Bers, A.

Ram, A.K.

Schultz, S.D.

DSpace@MIT

COUPLING TO ELECTRON BERNSTEIN WAVES IN TOKAMAKS*A. Bers, A. K. Ram, and S. D. SchultzPlasma Science and Fusion Center,Massachusetts Institute of Technology, Cambridge, MA 02139. U.S.A.Abstract The excitation and localized damping of electron Bernstein waves (EBW)in toroidally conned plasmas presents the possibility of ecient means for plasmaheating, current drive, and current prole control with external power in the electroncyclotron range of frequencies (ECRF). Such use of EBW is particularly interesting forhigh- plasmas (e.g., START, MAST, NSTX) but is also applicable to moderate andlow- tokamaks on which it can be tested.In this short paper, we focus on the excitation of EBWs from the outboard side byeither mode conversion of an X-mode or its direct coupling in an NSTX-type of plasma.Fig. 1 shows the distribution of critical frequencies in propagation across the magneticeld along the equatorial plane, and Fig. 2 gives the local kinetic dispersion relation forthe same direction; both use the plasma (density and temperature) and magnetic eldproles (Fig. 3) for NSTX. From Figs. 1 and 2, we note that the triplet of R-cuto|upper-hybrid-resonance (UHR)|L-cuto forms a mode conversion resonator (i.e., aresonator containing mode conversion to EBW as an eective dissipation) [1]{[3]. Insuch a triplet resonator, one can in principle obtain complete mode conversion of theincident power to EBW [4].We consider the simplest cold-plasma slab model in the equatorial plane, inhomoge-neous in x (the radial direction), and with arbitrary x-variations for both the (toroidal)z-directed and the (poloidal) y-directed magnetic elds:~B0(x) = y^Bp(x)+ z^BT(x). Thedetailed linearized eld analysis is straightforward. For numerically integrating the eldequations, it is found convenient to formulate them as a set of four coupled rst-orderdierential equations:d~Fcd= i$Ac~Fc(1)where~FTc= [EyEz( cBy) cBz], we have let (!x=c)  , and the variations in thedirections y and z (in which the equilibrium is assumed uniform) have been Fourieranalyzed with components, respectively, exp(ikyy) and exp(ikzz). The tensor (matrix)$Acis found to be given by$Ac=1Kxx266666666666664 nyxy nyxz nynzKxx  n2y nzxy nzxzKxx  n2z nynzKxx(yz+ nynz) Kxx(Kzz  n2y) nzxznyxz+xyxz+2xzKxx(Kyy  n2z) Kxx(yz+ nynz) nzxynyxy+2xy+xyxz377777777777775(2)where ny= (cky=!), nz= (ckz=!), and the susceptibilityijand permitivity Kij=ij+ijelements are as found from standard cold plasma perturbation theory for theconsidered equilibrium~B0(x) and n0(x). The solid line curve in Fig. 4 shows the modeconversion/resonant absorption as a function of frequency, obtained from a numericalintegration of (1) for NSTX plasma and magnetic eld proles, assuming ny= 0 andnz= 0:1. We note that high mode conversion eciencies, > 80%, are obtained over abroad range of frequencies of about 4 GHz around the peak in C  0:97 at f = 16 GHz.Accounting for kinetic eects in a nonzero temperature plasma removes the resonantabsorption at the UHR and replaces it with the kinetic EBW, which propagates theenergy away from the mode conversion region. An approximate description that includesthe kinetic EBW and the cold plasma modes, coupled near the UHR, follows fromgeneral WKB analysis [5], with due attention to conservation of kinetic energy owdensity. Thus to include the EBW, we setKKxxEx! KxxEx dd TdExd!: (3)T = 3!2p!2(vT=c)2(!2  !2ce)(!2  4!2ce)(4)is obtained from expanding the kinetic (Vlasov) susceptibilityKxxto second-order in(k?vTe=!ce) | appropriate for representing EBW between the rst and second electroncyclotron harmonics, and where damping can be neglected. The resulting set of coupledrst-order dierential equations ared~FKd= i$AK~FK(5)where~FTK= [ExEyEz(iTE0x) cBz( cBy)], we have let (!x=c)  ,$AK=266666666666666640 0 0  T 10 0ny0 0 0 1 0nz0 0 0 0 1Kxxxyxz0 nynz xyKyy  n2zyz+ nynz0 0 0 xzyz+ nynzKzz  n2y0 0 037777777777777775(6)The dashed curve in Fig. 4 shows the results from a numerical integration of (6) forthe same NSTX parameters used in the mode-conversion/resonance absorption calcu-lations given above. Up to about 16 GHz, the (approximate) kinetic mode-conversioncalculation veries very well the results from the exact mode-conversion/resonance ab-sorption calculation. For frequencies above 16 GHz, the UHR is above 2!ceand, unlessthe plasma extends suciently out, the power incident on the fast X-mode will onlyencounter a forward travelling wave (modied slow X-mode/EBW) propagating out ofthe plasma.Turning to direct coupling to EBW, we note from Figs. 1 and 2 that for plasmaparameters of interest and frequencies not too far above fce, the distances at the plasmaedge over which coupling takes place are shorter than a free-space wavelength. Thissuggests that eective direct coupling to EBW should be possible from an external slowwave structure that can be placed just inside of the conuence point of the EBW withthe slow X-mode. This coupling problem, which is similar to slow-wave coupling forlower-hybrid and ion-Bernstein waves, will be presented elsewhere.* Work supported by DoE Contract DE-FG02-91ER-54109 and by NSTX.[1] V. Fuchs et. al., Phys. Plas., 2, 1637 (1995).[2] A. K. Ram, A. Bers, S. D. Schultz, and V. Fuchs, Phys. Plas., 3, 1976 (1996).[3] A. Bers, A. K. Ram, A. Becoulet, and B. Saoutic, to appear in Phys. Plas.[4] K. C. Wu, A. K. Ram, A. Bers, and S. D. Schultz, in Proceedings of the 12th TopicalConference on RF Power in Plasmas, AIP, New York, 1997, pp. 207{210.[5] H. Berk and D. L. Book, Phys. Fluids, 12, 649 (1969).NSTX: R = 1:05 m, a = 0:44 m; ne0= nE+ (n0  nE)(1  x2=a2)1=2; n0= 4 1019=m3;Te0= TE+ (T0  TE)(1  x2=a2)2; T0= 3 keV; (nE; TE) = 0:02(n0; T0)

Coupling to Electron Bernstein Waves in Tokamaks

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Coupling to Electron Bernstein Waves in Tokamaks

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