Cairns, R.A.

Davies, C.N.

Decker, J.

Efthimion, P.C.

Ram, A.K.

Taylor, G.

DSpace@MIT

PSFC/JA-05-18   Current Drive by Electron Bernstein Waves in Spherical Tori  A. K. Ram, J. Decker,  G. Taylor, a P. C. Efthimion, a  C. N. Lashmore-Davies,b R. A. Cairnsc    August 2005     Plasma Science and Fusion Center, Massachusetts Institute of Technology Cambridge, MA 02139  U.S.A.  a Princeton Plasma Physics Laboratory Princeton, NJ 08543  U.S.A.  b Culham Science Centre, United Kingdom Atomic Energy Authority (UKAEA) Abingdon, Oxon  OX14 3DB   United Kingdom  c University of  St. Andrews Fife KY16 9SS   Scotland   This work was supported by the U.S. Department of Energy, Grant No. DE-FG02-91ER-54109, by the U.S. Department of Energy jointly with the National Spherical Torus Experiment, Grant No. DE-FG02-99ER-54521, UK EPSRC, and EURATOM.  Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.  To appear in Proceedings of the 20th International Atomic Energy Agency (IAEA) Fusion Energy Conference, Vilamoura, Portugal, November 1–6, 2004. Current Drive By Electron Bernstein Wavesin Spherical ToriA.K. Ram 1), J. Decker 1), G. Taylor 2), P.C. Efthimion 2), C.N. Lashmore-Davies 3), R.A. Cairns 4)1) Plasma Science & Fusion Center, M.I.T., Cambridge, MA, 02139, U.S.A.2) Princeton Plasma Physics Laboratory, Princeton, NJ , 08543, U.S.A.3) EURATOM/UKAEA Fusion Association, Culham Science Centre, Abing-don, Oxon, OX14 3DB, U.K.4) University of St. Andrews, St. Andrews, Fife, KY16 9SS U. K.e-mail contact of main author abhay@psfc.mit.eduAbstract. The high-β operating regime of spherical tokamaks (ST), such asin NSTX and MAST, make them attractive fusion devices. For access to such highβ regimes, it is necessary to heat and to drive currents in ST plasmas. While suchplasmas are overdense to conventional electron cyclotron waves, electron Bernsteinwaves (EBW) offer an attractive means toward this purpose. The applications ofEBWs in STs range from plasma start-up and heating of the ST plasma to modify-ing and controlling its current profile. The controlling of the current profile couldprovide better confinement as well as help suppress neoclassical tearing modes.This paper deals with two particular topics. The first topic is on the relevanceof relativistic effects in describing the propagation and damping of EBWs. Thesecond topic is on plasma current generation by EBWs.1. IntroductionIn order to achieve the high βs, STs generally operate at low magneticfields and high densities such that ωp/ωc >> 1 over most of the plasma. Hereωp and ωc are the angular electron plasma and cyclotron frequencies, respec-tively. Such an overdense nature of ST plasmas makes them unsuitable forheating and/or current drive by the conventional ordinary O and extraordi-nary X modes in the electron cyclotron (EC) range of frequencies. For lowharmonics of ωc the X and O modes are cutoff near the edge of the plasma.For high harmonics these modes do access the core of the ST plasma but areessentially undamped when they encounter the electron cyclotron resonances.However, EBWs offer an attractive alternative in the EC frequency range as1they have no density cutoffs, and damp strongly on electrons at the funda-mental, or any harmonic of the Doppler-shifted electron cyclotron resonance[1]. Since EBWs cannot propagate in vacuum (like the X and O modes) theyare excited, indirectly, by mode conversion of externally launched O modeor X mode [1, 2, 3, 4]. In this paper we do not discuss the mode conversionexcitation of EBWs as it has already been covered in the literature.In studying the propagation and damping of the traditional X and Omodes in the EC frequency range it has been noted that weakly relativisticeffects are important [5, 6]. We find that for EBWs the same, weakly rela-tivistic, formalism for the wave description cannot be used. Also, for EBWs,unlike the X and O modes, k⊥ρe can easily exceed 1 as the waves propagateaway from the mode conversion region into the plasma core [1]. (k⊥ is thewave vector perpendicular to the magnetic field and ρe is the electron Larmorradius.) Thus, we cannot do any Larmor radius expansions of the dielectrictensor elements. Consequently, we have developed a code R2D2 which solvesthe fully relativistic wave dispersion relation. Our initial results from R2D2indicate that relativistic effects are important for EBWs in a ST plasma awayfrom the mode conversion edge region.An important role for EBWs in STs could be to generate non-inductiveplasma currents. A study of the propagation of EBWs in a toroidal equilib-rium using a non-relativistic ray trajectory code shows that the parallel wavenumber n‖ = ck‖/ω can change from well below 1 to above 1 along the raypath [1]. (k‖ is the component of the wave vector parallel to the magneticfield, c is the speed of light, and ω is the wave angular frequency.) Thisis primarily due to the poloidal magnetic field. Consequently, the accessi-ble phase space for current drive by EBWs is richer than for the X and Omodes. For the X and O modes the resonance surfaces for the wave-electroninteractions are elliptic while the diffusion paths lie along hyperbolas. TheEBWs not only have these properties for n‖ < 1, but also the property thatthe resonance surfaces become hyperbolic and the diffusion paths become el-liptic for n‖ > 1. A drift kinetic Fokker-Planck code DKE [7] which includesa quasilinear RF diffusion operator and the effect of trapped electrons is be-ing used to study EBW current drive. Preliminary results show that in theouter half of a ST plasma, where trapped electron population is significant,EBWs effectively drive current through the Ohkawa mechanism [8, 9]. In thecore of the plasma where the fraction of trapped electrons is reduced, EBWseffectively generate current through the Fisch-Boozer scheme [10].22. Relativistic Propagation of Electron Bernstein WavesWe follow Trubnikov’s formalism for the derivation of the relativistic di-electric tensor [11]. From the linearized Vlasov equation, the perturbed dis-tribution function isf1 = − 1Ωexp{−iΩ(λφ− p⊥k⊥mγsin (φ− ψ))}×∫ φ−∞dφ′ exp{iΩ(λφ′ − p⊥k⊥mγsin (φ′ − ψ))}Pˆ ′ f0(p⊥, p‖)(1)where f0 is the unperturbed distribution function for the electrons,Pˆ−q =1√2(Ele−iφ + Ereiφ)Gˆ+ E‖(k⊥ωcos (ψ − φ) Hˆ + ∂∂p‖)(2)Gˆ =∂∂p⊥− k‖ωHˆ, Hˆ =p‖mγ∂∂p⊥− p⊥mγ∂∂p‖(3)λ = ω − k‖p‖mγ, γ =(1 +p2⊥m2c2+p2‖m2c2)1/2, Ω =qB0mγ, (4)px = p⊥ cos (φ) , kx = k⊥ cos (ψ) ,√2El,r = Ex ± iEy. (5)The plasma conductivity tensor is obtained from the current density~j = q∫d3p~pmγ(f0 + f1) = σ · ~E (6)There are two complementary approaches for obtaining σ [11]. The first isto perform the momenta integrals analytically and do the φ′ (time history) in-tegral in (1) numerically. This essentially works for a relativistic Maxwellianf0(p⊥, p‖). The second technique, is to perform the φ′ (time history) integralanalytically and do the momenta integrals in (6) numerically. This techniqueis valid for arbitray f0’s.For a relativistic Maxwellian the first approach leads to the following formof the conductivity tensor:σ =14piω2pωcc4v4t1K2(c2v2t) ∫ ∞0dξ{K2(R1/2)RT 1 −K3(R1/2)R3/2T 2}(7)3where ωp, ωc, vt are the rest mass electron plasma frequency, cyclotron fre-quency, and the thermal velocity, respectively, Kν is the modified Besselfunction of the second kind of order ν,T 1 =cos ξ − sin ξ 0sin ξ cos ξ 00 0 1 (8)T 2 =c2ω2ck2⊥ sin2 ξ −k2⊥ sin ξ (1− cos ξ) k⊥k‖ξ sin ξk2⊥ sin ξ (1− cos ξ) −k2⊥ (1− cos ξ)2 k⊥k‖ξ (1− cos ξ)k⊥k‖ξ sin ξ −k⊥k‖ξ (1− cos ξ) k2‖ξ2(9)R =(c2v2t− iξ ωωc)2+ 2(k⊥cωc)2(1− cos ξ) + k2‖c2ξ2ωc2(10)For any equilibrium distribution function f0(p⊥, p‖) the second approachleads to the following conductivity tensor:σ = − i2ω2pωc〈 ∞∑n=−∞1n− ω(1κTp⊥mγ)σN f0(p⊥, p‖) 〉(11)whereσN =n2ζ2p⊥J2n −inζ p⊥JnJ′n nζ p‖J2ninζp⊥JnJ′n p⊥J′2n ip‖JnJ′nnζp‖J2n −ip‖JnJ′np2‖p⊥J2n (12)ζ =k⊥p⊥mωc, ω =1ωc(ωγ − k‖p‖m), ωc =eB0m, 〈.....〉 =∫ ∞0dp⊥p⊥∫ ∞−∞dp‖(13)We have developed two separate numerical routines within the code R2D2which are based on the two relativistic formalisms discussed above. For avariety of cases we find that the two formalisms lead to numerically identicalresults. Since we are not aware of any similar code in existence, this allowsus to benchmark our code. In Fig. 1 we compare the results obtained fromthe relativistic description (red) with those obtained from the non-relativistic4description (blue) in the mode conversion region for NSTX model equilibrium[1]. We find that there is essentially no difference between the two cases inthe low temperature region where the mode coupling takes place. Thus,the mode conversion formalism developed in [1, 2, 3, 4] is not modified byrelativistic effects. However, the EBW part of the dispersion relation beginsto show some differences away from the mode conversion region. In Fig. 2 weshow the dispersion characteristics of EBWs as a function of n‖ for a uniformplasma with electron temperature of 3 keV, ωp/ωc = 6, and ω/ωc = 1.8. It isnow evident that there are significant differences between the relativistic andnon-relativistic properties of EBWs. This could have important consequenceswhen studying the propagation and damping of EBWs in ST plasmas.3. Electron Bernstein Waves Current DriveAs part of the overall scheme to use EBWs for driving plasma currentsin a ST, and possibly for achieving a steady-state operation, it is importantto understand the parametric dependence of EBW current drive efficiency.The relativistic code DKE [7] is being used for studying the EBW drivencurrent. This code solves in two-dimensional momentum space, on a givenflux surface, the neoclassical drift-kinetic equation with a RF quasilinear dif-fusion operator. In deriving the equations solved in DKE, we assume thatthe electron guiding center drift velocity across the flux surfaces is smallcompared to its streaming velocity along the field line. We further assumeweak collisionality so that the banana approximation is valid. This impliesthat the bounce time of trapped electrons is much shorter than the collisionaldetrapping time. The bounce time is also assumed to be shorter than the RFinduced diffusion time. Consequently, the electron distribution function isuniform along the field lines. To leading order in the small parameter definedby the ratio of the bounce time to the drift time, the electron distributionfunction f(p⊥, p‖) in momentum space is obtained from the bounce averagedFokker-Planck equation 〈C(f)+Q(f)〉, where C is the collision operator andQ is the quasilinear diffusion operator. DKE is used to solve this Fokker-Planck equation for f(p⊥, p‖). The quasilinear diffusion coefficient Q(f) inDKE is determined from R2D2. By taking the appropriate moments of f wecan solve for the parallel current and the power dissipated and, consequently,evaluate the current drive efficiency η = (J/enevte)/(P/νemenev2te). Here Jis the current density, P is the density of power dissipated, e is the electroncharge, me is the electron mass, ne,vte, and νe are the local electron density,543.85 43.9 43.95 440123456x (cm)Real(n⊥) EBW X OX 43.85 43.9 43.95 44024681012x (cm)Imag(n ⊥)Figure 1: Real (left) and imaginary (right) part of n⊥ = ck⊥/ω versus dis-tance on the equatorial plane for NSTX-type parameters [1]. The comparisonis between the relativistic (red) and non-relativistic (blue) characteristics ofECRF waves in the mode conversion region.60.2 0.4 0.6 0.8 1 1.2 1.4 1.62.833.23.43.63.84n||Real(n⊥)0.2 0.4 0.6 0.8 1 1.2 1.4 1.600.511.522.5n||Imag(n ⊥)Figure 2: Real (left) and imaginary (right) part of n⊥ versus n‖. The compar-ison is between the relativistic (red) and non-relativistic (blue) characteristicsof EBWs.7thermal velocity, and collision frequency, respectively. In our initial studiesdescribed below, we use the non-relativistic form of the EBW diffusion coef-ficient in Q(f). The non-relativistic evaluation takes substantially less timethan the relativistic calculation.As an example of EBW current drive, we consider a NSTX-type plasmawith major radius of 0.9 m, minor radius of 0.6 m, on axis magnetic fieldof 0.35 T, plasma current of 0.8 MA, peak electron temperature of 3 keV,and a peak electron density of 3 × 1019 m−3. The density and temperatureprofiles are assumed to be parabolic. In this high β regime, the poloidal fieldis comparable to the toroidal field on the outboard part of the plasma. Weassume an EBW wave frequency of ω/2pi = 11.8 GHz and n‖ = 1.5. Wealso consider a single EBW ray propagating along the equatorial plane withan input power in the ray of 1 MW. The left panel of Fig. 3 shows thatthe wave frequency matches the Doppler-shifted (second harmonic) electroncyclotron frequency where the total magnetic field is a minimum. The resultsfrom DKE are plotted in the right panel of Fig. 3. The peak current densityis Jpeak ≈ 0.64 MA m−2. At this location the density of power dissipatedis Ppeak ≈ 1.4 MW m−3. This leads to ηpeak ≈ 3.2. This current driveefficiency is significantly higher than is generally achieved by conventionalECRF waves in tokamaks. From the left panel of Fig. 3 the location wherethe EBW frequency matches the Doppler-shifted cyclotron resonance occurswhere the magnetic field is nearly flat (red line tangent to the green line).This leads to a large optical depth and the EBW interacts with energeticelectrons in the distribution function. Consequently, we also obtain highcurrent drive efficiencies. This shows that the non-monotonic magnetic fieldprofile has important implications for EBW current drive in STs. The leftpanel of Fig. 4 shows that the phase velocity of the waves is in a directionopposite to the direction of the driven current. The region of maximumdiffusion is situated near the trapped/passing boundary leading to a largeEBW-induced trapping of passing electrons. The right panel illustrates thismore clearly. The depletion of electrons due to trapping induced by EBWsin the region where the wave-particle resonance exists, and the accumulationof electrons due to detrapping on the opposite side (in p‖) contribute to thecurrent. This indicates that the current driven is the Ohkawa current [8, 9]rather than the Fisch-Boozer current [10]. Recent numerical simulations [12]have shown similar results.Figure 5 corresponds to Fig. 3 for a wave frequency of 19.5 GHz andn‖ = 0.5. The peak current density is Jpeak ≈ 2.6 MA m−2. At this loca-80 0.5 105101520ρf (GHz)2fce 2fce − 3.5k//vTefW 0.3 0.4 0.5 0.600.20.40.60.8ρJ (MA/m2)246810ηFigure 3: Left: wave frequency (fw), second harmonic of the electron cy-clotron frequency (2fce), and the Doppler-shifted (second harmonic) electroncyclotron frequency (in green) as a function of ρ (radial distance normalizedto the minor radius); right: current density (red) and efficiency (blue) as afunction of ρ.9−10 0 100510p///pTep ⊥/pTe−5 0 5−2−1012x 10−3p///pTeF // − F // MFigure 4: Left: contours of Maxwellian distribution (black) and distribu-tion function (red) in the presence of EBW diffusion coefficient (contours ingreen); right: the parallel component of the RF driven distribution function,with the Maxwellian contribution subtracted off, as a function of the parallelmomentum normalized to the thermal momentum.10tion the density of power dissipated is Ppeak ≈ 6.1 MW m−3. This leads toηpeak ≈ 1.9. From the two panels in Fig. 6 we note that this is the con-ventional Fisch-Boozer scheme of current drive. The EBW induced diffusionof electrons changes the resistivity non-symmetrically (in p‖) leading to thecurrent generation. However, the trapping effects reduce the current driveefficiency. Studies are underway to determine the conditions for optimizingthe Fisch-Boozer current drive and the Ohkawa current drive schemes in anST.In conclusion, our calculations show that relativistic effects are impor-tant in describing the propagation and damping of EBWs in ST plasmas.Furthermore, our calculations on EBW current drive show that the Ohkawacurrent drive is more suitable for the outer half of the plasma while theFisch-Boozer current drive works well near the core. Thus, the EBW spec-trum could be tailored, according to the needs, for current generation andcurrent profile control. Detailed studies on relativistic effects in EBW propa-gation and damping, and on the EBW driven current are continuing. Alongwith the previous results on coupling and excitation of EBWs the resultsfrom these studies will provide a general basis for defining the role of EBWsin present-day experiments and future ST power plants.This work was supported by DOE Grant Numbers DE-FG02-91ER-54109and DE-FG02-99ER-54521, and by UK EPSRC, and EURATOM.References[1] A. K. Ram and S. D. Schultz, Phys. Plasmas 7, 4084 (2000).[2] J. Preinhaelter and V. Kopecky, J. Plas. Phys. 10, 1 (1973).[3] A. K. Ram and A. Bers, Nucl. Fus. 43, 1305 (2003).[4] A. K. Ram, A. Bers, and C. N. Lashmore-Davies, Phys. Plasmas 9, 409(2002).[5] G. LeClair, I.P. Shkarofsky, Y. Demers, and J-F. Mercier, in Proc. 12thTopical Conf. on RF Power in Plasmas, A.I.P. Conf. Proc. 403 (eds.P. M. Ryan and T. Intrator), Woodbury, New York (1997), p. 219.[6] M. Bornatici, R. Cano, O. De Barbieri, and F. Engelmann, NuclearFusion 23, 1153 (1983).110 0.5 10102030ρf (GHz)n = 2n = 3nωcenΩ ± 3.5k//vTef = 19.5 GHzN// = 0.5 0.1 0.2 0.3 0.40123ρJ (MA/m2)0123ηFigure 5: Left: wave frequency (fw), second (2fce) and third (3fce) harmonicof the electron cyclotron frequency, and the corresponding Doppler-shiftedelectron cyclotron frequencies (in green) as a function of ρ; right: currentdensity (red) and efficiency (blue) as a function of ρ.12−10 0 100510p///pTep ⊥/pTe−10 0 10−0.01−0.00500.0050.01p///pTeF // − F // MFigure 6: left: contours of Maxwellian distribution (black) and distribu-tion function (red) in the presence of EBW diffusion coefficient (contoursin green); right: parallel component of the RF driven distribution function,with the Maxwellian contribution subtracted off, versus p‖/pTe. .13[7] Y. Peysson, J. Decker, and R.W. Harvey, in Proc. 15th Topical Conf.on RF Power in Plasmas, AIP Conf. Proc. No. 694 (ed. C.B. Forest),Melville, New York (2003), p. 495.[8] T. Ohkawa, General Atomics Report No. GA-A13847 (1976).[9] More recently Ohkawa current drive using waves in the electron cy-clotron range of frequencies has been discussed by: 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.[10] N.J. Fisch and A. Boozer, Phys. Rev. Lett. 45, 720 (1980).[11] B. A. Trubnikov, in Plasma Physics and the Problem of Controlled Ther-monuclear Reaction, edited by M. A. Leontovich (Pergamon Press, NewYork, 1959), Vol. III.[12] G. Taylor et al., Phys. Plasmas 11, 4733 (2004).14

Current Drive by Electron Bernstein Waves in Spherical Tori

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Current Drive by Electron Bernstein Waves in Spherical Tori

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