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PFC/JA-79-17 EFFECT OF MAGNETIC FIELD RIPPLE ON ENERGETIC IONS IN ALCATOR A J. J. Schuss November 1979 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 nonexclusive, royalty-free license in and to any copyright covering this paper. EFFECT OF MAGNETIC FIELD RIPPLE ON ENERGETIC J. J. IONS IN ALCATOR A Schuss Abstract The effect of toroidal magnetic field ripple on energetic ion confinement in Alcator A is theoretically investigated. A heuristic Monte Carlo particle simulation shows that for sufficiently large v /v the ion loss rate due to ripple trapping is independent of the toroidal fraction of the torus contained in the magnetic well. The resulting power loss is estimated to be large enough to limit the efficiency of rf heating on Alcator A. 2 The effect of toroidal field ripple on ion confinement in tokamaks has been previously considered,1- 5 and can significantly deplete the ion distribution function at high energies. would occur for vj, /v < /oB/B, where AB is the peak to peak change in BT due to the ripple and AB/B << 1. occurs at small values of v This Since this loss /v, it would most strongly affect the loss rate of the nearly perpenducular ions generated by rf heating. In Alcator A there is a ripple of 2.5% on axis and 7.5% at the plasma edge caused by the gap in the toroidal magnet at each of four access ports. This ripple, as graphed in Fig. (1), is localized to about 20% of the machine's toroidal angle, and is too large to be substantially reduced by the rotational transform. '3 In this letter we present calculations that show that for most values of v /v there is no reduction in the loss rate of initially toroidally circulating ions due to the fact that the ripple is localized only over a fraction of the torus. v Thus for small /v large loss rates for energetic ions can be present in Alcator A, and may have substantially lowered the efficiency of the recent lower hybrid heating experiment on this tokamak. 6 An ion is trapped in the ripple well for v /v < 6 S1/2 where 2 6 = AB/B. The ion has a vertical VB drift velocity VD ~ v /(2w ciR) and scatters out of the loss cone in a time Tout = 2T D6, where 7 TD is the 90* scattering time. drifts a distance d = 2 Thus an average trapped particle 6T D v 2 /(2wciR). When d '% a/2, where a is the plasma minor radius, the trapped ions are rapidly depleted by the VB drift, leaving a hole in velocity space. Since d a E 5/2 d = a/2 defines a sharp boundary in energy above which an isotropic 3 distribution would scatter into this loss cone in a time ' at which d = a/2 the loss time would vary Below this energy E as E. -7/2 E. < E T D' 3/2 since the slowing down time varies as E. 3, 1,2. , for the loss rate due to the VB drift would quickly diminish below the energy deposition time and not significantly affect rf heating efficiencies. E > E at In this letter we will consider ion energies which the loss cone will be depleted and for E. < E neglect this loss. we will We will investigate the effect of ripple localized to a small fraction of the torus, a situation uniquely relevant to the Alcator A tokamak. We can write down the orbit averaged Fokker-Planck equation for the energetic ion distribution f. as 8,9 3f. v + _t3 s + f 3 T v s 3 +v 2 f) Ov rBm + 1/2 V v W 1 c Ts 1 2 O 1> 2i E3 0 where 3J3/2 s (1 m 4 e 1/2 ne knA e 05 2 - i 4 2E vcc Ec = 2 /m (3/4) i /M) l/3T and where we have let Zeff = 1, let the energetic ions be of the same mass as the background species, and ignored charge exchange and energy diffusion. v is the ion velocity, E = vjj /v, EO and Bm are respectively E and B at the center of the ripple well, and <> denotes an orbit average. If we approximate the (1) by a constant over most of the torus magnetic field of Fig. and a cosine in the well, <B /B-l> can be calculated in terms m of elliptical integrals, - However, it is easy to see that <Bm/B-l> 6 when the ripple is located only over a small fraction of the torus. Thus for 02 >> ', <Bm/B-l>/E 02 << 1 and Eq. (1) reduces to the velocity diffusion equation in a homogeneous plasma. would therefore expect that for large E unaffected by the well. f(001 < 61/2) = 0; Equation We the diffusion would be (1) could be solved by requiring the resulting solution for a source at E > 0 would not allow any ions to cross the hole in velocity space and f i (0 0 < 0) = 0. sufficiently small ions However, in reality for can diffuse in velocity space as they travel between ripple wells so that they can acquire a AE = /2DAt = where D1 = v3T /v c3, between ripple wells. E c between ripples, Lt = RA$/vjj , and L$ is the toroidal angle We then obtain 5 / 2DRL# For E < c 1/3 ubstantial diffusion, including a v occur between ripple wells and Eq. reversal can (1) breaks down. This is because the orbit averaged Fokker-Planck equation assumes that an orbit time is much less than a characteristic diffusion time, which is not true for E < Ec. 14 ne = 2.4 x 10 10 keV -3 cm For Alcator A parameters, A4 during lower hybrid heating, and for E = v/2, = (an energy characterizing many rf produced energetic ions), nu 0.07. In order to obtain the solution to the ripple loss for small E, a simplified Monte Carlo scheme was employed. A particle on the magnetic axis was followed around the torus which had a ripple well at each of the four access ports; several times between ripple wells and within the wells the particle was scattered in velocity space by a Gaussian weighted scattering operator based on the local Fokker-Planck equation.9 in a ripple well it scattered into lost if E. > E . 1 W(j If when the particle was < 61/2, it was treated as In order to speed execution time each magnetic well was approximated as B B (0 BTB The wells occupy outside well (3) =3 (l-6) inside well a fraction of the torus = 1/N. This procedure does not take into account the possible effects of banana orbits 6 on this diffusion from ions located off the magnetic axis, or from highly energetic ions on axis with large radial orbit excursions. Figure (2a) shows the result of following 500 particles as they are lost while circulating around the torus; here = 2.4 x 10 n 14 cm -3 , E = 10 keV, initial ( = 0.05, N = 5, and each ion starts out between two wells in the ripple free region. In this case the velocity drag term is omitted and the number of ions remaining are graphed vs. time. ions left decays to l/e of its loss cone. Figure In 127 yisec the number of initial value due to scattering into the (2b) shows f(E) 19 yvsec after the ions are Substantial spreading of ( has occurred injected into the torus. and a sizeable fraction of the ions now have E < 0 by virtue of having a reversed v11 between ripple wells. These particles would not be treated properly by the orbit-averaged Fokker-Planck equation. Figure (3) shows the l/e loss time : T for the same plasma parameters as Fig. vs. initial ( (2) but for varying N. We see that for E = 0.3 changing N from 2 to 10 produces little change in T ; furthermore, while not go to zero as E -+ T k c2 for E > 0.1 it does 0, but asymptotes a finite value. can be explained heuristically as follows. For E >> This E , the c loss time will be the time for an ion with initial E to diffuse toE c plus the additional time for the ion with E = Ec to be lost. For >> = Ec c 2, this tire will essentially be ion is small. For < TZ 42TD, since the loss tine of a c the ion distribution function will spread by pitch angle scattering to Lr ^ c between wells; loss time would then saturate according to a loss rate the 7 RA$ D NE 0 ( k % - 61/2) 2 c RL0 NE-01/2 c 2) (1 - ( 02 ripple well when E = E < E 2 0 c CV where (4rD - 6) is the value of E inside the C outside. Equation (4) states that for the total loss rate will be the loss rate of ions inside the magnetic well multiplied by the fraction of time they spend there. c>> 6 1/2 , Eq. For c<< 61/2 we obtain T the variation of - . (4) 4Nc2 TD c/(46 saturates. Figure ). 4N c Figure 2 T For (4a) shows from the Monte Carlo code vs. 6 for v 6- 1/2 as expected unless 6 /2<< T -1 shows T (4b) shows T proportionality to N. /v = 0.05. , in which case T vs. N and obtains the expected We note the Eq. (4) is only an estimate of the absolute magnitude of Tk due to its heuristic derivation. We note that as 1/N -+ 0 and the fraction of the machine containing ripple goes to zero, v should vanish. From Eq. (4) we see that for E < Ec' \i a 1/N and satisfies this criteria. S>> 2 T -1 c' and is seemingly independent of N. ever, as stated above,this expression for v for valid if it is much smaller than the v, of Eq. 2 > N(o/ c be that of Eq. for N + (0 - 61/2) 2; (4) even for (4). > For How- c is only This requires for N sufficiently large vk will > Ec and will vanish as expected -. All of the previous results ignore ion slowing, which will 8 eventually cause E to this loss. of Eq. to drop below E and no longer be subject The first and second terms on the right hand side (1) describe this slowing; it has been shown that the ion velocity will vary as 8 ,9 v = is where v [(v 3 3 + vc) e the initial 3t/T velocity. taken into account in Fig. as in Eq. 3 1/3 S - VC] (5) ; when E. < E I (5). This slowing down has been Here each ion velocity varies the ion is no longer lost and its remaining energy is dumped into the plasma. that for E (5) Figure (5) shows = 10 keV, and for the Alcator A lower hybrid heating parameters almost half the ion power is lost through the ripple wells; for higher E the loss fraction is even greater. Here 1/ we let v11 /v initial = 0.3 -, (T /Ei) 1 / , as we assume that the rf creates energetic ions in times short compared to T D. perpendicular acceleration would result in larger v /v due to collisional scattering. Slower This example illustrates that when the fraction of the torus containing ripple is only 0.2, the loss rate is still large enough to seriously limit RF heating efficiencies. Again, it should be noted that particles orbiting off the magnetic axis may be affected differently due to their banana orbits. How- ever, it has been measured that the energetic ion and neutron 10 production is localized to r/a < 0.2 during rf heating in Alcator A. Finally, the rotational transform due to the plasma current will reduce the effective ripple. 1 ' 3 It can be shown that an ion vertically above the magnetic axis will see an effective ripple 6eff 9 6 0 = - (2 2 _sin- ( )) (6) where 60 is the ripple in the absence of plasma current and a = 2rARN'q(r)], cos N'). 6 0.1% for r/a 60 keV. and where we model BT = B0 (1 - r/R cos e - 60(r) has been estimated for Alcator C11 at less than '\ This E 0.3, which results in E being of the order of is much higher than that of Alcator A; we would therefore expect that the ripple loss process would be greatly reduced in Alcator C from Alcator A. -10- ACKNOWLEDGEMENTS The author is happy to acknowledge the valuable suggestions of Dr. Martin Greenwald and Dr. Robert J. Goldston concerning this work, and Wayne Hamburger, Magnet Design Group, M.I.T., for the calculation of BT c1() of Fig. 1. This work was: supported by U.S. Department of Energy Contract No. DE-AC02-78ET51013.A002. -11- REFERENCES 1. T.E. Stringer, Nucl. Fusion 12, 689 (1972). 2. D.L. Jassby, H.H. Towner, and R.J. Goldston, Nucl. Fusion 18, 825, (1978). 3. J.W. Connor and R.J. Hastie, Nucl. Fusion 13, 221 (1973). 4. J.N. Davidson, Nucl. Fusion 16, 731 (1976). 5. Fusion 17, 557 (1977). K.T. Tsang, Nucl. 6. J.J. Schuss, et al, Phys. Rev. Lett. 43, 24 (1979). 7. L. Spitzer, Jr., in Physics of Fully Ionized Gases, (John Wiley & Sons, Inc., New York) 1962. 8. J.A. Rome, D.G. McAless, Nucl. J.D. Callen, and R.H. Fowler, Fusion 16, 55 (1976). 9. R.J. Goldston, Ph.D. Thesis, Princeton University (1977). 10. D. Gwinn, J.J. Schuss, and W. Fisher, Bull. Am. Phys. Soc. 24, 998 (1979). 11. M. . Greenwald, J.J. Schuss, and D.Cope (to be published). -12- FIGURE CAPTIONS Fig. 1. Normalized BT vs. for the Alcator A tokamak; a=10 cm, R=54 cm and all values are calculated on the torus midplane. A-R=44 cm and 6=2.9%; B-R=49 cm and 6=2.3%; C-R=54 cm and 6=2.5%; D-R=59 cm and 6=3.5%; and E-R=64 cm and 6=7.5%. Fig. 2. (a) Number of ions vs. time from the Monte Carlo code, ignoring ion slowirig. ne= 2 .4x101 4 cm~1, E =10keV, v,./v initial=0.05, deuterium, N=5 and 6=0.02. (b) f( ) at t=19 lisec after launching. of AE=0.1. The axis is divided into blocks The ions having E<0 would not be correctly described by the bounce averaged Fokker-Planck equation. Fig. 3. T vs. E initial from the Monte Carlo code ignoring ion slowing; here E.=10keV, deuterium, ne=24x101 4cm-3 and 6=2%. L-N=5, X-N=10, and 0-N=2. T We see that for E initial=0.3, is nearly independent of N. Fig. 4. (a) Uc vs. 6 from the code ignoring ion slowing; n =2.4xl0'4cm~s, E =10keV, deuterium, and N=5. to 6-1/2. 6=0.02. The dotted line is proportional (b) T t vs. N for the same parameters as (a) except The dotted line is proportional to N. Fig. 5. Number of ions vs. time from the code including ion slowing. Here ne=2.4x101 4 cm-1, E1 =10keV, v/v initial = 0.3, N=5, 6=2%, Te=lkeV and E =5keV. At t>0.79 msec the number of ions is constant since now E <E and the ripple drift loss subsides. (b) Fraction of ion energy deposited in plasmas vs. time. A total of 46% of the initial ion energy is lost in the ripples. - I I,0 0 ro) 0 C\J 0 uLJ g I]ZI]lV\JON 01 0 0 00 00 LC) fO SNOI iO 0 0 Cu) 0 0 OJflN 0 i I I Go 0 .......- m4v (\' I Ln I 0 ro 0 Qa N~ C\i SNOI jo I C\J co j3-L fN I 0 I 7.0 6.0- 5.0 4.0 E 3.0 2.0x 1.0- 0.2 0.4 0.6 0.8 1.0 o o 0 0 0 0 c\J 0 01 0 0 0 C~I. 2/4 k. -J j Cj 00 0 0 0 SNOI jO 0 0 j-V\JfN 0 0 0 (Nj I (NJ I 7 0 q I I 0 Cd co D 0 d d VV\SV-ld.NI G]-iiSOdiG( ,k!dI N] NOI iO NOIIDVdi 0 E I 0 LC) 0 0 (\j 0 00 0 o 98 0C\j 0 GIZ 1]V1VN JON 0 0 0 0 0 0 0 0 0 0 SNOI iO di~VJfN 0 0 co -o~ 04 Lo C~C\J C(\J .SNOI~~-e n .o 7.0 I I iI I I I I I IL 6.0 -- 5. 0 - A 0 4.OF- U, E I~1 3.0k- 2.0 X 1.0 41 I i 0.2 I I II 0.4 I I II 0.6 II I 0.8 1.0 LOZ o o 0 0 (OaS TI)2' 00 0N ro CI. 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