PFC/JA-84-21 LOWER HYBRID WAVE PROPAGATION IN TOROIDAL PLASMAS J. J. Schuss t Plasma Fusion Center Massachusetts Institute of Technology Cambridge, MA 02139 June 1984 This work was supported by the U.S. Department of Energy Contract No. DE-AC02-78ET51013. Reproduction, translation, publication, use and disposal, in whole or in part by or for the United States government is permitted. By acceptance of this article, the publisher and/or recipient acknowledges the U.S. Government's right to retain a non-exclusive, royalty-free license in and to any copyright covering this paper. t Present address: Raytheon Co., Wayland, Massachusetts 01778 Lower Hybrid Wave Propagation in Toroidal Plasmas J. J. Schuss* Plasma Fusion Center Massachusetts Institute of Technology Cambridge, Mass. 02139 Abstract The electrostatic lower hybrid wave equation is solved in toroidal geometry by expanding the wave potential as a sum of poloidal eigenmodes and calculating their mutual resulting solutions These solutions are present coupling compared to first to toroidal a technique that can order ray in E - r/R. The tracing calculations. explicitly determine the evolution of the wave poloidal mode number spectrum over long trajectories that involve multiple passes of the plasma column. * Present Address: Raytheon Company, Wayland, Mass. 01778 - 1 - There is presently active in tokamaks heating. of an for driving toroidal plasma currents and also for electron These rf driven plasma currents can be sustained in the absence externally applied steady state tokamak used to interest in the use of lower hybrid waves enhance loop voltage and operation. the external could Lower hybrid transformer be employed wave to yield injection has driven plasma current been in the JFT-2, WT-2, Versator II, and the JIPP T-II 1-4 tokamaks, and to maintain the plasma current in the at-ence of an externally driven loop voltage in the PLT 5 and Alcator C6 , 7 tokamaks. has been used to initiate Recently, and maintain lower hybrid wave injection a low density ~ 1012 cm- 3 ) (He plasma discharge in PLT 8 . A complication of the lower hybrid wave-plasma interaction is the absence of a well defined resonance that causes the plasma electrons to absorb the wave energy on the wave's first pass of the plasma column. addition, the value k1 = of k-9./|11 into the plasma due to toroidal effects. varies as the wave In propagates The strength of the wave-electron interaction in an initially Maxwellian plasma is exponentially dependent on this same k 1 . The evolution of the lower hybrid wave as it propagates through the plasma has been studied using ray tracing, passes of the plasma 9 and for a downshift of k 1 on the wave's first field side of the torus, it single passl 0 pass when it . both for multiple Due to the toroidal is launched from the low must usually execute several loops through the plasma before it is damped out. These ray tracing calculations ignore any broadening in the wave's poloidal mode number m spectrum as the wave propagates through the plasma. Electron Landau damping and current drive are dependent on the k11 - 2 - spectrum, which depends on m as k l = m/r sinS Here tan6 = BE / B , where toroidal fi 1d, and n = Rk vary as exp (in4) where The strong dependence distort the wave's finite poloidal the wave being m BO is wave of (1) the poloidal a constant It * and damping spectrum. extent cos6 field, is the toroidal mode number, n of + k is on k 1l should B is the local (i.e., all fields the toroidal and also the waveguide launching angle). therefore on m can be that noted structure the results in characterized by both a spatial and an m spectral width that need to be taken into account in wave propagation. For these reasons there is a need to formulate a full wave solution to the lower hybrid wave equation in toroidal geometry. A solution to the electrostatic wave equation has been pursued using a ballooning mode formalism.11, 12 In this paper a straightforward method of solution to this wave equation is presented using an expansion of the wave in poloidal eigenmodes. This method is employed to generate solutions to the lower hybrid wave equation in plasmas to those Alcator C. While absorption, this the cases method can having similar parameters presented here be expanded are to simple and ignore of wave handle more V4(r,, ), where we expand complex wave heating scenarios. In this method the electric field E = - imO + in O(r,0,4) - Z m 0 m(r)e (2) - 3 - Here 0 is the poloidal electrostatic wave angle equation and r is is Vee.V the = plasma 0, minor where e = radius. AA AA xxe., The + ZZEZZ- We can expand this equation in the expansion parameter c = r/RO, where RO is the plasma major radius. Noting that, if we take the flux surfaces to correspond to constant r, we obtain exx = exo 1 + = tron frequency at 2 0 + 2ecoso) .... (3b) ) 2 and 5i'w1adwe 2 p~e ceo the plasma - (3a) center. ceo The is wave the electron equation then cyclobecomes 32 Da4' r2 e + r ar2 = (e2cos /2 1/RO (1 - e cosO + 1/R where 2 Sxxo+ - -- Dr ')m Om (rexxO+ ) e F1 (m-i'm+1) + C2 F 2 (4) Dr + where Kn (r) - - m 2 (exxo cos 2 6 + eZZ sin 2 6) - 2 kgrm (eZz k 2 r2 (eZZCo626 Here eZZ 1 - 2/w2 and ke = n/Ro. exxo) cos6 sin6 + exxo sin26) (5) - 4 - The left hand side of equation (4) is just the wave equation for a linear the plasma column; right hand side of equation (4) couples waves having different m numbers due to toroidal effects. together In this paper we consider the first order coupling term cFl, where F1 (Om-1, Om+1) = F..(m-I) + F+(Om+1) (6) and 2 F; (Dm+1) 5 KmT1 , n e 'ceo (7) -xxo 2 - kr cos sin 2 (m - 1) O'Zceo 2 32 1 +- -(m 2 T 1)(1 2 + cos 6 - - - sin 2 6) --2 2 2 ~eo i cos 2 6(m T 1)2 + k 2 r2 (1 - cos 2 6 -) pe 2 ceo 2 t kOr cosS sin6 [r2 + 2 Wpe - , mT1 ceoI a Dr r 5e 2+ 2 + 3 $i ( 2W2eo - 2 m 3r - 5 - For small e, F1 will dominate the poloidal mode coupling. Equation (4) has been solved on a VAX 11/780 computer using a RungeKutta integration with boundary wave. scheme; the integration is started at the plasma edge conditions Figure la plots on 0m and 4m that correspond I| (r,e)1 the resulting to 100 cm so as to illustrate the full wave method, maintaining e small. on the outer midplane extent of +46*. 0(r) I / krr = Kmn/r 2 e E(m) would )1/2. not change mean value of m, <m>, greatly as of the plasma having a poloidal finite (8) For a plasma having no dissipation and e =0, r varies; E(m) and the variance changes launching structure. account the still or that corresponds to a FxXO wave power flux in mode m at a given r. that E(m) while Figure lb plots the evolution of E(m), where E(m) and kr = Here RO is The edge boundary conditions are those for a uniform source at r - a = 16.5 cm that is 23 cm high, source centered inward over the plasma poloidal cross section for conditions similar to those of Alcator C. taken as an 2 would then correspond to the E(m) 2 is here used to define the in m, an. From figure lb we see shape as the wave propagates inward from the Figure lb illustrates the importance of taking into poloidal size of the wave launcher in calculating lower hybrid wave propagation. Figure 2a plots 10 (r,e)I over the plasma cross section for the same parameters as those of figure 1, reduced from 23 cm to 6 cm. The lower hybrid wave propagates into the plasma as a well defined cone, ment. except that here the launcher height is as might be expected from an e - 0 treat- - 6 - Figure 2b plots E(m) dominant change vs. r for this same in the poloidal mode number mean m value <m> during the wave's first a small broadening of the E(m) case. of of Am. figure As can be seen, spectrum is a shift pass into the plasma, the in its with only This figure shows that most of the broadening 1 could be obtained by ray tracing if several appropriately weighted rays were launched at different poloidal locations on the waveguide array. Of course, the full wave solution does this explicitly. Figure 3 shows a comparison of the radial behavior of <m> of a lower hybrid wave calculated using electrostatic ray tracing. high waveguide results, The launcher. illustrating the As that full the e optics and approach, and full wave method employs can the be full seen, wave behavior of the ray tracing calculation. the geometric wave full wave both methods solution that using the same yield predicts 6 cm similar the basic Some of the differences between solutions at high ne are due to expansion of the full wave solution which is not used for the ray tracing calculation. Finally, the previous solutions did not consider behavior of the wave near r=0. (and e r2 << the turning point This can be done by noting that for r << a 1) equation (4) becomes + r ' + (a2 r2 - m 2 )0 = 0 (9) where m ) (kg + 2 a5e w2 2 Roq . xxo 0 (10) - 7 - and where qo is the value of q at r=0. The solution to equation (9) is the Bessel function near r=O is of the first kind lm(r) = CJm(r), where Jm(X) of order m. Using this 4,m(r) as a solution at small r - m/a near the wave turning point, we can numerically connect the incoming and outgoing lower hybrid waves at the plasma $(r,O) that corresponds a complete pass of the wave core and generate the through the for the plasma same column. plasma evolution of <m> Figure parameters as for this wave 4a in shows figure both on its the 2. inward Besides exhibiting the expected wave behavior, Figure and further continued by 4b plots the outward passes. figure 4 illustrates this method of handling multiple passes of the plasma column. jectory could be I,(r,6)1 resulting connecting The wave tra- the incident and re- flected waves at subsequent turning points. Equation (4) can be straightforwardly orders of e in mode coupling. and the mode In addition, the plasma thermal dispersion conversion to the hot ion mode orating the appropriate of equation generalized to handle higher (4). The could be formulated 4 th order derivative term into the left hand side full electromagnetic in a manner lower hybrid wave similar to equation (4), but In any case, would be more complex. can be included by incorp- equation its such a modified equation (4) form would explicitly calculate the distortion or broadening of the wave's m spectrum over long wave trajectories. In summary, this paper presents a full wave treatment of lower hybrid wave propagation in toroidal plasmas. This technique should prove valuable in evolution trajectories. calculating lower hybrid wave over long wave - 8 - The author Bonoli, B. is happy Lloyd, supported by U. S. and Y. to acknowledge Takase the helpful concerning this suggestions work. This of work P. was Department of Energy Contract Number DE-AC02-78ET51013. I - 9 - References 1. T. Yamamoto, et al., Phys. Rev. Lett. 45, 716 (1980). 2. T. Maekaewa, et al., Phys. Lett. 85A, 339 (1981). 3. S. C. Luckhardt, et al., Phys. Rev. Lett. 48, 152 (1982). 4. K. Ohkubo, et al., Nucl. Fusion 22, 203 (1982). 5. S. Bernabei, C. Daughney, P. Efthimion, et al., Phys. Rev. Lett. 49, 1255 (1982). 6. M. Porkolab, J. J. Schuss, Y. Takase, et al., in Proceedings of the 9th International Conference on Plasma Physics and Controlled 1982, Paper IAEABaltimore, U.S.A., Nuclear Fusion Research, CN-41/C-4. 7. J. J. Schuss, Bull. Am. 8. F. Jobes, J. Stevens, Phys. Soc. 27, 962 (1982). R. Bell, et al., Phys. 52, Rev. Lett. 1005 (1984). 9. P. T. Bonoli, E. Ott, Phys. Fluids 25, 359 (1982). 10. D. W. Ignat, Phys. Fluids 24, 1110 (1981). 11. Alan 12. T. T. Lee, Z. G. H. Glasser, An, Report Y. C. PR3, Auburn Lee, Bull. Am. (Feb., University Phys. Soc. 27, 1980). 1104 (1982). - 10 - Figure Captions Fig. 1 a) I$(r,O)I plotted over the plasma column cross section for a lower hybrid wave launched by a 23 cm high waveguide at r = a - 16.5 cm. B = 100 kG, Here no f - 4.6 = (n/Ro)(c/w) GHz, "e - 400 kA in a deuterium plasma. Ro = 100 cm, - 3, 10 14 cm- 3 and Ip = Z points in the direction of the center line of the torus. b) E(m) vs. r for the conditions of (a). Here at r = 5 cm, m has increased from 3 to 22 as <m> has changed by a comparable amount to -25. Fig. 2 a) I0(r,B)I plotted over the plasma column cross section for a lower hybrid wave launched by a 6 cm high waveguide at r = Here no - 3, Ro - 16.5 cm - a. - 4.6 GHz, e = 1014 cm- 3 and IP 100 cm, Boo = 100 kG, f 400 kA in a deuterium plasma. b) E(m) vs. r for the conditions of (a). decreased to -34 as &n has only At r - 3 cm <m> has increased to 9 from a starting value of 7. Fig. 3 Comparison of ray tracing (electrostatic) vs. full wave calculations of <i> for a lower hybrid wave launched by a 6 cm high waveguide at r-a. Here we choose e - 2 x 1014 cm-3 , 1 x 1014 cm-3 and 3 x 1013 cm- 3 , and we fix Ro = 100 cm, a - 16.5 cm, Bo - 100 kG, f - 4.6 GHz, Ip - 400 kA in a deuterium plasma. Fig. 4 a) I0(r,O)I (RT - ray tracing, W - full wave). plotted over the plasma cross section for the conditions of figure 2, except that here the lower hybrid - 11 - wave enters and exits the plasma core and is followed until it reaches the plasma boundary. b) <m> vs. waves. r for both the inward and outward propagating 4e v Mod FIGURE la N I0 'U I .0 N FIGURE lb Jar TO0 000 FIGURE 2a EE %mo FIGURE 2b 0 I I co) E -Eo N 0 (0 x em I -1 Go E cc I' Cx I- cI I 0 cm 0 c0 0 04 -Jo 0 0 '-U A E V FIGURE 3 0 -er 1$0 FIGURE 4a 0 I 0 T- a YEo NM 0 wo q-0 (D .0 aNM C 0 0 co N4 I A E V FIGURE 4b a a