Bers, A.

Ram, A.K.

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PSFC/JA-01-37ELECTRON BERNSTEIN WAVE HEATINGAND EMISSION IN SPHERICAL TOKAMAKSA. K. Ram and A. BersDecember 2001Plasma Science & Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139, U.S.A.This 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 2nd IAEA Technical Committee Meet-ing on Spherical Tori and 7th International Spherical Torus Work-shop, Sao Jose dos Campos, Brazil, August 1–3, 2001.iiiELECTRON BERNSTEIN WAVE HEATINGAND EMISSION IN SPHERICAL TOKAMAKSA. K. Ram and A. BersPlasma Science and Fusion CenterMassachusetts Institute of Technology, Cambridge, MA 02139. U.S.A.E-mail contact: abhay@psfc.mit.eduAbstract. This paper summarizes the theoretical and numerical results we have obtained forthe excitation and emission of electron Bernstein waves in spherical tokamak plasmas.1. INTRODUCTIONThe high β operating regime of spherical tokamaks (ST), such as in NSTX and MAST,make them attractive fusion devices. To attain the high β’s there is a need to heat and todrive currents in ST plasmas. While ST plasmas are overdense to conventional electroncyclotron (EC) waves, electron Bernstein waves (EBW) oﬀer an attractive possibilityboth for heating and for driving plasma currents. EBWs, which have no density limits,can propagate into the plasma core for frequencies above the electron cyclotron frequencyfce [1]. Since EBWs are not vacuum modes, they are excited inside the plasma by modeconversion of the traditional X and O modes. From ray tracing analysis we ﬁnd thatEBWs are strongly absorbed by electrons in the region where the wave frequency matchesthe Doppler broadened electron cyclotron resonance frequency or its harmonics [1]. Thestrong and localized absorption implies that thermal emission of EBWs can occur forfrequencies corresponding to the local Doppler-shifted electron cyclotron frequency. Thisemission then converts, at the UHR, to the X and O modes which are then observed inthe vacuum region.2. EXCITATION OF ELECTRON BERNSTEIN WAVESThe excitation of EBWs have been studied in a slab geometry model where the xcoordinate is in the direction of the inhomogeneity, y is along the poloidal direction,and z along the toroidal direction. Assuming that the magnetic ﬁeld is along z, thepropagation of waves in an inhomogeneous cold plasma is given byK⊥d2Eydξ2+(K2⊥ −K2X −c2ω2k2‖K⊥)Ey =c2ω2k‖KXF (1)ddξ[1K‖dFdξ]+(1−c2ω2k2‖K⊥)F =KXK⊥k‖Ey (2)where F = ik‖Ex− dEz/dx, ξ = ωx/c, c is the speed of light, ω is the wave frequency, k‖is the component of the wave vector k along the magnetic ﬁeld B = B0(x)zˆ,K⊥ = 1−ω2peω2 − ω2ce, KX = −ωceω(ω2peω2 − ω2ce), K‖ = 1−ω2peω2(3)ωpe and ωce are the electron plasma and cyclotron angular frequencies, respectively.Let us consider the case when k‖ = 0. Then, Eqs. (1) and (2) are uncoupled anddescribe the propagation of the X mode and the O mode, respectively, across the magneticﬁeld. In this case the X mode equation (1) has a resonance at K⊥ = 0 correspondingto the upper hybrid resonance (UHR). Also Eq. (1) has two cutoﬀs corresponding tothe zeros of K2⊥ − K2X . The resonance power absorption of the X mode at the UHRcorresponds to the power mode converted to the EBWs [1]. From an analogous analysiscarried out in [2], we ﬁnd that, for the case when the left hand cutoﬀ is close to the UHR,the fraction of the power on the X mode that is converted to EBWs is given byCXB = 4e−πη(1− e−πη)cos2 (φ) (4)whereφ =π2+12cos−1(N2R −N2IN2R +N2I)(5)N = NR + iNI ,N = ln(iηη1)+ π[coth(π2η)+ cot(π2η1)]−1η1−iη+ψ(1 +η12)− ψ(1 +iη2)(6)ψ is the Psi function,η ≈[ωceLncα(√1 + α2 − 1)2]UHR(7)η1 ≈[ωceLncα(√1 + α2 + 1)2]UHR(8)In Eqs. (7) and (8) the right-hand sides are evaluated at the UHR, Ln is the densityscalelength, and α = ωpe/ωce. If φ is an integer multiple of π, then from (4) we note thatCXB ≥ 0.5 if 0.05 <∼ η<∼ 0.6. From (7) we ﬁnd that the necessary condition for the X-Bmode conversion to be better than 50% requires1.3× 10−2 T cm <∼ |BLn|UHR<∼ 0.16 T cm (9)This shows that the X-B mode conversion process is eﬃcient for sharp density gradients,i.e., short density scalelengths.From Eq. (2) it is clear that for k‖ = 0 the O mode does not couple power to theEBW since there is no resonance absorption of the O mode. For the O mode power tocouple to EBW it is necessary to have oblique propagation of the O mode, i.e., k‖ = 0.Then the O mode power can couple to the X mode which in turn can undergo resonanceabsorption. The necessary condition for optimum coupling of the O mode power to theX mode is [3]n‖,c =ck‖ω=[(1 +ωωce)−1/2]c(10)where the right-hand side is to be evaluated at the spatial location where ω = ωpe.However, in order for the O mode power not to get reﬂected back out on the X moderequires that η > 1 [1]. From (7) we ﬁnd that this condition is satisﬁed when the densityscalelength is long.From the above discussion we conclude that the X-B and the O-B mode conver-sion processes are optimized in diﬀerent regions of the two-dimensional parameter spacespanned by wave frequency and parallel wavelength. The X-B mode conversion processis optimized for k‖ ≈ 0 and wave frequencies which place the UHR in the short densityscalelength region. The O-B mode conversion process is optimized for k‖ given in (10)and for wave frequencies which place the UHR in a longer density scalelength region ofthe plasma. The optimum X-B process occurs closer to the edge of the plasma thanthe optimum O-B conversion. A more detailed analysis based on an approximate kinetictreatment of the EBW in sheared magnetic ﬁelds conﬁrms the basic results of the analysisgiven above [1].3. EMISSION OF ELECTRON BERNSTEIN WAVESFrom ray tracing analysis we have shown that EBWs are locally and strongly absorbedat the Doppler shifted electron cyclotron resonance or its harmonics [1]. The strong andlocalized absorption implies that thermal emission of EBWs can occur for frequenciescorresponding to the local Doppler-shifted electron cyclotron frequency. This emissionthen converts, at the UHR, to the X and O modes which are then observed in thevacuum region. The relationships between the scattering coeﬃcients for mode conversionexcitation and for mode conversion emission can be established based on global propertiesof the model description of the mode conversion region [4, 5]. The global properties ofour full-wave model equations for the mode conversion region are linearity, energy ﬂowconservation, and (Onsager-like) time reversibility.The wave ﬁelds are assumed to have a WKB form outside the mode conversion region.Referring to Fig. 1, towards the outside, low density region of the plasma (designatedas “right”) the X and O modes are propagating with energy ﬂow into or out of themode conversion region, while the EBWs are evanescent with no associated energy ﬂow.Towards the inside, high density region of the plasma (designated as “left”) the EBWsare propagating with energy ﬂow into and out of the mode conversion region, while the Xand O modes (due to their cutoﬀs) are evanescent with no energy ﬂow. Let the complexamplitudes of the propagating modes be such that |ai|2 is the energy ﬂow density into themode conversion region and |bi|2 is the energy ﬂow density out of the mode conversionregion. (i = X, O, B designating the X mode, the O mode, or the EBW, respectively.)Then, since the mode conversion equations are linear, the complex ﬁeld amplitudes aiLEFT RIGHTbaXXbaOOabBB MODELOSS-FREECONVERSIONREGIONFigure 1: The loss-free mode conversion regionand bi are related by a scattering matrixbBbXbO =SB SBX SBOSXB SX SXOSOB SOX SOaBaXaO (11)or, simply, in matrix notationb =↔S ·a (12)The matrix↔S is unique since the solution of the linear mode conversion equations withthe appropriate boundary conditions is unique.It can be readily shown [4, 5] that from energy conservation and from the energy ﬂowunder time reversibility that↔ST=↔S (13)where the superscript T indicates the transpose of the matrix. Thus,↔S is symmetric.This implies thatSBX = SXB, SBO = SOB, SXO = SOX (14)The symmetry of the scattering matrix↔S then leads to the following relationshipsthat have important experimental consequences [4, 5]EX = CXB, EO = COB (15)where EX and EO are the emission coeﬃcients that give the fraction of the emitted EBWpower that is mode converted to the X and O modes, respectively, and CXB and COBare the excitation coeﬃcients that give the fractions of the input power on the X andO modes, respectively, that is converted to the EBWs. These relationships show thatthe edge plasma characteristics that give maximum power emission from EBWs are alsothe optimum conditions for obtaining maximum power excitation of EBWs from sourcesexternal to the plasma. The monitoring of emission can be readily implemented experi-mentally. These observations can provide a useful guide for the design and optimizationof coupling power to EBWs for heating and current drive in ST plasmasACKNOWLEDGEMENTSThis work is supported by DoE Grant Numbers DE-FG02-91ER-54109 and DE-FG02-99ER-54521.References[1] A. K. Ram and S. D. Schultz, Phys. Plasmas 7, 4084 (2000).[2] A. K. Ram, A. Bers, S. D. Schultz, and V. Fuchs, Phys. Plasmas 3, 1976 (1996).[3] J. Preinhaelter and V. Kopecky, J. Plas. Phys. 10, 1 (1973).[4] A. K. Ram, A. Bers, and C. N. Lashmore-Davies, submitted to Phys. Plasmas (Au-gust 2001).[5] A. Bers and A. K. Ram, to appear in Bull. Am. Phys. Soc. (2001).

Electron Bernstein Waves Heating and Emission in Spherical Tokamaks

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Electron Bernstein Waves Heating and Emission in Spherical Tokamaks

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