Jassby, D. L.

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,:::.::·:::i:::::::::::::::::I~ .... •.•.·.·=·=·= 1080 +~-;<-, I . <:;L }..· • . ' . NOVEMBER 1974 MATT-1080 MAXIMUM POWER GAINS OF RADIO-FREQUENCY-DRIVEN TWO-ENERGY-COMPONENT TOKAMAK REACTORS BY D. L. JASSBY PLASMA PHYSICS LABORATORY PRINCETON UNIVERSITY PRINCE T 0 N, NEW JERSEY This work was supported by U. S. Atomic Energy Commission Contract AT ( 11-1 l -3073 . Reprod~ction, transla-tion, publication, use, and disposal, in whole or in part, by or for the United States Government is permitted . .... n~sTr' .. l'7lCN 'JP r::i-; DOCut.'l~ia ~NLIMlTED DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. • (' ... 'I Maximum Power Gains of Radio-Frequency-Driven Two-Energy-Component ~okamak Reactors.* D. L. Jassby Plasma Physics Laboratory, Princeton University Princeton, New·Jersey 08540 .. ABSTRACT Two-energy-component fusion reactors in which the suprathermal component {D) is produced by har-monic cyclotron "runaway" of resonant ions are consi-dered. In one ideal case, the fast hydromagnetic wave at w = 2w . CD produces an energy distribution f {W) ~ constant {up to W ) ·that includes all max . deuterons~ which then thermalize and react with the cold tritons. In another ideal case, f {W) = constant is maintained by the fast wave at w = wcD" If one neglects (1) direct rf input to the bulk-plasma electrons and tritons, and (2) the fact that many deuterons are not resonantly accelerated, then the .maximum ideal power gain is about 0.85 Qm in the first case and 1.05 Qm in the second case, wh~re Qm is ·the maximum fusion gain in the beam-injection scheme {e.g., Q = 1.9 at T = 10 keV). Because of nonideal . m e effects, the cyclotron runaway phenomenon may find its most practical use.in the heating of 50:50 D-T plasmas to ignition. *Pap~r ElO, pres~hted at·the Secon~ Topical Conference on RF Plasma Heating, Lubbock, T~xas, 20-22 June 1974. -~~STRIBUTIOM OF THIS DOCUMEtJT UNllMITEO ,~ -2-. I~ IJ'lTRODUCTION .·'.: In the two-energy-component,reactor scheme, 1 fast deuterium neutrals are injected into a relatively cold tritium plasma. -The resulting D ions slow down by Coulomb collisions with the.bulk-plasma electrons and tritons, and while thermalizin9 they produce Q times their injected energy in fusion reactions. For T > 4.4 keV,Q > 1, ·and Q can be as large as 3.i for a e-high-Te target. The 'nT' of the plasma need be only about 13 -3 . IO sec cm' , the exact value depending on the ratio 'r ::::: (sti"prathermal-ion energy density) I (plasma energy density) . In this paper we consider. how a two-component torus (TCT) may be operated with the suprathermal ion component produced entirely in situ by harmonic ion-cyclotron resonant heating that makes use of the fast hydromagnetic wave (rnagnetosonic wave) . In particular, we investigate two simple models that give limiting fusion gains without recourse to the detailed particle motion in the presence of rf fields. These ideal limiting gains are compa-rable.to the gains in the beam-injection scheme, but are bound to be seriously reduced after including rf inputs to the bulk plas~a. In any event, rf supr~thermal-ion production might find· applicatio~ in non-Maxwellian reactors where both bulk-plasma and ~earn-plasma fusion rea~ti6ns contribute to the energy output. 2 I I. E_XPERIMENTAL BACKGROUND Harmonic ion-cyclotron heating experiments have generally • demonstrated the formation of long "tails" on the transverse (,;• I . . . . . · d' ·b t' 314 1 · 3 d · h t ion-energy istri u ion. In ear y ST experiments, uring · ea -ing by the fast hydromagnetic wave at the second harmonic, the .._) -3-tail formed in 1 or 2 msec and was usually Maxwellian. These observations have been explained5 as th~ result of single-particle resonance heating of trapped ions; the calculations are summarized in Appendix A. In recent ST experiments at higher power levels (up to 600 kW) ~ 6 the observed tails make up a larger fraction of the total ion population~ but they ar~ still essentially Maxwellian. These results suggest that if sufficient rf power is applied, the two-temperature distribution will be replaced with a one-temperature distribution, although the 'banana orbits of very high energy ions may not be confined. where The Coulomb drag on ~ fast ion of energy W is n W ln A e e T3/2 e + and are·constants, n. ln A. 1 1 c2 wl/2 ln A e and ln A. .1 (1) are the Coulomb logarithms, and ne ~ ni are the bulk-plasma electron and ion densities. Equation (1) is plotted in Fig. 1. Consider the fast wave at w = 2wCD' where is the cyclotron frequency of the deuterons. The power input to a deuteron is dW/dt ~ W, so that there is a minimum energy W . (W3 in Fig. 1) at which deuterons will "run away~" For min _ cases of interest, Wmin lies on the ion-drag (low-W) side of Fig. 1, and one can easily show that for thos·e ions that remain resonant with the wave, ( 2) . I , -4-where Ex is the left-hand component of the transverse el~ctric field, and kx is the transverse.wave number. W . is generally min in. the range 0.5-2.0 keV, so that for· larg.e hot tokamaks, a large fraction of the deuteron population in the.resonant zone may 11 run away. 11 III. FAST HEATING AND RELAXATION We first consider a process whereby deuterons are rapidly accelerated to high energies, and then allowed to thermalize with the bulk plasma. The resonant·~one :f6r the. second h~rmoriic has width . . oR ::: 2k vt.R/w, II i where R is the major radius of · · the torus. This width can be made a large fraction of the plasma I radius by increasing kll • For very large rf power, deuterons will be trapped as their transverse energy becomes many times their initial thermal energy, and a suprathermal distribution formed before there is significant dtag, or before. the: trapped· ions can escape·· fr.om the. resonant interaction. by scattering· from·' .) trapped· orbits. This limiting distribution will be Maxwellian-iike in two dimensions (Appendix A), that is, f (W) ~ l/Tl exp (-W/T1 ),and for very large rf power can be approximated as f (W) ::: constant, up to a maximum energy Wm which may be that energy, for example, at which the ion banana width becomes . comparable to oR .. ( ··. Once this rectangular distribution is formed, the rf is terminated, and the fast ions thermalize. The fusion energy gain of this cycle ·c 2 2. 4 meV per reaction) is F' (W) w aw ( 3) w aw • .. -5-where F(W,Te) is the F-factor of Ref. 1. Here we have neglected direct rf input to the bulk plasma. The plasma temperature is maintained provided that ' r . -~ < "[ s >I"[ E' where <T > s is the average slowing-down time of the D ions. This relation deter-mines the required deuteron density n0 . As in the beam-injection scheme, n0/ne is in the range 0~03 to 0.15. Cyclic operation can be set up by applying·rf pulses at intervals of <"[ >. s Figure 2 shows F (W) for Te = 10 keV. If w = w ' m a then where F is the maximum value of F (W) ·. m This 'gain can be improved by increasing w. m Approximating F (W) by ~ I the triangle shown in Fig. 2, one can easily show that G1 is maximum when Wm= (W;Wb)l/3 = 360 keV. Then G1 = 0.84 Fm This relation holds rather well throughout the range Te = 4-20 keV. 0 .IV. ENERGY CLAMPING Next we consider a proc~ss whereby a rectangular energy distribution, f (W) =constant for ·w1 < W < w2 maintained by the fast wave at w = wco· (Fig. 1) , is First harmonic heating applies the same power input to all ions, regardless of their velocity, and this power input can thus be represented as the horizontal line in Fig. 1. If w1 is not too small, the drag is nearly constant for w1 < W < w2 , so that f (W) will remain fairly constant in this range. This process . f f 1 . 217 11 h f . is a orm o energy c amping. Actua y, t e r essentially scatters the ion energies between w1 and w2 . Ions .• ) tend to run to w2 , but cannot do so, since they do not experience a continuous power input. Ions at W < w1 receive the same power input Prf' but for these ions Coulomb drag is i:·· ' , ,i'· £. ' _-, .. ,f'_, .. ~~ fi• ' -6-dominant. If is the life~ime of the fast:ions~ the.maximum energy gain is attained when Th >> GTs>' or alternatively, if the fast-ion energy can be recovered. by rriagnetl,c decompression. 7 Thus where G 2max is the fusion power output, and ·p "f r . (4) is the average rf power input per .ion• • Note·. that .once w 1 is chosen, .the corresponding w2 must be set:by-the initial second h~rmonic heating; see Sec. III. If w1 is too smail {Fig. 1), most ions will have a strong . tendency to run toward w2, while for maximum gain with f {W) = o {W - WO) I w 0 should be near the trough of the dW/dt curve. 2,7 Furthermore, the low-energy ions have small fusion reactivities. On the .other hand, if w1 is too large, . . only a small fraction of the rf-heated ion~ have large fusion reactivities. Figure 3 shows G 2max as a function of w1 . The optimum w1 is 40 keV, with ·G /F - l 07 2max · M - · · · V. REDUCTION OF THE IDEAL GAINS In this section we briefly conside'r unavoidable phenomena that reduce the ideal gains. 1. In practice the distribution f {W) = constant will not include all deuterons in the wave interaction region. Thus, .. .'' the gain of the cyclic scheme becomes ' ....... "., .. _,_ ... -· .... __ ..;_.,_,_ -······ ~·- - ··' _ __':.-_. -. J ~/ • -7-= (5) where and a.re the. deute.ron d_ensi tie~. in the supra-thermal distribution a~d bulk-plasma'. respectively. When Ilob/ne ::: 0.05, G1 is not greatly reduced even if only .._,.1/3 of the deuterons are accelerated. On the other h~nd, because of the rf input to the cold ions when w = ~CD' the gain of the clamped scheme becomes G2 (n0b/n0) ·(nT/ne) , which may be a small ,fraction of G2 • Here we have ignored the contribution of bulk-plusma D-T reactions, which are important only if the plasma has a very long confinement time. 2 2. RF energy is supplied directly to plasma electrons. The gains of two-component reactors are derived assuming that all energy to the bulk plasma is supplied by the suprathermal ions. 1,2 Thus, both Gl and G2 are reduced by the factor PD/(Pe + PD)' where p and PD are the rf powers going to e electrons and deuterons, respectively~ For a one-ion-specie plasma, the magnetosonic wave tends to be damped equally by ions and electrons, but Adam 8 has shown that in two-ion-species' . . plasmas of the type relevant here, with nD/nT - 0.1, the D will be preferentially heated if w = wCD' and Pe/PD should be as small as 10- 2 • However, this ratio may_ be somewhat larger for w = 2wco· 3. RF energy is supplied directly to tritops, since the· ~l tail of the triton distribution tends to be heated resonantly by the fast wave when w = 2wCD = 3wCT" This effect reduces G 1 and G2 with composition 50% D, 50% T. but may be useful in a reactor .-8-The above effects may well reduce the gain of an rf--TCT -by a factor of 2, thus making energy break-even impractical. . \ VI. MISCELLANEOUS PROBLEMS There are other unfavorable features. of the rf~TCT that .are not found in the beam.;...injection scheme. These problems wili be only briefly mentioned. 1. The deuteron energy distribution is described by T.1. >>TO, unless scattering is very rapid. Scattering- of high-W ,ions should be negligible, however. This distribution is suscep~ible to a high-frequen6y shear Alfv~n.wave ins~ability. 9 2. Since practically, all the rf power goes into transverse ion motion, orbit confinement probl-ems. are· more·. severe: than in the beam-injection case with·tangential injection. Furthermore, the proportion of trapped· ions· may increase· substantially,. thus ag.gravating .. trapped-ion· insta:bi.li,ties .. 3 ·~. The us.e;· 0f;.- r.f .. coil-s." inside'.. the• •. :v::ac.uum; v:esse--1 ·:may.· -be.:, ... prohibitive in a neutron environment, may be damaging to plasma p~rity, and poses serious cool{ng and insulation problems. ACKNOWLEDGMENTS This work was supported by U.S. Atomic Energy Commission Contract AT(ll-1)-3073 •. The author acknowledges helpful conversatiorts with Drs. W. M. Hooke, F. W. Perkins, T. H. Stix, and F. Troyon. {' ~) -9-REFERENCES 1J. M. Dawson, H.P. Furth, and F. H. Tenney, Phys. Rev.· Lett. 26, 1156 (1971); F. Powell, California Research and Development Company Report LWS-24920 (1953) unpublished. 2 H. P. Furth and D. L. Jassby, Phys. Rev. Lett. g,. 1176 {1974). 3w~ M. Hooke and J. c. Hosea, Proceedings of the Fifth European Conference on Controlled Fu~ion and Plasma Physics {Grenoble, ' France, 1972) _!, 107. 4o. G. Swanson, et al., Phys. Rev. Lett. ~' 1015 {1972). 5w. M. Hooke and D. L. Jassby, Princeton Plasma Physics Laboratory Report TM-270 (1973). 6w. M. Hooke and J. C. Hosea, private communication. 7 D. L. Jassby and H. P. Furth, Princeton Plasma Physics Laboratory Report MATT-1048 {1974). 8 d . 1 . J. A am, Symposium on P asma Heating and Injection (C.N.R. Lab .di Fisica del Pl~sma, Milano, Italy, ·1973), 83. 9H. L. Berk, private communication. ··. ·,. '· ; -10-·APPENDIX A. SINGLE-PARTICLE HARMONIC RESONANT HEATING The equation of motion of· anion.in the·field of the fast wave 'at w = ·2w · c is where E x refers· to the left-hand component, .· aL .. ·' i<t>l (Al) is the ion gyroradius, arid k a · << l. Considering only ions trapped in .x. L the wave field, the solution of ~q~ (Al) is aw· dt = aWt where W is t}'le ion energy and a = (ckxE~/8B) 2 • ·. Equation . (A2J. pr~serves the nature of energy distr~butions in the absence of collisions or escape from the.resonance zone. If the initial f (W) is Maxwellian. at temperature T o' then\the solution of . the equation of continuity . in ene'rgy space is. f (W, t) = [exp (-O~Sat2 ) ]/T {exp· '[(-W/T ) exp (-O.Sat2 )]} 0 0 .(A3} Details are given in Ref. 5. \ -~- _·_...: ''-·-~- ... ~'- -··- ~ . ." .. ;:_, /• ... . -~ -i·• 0 • -11-; Ill• ' ., .. .: • ~. 14------..----...... ----..... ----.,.....----.----...... -----. 12 10 s dW -dt ' 6 4 0 w, TOTAL COULOMB DRAG ·/ ~ I / W3 :/ / ,, .W2 ,,,,"' / / / / / /~ .. ' · ,,, -""" ELECT~ . / . . . ,,,. ·OR)\G ~. .. / / ,,,. /. / ~ /. /. / / ·. / ~ / . . /. 50 100. ·Te= IOteV 150 . 200 250 / 300. 350 W-CkeV) 743300 Fig. 1. Coulomb drag on fast deuterons vs deuteron energy, at Te = 10 keV. The ordinate units are arbitrary . . . .. F , \ -12-w 0 Te= 10 keV 200 400 600 800 . 1000 . 1200 1_400 ' W(keV) 7,43301 Fig. 2. Energy multiplication f~6~oi-F fr6m Ref. 1, fci~ D on T~ - w is the deuteron~energy~ ne = iol4 cm..:.3 . . ; •\ .. . . . . : .~ . '.2 ·..--....... --...--....... --..-.-.... LO G2 0.8 , F 0..6 M. 0.4 0.2 0 Te= 10 keV 20 40. 60 80 100 W1 (keV) 743842 Fig. 3. Gain of the rf-TCT with a uniform fast deuteron population clamped in the energy range Wi < W < W2, where W1 and W2 have the same Coulomb drag (see Fig. 1). Target_plasma is 100% T. FM is· the maximum F-factor at Te = 10 keV .. • This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commi.ssion, nor any person acting on behalf of the Commission: A. Makes any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any infor-mation, apparatus, method, or process disclosed in this report may not infringe .privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or prooooo dioclosed in this repoLt. As used in the above, "person acting on behalf of the Commission" includes any e~loyee or contractor of the Commission to the extent that such employee or contractor prepares, handles or distributes, or provides access to, any information pursuant to his employment or contract with the Commission. 

Maximum power gains of radio-frequency-driven two-energy-component tokamak reactors

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Maximum power gains of radio-frequency-driven two-energy-component tokamak reactors

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