Cairns, R.A.

Davies, C.N.

Efthimion, P.C.

Ram, A.K.

Schultz, S.D.

Taylor, G.

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PSFC/JA-00-41HEATING AND CURRENT DRIVEBY ELECTRON BERNSTEIN WAVESIN NSTX AND MAST-TYPE PLASMASA. K. Ram and S. D. SchultzPlasma Science & Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139 USAC. N. Lashmore-DaviesEuratom – UKAEA Fusion AssociationCulham Science CentreAbingdon, Oxon OX14 3DB U.K.R. A. CairnsUniversity of St. AndrewsSt. Andrews, Fife KY16 9SS U.K.P. C. Efthimion and G. TaylorPrinceton Plasma Physics LaboratoryPrinceton, New Jersey 08543 USADecember 2000This work was supported by the U.S. Department of EnergyContract No. DE-FG02-91ER-54109 and DE-FG02-99ER-54521,and by UK DTI, and EURATOM . Reproduction, translation,publication, use and disposal, in whole or part, by or for theUnited States Government is permitted.To appear in Proceedings of the 18th International Atomic En-ergy Agency (IAEA) Fusion Energy Conference, Sorrento, Italy,October 4–10, 2000.iHeating and Current Driveby Electron Bernstein Wavesin NSTX and MAST-Type PlasmasA. K. Ram, A. Bers, S. D. SchultzPlasma Science & Fusion Center, M.I.T., Cambridge, MA, 02139, U.S.A.C. N. Lashmore-DaviesEuratom – UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxon,OX14 3DB, United KingdomR. A. CairnsUniversity of St. Andrews, St. Andrews, Fife, KY16 9SS United KingdomP. C. Efthimion, G. TaylorPrinceton Plasma Physics Laboratory, Princeton, NJ, 08543, U.S.A.E-mail contact of main author: abhay@psfc.mit.eduAbstract. The high β operating regime of spherical tokamaks (ST), such as in NSTX andMAST, make them attractive fusion devices. To attain the high β’s, there is a need to heat andto drive currents in ST plasmas. While ST plasmas are overdense to conventional electroncyclotron (EC) waves, electron Bernstein waves (EBW) oﬀer an attractive possibility bothfor heating and for driving plasma currents. We consider techniques for the excitation ofEBWs on NSTX and MAST-type plasmas. Emission of EBWs from inside the plasma andits conversion to the conventional EC modes at the plasma edge are also considered.1. IntroductionIn STs like NSTX and MAST, fpe/fce >> 1 over most of the plasma cross section (fpeand fce are the electron plasma and cyclotron frequencies, respectively). Consequently,in the frequency regime for which the ordinary O-mode and the extraordinary X-modedamping is appreciable inside a plasma, these modes are cutoﬀ at the plasma edge.However, EBWs, which have no density limits, can propagate into the plasma corefor frequencies above fce, and damp eﬀectively on electrons near the Doppler-shiftedelectron cyclotron resonance or its harmonics [1]. The localized, strong absorption ofEBWs also makes their emission from the electron cyclotron resonance, or its harmonics,a possible means for diagnosing the electron temperature proﬁle [2]. Since EBWs are notvacuum modes, the coupling of power to EBWs is indirectly through mode conversion ofthe slow X-mode (SX) to an EBW at the upper hybrid resonance (UHR). The couplingto the SX-mode is either through the fast X-mode or the O-mode, which can be directlyexcited from outside the plasma. The former is referred to as the X-B mode conversionprocess and the latter as the O-X-B mode conversion process. Results from theoreticaland computational studies on mode conversion coupling to EBWs, and the emission ofX- and O-modes via EBWs, are summarized below.2. Theoretical Modelling of Mode ConversionFrom an analytical treatment of wave propagation in an inhomogeneous, cold plasma[1], we ﬁnd that the mode conversion eﬃciency is dependent on an eﬀective tunnelingparameter η given by:η ≈ ωceLncα[√1 + α2 − 1]1/2(1)where all the quantities on the right-hand side are evaluated at the location of the UHR,ωce = 2πfce, Ln is the density scalelength, α = fpe/fce, and c is the speed of light. Forα ∼ 1:η ≈ 12[ωceLnc]≈ 293.5 |BLn|UHR (2)where B is the local magnetic ﬁeld in Tesla and Ln is in meters.For maximum X-B mode conversion, the X-mode should be propagating essentiallyacross the magnetic ﬁeld (i.e., n‖  1 where n‖ is the wave index parallel to the magneticﬁeld). Then, for a given plasma conﬁguration, the maximum power mode conversioneﬃciency (at n‖ ≈ 0) is:Cmax = 4e−πη(1− e−πη) (3)For Cmax >∼ 0.5, we require that 0.05 <∼ η <∼ 0.6; for η ≈ 0.22, Cmax = 1 giving completemode conversion.The O-X-B mode conversion is most eﬃcient when the O-mode cutoﬀ coincideswith SX-mode cutoﬀ. This occurs at a critical n‖ [3]: (n‖)crit = 1/√1 + α. However,in addition, one requires η > 1 in order to avoid coupling appreciable power to theoutgoing fast X-mode [4]. From these conditions, we ﬁnd that the X-B and the O-X-Bmode conversion processes optimize in diﬀerent regions of frequency and n‖ space.43.7 43.8 43.9 4402550x (cm)Real(k⊥) (cm−1 )(a)43.4 43.6 43.8 44050100150200250x (cm)(b)EBW SX FX O SX EBW Figure 1: (a) fUHR < 2fce, (b) fUHR > 2fce.For NSTX-type plasmas the wave frequency has to be such that the UHR frequencyfUHR ≈ 2fce for optimum X-B mode conversion. For optimum O-X-B mode conversionfUHR ≈ 4fce. The dispersion characteristics of the EBW change signiﬁcantly as thefUHR is changed from below 2fce to above. This is illustrated in Figures 1(a) and 1(b)for NSTX-type parameters [1] where the fully kinetic dispersion characteristics of thevarious electron cyclotron modes are plotted as a function of the radial distance alongthe equatorial plane; x = 44 cm is the outside edge of the plasma (x = 0 is the magneticaxis). The UHR is located at x ≈ 43.96 cm in the ﬁrst case and at x ≈ 43.82 cm in thesecond case. In the second case, the SX mode propagates through the UHR and thennear the edge of the plasma it turns into an EBW with a much shorter wavelength thanin the ﬁrst case.The results of Eq. (2) can also be used to determine the required density scalelengthsfor optimum mode conversion for a ﬁxed frequency. Such would be the case, for instance,for MAST where sources and couplers at 60 GHz are already installed on the machine.For MAST-type parameters with an edge magnetic ﬁeld of 0.37T and an edge densityof 4× 1019m−3, complete mode conversion (for n‖ = 0) is possible if Ln ≈ 1.7× 10−3mat the UHR. This corresponds to a density proﬁle of the form (1 − r2/a2)0.25. Foroptimizing the O-X-B mode conversion, a scale length of Ln ≈ 6×10−2m and a critical(n‖)crit ≈ 0.4 are required. These conditions also ensure that η > 1.3. Mode Conversion and EmissionWe have developed a numerical code that solves for mode conversion in an inhomo-geneous slab plasma with a sheared magnetic ﬁeld [1]. The code uses an approximatekinetic (Maxwellian) plasma model in which the EBW can be clearly identiﬁed. This isa sixth order ordinary diﬀerential equation and the mode conversion coeﬃcient is deter-mined from the actual power ﬂowing in EBW. This code can also be used to determinethe emission of EBWs from the electron cyclotron resonance. The emitted EBWs wouldmode convert to the X-mode and/or O-mode radiation at the UHR and be detected inthe vacuum region of a plasma device.Figures 2(a) and 2(b) show the results obtained from this numerical code for thecases of mode conversion heating and emission, respectively, for NSTX-type parameters[1] for a frequency of 14 GHz. In Fig. 2(a) we assume that the power is coupled intothe plasma through the X-mode. Then C is the fraction of this power that appearsin EBW, and RX and RO are the fractional powers being reﬂected back out on the Xand O modes, respectively. These fractional powers are plotted as a function of thenormalized toroidal mode numbers. (The poloidal mode number is set to zero.) InFig. 2(b) we assume that the EBW is emitted inside the plasma and propagates outtowards the edge. Near the UHR the EBW can couple to the X- and O-modes and canalso be reﬂected. CX and CO are the fractions of the input EBW power that are modeconverted to X and O modes, respectively. RE is the fraction of the input power thatis reﬂected back into the plasma along the EBW.4. RF Current Drive With Bootstrap CurrentSteady state operations of tokamaks will require non-inductive current generation.RF current drive in combination with the bootstrap current may achieve this goal; RFdriven current also oﬀers a means to control the current proﬁle. We have initiated astudy on the self-consistent interaction of RF current drive with bootstrap current in atoroidal plasma. The RF quasilinear diﬀusion coeﬃcient is included in the relativistic0 0.2 0.4 0.6 0.800.20.40.60.81nz=ckz/ω(a)0 0.2 0.4 0.6 0.800.20.40.60.81nz=ckz/ω(b) C  RX ROCX RE  COFigure 2: (a) X-B mode conversion heating, (b) emission from EBWneoclassical drift-kinetic equation for plasma transport. The formulation is limitedto the thin banana regime of electrons. A numerical code FASTFP-NC [5] in two-dimensional momentum space for solving the drift kinetic equation has been developed.We ﬁnd that synergistic eﬀects increase the total current beyond the linear combinationof RF generated current and bootstrap current. Bootstrap current in the presenceof lower hybrid current drive or electron cyclotron current drive has been calculatednumerically and with analytical approximations. The scaling of the synergistic eﬀectswith plasma parameters and with the bootstrap current are obtained from the analyticalformulation and are in reasonable agreement with the numerical results. The eﬀect ofEBW current drive on bootstrap current will be incorporated in the near future.5. AcknowledgementsThis work is supported by DoE Grant Numbers DE-FG02-91ER-54109 and DE-FG02-99ER-54521, and by UK DTI, and EURATOM.6. References[1] A. K. Ram and S. D. Schultz, to appear in Phys. Plasmas (October 2000); A. Bers,A. K. Ram, and S. D. Schultz, in Proc. 2nd Europhysics Topical Conf. on RF Heatingand Current Drive of Fusion Devices (EPS, Petit-Lancy, 1998), Vol. 22A, pp. 237–240.[2] G. Taylor et al., Princeton Plasma Physics Laboratory Report PPPL-3476 (July2000); to appear in Rev. Sci. Instr. (2000).[3] J. Preinhaelter and V. Kopecky, J. Plas. Phys. 10, 1 (1973).[4] S. D. Schultz, A. K. Ram, and A. Bers, paper IAEA-F1-CN-69/CDP/13, in Proc.17th IAEA Fusion Energy Conference (1998).[5] S. D. Schultz, Ph.D. thesis, Department of Physics, M.I.T., Cambridge, MA 02139USA. (1999).

Heating and current drive by electron Bernstein waves in NSTX and mast-type plasmas

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Heating and current drive by electron Bernstein waves in NSTX and mast-type plasmas

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