- - - - - do:'. | OF I ORNL P 1338 con Search - venta . . * le; , . PEETEEET .. 1 MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS -1963 . . ORMULP, 1338. CONF-6509350442965 : Anomalous Plasma Diffusion in Magnetic Wells* T. K. Fowler and G. E. Guest Oak Ridge Natia al Laboratory .. Oak Ridge, Tennessee, U.S.A. 1. Introduction It has been demonstrated that interchange instability is inhibited in plasmas conrined in a magrietic well, that is, at a field minimum (1), and theory predicts interchange stability if the field has well shape only on the average 2). After interchange, the instabilities expected to pose the most serious threat to thermonuclear confinement if they occur are those at frequencies beloir the ion gyrofrequency caused by the prossure gradient 3). They have been variously called "universal." instabilities because the gradient is synonymous with confinement and "dri It" instabilities because the diamag- netic drift implied by the gradient drives the instability. We shall, the former name to include the whole family. Universal instabilities, in this broad sense, have been observed in low temperature discharges (4), and they offer one possible explanation of "pumpout" in stellarators (5). Possibly, universal instabilities can be controlled by magnetic shear in torii [67 and by short length in magnetic mirror systems [7]. Also, it has been shown that magnetic wells should prevent low-frequency phenomena including certain universal instabilities for special plasma distributions [8]. I use *Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. PATENT CLEARANCE OBTAINED. RELEASE TO THE PUBLIC IS APPROVED. PROCF,DURES ARE ON FILE IN THE RECEIVING.SECTION. Here, we show that magnetic wells and also average wells at least reduce universal instability to a tolerable level for quite general plasma distributions 11 the electron temperaturo is hold low. We examine three points. We first show that the free energy driving universal instabilities is proportional to the electron temperature, Te. Next, we argue that a magnetis ratuito well introduces a compensating energy torm yielding stability 11 To 18 small enough and the well depth is large enough, everywhere or on the average. Finally, we explore briefly the practicality of controlling electron temperature in mirrors and torii. We conclude that, at least in mirrors, Te can indeed be held low enough to prevent universal instability. The identification of the free energy driving universal instabilities is supported by measurements of oscillations in stellarators. The stability criterion derived from free energy arguments is supported by models employing conventional dispersion analysis and linearized equations of motion. g i 2. Free Energy Driving Universal Instabilities We have previously obtained a thermodynamic estimate of free energy supporting unstable growth of electric fields in plasmas (97. The main difficulty is that, without care, one gets the trivial. answer that the maximum free energy is the total plasma energy. By noxpanding to fill the universe, an initially confined plasma can eventually cool to zero temperature IN or w h oice ,...". . . . . . ... . . . . . . . . . . . without violating the principles of energy and momentum conservation and the monotone increase of entropy on which thermodynamics rests. Since we know that invariably coherent instability processes have finite extent, the wavelength, we avoid the above difficulty by introducing the wavelength a perpendicular to the magnetic field as a parameters (Fortu- nately, 2 drops out of our main results, Sect. 3.) We divide the plasma into sample zones, cylindrical shells of thickness concentric to the magnetic axis. We assume that growth of fields within any one zone is supported primarily by plasma energy within that zone. Energy transport between zones, over distances > ? , is incoherent and slow. Thus, the free energy A should be calculated in each zone separately and energy flow between zones should be noglected in connuting dA/at. With this assumption, the Helmholtz free en orgy Alt) for any zone satisfies da/at so, the inequality representing collisions. Another property is A - 520, (2) where is the energy in fields with in the zone; $ and the particle kinetic energy make up the internal energy U in the Helmholtz function A = U.TS. Together, properties (1) and (2) insure that in the course of time the field energy does (1) not exceed the initial free energy, I(t) = A10). (3) Consider the following model, treated in detail in Ref. 197, which exhibits universal instability if dBc/ar = 0, B. being . the external magnetio Mold. Lot ion and el octroa distributions have the form foc exp{ -sma2 (impe + pp +90). (4) Here Po is the particle canonical angular momentum in the field. Bc assumed to be axially symmetric and constant in time. These are Maxwellin distributions in rigid-body rotation at angular frequencies d'1 and ce for ions and electrons respectively. The radial den sity profiles are roughly Gaussians (exactly so ir Ēc is unifor..) the plasma potential which can be small or zero if de and do are properly related. For any one of the sample zones, the appropriate Helmholtz function relative to a state of thermal aquilibrium within the zone, state 8, is given by the following integrals over the zone in questio, volume V, Alt) = Črt)+ Efrat doſ G() -G(8)+(-8)(fav? tupost (5) G(x) = T(x in c-Ix - x) 8 = 6 9x02-7-2 (Bonv2 + upor} $(t) = (871-7560*{+ (B-B.)2+290-lêof x (Ex (E.)}.(8) (6) .. wow.nice ..... In Eq. (2), the sum runs over each particle spocies, distributions 1 (5,7,t). The integral of G will be recognized as temperature T time negative entropy, the roma inder of A being the intemal energy including the energy in the total electric field Ē and magnetic field B - Bc. In Eq. (5), Î 18 the unit vector along the cylinder axis. Observing that curi Bo= 0, one obtains Eq. (1) by differentiating A As prescribed by the non-linear Vlaso V-Bolt zmann equation and Maxwell's equations and throwing away power 11 ow through the surface bounding V as a 8 sunod above. Eq. (2) follows from the choices or g and G. Since energy is not an invariant, for generality we have written A in a reference frame rotating at an unspecified angular frequency y by adding to the laboratory free ono rgy D times the total angular momentum in fields and particles within the zone considered. Since Eq. (3) holds for all such A, the best estimate of free energy is obtained by minimizing A with respect to w and parameters T and C in G for each species. Here, 11 T. $ T's, the best estimate is obtained by rotating with the ions, N=0. Then, at t = 0, the Maxwellian ions appear to be stably confined by a potential well and the only free energy is that of tho el ectrons. Neglecting the enore in initial disturbances, A(0) for our model is thus obtained by introduoing distributions (4) - ... into Eq.(5) with V = Ag, T being taken the same in 1 and 8. and C being chosen so that f and g represent the same number of perticles: A(0) = avry [ + + +R]. (9) This is the representative value for all zones for our model with smooth density profile, Hore n is density, Ps is the ion byroradius, H is the plasma radius, and B ie the usual ratio of thermal and magnetic energy densities. We have omitted energy in o, nogligible for Debyo longth << R. As is di.scu 98ed more rully in Ref. 297, the first term in Eq. (9) 18 kinetic energy 1: electron average drift relative to ions, mainly due to diamagnetic currents, and the third term is diamagnetic field enorgy. The more interesting term 18 the second, representing coolu.g of the plasma as it expands In volume within the zone and does work on the fields wick grow thereby. Since universal instabilities are recovered in approximato theories assuming -0 and the fluid limit me → 0 5), the energy driving universal instabilities can be identified by taking these limits in Eq. 19). As one vould expect, the surviving tarm is the expansion cooling contribution arising from the pressure gradient, A10) universal. (DVT.)(2 22/3R2). (10) instabilities In most situations of interest, Ec. (10) is in any case the dominant [roo energy contributing to olectric Melds. The drift enorsy 18 small because mg.mg, and it is probable that most of the diamagnetic field energy does not transfer to eloctric fields at low frequency. As is indicated by the proportional to cu. E, the magnotic onorgy transform (d1jøctly) only to the transverse component of Ê, Et - 12W/C)B wore ) 18 the frequency. Thus, oncept in computing enorcy available to bend magnetic field lines, as in interchange, tho diamagnetic Tres energy contributing to electric fields causing transport across field lines probably should be reduced by (Pt AB). At frequencies belon the son syrofrequenoy and H > Pse the wiversal instability range, we find aW5Vz. Thus, even at large ß , this estimate of diamagnetic froe energy would generally be 1988 than the expansion energy retained in Eq. (10). To compare our result with experinent, we est imate the average electric Meld, E. By Eq. (3) and properties of $ , VE S 877 (t) S A10). However, though a propos upper bound on the instantaneous field energy, this is an overestimate of energy at frequenoles Wcwoi, the ion syrofrequenoy, as in universal instabilities. For the latter, the field endures long enough to share energy with "non-resonant" particles by accelerating them. mis is expressed through a dielectric constant € by roplacing EP abovo by €82. On s teplo g'ounds [9;', € = 1 + w werd in wa wago monco (1+ wfs/W.! (2/87) s vlaso). (11) mat is actually woasured by probing an unstablo plasma 18 the potential fluctuat 1on Sd and the corrolation length, corresponding to a , in tons of Whiob Es-(dpa). with this substitution, and Wpe » Wado Eq. (11) with Aloj in Eg. (10) yiolds tbo rolation (302542/61013)*(P2R). (12) This formula compares well with experimental results. Por example, or the lodol C stollara tor, WowWee ~ 10 and R = 200 Ps, and exporimontally o&b ~ To ~ Tj. Ta on, by Eq. (12), 2 = 11 P1 = ..3 cm, Tho obsorved correlation length 18 ~ 1 cm. As the observed and calculatod values of , are closer to each otbor than to othor caractoristio longtns, such as f or the Debye longin, we consider the rough agroomont botween thom to be significant (10). We concludiº that our ostimats of froe onorsy at froguonoios Wawat, B4. (10), 18 reason able. Noto taat A10) 18 proportimal to To 3. Stabilization of Universal Bod os The f'roe mergy estimate of the previous section requiros moc.im cation in magnetic wolls woon appllod to wiv orsal instabilities. The reason is that our ostinato, applloablo hoovor rapidly oners is trans:erred, does not take account of a tondonos toward consertation of the magnetic moments M's and tie ir 8 W. Further, since the constraints from which A(0) is calculated are not adiabatic, ve sumise that A(0) reprosents the maximum possible non-adiabatio mergy transitions, just the higher order terms in Sw, but A10) omits sw. For instance, in varying configurations from a magnetic well to a uniform field, A(0) is unchanged in form, mile Sw. >0. Thus, dropping SW in A(0) = SW, we make the conjecture that A(0) is a bound on higher order terms, Svig < A(0) = (n V T.) (15) 32 with A(0) from Eq. (10). Note that, while in minimizing 1(0) in Sect. 2 we appear to allow only electron expansion to derive Eq. (10), this is merely a correct but artificial mathematical device. For real plasma expansions appropriate in computing do, ions and electrons must expand together to avoid enormous charge separation energy; hence the appearance of both Te and Ti in Eq. (14.), though only Te appears in Eqs. (10) and (15). As usual, negative di implios stability. Thus, crwring Egs. (14) and (15), we conclude stability with respect to modos with w P 1 if (16) Note that the union wavelength a dropped out so long as a pio Note also that Eq. (15) does not apply to high frequency instabil. ities, w Woly since generally for such instabilities expansion (13) in powers of W and pe would not converge. Te for su , lor stability. 3 Te + T According to Eq. (16), assuming TeIfsystems are more likely to be stable when To/T4 is small. In Sects. 5 and 6, we shall examine under waat circumstances this fact can be exploited. The above argument can be extended to average wells when the distance I between regions of positive dB/dr is not too large compared to the plasma radius, R. To do so, we modify the sw expansion to expand also in the transit time, L/V1, where ví = 2T2/me• Then, as a leading term, Eq. (14) is replaced by Taylor's calculation of Sw. assuming the conservation of both H and the adiabatic invariant J 117. Equivalently, we may retain the convenient forms, Egs. (14) and (16), with AB replaced by A Bavg defined by equating Eq. (14) to Taylor is result, namely, ABavg= - RB2 (1943-1/ -2 altern. (17) Here, the integral is a long field lines over one period for a pe riodic average well, or between mirrors for a short system, AZ SO, X is a distance perpendicular to lines, B and R are Whatever representative average values we choose to employ in writing Eq. (14), and p is the plasma pressure. We have used the approximation to Taylor's Sw. which is the result given by Furth, valid for gentle magnetic curvature (12). The appearance of I in the formula is also approximate. .. wewe . .--..... ... ... . 12 Validity of the stability criteri on for average wells, again a question of convergence of the new expansion of Sw, requires roughly that the transit time 1/v4 be shorter than Eng the time required for release of the expansion energy capable of driving instability. This is the condition that an ion "See" the average well rather than the negative dB/ar region alone. Now, E = ( B/0E) 2 (3T3/2016)*(R/v4] (18) where we took E from Eq. (11), which is the maximum electric field which plasma expansion over wavelengths ~ can support. In Eq. (18), we have assumed the interesting case Wpi » wcze. Then, stability criteria (16) may be applied to average wells with AB reple ed by ABavg 1f, in addition to wewei and 1 > Pi, also [/v4 < Ed, that is, ir I/R < (3T4/2T.). (19) Note that once again the unknown wavelength a has dropped out of the result. A st ab ility criterion which is essentially Eq. (16) can be obtained from a conventional dispersion analysis of systems which exhibit resistive universal in stabilities. We shall adopt the viewpoint of Jukes [13], tiho used a microscopic vlasov description of ions and a fluid description of electrons. However, in the limit of long perpendicular wavelengths relative to the lon gyroradius, the resulting dispersion -... .-.. - - ..- chwiliada - - - relation is identical to that found by Chen using a two-fluid model 257. The effect of the magnetio well is simulated by an effective gravity causing ion drift velocity ūoi. One obtains a disporsion relation of the following form, valid for smal i Debye length: 0=425 + YEx{ye + F (1627 - Exp + 0x80] +0 7 x28(1KY-p), (20) where y = ww- Foão1)/W61 of =1 – +k%AF/2 , Q = exp(-3k3p})I. (kk{p) 20 = lon Debye length, K2 = k?Te/mgwai , Y=k& B onn, n = resistivity, O=T_/To, ő = 0+1, p=R+(To/mpcima*, S = R (7./mauriit, RTE 2+2 (an/ax), Rc = radius of curvature of Meld lines. New syinbols follow Chen 157, except for o, fini ch he calls a. The limit of stability (Im w = 0) is given by the following expression: B/B = FT/IŠTE + Tz), for stability. (21) For short wavelength, 5 > 1, whereas for 1>Pleso Thus, deeper wells are required to stabilize shorter wavelengths. Apart from the factor 2/3, che short wavelength limit of Eq. (21) is just Eq. (16). When resistivity is negligible, Jukes gives a stability criterion which in our terms is 4B/B + exp(-3k>pn) 1. (šk}}}) - (To/Tz) 20. (22) Again, the wavolength dependence is such that short wavolongths are the more difficult to stabilize, the requirement then being AB/B = T/T4 for stability. . (23) Once more, this is essentially Eq. (16). 4. Anomalous Diffusion If either woWay or 2 < pe, there is little tendency to . -- - - - - - - - - .. conserve the adiabatic invariants and a power expansion of Sw with a conserving leading term, as in Eq. (13), probably diverges. Then we must resort to Eq. (9) as the best estimate of free energy and admit the possibility of instability. For as pe but frequencies wzwei, and for long wavelengths along B, particles drift radially in the electric field stead:lly for a time <= (2B/CE) 11 I .07 B and L/R 54 Even classical electron heating by binary collisions wita ions demands large ion temperatures to maintain Tz/T= 10. An approximate stoady- state relation equating onergy removed by electrons to that transferred from ions, energy Ti, to Maxwelllan electrons, temperature To, 18: Ivo @ 14 (767BOTOM (4m/15 mg )* (291- 17. orto (2m. Telor" 3T (25) : te doen for Here C is the lon 11 retine, It are ion and electron Injection rates, and E. is the energy of escaping electrons before encountering the plasma potential difference. Typioally, Eg v 1-2 Te in torii and 47. in mirrors. With It = In for neutral injection and no = 1014cm-3sec for D-T power production (167, T4/To = 10 corrosponds to T1 = 300 Kev. At constant the C, roughly T1/T. T23/5. For true magnetic wells, AB can be large and the appropriate stability criterion, just Eq. (16), can be satisfied for smaller Ti/Te. However, with solid conductors magnetic 'vells must be open-ended and rely on mirror confinement. Mirror confinement requires large Tz, corresponding to large 14/12, to achieve ne sufficient to produce net power. (Efforts to employ the field or gyrating relativistic electrons might bypass this difficulty.) . Lurror 108808 by classical collisions have been calculated by Miss llozelle Rankin of the Oak Ridge National Laboratory using our previously developed code solving FokkerPlanck equations for lon and electron energy distributions (17. Wo considor a DoT systom maintained by d.c. injootion of deuterons at energy E.. Reaction products are assumed to escape. We omit ionization, charge exchange and radiation, which do prove negligible. As a function of E., we compute an effective (50)7osg, defined as the escaping current (EIL) divided by 12 times volume. We take the mirror loss formula of Bing and Roberts [187, rather than that in Ref. [17], with an effective mirror ratio depending on the computed plasma potential as in Ref. (17. In Fig. 1, we display (Vlogs through A, the ratio of nuclear power to injected power, A = LEN(T)DT/E.)loss]. (26) The experimental D-T cross section (197 times velocity is averaged over the calculated ion distribution. For Dar, Ev = 17.6 Mev. Curves are given for IL/I+= 1 (To/To= 10) and various values of Te held fixed by adjusting Id/IX21: neutral injection plus electron sources). We see that the optimum injection energy is E. = 300 Kev (and T, = 300 Kev) corresponding to A = 4, n = (5 vlogs = 5 x 1013 cm-3 sec (similar to Ref. [167), and Tz/T= 10, adequate for stability and in agreement with approximation (25). The above results are altered if electrons are heated anomalously, by high froquency instabilities. The drift-cyclotron instability (147 may cause such difficulties in torii. But for short mirror systems, it now seems that such instabilities may be controlled. Aside front possible residual universal instabilo ities discussed in Sect. 4, in a magnetic well with central field B = 60 kg and E. -- 300 Kov, the known Instabilities can be avoided by: mirror ratio 3.3 (Harris instability (207); this ratio and B=0.3 (mirror modo (217); this B and AB=0.3B (interchange [227); this AB and T2/T. = 10 as calculated above (universal modos, criterion (16)); 1 = 2m (1088 cone instability propagating along B, 157); and R = 2m (loss cono flute mode (237). Correspondingly, nt=1014cm-3, the nuclear power is 700 MI, the injected current is 600 ap, and neglected losses (syn- chrotron radiation [247, etc.) are 5 8 MW, indeed negligible compared to the 80 MV transferred to electrons by collisions. If residual universal instabllities occur with diffusion cooriicient Eq. (24), wita B = 60 kg the required radius is R= 5m. 6. Conclusions Aside from.:possible short wavelength modes causing anomalous diffusion discussed in Sect. 4, we have obtained criteria for preventing universal instabilities in magnetic wells (Eq. (16)) and average yells (Eqs. (16) and (19)). Stabilization is easiest when To << Ti. We find that, to satisfy our criteria, for all marinetic geometrios (with solid conductors) the 19 preferred operating range is about the same, Tq ~ 300 Kov ~ 10 T. Operation at lower temperatures (in tori!) must rely on other means of stabilizing univorsal modes. 20 Rererences 1. Yu, B. Gott, bil. S. Iof fo and V. G. Tolkovsky, Nucl. Fusion Suppl. Pt. III, 10145 (1962). 2. H. P. Furth and M. N. Rosenbluth, Phys. Fluids 2, 764 (1964); A. Lenard, Phys. Fluids 21875 (1964). 3. For example, L. I. Audakov nd R. 2. Sagdoov, Mucl. Fusion Suppl. Pt. II, 1,81 (1962); B. B. Kadontsov and A. V. Timofeov, Dokl. Akad. Nauk SSSR 106, 581 11962) (translation: Sov. Pays, Doklady I, 826 (1963)]; :-N. A. Krall and M. N. Rosenbluth, Phys. Fluids 6, 254 (1963); S. S. M0188ov and R. Z. Sagdoov, J. Expti. Theoret. Phys. (0.S.S.R.) W, 763 (1963) (transla- tion: Sov. Phys. JETE .. ] 4. H. Lashinsky, Phys. Rev. Letters 12, 121 (1964). 5. F. F. Chen, Princeton University Plasma Physcics Laboratory Reports, "Resistive Overstabilities and Anomalous 'Di fusion", MATT-306, Oct.,1964; "'Universal' overstability of a Resistive, Inhomogeneous Plasma," MATT-311, Nov., 1964. 6. H. N. Rosenbluth, summary talk, Conference on Closed Configurations, Harveli, England, Sept. 1962, Culham Laboratory Report CLM P-21. 7. H. Lashinsky, Phys. Rev. Letters 13, 47 (1964). 8. The equilibria described by J. B. Taylor, Phys. Fluids : 6, 1529 (1963). Stability with respect to universal modes was iu rther cla rified recently by N. A. Krall, to be published. 9. T. K. Fowler, Phys. Fluids 8, 459 (195). MA 20. We aro indebtod to Drs. K. Young and 11. Harrios for suplying data on a uctuct ions in Princoton ozoriments, and so wish to thank then and other members of the Llodel C Stellarator group for help:ul discussions, 11. J. B. Taylor, Phys. Fluids 2, 767 (1962). 12. H. P. Furth, Phys. Rev. Letters 11, 308 (1963). 13. J. D. Jukos, Pays, Fluids 2, 11;68 (1964). 14:.. A. B. Ikhailovsky, Nuclear Fusion, to be published. 15. 2. N. Posonbluth and R. 7. Post, pays. Fluids 8, 547 (1965). 16. J. D, Larsoit, Proc. Phys. Soc. (London) B 20, 6 (1957). 17. T. K. Forder and M. Rankin, Plasma Phys. (J. Mucl. Energy:C) 4, 311 (1962), 18. G. 7. B.ing and J. E. Roberts, Phys. muids 4, 1039 (1961). 19. "Charged Particlo Cross Sections," Los Alamos Scientific Laboratory Report LA 2014 (1956), 20. Yu. H. Dnestrovsky, D. P. Kostoma rov and V. I. Pistunovion, Huci, fusion 3, 30 (1963); L. S. Hall and Y. Shima, to be published; they show stability if (T, /1,,/e<2, true with *.. a large mirror ratio and high density. 21. R. 7. Post, Nuci. Fusion Suppl. Pt. I, 99 (1962). 22. J. B. Taylor and R. J. Bastie, Phys. Fluids 8, 323 (1965). 23. ¥. Shima, to be published. 24. ¥. E. Drummond and M. N. Rosenbluth, Phys. Fluids 6, 276(1963). Here, the quantity (2*73 defined in their paper is ~ 200. 22 FIGURE CAPTION ..... Fig. 1. The ratio A of nuclear power to injected povor is shom as a function of injection energy E..for various values of electron temperature T, and for neutral injection, I./I4 = 1. ... . . ) .. ... 0:10 DICT101N GRADN PARCR .: 10 x 10 PER.INCM . . ..... :. GENE SIETIGEN CO KADI IN V. I. de :::: :::: ::::::: ... لالالالالالالا ECKE VON 600...:::;..800 ..:' L . . . * BLANK PAGE 9 END DATE FILMED 11/ 9 /65 43 "35273 IM ' . 235....: . .