

jmn Ion cyclotron system Tube- amplifier typ. 2 MW MHz 50  Koax-Ltg  Matching-Tuner 50   1··3  Plasma: Re(N) >>1 Dipole-antenne Preamplifier

jmn Wave propagation and absorption  UH  k LL Alfvèn-Wave 0 O-Mode X-Mode Lower Hybrid Wave Upper Hybrid Wave  =ck  pe RR  LH  ci ECRH LH ICRH 3 frequency regions for plasma heating:  ce

jmn Absorption wave - particle ions : cyclotron resonance fundamental  need correct polarisation  E +  minority heating harmonic  need gradient in E +  preferentially fast particles wave  other wave  particle mode conversion

jmn fundamental  need correct polarisation  E +  minority heating harmonic  need gradient in E +  preferentially fast particles wave  other wave  particle mode conversion

jmn Larmor radius  = m v perp / Z e B Cyclotron frequency  = Z e B / m = Z/A * e/m H * B Cyclotron motion v perp B 15 MHz per Tesla * Z/A

jmn Ion Cyclotron Resonance Heating

jmn particles are equally –accelerated and –decelerated by the wave when –more particles at low velocity then –net transfer from wave to particles Diffusion in velocity space f(v) v

jmn H to absorb but no correct polarisation for pure H at  =  cH the left hand polarization is = 0 n H /n e  = 100 %

jmn Correct polarisation but no H to absorb n H /n e  = 0 % in pure D at  =  cH the left hand polarization exists but there is „no“ resonance

jmn What happens if we add a bit of H n H /n e  = 0 % in plasmas with dominant D and a bit of H the left hand polarization is set by the dominant D but now we have the H to absorb! n H /n e  = 0.1 %

jmn How much is a bit before it starts to affect the polarisation dependence on concentration at  =  ci the left hand polarization decreases with increasing H concentration n H /n e  = 2 % n H /n e  = 15 %

jmn Need the correct polarization AND the species to absorb  amplitude with correct polarisation  0 !! n H /n e   10 %

jmn Need to check the propagation of the wave

jmn Propagation of the wave

jmn At even higher n H concentration: cut-off

jmn fundamental  need correct polarisation  E +  minority heating harmonic  need gradient in E +  preferentially fast particles wave  other wave  particle mode conversion

jmn Resonance/ Cut-off, Tunnelling, Mode conversion x k 2 perp k 2 inf Resonance Cut-off

jmn In the vicinity of the ion-ion hybrid layer, mode conversion to shorter wavelength waves occurs. IBW : Ion Berstein Wave Propagates towards the high field side ICW : Ion Cyclotron Wave Propagates towards the low field side F.W. Perkins, Nucl. Fusion 17, 1197 (1977) ICRF Heating in DIII-D: Mode conversion M. Brambilla, Plasma. Phys. Cont. Fusion 41, 1 (1999) Mode conversion to ion Bernstein wave, electrostatic ion cyclotron wave

jmn Ion-ion cut-off and resonances from low field side, sequence is always cut-off, then ion-ion resonance lies between cyclotron resonances of both ions lies closest to the cyclotron resonance with the lower concentration location of the cyclotron resonance wrt to pair thus varies HFS LFS R B B1 B2 low H concentration low D concentration

jmn Multiple ions cut-off /resonance pair between each of ion cyclotron resonances location depends on the relative concentrations HFS LFS R B B1 B2

jmn fundamental  need correct polorisation  E +  minority heating harmonic  need gradient in E +  preferentially fast particles wave  other wave  particle mode conversion

jmn “Second” harmonic 2  cH, also in H plasma -> amplitude with correct polarisation  0

jmn Cyclotron motion B v perp B B B

jmn Second Harmonic Heating m d (v perp 2 ) / 2 dt = Z E v perp cos  Ev.7Ev 0 -.7Ev - E v -.7 E v 0.7 E v netto Ev- E v = (E - E ) v -> gradient in electric field

jmn Second Harmonic Heating

jmn particles are equally –accelerated and –decelerated by the wave when –more particles at low velocity then –net transfer from wave to particles Diffusion in velocity space f(v) v

jmn nd and higher Harmonics damped if wave-field non-uniform on length-scale of Larmor radius t=0 t=T/2 t=T B B B v v v E gradient of electric field T = period of the wave

jmn How do we get this variation of E along the orbit ? E constant maximum effect 2  = /2 4 zero effect again

jmn maximum effect 2  = /2 k  =  /2

jmn At the second harmonic, the net acceleration depends on  L / Acceleration and deceleration by the wave along the ion orbit. Net effect only if E + is not constant, ie if  L /   L k  is finite. Effect becomes weak for  L /   L k  near 1

jmn FLR effects play a key role in determining the fast ion distribution function, in agreement with theory no more acceleration Wave-particle interaction becomes weak for  L /   L k  near 1 with k  = F(n)

jmn ICRF-acceleration of 4 He beam ions for simulating fusion  ’s ITER: fusion-born 3.5-MeV alpha particles ( 4 He ions) JET simulation:  3  ( 4 He) in 4 He plasma. For n = 3, damping on thermal ions is small  boost with high-energy 4 He beams with finite  L. Strongest 4 He tails with ICRF- acceleration of highest energy (120 keV) 4 He beams. (Mantsinen, PRL 2002) JET

jmn Possible absorption scenarios fundamental  need correct polorisation  E +  minority heating harmonic  need gradient in E +  preferentially fast particles wave  other wave  particle mode conversion in the ion cyclotron range of frequencies

jmn Wide range of possibilities, beyond heating Power to ions to electrons to thermal particles to fast particles (tails) to particles going one way or another on axis off-axis

jmn One of the first fast-ion effects: Sawteeth stabilisation Sawteeth = periodic relaxations of plasma core temperature. Central ICRH  Strong sawtooth stabilisation due to ICRH-heated ions with  Drift >  mode. Relevance for ITER: sawtooth stabilisation by fusion  ’s. Later: sawtooth destabilisation by ICRF. (Campbell, PRL 1988; Phillips, Phys. Fluids 1992; Porcelli, PPCF 1991) JET

jmn Power deposition with  = n  c depends on the finite orbit widths of fast ICRF- accelerated ions. Fast ions orbits, and thus the power deposition, can be modified using toroidally directed waves. Absorption of wave toroidal angular momentum  ICRF-induced pinch of fast ions. Turning points of trapped fast ion orbits are driven either outwards or inwards, depending on the direction of wave propagation. Wave  I p Wave  I p Inward pinch Outward pinch (Hellsten, PRL, 1995) Eventually co-passing orbits residing on the low-field side of R res Control : ICRF-induced pinch modifies fast ion orbits

jmn Experimental evidence for ICRF-induced pinch (Kiptily, Nucl. Fusion, 2002; Mantsinen, PRL, 2002) 3 He cyclotron resonance

jmn Basics of ICRF power deposition localisation The most narrow ICRF power deposition profiles can be obtained with ICRF mode conversion (MC). Most common: MC in the vicinity of the ion-ion hybrid resonance of two ion species with comparable concentrations. Real-time control of ion species mixture in routine use on JET to keep MC power deposition fixed in time. (Mantsinen, Nucl. Fusion 2004) Plasma Centre

jmn Toroidally directed waves couple asymmetrically to ions and electrons in the v || -space due to their average finite k ||. This gives rise to –ion cyclotron current drive (ICCD) –fast wave electron current drive (FWCD) –ICRF mode conversion current drive Current drive with ICRF waves

jmn Ion cyclotron current drive (ICCD) Ion cyclotron current with passing ions is dipolar, i.e. its sign reverses when crossing  = n  ci (Fisch, Nucl. Fusion 1981).  Flattening or peaking of the current profile depending on - location of R res versus R 0 - toroidal direction of the wave. While net current is small, local effect can be rather strong.

jmn Sawtooth control with ICCD ICCD at q = 1 on JET is used modify the current profile and thereby to stabilise or destabilise sawtooth activity. Recent JET experiments:  =  cH (Mayoral, Phys. Plasmas 2004; Eriksson, Nucl. Fusion 2006)  = 2  cH (Mantsinen, PPCF 2002) Expanded capabilities and heavy use. JET (Start, EPS1992; Bhatnagar, Nucl. Fusion 1994)

jmn Coupling with antennas Electricity -> other form (electromagnetic oscillations) Transport to plasma (outside part) transmission lines antenna (inside part) waves Thermalisation Resonance zone Antenna Wave to particles

jmn Waves wave excitation, propagation, absorption –absorption in plasma –wave excited in plasma and propagates –in density layer in front of antenna fields decay (evanescence) –antenna creates E and B fields

jmn Fast wave evanescent at the edge

jmn Absorption... Coupling zz Absorption Wave Coupling Cut-off excited by Antenna boundary condition for Antenna Evanescent Wave E co = f(k // ) E plasma/vacuum E plasma/vacuum ~ B z (relation in plasma) E y / B z x z y

jmn Depth of the evanescent layer depends on k // in the tenuous plasma, below the cut-off density the fast wave (k vector mostly perpendicular) is evanescent with we obtain thus relation between coupling and antenna spectrum due to the cut-off

jmn

jmn Antennas

jmn ICRF Antennas on JET ICRF antenna launches wave , k ||

jmn Dynamic matching or load isolation necessary 50 Ohm matching 1 ms DD R  R Ohm + j X +  X Antenna timescales of variations –particle/energy confinement time ms to s –MHD events100  s

jmn HH Power to antenna Reflected power Reflected power at generator 5 kW 100 kW 1 MW 3 dB couplers for ELM resilience

jmn Overview steps: generation, transport, absorption, thermalization ICRF: minority, mode conversion, harmonic not just heating 3 dB couplers Electricity -> other form (electromagnetic oscillations) Transport to plasma (outside part) transmission lines antenna (inside part) waves Thermalisation Wave to particles

jmn Summary for power to be absorbed into the plasma it must first get there wave propagation: range of frequencies –Electron cyclotron –Lower Hybrid –Ion cyclotron absorption in plasma: wave - particle interaction –cyclotron damping, also at harmonics –Landau damping different types of approximations –cold plasma: typically two waves –CMA diagram, parallel, perpendicular propagation very large number of possibilities, not just heating –current drive –control of instabilities –…

jmn Literature M. Brambilla Kinetic Theory of Plasma Waves, R. Koch Physics and Implementation of ICRF of Fusion Reactors, Lab. Rep. 108, 1997, Brussels Stix, Thomas H. Waves in Plasmas, New york, American Institute of Physics, 1992 Wesson, John Tokamaks, Oxford, Clarendon Press 1997, 2nd Ed. The Oxford Engineering Science Series, Vol. 48 Swanson, D.G. Plasma Waves, Boston,MA, Academic Press 1989 Cairns, R.A. Radiofrequency Heating of Plasmas, Bristol, Hilger 1991