Electron Cyclotron Heating Calculations For ATF
.-.P;inted i i l tkfe 1Jniii.d Statcs of Americe ,A.vailn! e fi-oiiiNationa! -Ie:ckr;ical lrlforiT3&ticjic ServiczII S. i i e o a i t i - w i t of Conmcrc::5385 Pori Sayal S o x i , Srjr iticliieid, Virginia ?2161NT IS price codes.- - P m t s Copy: ,4W.44nicroficIir A01. --. . --.This report was preparcd as an accoi.int of x.A!oiX sponsored by an agency of ihsUnited StatesGnvcrnment. Neithai iheU nited StatesGovnrnirient nor any ageniytherm!, nor any of i1h611'e;nphyess, iildkes 3fly i:arranty. ex pie's 'Jr implied, orassuniet any !ege! liability or rzsponsihility for the atracf, co;nplntmer,s orusefulness of any inforiiiaiisn, apparatus, piOdtSt, or process disciosed. orreprcsczis that its usr w?uIdnot infringe privatsl lwnec! rights. Rcfsrence heieinto any speciili COiiiindiiial product, proccss, or service by trade name, traderark,manufas!ure:, or uii c?rrlc?,docs not neceasanly canstitutc or irilply I&codnrsxient. remnmendatior,, 0 1 f i i ing i by the United SidieS Governmeiii orany agency thoreof. The vieti(:'; and opinions of auii-iors cxpressed hsrei:: d v riotnecessarily sta!e cr reflect those c ! :he 1Jnitcd States Gove:n;-,en;or any a-,c:,cythcico!.
ORNL/TM--9869Dist. Category UC-20 gFusion Energy DivisionELECTRON CYCLOTRON HEATING CALCULATIONS FOR ATFR. C. GOLDFINGERComputing and Telecommunications DivisionD. B. BATCHELORDate Published - March 1986NOTICE: This document contains information of a preliniharynature. It is subject t o revision or correction and therefore does notrepresent a final report.Prepared byOAK RIDGE NATIONAL LABORATORYOak Ridge, Tennessee 37831operated byMAR,TIN MARIETTA ENERGY SYSTEMS, INC.for theU.S. DEPARTMENT OF ENERGYunder Contract No. DE ACO5-840RZ1400'3 4456 0137544 5
CONTENTS. . . . . . . . . . . . . . . . .ABSTRACT . . . . . . . . . . . . . . . . . . . . .1. INTRUDTJCTION. . . . . . . . . . . . . . . . . .ACI-CNOW LED GMENTS.2.1. THE RAYS CODE . . . .2.2. WAVE ABSORPTION . . .2.3. PLASMA MODEL . . . .2 . THEORETICAL BACXGROTJND3 . RAY TRACING CALCULATIONS3.1. INTRODUCTION.vii13357. . . . . . . . . . . . . . . . . . . . . . . . . . . . .17. . . . . . . . . . . . . . . . .183.2. LOW-FIELD LAUNCH, FUNDAMENTAL RESc)NAWCE, FIRST-PASSCALCULATIONSv173.3. HIGH-FIELD LAUNCH, FUNDAMENTAL RESONANCE, FIRST-PASSCALCULATIONS. . . . . . . . . . . . . . . . .253.4. LOW-FIELD LAUNCH, SECOND HARMONIC RESONANCE, FIRST-PASS. . . . . . . . . . . .3.5. FIRST.PASS, HIGH-DENSITY CALCjULATIONS . . .3.6. WIDE BEAM, DISPLACED AXIS CALCULATIONS . .3.7. WALL REFLECTION CALCULATIONS. . . . .4 . STJMMARY AND CONCLUSIONS. . . . . . . .CALCULllTIONSItEFER.ENCES. . . . . . . . . . . . . . . . . . . .iii.272332353943
ACKNOWLEDGMENTSThe authors thank J. H. Harris, T. L. Whit,e, and J. C. Glowienka for valuable discussions c,onnec.t,edwith this work. V. E. Lynch supplied t,he AVAC c d e and help with theplasma model.v
ABSTRACTThe RAYS geometrical optics code has been used to calculate electron cyclotron wavepropatgation and heating in the Advanced Toroidal Fxility (ATF) device under eonstriic,tionat Oak Ridge National Laboratory (ORNL). The intent of this work is to predict theoutcome of various heating scenarios and to give guidance in designing an optimum heatingsystem. Particular attention is paid t o the effects of wave polarization and antenna location.We investigate first and second harmonic cyclotron heating with the parameters predictedfor steady-state ATF operation. We also simulate the effect of wall reflections by calculatinga uniform, isotropic flux of power radiating from the wall. These results, combined with thefirst-pass calculations, give a qualitative picture of the heat deposition profiles. From theseresults we identify t h e compromises that represent the optimum heating strate@es for theATF model considered here. Our basic conclusions are that second harmonic heating withthe extraordinary mode (X-mode) gives the best result, with funda.mentd ordinary mode(0-mode) heating being slightly less efficient. Assuniirig the antenna location is restrictedto the low magnetic field side, the antenna should be ptaced at # 0" (the toroidalangle where the helical coils are a t the sides) for fundamental heating and a t4 15"(where the helical coils are a t the toy and bottom) for second harmoiiic heatiag. Theserecommendations come directly from the ray tracing results as well as from a theoreticalidentific,ation of the relevant fac,tors affeding the heating.vii
1. INTRODUCTIONThis paper presents results for electron cyclotron heating using a 53.2-CHz microwavesource in the plasma regime that should provide a target plasma for neutral beam injection[T, 500 eV,neN1013 crnw3, \BO\N1.9 T and 0.95 T (first and second harmonic,respectively)]. The main objective of the study is to determine how t o maximize the first-pass absorption of the beam and how t o have this heating occur as near as possible to themagnetic axis. When the magnetic field level is adjusted to place the cyclotron resonancea t the axis (fie eB/2rMC' 53.2 GHz and 2f,, 53.2 GHz on axis for the first andsecond harmonic absorption, respectively) the absorption will occur near the axis, but itmay not, be total. The damping strength is a function of the plasma parameters for thedesired polarization and of the local scale length in{BI.Thus the optirnum power depositionprofile will be a compromise between maximizing the absorption and having the heatingoccur near the axis. Other considerations in choosing the optimum launch configurationresult from sensitivity of the results to antenna characteristics and plasma parameters. Foresample, finite bPta effects and small chaages in vertical field currents can radially displacethe magnetic axis location. These factors can help point to a preferred launch position.Our calculations are based on one assumed set of coil currents that gives rise to anominal ATF configuration with the magnetic axis located a t 210 cm. The rotationaltransform profile is a ( 0 ) 0.33 and &(a) 1.0, where a 30 cm is the plasma radius.Since other current configurations significantly change the topology of the magnetic fieldand the flux surfaces, the recommendations we make concerning optimum launching areHowever, the same general principles andvalid only for this particular configuration.analysis can be applied t o other field configurations, which can be readily analyzed withour code. The complicated topology of the mod-B resonance surfaces and their relationshipto the location of the magnetic axis are shown to be the primary factors in determiningthe wave absorption. Specifically, the maximum absorption occurs when the resonanceoccurs a t the saddle point of the magnetic field. Other considerations, such as: naturalsymmetries in the system, the separation between the saddle point and the magnetic axis,and the gradient perpendicular to the ray trajectories, all help to point to a preferred launchlocation for any given equilibrium.The ray tracing is done using cold plasma theory t o propagate the rays according tothe results of geometrical optics [1,2]. We treat each case by doing two runs: a singlepass calculation of the incident beam and a calculation that simulates the effect of theremaining power. This remaining power is trea,ted by assuming. that the part of the beamnot absorbed on the first pass is quickly randomized in direction, location, and polarization1
2Introductionthroughout the device. We base the model on our experience with the ELMO Bumpy Torus(EBDT) experiment. Rence, we can estimate the effect of the initid beam plus subsequentwall reflections by a superposition of the two cases. We present results for both bigh-and low-field launch, although only low-field operation is planned on ATF a t this time.It will be shown that the 0-mode polarization(.6 11for perpendicular propagation)is st,rongly absorbed at the fundamental cyclotron resoiiaiice. The X-mode cannot reachthe fiindament,al resonance from the low-field side because of the presence of the righthand cutoff. When the magnetic field level is decreased t o the point at which the plasmacenter is resonant at the second harmonic of the wave frequency, the roles of the 0- andX-modes are reversed. For this case, the X-mode can reach the second harmonic [as long asfpe(0) (53.2 GHz)/&]and i s very strongly absorbed. The 0-mode is weakly absorbed atthe second harmonic and makes an insignificant contribution t o second harmonic heating.The organization of the paper is as follows. Section 2 presents some of the theoreticalbackgroiind of the RAYS code. Specifically discussed are cold plasma ray tracing, thefinite temyerat ureapproximations used in cakulating the damping, and d e t d s about theplasma model. Section 3 displays the ray tracing calculat'ions, and Section 4 presents theconclusions.
2. THEORETICALBACKGROUND2.1. THE RAYS CODERAYS, a code first developed a t ORNL for the study of electron cyclotron heating inEBT [ 1-31, traces rays in an arbitrary magnetized plasma configuration. The ray tracing iscarried out by integrating the Hamiltonian form of the geometrical optics equations usingthe two-component, cold plasma dispersion relation:dF-1 dF- -sgnIFpldtIId8aD/a(aDaw-) pD/aEy-(1)andD [ ( E 1 - ni)whereWee (E1-94- 6-42 (E3 - a ; )eB --Here, nl c&'w is the real refractive index,q E . k,and dq.For the caseof weak absorption considered here the effect of finite temperature on the ray trajectoriescan be neglected 141. We can develop insight into this dispersion relation for the regime ofinterest if we consider a simplified form that comes from assuming thatand nnl nil. Under these approximations, Eq. (3) becomes31;1Nw,, u,i
4Theoretical BackgroundT h e two roots of this biquadrat,ic equa,tion in n: correspond to the 0- and X-polarizatLionmodes, A resonance occurs in the X-mode (the upper hybrid resonance) when the coefficientof t h eterm vanishes: 1 - d i , J w 212:- w?Jw2 0. Cutotfs occur in both branches whenthe constant term vanishes:21 - WpeW2II0(0-mode cutoff)-0(X-mode cutoff)For the 53.2-GHz wv;p;vebource, the O-snode will be uiiable t o penetrat,e t h e plasma centerwhpn the density exceeds n, 3.5 x 1 O I 3*4n X-mode ray from the low magneticfield side will encounter the right-hand cutoff and be reflected beforc the ray can reach thelundamental cyclotron rebonance surface,Wce/W I (except for a small fraction of X-modepower that can tunnel through the evanescent region on the outer edge of the plasma). TheqnrJstioii of whether the X-mode can reach the second harmonic before hitting the cutoffw h m launched from the low-field side dependb on the density: when wz,/w2 0.5 (whichcorresponds t o a density of 1.75 x l O I 3 cme3 for 53.2 GBz), the ray can reach the Yecondharmonic.For high-field launch the situat,inn is as follows. -4pi 0-mode ray incident from thehigh-field side will he able to reach the fundame1xta.l resonance as long as the density isbelow @-mode cutoff,ti, 3.5 xlOI3A high-field X-mode ray will first encounterthe fundamental resonance, W,e/W I , and then possibly the upper hybrid layer, u;,/d-t@?,/w2 I. There is very weak absorption of the X-mode at the fundamental resonancefor rill, but strong absorption near the upper hybrid resonanceTIL- J Ymode conversionto elect,ron Bernstein modes [5-71. If oblique propagation is done, rillnl, then verystrong high-field X-mode absorption occurs near the fundarxlentd resonance. The densitycutoff for this case is wi,/w2 2 or12, 7 x 1013 r n " Bile . t,o the difficulty of high-geldoblique launching, no such experiments are planned, and hence no cdculat,ions amredonein this paper for high-field X-mode launch. As is shown below, it turns out that, the Xmode is milch more strongly absorbed at t,he second harmonic than .the 0-mode. Thus,-in suinmaiy, the best strategy for low-field launch is to use X-mode polarization whent h e central magnetic field makes the second harmonic resonant there-(lzoI0.95 11).When t8hefield is increa,sed t o cause the fixnda.mentd resonance to appear near the center(lpul1.9 T), then accessibility requires 0-mode launch.
Theoretical Background52.2. WAVE ABSORPTIONThe fractional absorption of wave energy due to cyclotron damping is given bywhereIcdis the imaginary part of the wave vector and s is the path length along the ray. Inorder to perform the integration in E¶. (6), it is necessary to determine & . ds as a functionof8.The value of4 thatcorresponds to damping at the fundamental cyclotron resonanceis obtained from a, weak damping approximation for oblique propagatmionsimilar tso thatgiven in Ref. 181. In this application, we expand about the local value ofby the RAYS code. The component ofE; in the direction of(rill, n l ) as givenis given byHere DC is the cold plasma dispersion relation, ni and n l are the zero-order refractiveindices obtained directly from the integration of the ray equations, and Dw is a warmplasma correction term of the formwhereD1 (1 - q ) n2nI (1 - P ) n2ni - (1 - q) (1 - P) (1 - 2q) (1- P)D3 P[.21%ir - .e - (1 - 4) (2n2 (1 - 2(4 -nr -*.;iI2 (1 - 2q) -and Z(q) is the plasma dispersion function.For nearly perpendicular propagation, nilthis rpgime our code approximatesnumerically solve'2115v / c , relativistic effects predominate. In 0 and uses the Shkarofsky formulation [Q] t o
6Theoretical BackgroundwhereCUsing these approximations, Fig. 1 shows the 0-mode ki as a function of magnetic fieldstrength (normdized to the wave frequency) around the fiindamental resonance for variousvalues of 1111.It is seen that as the wave vector becomes more nearly perpendicular t othe magnetic field (small values of nil), the damping is confined to an increasingly narrowregion in spase on the high-field side of the resonance.The X-mode damping near t,he second harmonic is obtained using Poynting’s theorem,where S Re[E* x(E xE ) ] is thePoynting’s vector andthe relativistic conductivity tensor. The model for H Ht is the Hermitian part ofassumes an isotropic M w e l l i a ndistribution and a.llows contributions from a.n arbitrary number of cyclotron harmonics.(For this application, only the n 2 term contributes.) The real refractive index is determined from the ray tracing code; the electric field eigenvectors,z,used in the absorption- . equat,ion are determined from the cold plasma, dispersion tensor, D . E 0. Details on the Hc.aJlc.iila,t,ioiisfor nmay be found in Ref. [ 101. Figure 2 shows k; as a funct,ion of niagnet#icfield around the second harmonic. A compa,rison of Figs. 1 and 2 reveals that the X-modeahsorpt,ion at the second harmonic is between two and five times great,er than the 0-modeabsorption at the fundaxnmtal, with the line shapes quite similar for t h e two cases.
Theoretical Background0.5 10.4IO R N L - D W G 8 5 C - 3 4 8 2 FEDII\1III0.01-- 0 .1 0 0.10---- 0.300.20-*-h.IE0*0.20.98 0.991.001.01Oce(0)/W1.021.03k*,as a function of normalizedFIG. I. The imaginary part of the 0-mode wave vector, k;,magnetic field strength, LI,,/w.2.3. PLASMA MODELFor the magnetic field model we have used the three-dimension4 vitcuuin field code,AV44C 111-131, developed by V. E. Lynch and others. AVAC uses the Biot-Savart law t ocalculate the magnetic field that results from an arbitrary configuration of external currents.Its considerable generality and flexibility allow the accurate calculation of vacuum fieldsin torsatrons, stellaratnrs, helical-axis devices, and other machines. It is shown here thatmagnetic topology is the significant factor in dPtermining the power deposition profile aswell as the total absorption. This sensitivity implies that it is essential t o usc an accurate,fully three-dimensional magnetic field for these calculations to get meaningful results.We approximate the magnetic field in ATF by using AVAC with two filamentary helicalfield (HF) coils and four vertical field (VF) coils in a configuration appropriate for ATF. The7
8Theoretical BackgroundO R N L - D W G 85C-3483 FED2.8II1III2.4- I-a .500.51W,,(O)/C3FIG. 2. The imaginary part of the X-mode wave vector, k;, as a function nf normalizedmagnetic field strength, w,,/w.
Theoretical Backgroundtwo HF coils carry equal current with major radius of 210.0 cm and minor radius of 48.0 cm.The VI" coils are locaked a.t R (cm), 2 (cm) (281.0,61.7), (281.0, -61.71, (133.0,20.0) and(133.0, -20.0).The outer VI" coils ca,rry 47% of the H F coil current anti-parallel to thedirection of the HF current, while the inner W coils carry 15% of the HF current in the samesense. Ampkre's law applied to the HF coils shows that a current of 0.875 M A produces a.magnetic 5eld on axis ( R 210.0CUI)of 1.0 T. The plasma density and electrontemperature are assumed constant on a flux surface and therefore are parameterized by thetoroidal magnetic fiux function, . The dependence of density and electron temperatureon minor radius, n e ( ) , is illustrated in Fig. 3. For the toroidal flux function we haveused an ad hoc UrotatiJigellipse" model that is a reasonable representation (at least for thepurposes of ray t,racing) o f the va,cnnrn flux function. In this model, the flux surfaces arc4, and rotate about the Illinor axis with the 210 em.helical windings. These ellipses a,re concentric and centered at R RMAJORelliptical in a plane of constant toroidal angle,In Fig. 4 we show (a) the resormnce and (b) the cutoff surfaces described in Section 2.1. We have plotted the electron cyclotron resonance, the upper hybrid resonance,.an 0-mode rayand the right-hand cutoff for lBol 1.9 T and ne 1013 r n - Becauseexperiences no resonances or cutoffs for this density, the entire chamber is ac,cessible t o it.An X-mode ray is evanescent in the region between the right-hand cutoff and cyclotron resona,nce so that it cannot reac,h the plasma center when launched from the low-field side withthese pararueiers. When a n X-mode ray is launched from the high-field side, it reaches thecyclotron resona'nce but does not encounter the upper hybrid resonance unless it is propagating a t considerably oblique angles. This leads us to conchde that mode conversion andabsorption near the upper hybrid resonance play an insignificant role for these parameters.Figures 5-7 show mod-B and flux contours on various surfaces for ATF. EacA figurehas three contours of constant flux, indicated by dotted lines that c,orrespond to plasmadensities (or temperatures) in our model of 90%, SO%, and 20% of the maximum value at!the magnetic. axis, 3 0. The solid lines are contours of const,ant1B1 thatof 53.2 GHz, this surface is the location of the cyclotron resonance if\Dl a,re labeledon each contour. The contour with value 1.0according to the normalized value ofpasses through the magnebic axis, 11, 0, in the 4 0" plane. Thus, for a wave frequencypoint, ( x , g , 2 ) (210,0,0).planes 0" atid41.9 T a t thisFigures 5 and 6 are poloida,l cross sec.tions in the toroidal 15", respectively.In the4 0" plane the helied windingsare at the sides of thc v i t c i i u chamber, while at (p 15" t,he windings are a t the topand bottom of the chamber. Figure 7 displays the contours on the orthogonal surface, . I .-R t/(z2 g2] 210 cm. (This is the cylindrical sheet containing the minor axis.) We9
10Theoretical BackgroutadO R N L - D W G 8 5 C - 2 1 1 2 FED1.000.75IIIIIIIII\IIII\,"0 . 5 00.250160III1802001220240R (cm)J260FIG. 3. Normalized electron density, or temperature, as a function of major radius orQ, 0" and 4 15". The maximum density occurs on the rninor axis, R 210 cm;the asymptotic density is 10% of the maximu
identific,ation of the relevant fac,tors affeding the heating. vii . 1. INTRODUCTION This paper presents results for electron cyclotron heating using a 53.2-CHz microwave source in the plasma regime that should provide a
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
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
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
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
Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original
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.
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