RELATIVISTIC EFFECTS IN ELECTRON CYCLOTRON RESONANCE HEATING AND CURRENT DRIVE

RELATIVISTIC EFFECTS IN ELECTRON CYCLOTRON RESONANCE HEATING AND CURRENT DRIVE Abhay K. Ram Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge, MA 02139. U.S.A. Joan Decker CEA Cadarache F-13108 St. Paul lez Durance, France Supported by DoE Contracts DE-FG02-99ER-54521, DE-FG02-91ER-54109. 4 th IAEA Technical Meeting on ECRH Physics and Technology for ITER. Vienna, Austria. June 8, 2007.

SYNOPSIS A brief description of the evaluation of the relativistic conductivity tensor. The effect of relativity on the propagation and damping of waves in the electron cyclotron range of frequencies. The relevance, to ITER, of studies on electron cyclotron heating in spherical tori.

RELATIVISTIC CONDUCTIVITY TENSOR [Trubnikov 1959] There are essentially two steps in determining the homogeneous plasma conductivity tensor. Linearizing the Vlasov equation f t + p mγ f + q ( where γ = (1 + p 2 /m 2 c 2 1/2. E + p mγ B f f ( r, p, t = f 0 ( p + f 1 ( r, p, t, B B ( r, t = B 0 + B 1 ( r, t, p f = 0 E E ( r, t = E 1 ( r, t

RELATIVISTIC CONDUCTIVITY TENSOR df 1 dt = q ( E 1 + p mγ B 1 p f 0 on the left hand side is the convective derivative. Formally: f 1 ( r, p, t = q t dt ( E 1 + p mγ B 1 p f 0 where the integration is along the unperturbed orbits.

RELATIVISTIC CONDUCTIVITY TENSOR The wave fields are given by the Faraday-Ampere system of equations ( E + 2 t 2 E = J where J is the current density ( p J ( r, t = qn 0 d 3 p f 1 ( r, p, t mγ The plasma conductivity is obtained from J = σ E = σ t dt d 3 p...

RELATIVISTIC CONDUCTIVITY TENSOR For a Maxwellian f 0 ( p only one of the integrals can be done analytically. For a non-maxwellian f 0 ( p, in general, only the time history integral can be done analytically. A numerical code R2D2 has been developed to evaluate fully relativistic σ (Ram et al. - 2002, 2003 do the time history integrals numerically (highly oscillatory integrands; do the momentum space integrals numerically (singular integrands do the momentum space integrals numerically (significant analytical massaging one to two orders of magnitude reduction in time.

COMPARISON BETWEEN RELATIVISTIC AND NON-RELATIVISTIC WAVE CHARACTERISTICS 1.4 0.08 dashed: O mode 1.2 X mode 0.06 Re(n 1 Im(n 0.04 0.8 0.02 O mode 0.6 0.8 0.9 1 1.1 1.2 ω / ω ce solid: X mode 0 0.8 0.9 1 1.1 1.2 ω / ω ce ω/2π = 170 GHz, ω pe /ω = 0.75, n = 0.1, T e = 5 kev

COMPARISON BETWEEN RELATIVISTIC AND NON-RELATIVISTIC WAVE CHARACTERISTICS 1.5 0.12 1.25 X mode 0.1 0.08 dashed: O mode Re(n 1 Im(n 0.06 0.75 O mode 0.04 0.02 solid: X mode 0.5 0.8 0.9 1 1.1 1.2 ω / ω ce 0 0.8 0.9 1 1.1 1.2 ω/ω ce ω/2π = 170 GHz, ω pe /ω = 0.75, n = 0.1, T e = 10 kev

DAMPING PROPERTIES OF THE O MODE 0.06 0.02 0.05 0.04 10 kev 5 kev 0.015 10 kev Im(n 0.03 Im(n 0.01 0.02 0.01 0.005 5 kev 0 0.8 0.85 0.9 0.95 1 1.05 ω / ω ce 0 0.8 0.9 1 1.1 1.2 ω / ω ce n = 0.1 n = 0.5 Dashed line: ω pe /ω = 0.75; solid line: ω pe /ω = 0.92

ELECTRON CYCLOTRON WAVES IN PLASMAS The electron cyclotron plasma waves usually used in experiments are the X and O modes. In the same frequency range there also exist electron Bernstein waves heating and current drive demonstrated, e.g., in Wendelstein 7-AS and Compass-D; emission and coupling studies in spherical tori (NSTX, MAST, reversed field pinch (MST, and tokamak (TCV.

ELECTRON CYCLOTRON WAVES IN PLASMAS The differences between X & O modes and electron Bernstein waves are vacuum waves, plasma waves; directly coupled to in a plasma, coupling to a plasma through mode conversion; can have density cutoffs in a plasma, no density cutoffs in a plasma; strong cyclotron resonance damping at low ( 3f ce harmonics, strong cyclotron resonance damping at all harmonics; not useful in plasma regimes where f pe /f ce 1, useful waves in overdense plasmas (NSTX, MAST.

ADVANTAGES OF STUDYING EBW PHYSICS IN SPHERICAL TORI Electron Bernstein wave physics in a spherical torus can be of help in understanding electron cyclotron wave physics in general high power sources at low frequencies are more readily available; the physics of wave damping and of the waveparticle interaction in the vicinity of the electron cyclotron resonances has similar features to those of X and O modes; the current drive physics also has similar physical characteristics.

COMPARISON BETWEEN RELATIVISTIC AND NON-RELATIVISTIC EBWs 0.4 0.3 Im (n 0.2 0.1 0 1 2 3 T (kev e ω pe /ω ce = 6, ω/ω ce = 1.9, n = 0.2 Relativistic effects become important for T e > 500 ev.

IMPLICATIONS OF RELATIVISTIC EFFECTS 1 0.15 Imag ( k ρ e 0.8 0.6 0.4 0.2 B LOW FIELD Imag ( k ρ e 0.1 0.05 B HIGH FIELD 0 1 1.05 1.1 ω / ω ce 0 1.7 1.8 1.9 2 ω / ω ce Approaching the electron cyclotron resonance from the low field side (ω > nω ce : relativity narrows the absorption profile; from the high field side (ω < nω ce : relativity broadens the absorption profile.

COMPARISON BETWEEN RELATIVISTIC AND NON-RELATIVISTIC WAVE CHARACTERISTICS 1.5 0.12 1.25 X mode 0.1 0.08 dashed: O mode Re(n 1 Im(n 0.06 0.75 O mode 0.04 0.02 solid: X mode 0.5 0.8 0.9 1 1.1 1.2 ω / ω ce 0 0.8 0.9 1 1.1 1.2 ω/ω ce ω/2π = 170 GHz, ω pe /ω = 0.75, n = 0.1, T e = 10 kev

WAVE-PARTICLE INTERACTIONS AND OPTICAL DEPTH FOR ELECTRON CYCLOTRON WAVES p /p te 5 4 3 2 ECW EBW 1 1 100 10000 τ n For EBWs: 200 < τ n < 1000. EBWs interact with electrons in the range 3 < p /p te < 4.

NONLINEAR WAVE-PARTICLE INTERACTIONS Various physical time scales Landau damping time: τ LD Decorrelation time: τ d 1/ (v te k Transit time: τ TR b /v te Quasilinear time scale: τ QL τ LD τ d τ TR Nonlinear time scale: τ NL me e E k Validity of quasilinear theory: τ QL τ NL this could be easily violated by electron Bernstein waves in spherical tori.

KINETIC EVOLUTION OF THE ELECTRON DISTRIBUTION FUNCTION R2D2 has been coupled to LUKE - a drift kinetic Fokker-Planck solver (J. Decker & Y. Peysson f t + v GC = C(f + Q(f f is the guiding center distribution function v GC = v ˆb + vd. This set of codes is used to calculate current drive by electron Bernstein waves.

FISCH-BOOZER CURRENT DRIVE 3 1.5 2D Distribution function f 0 P RF (MW/m 3 2 1 1 0.5 J RF (MA/m 2 ref p /p Te 10 n=2 5 0 0.4 0.45 0.5 0.55 0 ρ 0 10 5 0 5 10 ref p /p Te P = 1 MW, I = 100 ka. η = m e ν e v te e J P, ξ CD = 32.7 I R n e[20] T e[kev ] P. η peak = 3.7, ξ CD = 0.67 (for ECCD: ξ CD 0.3.

CONCLUSIONS Relativistic effects play a significant role in the propagation and damping of electron cyclotron waves modification to the deposition profile and interaction with electrons. A set of codes (beam and ray tracing with relativistic conductivity, Fokker-Planck for the evolution of the distribution function is being developed. Heating and current drive experiments with electron Bernstein waves on spherical tori can provide insight into the interaction of high power electron cyclotron waves with electrons these experiments will also be a good test of the analytical and computational models.