Current Drive By Electron Cyclotron Waves In Stellarators.
693SplSSN614-087-XCURRENT DRIVE BY ELECTRON CYCLOTRONWAVES IN STE LLAR ATO RSpor:F. CastejónC. AlejaldreJ. A. CoarasaCENTRO DE INVESTIGACIONESENERGÉTICAS, MEDIOAMBIENTALES Y TECNOLÓGICASMADRID.1992
CLASIFICACIÓN DOE Y DESCRIPTORES700350MAGNETIC CONFINEMENTELECTRON CYCLOTRON-RESONANCENON-INDUCTIVE CURRENT DRIVEPLASMA HEATINGSTELLARATORS
Toda correspondencia en relación con este trabajo debe dirigirse al Serviciode Información y Documentación, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas, Ciudad Universitaria, 28040-MADRID, ESPAÑA.Las solicitudes de ejemplares deben dirigirse a este mismo Servicio.Los'descriptores se han seleccionado del Thesauro del DOE para describir lasmaterias que contiene este informe con vistas a su recuperación. La catalogación se hahecho utilizando el documento DOE/TIC-4602 (Rev. 1) Descriptive Cataloguing OnLine, y la clasificación de acuerdo con el documento DOE/TIC.4584-R7 Subject Categories and Scope publicados por el Office of Scientific and Technical Information delDepartamento de Energía de los Estados Unidos.Se autoriza la reproducción de los resúmenes analíticos que aparecen en estapublicación.Este trabajo se ha recibido para su impresión en Febrero de 1.992.Depósito Legal nQ M-9487-1992ISBN 84-7834-142-0ISSN 614-087-XÑIPO 238-92-013-5IMPRIME CIEMAT
I-INTRODUCTIONCurrent drive by microwaves in the range of the enexperimentally demonstrated in tokamaks [1,2,3] and stellarators[4], with results that allow for a modérate optimism on thecapability of this method to genérate non-inductive currents in"next step" fusión devices. The motivation to use ECCD inTokamaks is clear, microwaves in this frequency range canpenétrate to the core of a reactor grade plasma and noninductively drive current where it is needed. Stellarators, being"current-free" devices, do not need any such scheme to obtainparticle confinement, all needed currents are produced byexternal coils. The usefulness of ECCD on stellarators, specially inshearless- devices, lays on the possibility of using these waves tocompénsate intrinsic plasma currents like bootstrap that can bedeleterious for confinement in the device. The clear experimentson Wendelstein VII-AS show how confinement is greatlyimproved when ECCD is used to compénsate bootstrap and PfirschSchlüter currents making the device a true current free stellarator[5]. This is particularly important in configurations, like the Heliacunder construction in Madrid, TJ-II, with magnetic configurationshaving an iota -or q- profile practically shearless, where pressuredriven currents can push the iota profile towards rational valúeswith the corresponding consequences for confinement. Also, dueto the extreme localization of EC waves power deposition, thepossibility opens to use ECCD to locally control q-profíles [6],although this awaits experimental proof.Current drive by EC-waves consists of the asymmetricmodification of the electrón resistivity in the momentum space [7].Even a diffusion in the perpendicular momentum direction causesa net current, provided it is asymmetric in the parallel direction,by the modification of the colusión frequency of the electrons. Theresonant absorption modifies the distribution function which canbe computed solving the Fokker-Planck equation (see e. g. [8, 9,10]). Once the distribution function is known, the current densityparallel to the magnetic field can be evaluated.Such a calculation is expensive in terms of computationaltime. There are alternative methods which allow the estimation ofthe current without knowing the distribution function. Thosemethods are either based on the adjoint approach as presented in[11, 12], or on the Langevin equations [13], as the calculation
presented in this work. The last one gives the current driveefficiency for a test particle as:5P(s-V)E(p)where E is the kinetic energy of the resonant particle, the vector sis parallel to the diffusion direction in the momentum space, andis given by:co(2)and % is the so called response function, i. e., the total contributionof the particle to the current:To calcúlate the response function one must know the testparticle evolution in momentum space, that is obtained averagingLangevin equations for a particle ensemble embedded in athermal bath [14].This method gives a linear estimation of the induced currentand is accurate if the absorbed power density is not very high andthe distribution function not far from Maxwellian. The method isfast enough to allow the inclusión of the efficiency function in aray tracing code, which calculates the absorbed power densityand, therefore, the induced current in a complex magneticconfiguration can be computed.In the present work the induced current by EC waves iscomputed for the heliac TJ-II. The complex geometry of themachine is introduced in the ray tracing code RAYS [15] tocalcúlate the absorbed power density profile. The width andstructure of the microwave beam are taken into account sincethey are not negligible respect to the plasma size.
II-THE EFFICIENCY FUNCTIONThe induced current density parallel to the magnetic fieldcan be written in terms of the absorbed power density in phasespace and the microscopic efficiency as follows :(4)The constant A is:A — meAewhere n, m and e are the electrón density, mass and charge, c isthe speed of light and A is the Coulomb logarithm.The, efficiency can be obtained from the Langevin equationsaveraged for a Maxwellian distribution function in the highenergy limit (see [13]). For the relativistic case and, in general, foroblique propagation, the efficiency is :Tl(?) L G(v) Nllf[ " (Y 1 z) ]2vv,,(5),where the function G(v) is given by:2G(v) vfVri31 Z2f dxJOJo(6)and we have adopted the following normalization:/.pv 9\i/2meFor Zeff l the function G(v) can be computed analytically:(7)For arbitrary valúes of Zeff the integral in (6) must be performednumerically. In figure 1 the efficiency is presented for somevalúes of Zeff. It is shown that the efficiency decreases with Zeff,
which means that the impurities have a negative influence oncurrent drive efficiency and so do múltiple charged ion plasmas.This is not surprising since the colusión frequency rises with Zeffand the particles are thermalized faster.When varying the parallel refraction index, Nu, theefficiency rises for the momenta of the sign of the index, but fallsin the opposite direction, as can be seen in figure 2. The efficiencyis an odd function in vn when Nn 0 and is an even function in theproduct V N .Rising the perpendicular momentum is in generaldeleterious for the efficiency, but for the valúes we consider here(pj /mc«0.01) it has no appreciable influence.We consider the absorbed power density in phase space atharmonic s, w s , which is different from zero only for the resonantelectrons:w s (v)oc8 y Vco.,„.„,;(g)This guarantees that we obtain zero current for Nn 0, since ws isan even function and the efficiency an odd one. Moreover, as theefficiency is an even function in the product VHNH, changing thesign of Ñu will change the sign of the induced current density.After integrating in v¿ in (4), the current density parallel tothe magnetic field can be written as, ¿Jv Bs i-n(H)i l l(9)The absorbed power density for a Maxwellian distributionfunction, at lowest order in Larmor radius, is given by (see e. g.[16]) :(v ) 64coK2( X)lv2coc2-¡¿Y )(10)
In this linear theory, the efficiency has been calculated as anaverage in time and for a test ensemble of partióles embedded ina Maxwellian distribution function, therefore it does not dependon the absorbed power density.The result of making the integration for the O mode at firstharmonio is shown in figure 3 a, and for the X mode at secondharmonio is shown in figure 3b. In both figures, the inducedcurrent is plotted versus frequency for several valúes of Nn,keeping constant the other plasma parameters. The absorbedpower density is the same for all the cases in each plot. It can beseen that the current has opposite signs at upshifted anddownshifted frequencies. The point where the current changessign does not depend on Nn, but on the valué of YS SCOC/CD. WhenN ii is increased the induced current density becomes moreunlocalized and falls. The difference between both figures, nearthe point where the current density is 0, can be explainedconsideriñg the different expressions of the absorbed powerdensity for X and O modes.The deformation of the distribution function can beconsidered a perturbation of the Maxwellian, that we can assumethat is well localized in the momentum space. The position of thedeformation will be near the so called Collective ResonantMomentum [16]. This momentum is defined as the point inmomentum space where the deformation in the distributionfunction is máximum or, equivalently, where absorbed power ismáximum:dw*(v') 0This condition is fulfilled by these valúes of VH:(11)Only one of the two valúes is inside the resonant momentuminterval, and it depends on the sign of Nn:N,, 0 v llR v R ; N,, 0 vllR vR
The induced current density parallel to the magnetic field at apoint in the plasma can be approximated by this expression [16,17]:jB( r )«Aii(vR) P (12)where the efficiency is calculated at the Collective ResonantMomentum. This approximation can be avoided calculatingnumerically the momentum integral (9). In figure 4 the currentdensity obtained by both methods at the second harmonic for Xmode is shown. The Collective Resonant Momentum approximation(curve marked with squares) is accurate only at downshiftedfrequencies, but this estimation is not so good at upshiftedfrequencies and near the point where the current changes the itssign.Moreover, the numerical integration allows the introductionof trapped particles effects, that are not evident using theCollective Resonant Moment approximation, because it gives nuilcurrent when such momentum is in the trapping región.
III- INCLUSIÓN OF TRAPPED PARTICLESThe effect of trapped particles on current drive can beimportant in stellarators, especially when ECRH is in consideration,since the diffusion in momentum space is mainly in theperpendicular direction and the stellarator configuration allows abig variety of trapping mechanisms.As it is well-known, a particle with momentum v, whoseguiding centre moves along a field line with momentum vn(t), istrapped if:l/2where fit is the trapping parameter, B is the local magnetic fieldand B m a x ' is the máximum magnetic field on a field line. Since thefield lines are dense on a magnetic surface, B m a xcoincidesapproximately with the máximum of the magnetic field at themagnetic surface. In a stellarator configuration, both fields mustbe calculated numerically using the Biot-Savart law [18].Let us consider how trapped particles modify the efficiency(5). Trapped particles do not contribute to the current in thebounce average, so the efficiency is zero in the trapping región:,) 0 ; v, v l l v 2Additionally, circulating particles can become trapped bythe effect of the diffusion in momentum space and, from thatmoment on, they will not contribute to the current, which must betaken into account in the calculation. The contribution of this kindof particles modifies the response function as follows [19]:V" X(vT)V,(15).is the momentum when the particle starting with momentum vbecomes trapped. That momentum is calculated using theLangevin equations, averaged over an ensemble of particles. The
results we have obtained for its parallel component and moduleare:andN"'l-g(v)(16),where we have introduced the function:(17).Using equation (1) we obtain the efficiency in presence oftrapped particles, outside the trapping región:TlT(v) r ( v ) -lJv2(18),where the derivative of the response function is given by:3v T(19)and r\(\) is the efficiency without trapped particles, given byequation (5). The sign of T\T(V) can be opposite to the sign of TJ(V).Then, we evalúate the integral (9) introducing the efficiency(18) and obtain the induced current density in presence oftrapped particles. In figure 5a the influence of trapped particleson EC current drive efficiency is shown. The induced currentdensity parallel to the magnetic field is plotted versus frequencyfor X mode at second harmonic and different valúes of thetrapping parameter. The nature of EC diffusion in momentumspace, which is mainly in the perpendicular direction, makes theeffect of trapped particles deleterious, and the module of theinduced current density falls when it rises. Even a change in thecurrent direction is possible when trapping parameter is largeenough.
The effect of trapped particles changes with N . This isbecause the number of resonant electrons which are trappedvaríes and their contribution to the current also varíes. This canbe understood considering the expression inside the integral of(9): When N\\ rises, the resonance curve in momentum spacemoves in the positive direction of vn and the máximum of theexpression inside the integral goes into the trapping región andthen goes out. The induced current versus frequency, for differentvalúes of N , is plotted in figure 6, we have taken a médium valuéfor the trapping parameter \it 0.2. It is shown that, when N\\ rises,the peak of positive current which appears at upshiftedfrequencies moves to the right and its máximum is notmonotonous.
IV - ADAPTATION TO A RAY TRACING CODETo take into account refraction effects and the complexmagnetic geometry of stellarators, we included the expression (9),with the efficiency (18), in the ray tracing code RAYS [20]. Theparameters needed to calcúlate the efficiency are given by thecode. We assume that the resonant electrons move on themagnetic surfaces and are quickly mixed with the rest. Thecurrent is then uniform on each magnetic surface and we calcúlatethe averaged absorbed power density inside a given magneticsurface [21], considering the differential volume of the wholemagnetic surface.The induced current at a point in the plasma can be writtenin terms of the absorbed power density, at the consideredharmonic s, as:J,,(r) Y(r)W s (r)(20)fwhere the global current drive efficiency has been introduced:Ajdvr T(v)5 s(v)y(r) Iw s (v)W s ( r ) , the averaged absorbed power density at a magneticsurface, is calculated as follows:W(r) AP(Í)22jt R 0 (2p(r)8 8 2 )(22)We have defined:(23)and8 max A r , b sinIIjkllWljj10
AP is the power absorbed in a ray step, given by the code, Ro isthe major radius, r is the mean minor radius of the magneticsurface, b is the width of a single ray, k is the wave vector, *P isthe toroidal flux and A r is the variation of the mean radius in theray step, calculated from flux coordinates. The parameter 8 ischosen to take into account the fact that when the ray is notnearly perpendicular to the magnetic surface (the direction givenby the flux gradient) only a fraction of the power carried by theray is deposited at such surface.To obtain the toroidal component of the current density onemust perform:DV (25),where B§ is the toroidal component of the magnetic field and B itsmodule.From this expression, one sees that the averaged powerdensity can be raised when the ray path is almost parallel to thesurface. It is also possible that a ray crosses twice the samemagnetic surface. In the last case the induced current densitydepends on the local plasma parameters of the crossing points andthe total current density at the surface is obtained adding all thesingle surface cross contributions. In this way we obtain anexpression for the current density induced by one ray, which isfunction of the magnetic surface, i. e., of the mean minor radius.The microwave beam structure must be taken into account,because its width in TJ-II is d 5 cm, which is not negligiblerespect to the plasma dimensions. We take a squared Gaussianbeam simulated by a number L 2 of rays and we disregard the fourcomer rays. The width of a single ray is b d/L. This will make theabsorption área more unlocalized in the plasma than when onlyone ray is considered [17], so the current profiles we obtain arewider. Additionally, the several ray paths can be very differentand so do the plasma characteristics where the absorptionhappens, namely the valúes of the density, temperature, parallelindex and magnetic field. Henee, the absorbed power and theinduced current densities are different for the several rays of thebeam. The averaged absorbed power density, which depends onthe angle between the wave vector and the toroidal flux gradient,can change, especially when the ray beam diverges.11
The total toroidal current density at each magnetic surface isobtained adding the currents induced by every ray:(26)i raysIn the same way, the global absorbed power density can becalculated:W()(27)The current intensity is calculated in terms of the meanradii of the magnetic surfaces. To perform the integration it isnecessary to consider the current perpendicular to the toroidalsurface, i. e. :I f J . dS 2TC f r J ( r ) d r (28)The amount of induced current depends on many factors,which vary from a ray to another. The fraction of absorbed powerat downshifted or upshifted frequencies changes, which modifiesthe sign of the current density. Generally speaking, a goodefficiency will be obtained when one has strong well-localizedabsorption because it happens only at upshifted frequencies, forlow field side injection, or downshifted frequencies, from highfield side injection.The influence of trapped particles depends on the chosenposition for the injection. The role played by them is moreimportant when absorption happens at external magneticsurfaces. This is because the máximum magnetic field on amagnetic surface rises for increasing mean minor radius. Thetoroidal positions of the beam for which the plasma is far from thedevice axis present also important trapped particle effects, sincethe local field is lower. In some cases trapped particle effects caneven change the sign of the current.12
V - NUMERICAL APPLICATION TO TJ-IIThe TJ-II heliac [22, 23] is a médium size, ¿ 1, M 4stellarator under construction in Madrid. Its minor radius canvary depending on the magnetic configuration between a 0.1-0.20m, its mean major radius is Ro 1.5 m, the magnetic field in thecentre of the plasma is about 1 T. Its magnetic configuration ischaracterized mainly by having a helical magnetic axis and beamshaped magnetic surfaces. The microwave injection will be doneby two 200 kW gyrotrons which emit at 53.2 GHz, i. e., at thesecond harmonic frequency. For the present work we choose aconfiguration with a 0.17 m.The total amount of induced current is calculated simulatingthe microwave beam by 144 rays (L 12) and disregarding thefour comer rays. As the beam is 5 cm wide, the distance betweenthe rays will be b 0.4166 cm, narrow enough to simúlate acontinuum. The difference between the mean radii of contiguousmagnetic surfaces is taken to be 0.5 cm, which is a goodapproximation, taking into account the transversal separation ofthe rays. The induced current is calculated by linear interpolationin the mean minor radius.The cases presented here correspond to the toroidal positionof 16.8 for only one gyrotron. The geometrical position for theinjection and the TJ-II plasma are shown in figure 7. The effect ofboth gyrotrons can be obtained by adding the density currentsinduced by each one. The azimutal and poloidal angles of injection,cp and 0, can be varied to study the behaviour of the inducedcurrent. By varying these angles one can change the amount andthe sign of the current and the current density profile. Theelectrón density on axis is no 1.5 x 10 1 9 nr 3 and the electróntemperature on axis is To 0.8 keV, which are typical valúes weexpect to reach in TJ-II. The toroidal magnetic field is in thenegative direction, i. e., in the clockwise direction.The control over
Current drive by microwaves in the range of the Electron Cyclotron Resonance frequency (EC-waves) has been experimentally demonstrated in tokamaks [1,2,3] and stellarators [4], with results that allow for a modérate optimism on the capability of this method to genérate non-inductive curr
VII CONTENTS Preface v ELECTRON CYCLOTRON THEORY 1 Summary on Electron Cyclotron Theory 3 E. Westerhof Electron Cyclotron Radiative Transfer in Fusion Plasmas (invited) 7 F. Albajar, M. Bornatici, F. Engelmann Electron Bernstein Wave Experiments in an Over-dense Reversed Field Pinch Plasma
Free electrons follow cyclotron orbits in a magnetic field. Electron has velocity v then it experiences a Lorentz force F -ev B The electron executes circular motion about the direction of B (tracing a helical path if v 0) Cyclotron frequency f c v /2πr f c eB/2πm e Electrons in cyclotron orbits radiate at the cyclotron .
Appendix D-7, Sub Panel VII Report on In-Vessel Components . EC Electron Cyclotron ECCD Electron Cyclotron Current Drive ECH Electron Cyclotron Heating EDA Engineering Design Activities of the ITER program EM Elec
VII. ELECTRON CYCLOTRON HEATING M. POFtKOLAB (MIT), P. T. BONOLI (MIT), R. C. ENGLADE (MIT), A. KRITZ (Hunter College), R. PRATER (GA), and G. R. SMITH (LLNL) WA. INTRODUCTION Electron cyclotron resonance heating (ECH) in BPX is planned as a possible upgrade to s
Spiral Cyclotron The pole inserts in a spiral cyclotron have spiral boundaries. Spiral shaping is used in both standard AVF and separated-sector machines. In a spiral cyclotron, ion orbits have an inclination . The longitudinal dynamics of the uniform-field cyclotron is reviewed in Section 15.2.
Ion Cyclotron heating i. Introduction ii. Linearity iii. Cyclotron Absorption Methods iv. Scenarios v. Database and Applications b. Lower Hybrid Heating c. Alfven Wave Heating vii. Electron Cyclotron Wave Heating a. ECRF generation b. ECFR transport c. ECRF launching d. ECRF accessib
Electron heat difrusivity in the sawtoothing tokamak core 435 G. Cima, A. Wootton, B.H. Deng, C.W. Domier, N.C. Luhmann Jr. and D. Brower Applications of electron cyclotron waves in ITER 443 B. Lloyd ELECTRON CYCLOTRON TECHNIQUES 455 Summary of the techniques session 457 W. Kasparek Dev
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