Radio Frequency Plasma Heating

Radio Frequency Plasma Heating Giuseppe Vecchi Credits/thanks: Riccardo Maggiora & Daniele Milanesio 1

Ohmic Heating The plasma current is driven by a toroidal electric field induced by transformer action, due to a flux change produced by current passed through the primary coil Initial heating in all tokamaks comes from the ohmic heating caused by the toroidal current (also necessary for plasma equilibrium) PΩ = ηj : ohmic heating density Limitations: on current density to avoid instabilities and disruptions by plasma resistivity 3 η T Additional heating needed

Auxiliary Heating and Current Drive (H&CD) Methods Method Principle Heated species Neutral Beam Injection Electromagnetic Waves Injecting a beam of neutral atoms at high energy across magnetic field lines Exciting of plasma waves that are damped in plasma Alfven waves ion cyclotron waves lower hybrid waves electron cyclotron waves Electrons, ions Electrons Electrons, ions Electrons Electrons α-particles Collisions Electrons, ions At ignition, only α-particles sustain the fusion reaction 3

Electromagnetic Wave H&CD How does this work? Excitation of a plasma wave at the plasma edge Wave transports energy into the plasma At a resonance the wave is transformed into kinetic energy of resonant particles Collisions distribute the energy Courtesy of D. Hartmann Method Advantages Disadvantages Ion Cyclotron Resonance Heating (ICRH&CD) Lower Hybrid Current Drive (LHCD) Direct ion heating, possible current drive, high efficiency, low cost Localized current drive useful in current profile control, waveguide antenna Internal solid antennas, minority heating, low plasma coupling Low power capability, low plasma coupling The Ion Cyclotron, Lower Hybrid and Alfven Wave Heating Methods R. Koch - Transactions of Fusion Science and Technology 53 (8) 4

E= B t H = D+ t D(E) J (E) J + J src Accounts for bound charges (dielectric) Accounts for free charges (conduction) In a (fully ionized) plasma: free charges dominate J (E) Maxwell Equations D =ε E Couples kinetic effects (Coulomb+Lorentz) to EM fields

Linearity typical parameters of an ICRF system: frequency: f 1-1 MHz Power: MW/antenna strap Voltage: 1-5 kv at the antenna Antenna current: IA 1 ka Central conductor: width.m, length 1m, distance to the plasma 5cm, to the wall cm Typical RF electric field: kv/m Typical RF magnetic induction: 1-3T BRF 1-3 T «B 3 T. RF electric field kv/m << Vti B 1.5MV/m F = q( E+ v B) B = B + E= E RF B RF Likewise one can show that also RF perturbation on // motion of particles is << thermal velocity (We can use the unperturbed trajectories) (Koch 8)

Non-collisional (1) Typical machine size: JET-type machine R = 3m, πr m; ap=1.5m, πap = 1m ion an electron collision frequencies: νe 1kHz, νi 1Hz. electron mean free path: 3km or 15 toroidal revolutions. ion mean free path: 5km or 5 toroidal revolutions. (Koch 8) J can be approximated as contribution from (average) charge motion of all species (electrons, one or more ion species) Motion can be considered single particle (collective effects neglected at first order)

Non-collisional Wave energy absorption is not by collision drag In bulk of a hot plasma, e.g. Te Ti 5keV, n=5 1^19m-3 collision frequency ν khz RF frequency f above 3 MHz, v/f<<1 B-lines are guiding νe 1kHz, νi 1Hz electron cyclotron gyration: 1ps ion cyclotron gyration: 4ns During one gyration: electron travels.4mm in the toroidal direction and the ion cm. Electron: 1µs for one toroidal turn= 5, cyclotron gyrations, ion: 4µs= 1, cyclotron gyrations (Koch 8, 6)

Particle motion linearizaton Unperturbed: thermal motion (equilibrium) Perturbed: RF fields (much smaller fields or effetcs)

Time-harmonic Maxwell equations For (small perturbation) linearized RF field E( r, t) = Re[ E( r; ω)exp( iωt)] H E = iωµ H = iωε E+ J + J src Important notes: 1) the RF field here is strictly sinusoidal (time-harmonic), it is so produced by the RF generators (in radio communications, it is nearly sinusoidal) ) Since the problem is linear, the frequency is the same everywhere and no matter what

Cold Plasma Approximation J ( E) = σ ( ω) E For a static magnetic field (B) along z axis σ ( ω) = σs σ yx σ σ xy H = iωε E+ J + s σ J src H = iωε E+ J src ε = ε + 1 σ iω

Cold Plasma Approximation The dielectric tensor results as: ε = ε R 1 S id id S + s ω ps P Stix parameters are defined as: ( ω ) ω ω cs ( R L) 1 S + ps ωps L 1 P 1 ω ω ω s D The cold-plasma approximation provides a good description of wave propagation even in quite hot plasmas, except for the reason where absorption takes place ω ( ω ) 1 cs ( R L) s 1

Look for a solution of the kind Plane wave solution E( r, t) = Re[ E( ω)exp( i[ ωt k r]) To be determined in such a way that the solution satisfies (sourcefree) Maxwell eqs. f ( r) = exp( ik r) f ( r) = ikf ( r) E= iωµ H k E= ωµ H H = iωε E k H = ωε E normalize k = k n ê x ê y k ϑ ê z B= Beˆ z static magnetic field

Wave Equation and Dispersion Relation n ( n E) + ε E = : wave in homogeneous plasma M ê x ê y k, ω E S = n n id cos ϑ cosϑ sinϑ id S n n cosϑ sinϑ E E P n sin ϑ E 4 detm ( k, ω) = A( θ ) n B( θ ) n + C( θ ) = k ϑ ê z A= S sin C= PRL ϑ+ P cos ( ) where: B= RLsin ϑ+ PS 1+ cos ϑ x y z ϑ = 14

Plane waves Note: setting k and ϴ means choosing the wavevector k = k n Recall: frequency is a constant everywhere (enforced by generator, linear problem) Consider first vacuum (or air) ( n E) + E = n n n= n = 1 n=±1 Observe: There is ONE solutions for n^ There are two solutions for n and k, corresponding to counter-propagating waves If you fix frequency and angle, then n is chosen by the physics and this gives the wavelength (spatial period of wave oscillations) n=1 means k=k f ( x ) = exp( iknx ) π λ = k n

Example: Consider simple medium with (slowly) varying material properties ε ( x) = εp( x) n ( n E) + p( x) E = n n= n = λ ( x) = p( x) f ( x) = exp( ikn( x) x) π k n( x) Plane waves n(x) f(x) 1.8 1.6 1.4 1. 1 4 6 8 1 x f(x)=cos(π n(x) x) 1.5 -.5-1 4 6 8 1 x

Wave Equation and Dispersion Relation n ( n E) + ε E = : wave in homogeneous plasma 4 detm ( k, ω) = A( θ ) n B( θ ) n + C( θ ) = Observe: There are TWO solutions for n^ (only one in vacuo) If you fix frequency and angle, then n is chosen by the physics ê x ê y k ϑ ê z A= S sin C= PRL ϑ+ P cos where: ( B= RLsin ϑ+ PS 1+ cos ϑ) Recall: frequency is a constant everywhere (enforced by generator, linear problem) ϑ 17

Dispersion Relation Solutions ϑ= ϑ= π : parallel propagation : perpendicular propagation ê x ê y k ϑ ê z B Beˆ Langmuir wave Ionic whistler Electronic whistler Slow (O) wave Fast (X) wave = : static magnetic field z ( E // B) ( E B) 18

Note: setting k and ϴ means choosing the wavevector Who chooses k and ϴ? Consider first vacuum (or air) ( n E) + E = - The RF generator chooses (enforces) the frequency - The physics chooses k (i.e. n), i.e. the wavelength - The antenna chooses angle ϴ (if very directive..) k = k n n n= n = 1 Actually, we never launch a single plane wave, we launch a field with some plane-wave spectrum e.g. we consider its Fourier transform e.g. 1D case = a( x) A( u)exp( ikux) dxu u= cosθ Plane waves and plane wave spectrum n

Plane waves and plane wave spectrum Who chooses k and ϴ? k = k n Any source distribution corresponds (can be represented as) a collection of plane waves with different wavenumber (PW spectrum) Each component (each individual PW) will travel its own way At a first approx, we consider only the peak of the plane wave spectrum (like the dominant tone in a sound or color in light) In fact, all ICRH antenna have a pretty broad spectrum Plasma propagation acts as a filter, some plane waves pass through better than others, some get absorbed well etc. We d like to put all our power in those that get well absorbed

Wave Propagation Dispersion relation for plane waves: k =k( ω) Phase velocity: Index of refraction : (wavenumber normalized to vacuum value) Cutoff: v ph = ω Group velocity: k n= ck At which ω Note: when frequency or angle is such that f ( r) exp( αx) = Evanescent wave n = v ph Resonance : n, v ph v g = ω k energy and information travel n < n=iα Wave slows down enormously, filed can now interact with thermal velocity (intuitive), absorption mechanisms favored 1

Wave Propagation Dispersion at fixed frequency and non-homogeneous plasma (density and/or B field vary in space) n Cutoff: k =k( ω) n, v n, v propagation ph evanescence n Resonance: propagation ph propagation Space Space

Ion Cyclotron Resonance Tore Supra ICRH antenna ω cs Frequency range: 4 8 MHz ω ω, ω << ω, ω = s ci qsb m pi ce ω pe Generators: tetrode tubes nsqs mε Principle: absorption of the wave by ions (cyclotron resonances) or by electrons (ELD - TTMP) ps = s Courtesy of CEA-Cadarache: http://www-cad.cea.fr 3

Improved resonance condition in IC range ω n h ω ci + k // v// = Adding effect of parallel motion due to RF field (v ) It is a Doppler effect n n h h = 1 : first harmonic heating : second (or higher) harmonic heating 4

ω= n hω + k v ci // // First harmonic heating Slow wave: sensitive to the fundamental resonance not excitable in toroidal geometry (evanescent) Fast wave: excitable in toroidal geometry not sensitive to the fundamental resonance Single Ion H&CD n n h h = 1 : first harmonic heating : second (or higher) harmonic heating Second harmonic heating FW is sensitive to the harmonics of the cyclotron frequency, but damping strength strongly decreases with harmonic number High density and high temperature needed NOT WORKING!!! NOT EFFICIENT!!! 5

Minority H&CD (Multiple Ions) ω ω ω ω nh ci k// v// = = ci+ k// v// ω ω ci Propagation and polarization are determined by the majority ions Good cyclotronic absorption on the minority ions (< 1%) Possible mode conversion to Ion Bernstein Waves (IBW) Ion Bernstein Waves: Perpendicularly propagating warm plasma waves with solutions near each harmonic of the cyclotron frequency of each species Higher percentage of minority species (~ 15-%) Landau damping on electrons 6

Main Collisionless Wave Damping Mechanisms Landau damping Strong interaction if v ω k Transit time magnetic pumping (TTMP) Force on magnetic moment: F = µ B similar to Landau damping with substitution: µ q B Slower particles are accelerated and faster particles are decelerated E 7

ICRF Power Scheme ICRF power FW + cycl. res. Abs. fund. cycl. Abs. harm. cycl. Fast Wave Abs. Landau TTMP Ion Bernstein Wave Abs. Landau Ions Fast ions Fast electrons Ionic heating Electronic heating 8

Tore Supra LH antenna Courtesy of CEA-Cadarache: http://www-cad.cea.fr Lower Hybrid H&CD Frequency range: 1 8 GHz with ω << ω << ω ci ω LH LH ω ω 1+ pi pe ω Generators: Klystrons Principle: Landau absorption of the wave by fast electrons ce ce Alcator C-Mod LH antenna Courtesy of PSFC (MIT): http://www.psfc.mit.edu/ 9

Lower Hybrid H&CD Original use: ionic heating by conversion of LH wave to a compressional wave Modern use: electronic heating by Landau damping on fast electrons Propagation on a narrow cone of resonance almost parallel to magnetic field when n >n // Group velocity: v g k Accessibility criterion : Polarization : n // >> n //, acc the best, experimentally proven, current drive method In ITER: controlling current profile (in addition to EC) E // k = 1 ω 1 ω ci ω ce 3

Generator ICRF Overall Scheme ITER IC antenna T&M scheme ~ DC breaker Tuning and matching systems Feed through Launcher T&M solutions (two elements): Resonant loop: the two feeding arms are set to the proper length to achieve the desired phasing Hybrid: the two feeding arms are connected to the two output ports of an hybrid device Conjugate T: the two feeding arms of equal length are connected in order to minimize the imaginary part of the active input impedance of the elements 31

Issues with Antennas Plasma facing antennas are used in experiments towards controlled nuclear fusion with magnetically confined plasmas to transfer power to the plasma and to control plasma current LH antennas ICRF antennas Courtesy of JET: http://www.fusion.org.uk These antennas are very complex geometries in a very complex environment and they can not be tested before being put in operation A numerical predictive tool is necessary to determine the system performances in a reasonable computing time and to properly optimize the antenna Courtesy of CEA-Cadarache: http://www-cad.cea.fr 3

Example : the Tore Supra ICRH Antenna Courtesy of CEA-Cadarache: http://www-cad.cea.fr Some features: adjacent cavities center-fed straps 4 loading capacitors to resonate the straps (resonant double loops) Main parameters: Major radius:.355 m Minor radius:.75 m Toroidal magnetic field: 3.13 T Generator frequency: 48 MHz Scenario: D(H) with 1% H minority Loading capacitors Analysis of Tore Supra ICRF Antenna with TOPICA D.Milanesio, V.Lancellotti, L. Colas, R.Maggiora, G.Vecchi, V.Kyrytsya Plasma Physics and Controlled Fusion 49 (7) 33

Example : the JET ITER-Like Antenna Courtesy of JET Task Force H Some features: Single cavity 8 straps with coax cable excitation, grouped in 4 resonant double loops Main parameters: Major radius:.96 m Minor radius: 1.5 m Toroidal magnetic field: 1.9 T Generator frequency: 4 MHz Scenario: D(H) with 3% H minority Measured density/temperature profiles Jet ITER-like Antenna Analysis using TOPICA code D. Milanesio, R. Maggiora, F. Durodié, P. Jacquet, M. Vrancken and JET-EFDA contributors 51 st APS-DPP meeting, Atlanta (9) 34

Example : the ITER IC Antenna Side views Proposed reference launcher Some features: 4 straps grouped in poloidal triplets Complex antenna structure and matching scheme (never experienced before) Main parameters: Major radius: 6. m Minor radius:.1 m Toroidal magnetic field: 5.3 T Generator frequency: 4 55 GHz Main scenario: 5%D-5%T Expected density/temperature profiles 35

Large Plasma-Antenna Distance Dependence Several plasma profiles have been loaded to predict the antenna performances in a wide range of input conditions By increasing the distance between the antenna mouth and the plasma, results converge to the vacuum case TOTAL power to plasma (MW) Max. voltage in coax: 45kV 36

Plasma-Surface Interactions Why rectified potentials are so important? RF-induced drifts accelerate ions that can hit the tokamak first wall, causing: hot spots sputtering (impurities) fuel dilution disruption The heat flux attributed to accelerated ions is directly proportional to the DC sheath (rectified) potential. Solutions? By accurately knowing the DC potential map resulting from the rectification process due to RF fields in front of the antennas, one can try to mitigate this effect modifying the antenna geometry itself. 37

y (m) Re(E // ) (V/m for 1V @ feeder), x=5mm 1.5 -.5-1 -1.5 Electric Field Map and Rectified Potential Electric field maps can be evaluated at every radial position in front of the antenna mouth 1.5 Upper box corner zone Lower box corner zone -1 -.5.5 1 z (m).8.6.4. -. -.4 -.6 -.8 y (m) 1.5 1.5 -.5-1 -1.5 V RF (V for MW coupled).1..3.4.5 x (m) Rectified potentials are influenced by plasma scenarios, by input phasing and by the geometry of the front part of the launcher 8 7 6 5 4 3 1 38

TOPICA as an Optimization Tool Reference antenna TOTAL power to plasma (MW) Max. voltage in coax: 45kV Optimized antenna The optimization process has been focused on the shape of the horizontal septa and their position, on the dimension of the feeder and its transition with the coaxial cable and on the wideness of the straps A significant increase in the antenna performances has been reached by optimizing some geometrical details 39

Proposed Design II: the ITER LH Launcher Detailed view of a single module Proposed reference launcher Some features: 35 waveguides, grouped in 4 blocks of 1 rows Based on the PAM concept, i.e. on the alternation between active and passive waveguides Main parameters: Major radius: 6. m Minor radius:.1 m Toroidal magnetic field: 5.3 T Generator frequency: 5 GHz Main scenario: 5%D-5%T Expected density/temperature profiles Courtesy of ITER-LH working group 4

To fill in the gaps/to probe further Tutorial, with nice application to RFH R. Koch, The Ion Cyclotron, Lower Hybrid and Alfven Wave Heating Methods, Transactions of Fusion Science and Technology 53 (8) Tutorial, tries to explain wave penetration in a Tokamak-like geometry R. Koch, The Coupling of Electromagnetic Power to Plasmas, Transactions of Fusion Science and Technology 49 (6) All-time classics T.H. Stix, The Theory of Plasma Waves, McGraw-Hill, New York, 196 T.H. Stix, Waves in plasmas, American Institute of Physics, New York, 199