Decker, Joan

Ram, Abhay

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PSFC/JA-08-23RELATIVISTIC PROPAGATION ANDDAMPING OF ELECTRON CYCLOTRONWAVES IN TOROIDAL PLASMASAbhay K. Ram and Joan Decker∗July 2008Plasma Science & Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139, U.S.A.∗CEA CadaracheF-13108 St. Paul lez Durance, FranceThis work was supported by the U.S. Department of Energy Con-tracts No. DE-FG02-91ER-54109 and DE-FG02-99ER-54521. Repro-duction, translation, publication, use and disposal, in whole or part,by or for the United States Government is permitted.To appear in Proceedings of the 15th Joint Workshop on Electron Cy-clotron Emission and Electron Cyclotron Resonance Heating, YosemiteNational Park, California, 2008.1RELATIVISTIC PROPAGATION AND DAMPING OFELECTRON CYCLOTRON WAVES IN TOROIDALPLASMASABHAY K. RAM∗Plasma Science and Fusion Center, Massachusetts Institute of Technology,Cambridge, MA 02139. USA∗E-mail: abhay@mit.eduJOAN DECKERCEA CadaracheF-13108 St. Paul lez Durance, FranceIn the electron cyclotron range of frequencies (ECRF), the extraordinary Xwaves or the ordinary O waves have been successfully used in many conven-tional tokamaks for generating plasma current and for modifying the currentprofile. ECRF waves are expected to play a similarly important role in ITERfor current profile control. Also, in the EC frequency range, electron Bernsteinwaves have been used for heating and for current generation in stellarators,conventional tokamaks, and a levitated dipole. EBWs are also highly suit-able candidates for current profile control in overdense plasmas encountered inspherical tokamaks (ST). It is well-known that relativistic effects need to beincluded in a proper description of the propagation and damping of EC waves.We have developed a code R2D2 which numerically solves the fully relativisticdispersion relation for all EC waves. The results from R2D2 provide an insightinto the properties of EC waves and the effect of relativity on the dampingof these waves. We present similarities between the damping of O waves andEBWs. The physics of the interaction of O waves with electrons in ITER-typeplasmas bears similarity to the interaction of EBWs with electrons in STs. SoEBW current drive in STs could provide useful insight into O wave physics inITER.1. IntroductionWe have developed a fully relativistic code R2D2 which solves for wavepropagation and damping of all waves in the electron cyclotron range offrequencies (ECRF) [1]. This code has been used in a number of studieson electron cyclotron waves and electron Bernstein waves (EBW) [1–5]. Inthis paper we present results on the damping of the ordinary O waves in2plasmas which are ITER-like and the damping of EBWs in plasmas whichrepresent present day STs.2. Relativistic Effects in Wave Propagation and DampingIn Fig. 1 we plot the real and imaginary parts of n⊥, calculated from R2D2,as a function of ω/ωce for the O wave in ITER-type of plasma conditions.Here n⊥ is the wave index perpendicular to the magnetic field, (n‖ beingthe parallel wave index), ω is the wave angular frequency, and ωce is theelectron cyclotron angular frequency. The real part of n⊥ describes thepropagation characteristics of the O wave. The imaginary part of n⊥ isa measure of the damping of the wave. In these figures we compare therelativistic (solid red) and the non-relativistic (dashed blue) results. It isevident that the relativistic results are significantly different from the non-relativistic results. The differences are more pronounced in the vicinity ofthe cyclotron resonance for both the propagation and damping parts of thewave dispersions. In approaching the cyclotron resonance from the high-fieldside (ω/ωce ≤ 1), the relativistic damping starts to occur before the non-relativistic damping. In the approach to the resonance from the low fieldside, relativity tends to narrow the deposition profile so that depositionstarts closer the cyclotron resonance. Overall, relativity tends to broadenthe deposition profile and reduce the maximum value.Figure 2 displays the O mode dispersion characteristics for a plasmatemperature of 10 keV. The higher temperature seems to broaden the wavedamping profile while the maximum value of the imaginary part of n⊥remains essentially the same as for a temperature of 5 keV. In approachingthe cyclotron resonance from the low field side, the damping occurs towardthe high field side of the resonance with the deposition of wave poweroccurring closer to the resonance for low temperatures.The propagation and damping physics of O waves is being studied inpresent tokamaks. However, an ST plasma is not a suitable candidate forO waves since the plasma is overdense. At low harmonics the O wave iscutoff near the edge of the plasma while at high harmonics the plasma isessentially transparent to the wave. Since EBWs do not have density cutoffsand are well absorbed by electrons in the Doppler-shifted vicinity of loworder cyclotron resonances, they are well suited for ST plasmas [6] such asthose encountered in NSTX [7] and MAST [8]. The differences between non-relativistic and relativistic dispersion characteristics in present day STs aresimilar to those for the O wave in ITER-like plasmas. In Fig. 3 we plot theimaginary part of k⊥ρe as a function of ω/ωce for parameters relevant to a30.85 0.9 0.95 1 1.050.60.650.70.75ω / ωceRe(n⊥)0.85 0.9 0.95 1 1.0500.020.040.060.08ω / ωceIm(n ⊥)(a) (b)Fig. 1. (a) Real-n⊥ versus ω/ωce for ω/2pi = 170 GHz, electron temperature Te = 5keV, ωpe/ω = 0.75 (ωpe is the electron plasma frequency), and n‖ = 0.1. (b) Thecorresponding imaginary part of n⊥.0.85 0.9 0.95 1 1.050.550.60.650.70.75ω / ωceRe(n⊥)0.85 0.9 0.95 1 1.050   0.020.040.060.080.1 0.12ω / ωceIm(n ⊥)(a) (b)Fig. 2. The same as Fig. 1 except that Te = 10 keV.NSTX plasma. Here k⊥ is the perpendicular component of the wave vectorand ρe is the electron Larmor radius. We see differences, similar to thosefor the O waves, between the relativistic (solid red) and the non-relativistic(dashed blue) results when approaching the resonance from the low fieldside and from the high field side. When approaching the resonance fromthe low field side, relativity tends to narrow the absorption profile, whilefor the high field side approach relativity tends to broaden the absorptionprofile. In a high-β NSTX-type plasma, there is a dip in the magnetic fieldalong the equatorial plane [6]. By an appropriate choice of EBW frequency,the wave can approach the cyclotron resonance either from the low field4side or from the high field side while keeping the launching position of thewave same. Thus, an ST offers an extended test of waves in the EC rangeof frequencies.1 1.05 1.100.20.40.60.81ω / ωceImag ( k⊥ ρ e )1.7 1.8 1.9 200.050.10.15ω / ωceImag ( k⊥ ρ e )(a) (b)Fig. 3. Imaginary part of (k⊥ρe) versus ω/ωce for ωpe/ωce = 6, Te = 3 keV, andn‖ = 0.2: (a) for low field side approach and (b) for high field side approach to thecyclotron resonance.In Fig. 4(a) we compare, as a function of the plasma temperature, therelativistic (solid red) and the non-relativistic (dashed blue) values of theimaginary part of n⊥ for EBWs in typical NSTX-type plasmas. From thisfigure it is evident that relativistic effects become important for EBWsfor temperatures above 1 keV. From Fig. 2, we note that, in ITER-typeplasmas, the damping of the O mode and its interaction with electrons issignificantly affected by relativity for temperatures near 10 keV. Relativisticshifts in the damping profile can be studied in present day STs. Thus,the dispersion characteristics of EBWs in STs will provide an insight intorelativistic modifications to the characteristics of O waves in ITER.3. Current Drive by Electron Cyclotron WavesThe primary role of EC waves in ITER will be to drive localized plasma cur-rents. The current drive physics depends on the momentum of the electronsin the distribution function that interact with the EC waves. A measure ofthis interaction is the optical depth of the EC waves. The optical depth canbe determined from the linear theory of EC wave propagation and damp-ing [9]. In Fig. 4(b) we plot the momentum of the electrons, normalized tothe thermal momentum, as a function of the optical depth of EC waves.50   1 2 300.10.20.30.4Te (keV)Im (n⊥)1 100 1000012345τn|p |||/pteECW EBW (a) (b)Fig. 4. (a) Im(n⊥) versus electron temperature for ω/ωce = 1.9 for ωpe/ωce = 6, andn‖ = 0.2. (b) The parallel electron momentum normalized to the thermal momentumversus the optical depth for the electron cyclotron waves. ECW represents the X and Owaves.The optical depth of X and O waves is around 10 (represented by ECWin the figure) so that these waves interact with electrons near the thermalmomentum. For EBWs the optical depth is in the range 200 <∼ τn <∼ 1000,so that EBWs interact with electrons in the range 3 <∼ p‖/pte <∼ 4. Thus,EBWs interact with electrons which are approximately an order of mag-nitude more energetic than the electrons with which the O wave interactsin present day tokamaks. If the plasma temperature in the region whereEBWs damp in an ST is 3 keV, the effective energy of the electrons withwhich the EBWs interact is around 30 keV. In ITER the O mode will beinteracting with electrons in this approximate energy range. Thus, EBWexperiments in a ST can provide insight into the physics of the interactionof EC waves with highly energetic electrons.The EBWs can drive plasma currents in a ST either through the Fisch-Boozer scheme [10] or the Ohkawa scheme [11,12]. The latter means ofcurrent drive is possible since, on the outboard side, a large fraction ofthe electrons in a ST are magnetically trapped. Even though the Ohkawacurrent drive is not envisioned for ITER, experiments on a ST can be usefulin extending our understanding of the wave-particle interactions.The code R2D2 has been coupled to a code LUKE [13] which solvesfor RF driven current using a quasilinear diffusion operator. Some resultson EBW current drive obtained from this combination of codes have beendiscussed in [1].64. ConclusionsThe implication of the results presented in this paper is as follows. Wecan study the importance of relativistic effects on EC wave propagationin present day STs. The relativistic modifications to the propagation anddamping of EBWs in STs will provide an insight into the effect of relativityon O wave propagation and damping in ITER. The ECRF waves in ITERwill be used for stabilizing the neo-classical tearing mode. For this to beaccomplished successfully we need to account for any changes in the spatiallocation of wave damping. The EBWs in STs will damp on electrons whoseenergies are similar to those that will interact with O waves in ITER. TheO waves and EBWs interact with electrons via the cyclotron resonanceinteraction. Consequently, we can study the interaction physics of EC wavesat high ITER-like temperatures in present day ST plasmas.5. AcknowledgmentsThis work was supported by DoE grants DE-FG02-91ER-54109 and DE-FG02-99ER-54521.6. ReferencesReferences1. A. K. Ram, J. Decker, and Y. Peysson, J. Plasma Phys. 71, 675 (2005).2. J. Decker and A. K. Ram, Phys. Plasmas 13, 112503 (2006).3. E. Nelson-Melby, R. W. Harvey, A. P. Smirnov, and A. K. Ram, Plasma Phys.Control. Fusion 49, 1913 (2007).4. R. Prater et al., Nuclear Fusion 48, 035006 (2008).5. R. W. Harvey, A. P. Smirnov, E. Nelson-Melby, G. Taylor, S. Coda, and A.K. Ram, Fus. Sci. Tech. 53, 237 (2008).6. A. K. Ram and S. D. Schultz, Phys. Plasmas 7, 4084 (2000).7. M. One et al., in Proc. 17th International Conf. on Fusion Energy1998 (Yokohama, 1998) (Vienna: IAEA) CD-ROM file ICP/01 andhttp://www.iaea.org/programmes/ripe/physics/fec1998/pdf/icp 01.pdf.8. A. Sykes, ibid., CD-OM file OV2/5.9. J. Decker, Ph.D. Thesis, Department of Electrical Engineering and ComputerScience, M.I.T., Cambridge, MA 02139. USA (2005).10. N. J. Fisch and A. Boozer, Phys. Rev. Lett. 45, 720 (1980).11. T. Ohkawa, General Atomics Report No. GA-A13847 (1976).12. See also J. Decker, in Proc. 15th Topical Conf. on RF Power in Plasmas,AIP Conf. Proc. No. 694 (ed. C.B. Forest), Melville, New York (2003), p. 447.13. J. Decker and Y. Peysson, Report EUR-CEA-FC-1736, Cadarache, France(2004).

Relativistic Propagation and Damping of Electron Cyclotron Waves in Toroidal Plasmas

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Relativistic Propagation and Damping of Electron Cyclotron Waves in Toroidal Plasmas

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