Kesner, J.

DSpace@MIT

PFC/JA-96-28Stability of Electrostatic Modesin a Levitated DipoleJ. KesnerNovember 1996MIT Plasma Fusion CenterCambridge, Massachusetts 02139 USAThis work was supported by the US Department of Energy under contract DE-FG02-91ER-54109.Reproduction, translation, publication, use, and disposal, in whole or in part, by or for the USGovernment is permitted.To be published in: Physics of PlasmasPFC/JA-96-28Stability of Electrostatic modes in a Levitated DipoleJ. KesnerPlasma Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139Cambridge, MassachusettsAbstractPlasma confined in a magnetic dipole is stabilized by the expansion of the magnetic flux.The stability of to low beta electrostatic modes in a magnetic dipole field is examined whenthe distribution function is to lowest order Maxwellian. It is shown that for sufficientlygentle density and temperature gradients the configuration would be expected to be stableto magnetohydrodynamic interchange, as well as to dissipative trapped ion and collisionlesstrapped particle modes. These results are applicable to any magnetic configuration forwhich the curvature drift frequency exceeds the diamagnetic drift frequency.PACS 52.35.P, 52.35.Q, 52.55, 94.30.F,G1I. IntroductionThe dipole magnetic field is the simplest and most common magnetic field configu-ration in the universe. It is the magnetic far-field of a single, circular current loop, andit represents the dominant structure of the middle magnetospheres of magnetized planetsand neutron stars. The use of a dipole magnetic field generated by a levitated ring toconfine a hot plasma for fusion power generation was first considered by Hasegawa'. As aconfinement configuration for magnetic fusion a dipole possesses uniquely good properties.The coil set is simple and axisymmetric. Operation is inherently steady state and the largeflux expansion is expected to simplify the divertor design. Vis-a-vis a tokamak there is noneed for current drive and no disruptions. It is expected to have good magnetohydrody-namic (MHD) properties, including plasma pressures that can locally exceed the magneticpressure, i.e. # > 1, and excellent confinement properties. By levitating the dipole magnetin order to prevent end losses, conceptual reactor have studies supported the possibilityof a dipole based fusion reactor 2- 4 . In this paper we will focus on the stability of driftmodes that are thought to degrade confinement in fusion grade plasmas.For a plasma confined in a levitated dipole the pressure falls off (moving away fromthe internal coil) in a region of "bad" curvature. In this situation it is well known thatstability can be obtained due to the so-called compressibility and there is a critical valueof the pressure gradient that can be confined stably. The dipole reactor concept is basedon the idea of generating pressure profiles near marginal stability for low-frequency MHDfluctuations. From ideal MHD, the marginal stability of interchange modes results whenthe pressure profile satisfies the adiabaticity condition, 6(pV' ) = 0, with p the plasmapressure, V is the flux tube volume and -y = 5/3. We derive the equivalent conditionfrom the drift kinetic equation. In the derivation of the dispersion relation for low fre-quency interchange modes from the drift kinetic equation the stabilizing term derives fromthe square of the curvature drift frequency (Eq. [8]) and there is no need to make theassumption of an equation of state. This derivation therefore derives from first principlesand clarifies the origin of compressibility in MHD.2The ability to confine plasma at high beta makes the dipole configuration particularlywell suited as an advanced fuel reactor. Ignition in advanced fuel plasmas such as D3He,requires particularly good confinement properties. Since the magnetic field is entirely inthe poloidal plane there are no particle drifts off the flux tubes (which in a tokanak result ina "neo-classical" degradation of confinement). In this paper we show that plasma confinedin a levitated dipole may be expected to be free of drift wave turbulence and therefore adipole based reactor may be expected to attain classical confinement.Hasegawa has pointed out', that when the plasma is sufficiently collisionless, theequilibrium distribution function may be by Fo = Fo(p, J, 4), with p the first invariant,P = v'/2B, J the second invariant, J = f ds v11, and 0 the flux invariant. For fluctuationsin the range of the curvature drift frequency, flux is not conserved and a collisionlessplasma can approach the state &F/(9 -- 0. Furthermore when 8F/&4' = 0 the plasmacan be shown to be stable to drift frequency fluctuations. This condition leads to dipolepressure profiles that scale with radius as r- 2 0/3, similar to energetic particle pressureprofiles observed in the planetary magnetospheres- 6 .In a conceptual reactor confinement must be maintained on a collisional time scale.Therefore we would expect the distribution function to be, to lowest order, Maxwellian, i.e.Fo(p, J) -+ Fo(E, 4) and therefore OF/o4 5 0. In this article we assume to lowest order aMaxwellian distribution function for both ions and electrons and we derive the conditionfor marginal stability to MHD interchange modes as well as collisionless and dissipativetrapped ion modes. We find that each of these collective modes becomes stable when thedensity and temperature gradients are sufficiently gentle. Therefore a plasma confined ina levitated dipole field may be expected to be particularly stable to collective modes andfor sufficiently gentle gradients may exhibit classical confinement.These results are applicable to any magnetic configuration for which the curvaturedrift frequency is comparable to the diamagnetic drift frequency such as a low aspect ratiotokamak.3II. Electrostatic, Trapped Particle ModesTo derive the stability criterion for electrostatic modes we consider a fluctuating elec-trostatic potential, # and ignore any equilibrium electrostatic potential. From Faraday'slaw it is possible for a perturbation to leave the magnetic field undisturbed if E = -VO,which is consistent with # < 1. If q varies along a field line, there will be a finite Ell (asituation not possible in ideal MHD theory).We analyze the stability of such a perturbation under the assumptions that the wavefrequency w is less than the cyclotron frequency Q, and that the ion Larmor radius pi isshorter than the perpendicular wavelength A = 27r/k± which is, in turn, short comparedto a parallel wavelength, 27r/klj. The appropriate equation for the distribution function fis 7,81= qOFo, + Jo(k- p)h (1)and h satisfies(w - Wd + ivjlb - V') h = -(w - w.)qFoJo(kip) + iC(h). (2)Fo(e, ) is the equilibrium distribution function and V' is the gradient at constant e andpI.Foe = -- ,o (3a)b x k± -V'F (3b)w.~~ =ob-m~c Fo6(mV 2b -Vb + PVB)Wd = k -b x (3c)B = V x VO, (3d)b = B/IBI. (3e)Jo(kip) is the Bessel function of the first kind and 0 is the azimuthal angle.4We consider a perturbation whose growth time is long compared to a particle bouncetime and obtain the result that h is a constant along a field line h = ho(e, p, 0). We canthen determine the constant by taking the time average of Eq. (3),- (w - w*) qFoJoho (W d+iq) (4)The overbar indicates a time average and V' is the gradient at constant e and p.1 dl (5a)TB |vil|ITB = . (5b)For simplicity the collision operator has been replaced by a Krook model in Eq. (4), i.e.C(h) -- -vh with v3 the appropriate collision frequency.III. Fast Growing ModesTo explore modes that grow on the MHD timescale we assume that w > Od > v3and expand the denominator of Eq. (4) to obtain for the perturbed particle distributionfunction[b x k. -V'Fo Wd b x kL -V'Fo d 1df=qOFo, - F& J0+ I +WW 2'Fo,- - qOJO2. (6)f=q4o-qb~e~+[ m~w w m~cw w w0JWe determine the eigen frequency w by requiring that the mode be quasi neutral. Weexpand JO as J2 oc 1 - (kLpi) 2 in the first term, but neglect the k2p? correction in thesecond term. With these assumptions the quasi neutrality condition becomes= q ) k + (W*&d - . (7)The terms proportional to 1/w have cancelled in the sum over species. Consider first thecase when the eigenmode is flute-like, i.e. l, v = 0. In this limit we can solve for w 2to obtain:522 d3v Fo,3d (aid -(w)q2 fd 3 v Fork v /2 . (8)In out convention w > 0 and for a dipole there is "bad" curvature, i.e. Wd > 0. Flute-like(interchange) modes are nevertheless stable when f d3v Fo(Od 2 - Iw.Owl) > 0. DefiningS= f Fow,. d3v , C0 = f FoaiV d3v and approximating f Fow.Od d3v ; (1 + )QCOd with7= Vln(T)/VIn(n), we see that stability requires:Od > (1 + 77)CO, (9)i.e. r > Rc with r,. p/Vp and Rc is the radius of curvature.Under some circumstances non-flute-like "trapped particle" modes can grow on theMHD timescale'. These fast growing modes can be investigated by constructing a quadraticform. We multiply Eq. (7) by */B and perform a flux tube (f dl/B) integration. Writingjdv=27rB dedyfd~v=m2Ivinterchanging the order of integration, and solving for w2 gives:2 Eq q 2 $ f ded-rB&2Fo( - w*) (10)-L0 ==_ (10)E, q22 , dedpIB (-F 6) ( 2 - ) +42 k p}The quadratic form, Eq. (10) is variational with respect to 0 and since the denomi-nator is positive definite, if we can find a trial function for 0 such that w2 < 0 the trueeigenfunction will give an even larger growth rate. The numerator of Eq. (10) can be pos-itive for regions where the magnetic field curvature is concave with respect to the plasmaif w, > Od. When the drive is localized the eigenfunction tends to concentrate in theseregions. The denominator, however, is small for trial functions which are spread out. Theactual eigenfunction is determined by the balance between concentrating in regions of highcurvature to make the numerator larger and spreading as much as possible to make thedenominator smaller. For a mode that localizes on the outer midplane of the torus the6deeply trapped particles feel the full strength of the mode while the passing particles feela reduced "average" fluctuation.For a dipole we have seen from Eq. (9) that interchange stability requires that Cod > w.Since for a dipole, wd is largest at the outer midplane, we see that when a levitated dipoleis stable to interchange modes it will also be stable to fast growing trapped particle modes.IV. Drift Frequency ModesLower frequency modes can be destabilized by the resonance at w = Od (Eq. 4).To evaluate the stability of the resonant modes we begin with Eq. (4) and apply quasi-neutrality:q2  d3 v Fo00 = q2 J d3 v J2Fo w - b x k V'Fo/(mQFo,)q qIMultiplying Eq. (11) by 0* and taking a flux tube average yields:+ J V JF 2 w - b x k .V'Fo(meFo 2(1 + Te/Tijfd v Fo k2  d qo W~ OOc +.~ o (12)The bounce-averaged quantities, T and 2 are in general functions of the pitch angle. Fora flute-like mode F= ;2 and we will show below that such modes are not unstable. Atrapped particle driven mode is localized to the outer midplane so that trapped particlesexperience the mode more strongly than passing particles and 4 > 2. To obtain anapproximate dispersion relation for the collisionless trapped particle modes we extract anaverage value of the bounce average quantities, q and 2 and rewrite Eq (12) as(1 + Te/T) = J d3 v J0FOe o - b x k_ -V'Fo/(mQeFo,)] (13)ft q W - O + ZVI7For constant q (# = = 4 ), ft = 1. For a mode that is localized near the outermidplane or which changes sign from the outside to the inside the trapped particles feelthe mode more strongly (ktapped ' c,.na) than the passing particles ('kas, ~ 0) and it canbe seen that ft is an estimate of the trapped particle fraction.For a dipole the curvature is fairly uniform along the field line and we can approximatead -+ (e/T) * Wd in Eq. (13) to obtain:W - W*n[1+ 77e (E/Te - +j)] + ±_[- r(e/T= ( + r)/ft (14)\ O w- (e/T,) +iv,,-/ \ + ((d/r)(E/T) + iVi = +)/ (4where (A) = (2/ir1/2T3/ 2) f I dee'/ 2 exp(-e/T)A, 'r Te/Ti, W*n -*nir = -kepivi-r/2rn,= (2Ti/mi)/ 2 , pi = v/i, = eB/mic, ko = m/r, rn = -(din n/dr)-', Wd (wd), =-(Wd)ir = kOpivi7r/2RC > 0, dln T/dln nj, and w~n 0 for dn/dr>0.For Te = T Eq. (14) simplifies:1O - w.n[1+ (e/T - )]\ /w + w.[1+ 7(e/T - )] 2ft (15)w - O2d(e/T) + ive W + Od(E/T) + ivi /V. Collisionless Resonant ModeEquation (14) was analysed by Tagger et al. 10 and by Tang et al. 11 for the tokamakcase, i.e. when c% > d. We have seen that for a dipole MHD stability requires C. < Cd.Consider first the collisionless mode (ve - vi - 0). If we define Q = w/wd, Q* =W*n/d and *T = r/Q Eq. (15) becomes:Q + -.(3/2-1/r- e/T)\ /Q- *T( 3 /2 - 1/-e/T) \ 12K ( - E/T - + \ Kl E/T / (1)Noting thatK +A >=l+/A-e/T\\ + E/T| \+ -/T|8Eq. (16) can be written as9*T(3/2 - 1 r7)/(1 - KT) + E *T(3/2 - 1/rl)/(1 - ,*T) + e/T 2/ft - 2\ G- eT Q \ C elT Q -,T'(17)Equation (17) has complex solutions, Q =, + -y. Consider marginal stability, f2 = ,=Qo, with Ro > 0. The singularity in Eq. (17) is removed when-, __T(3/2 - 1/r7) (18)1 - Q*TSubstituting into Eq. (17) we obtain an equation for Qo:Ko - e/T 1 + (2/ft - 2) (19)\go+ E/T/ (1 - n*)'Since ((no - e/T)/(Qo + e/T)) < 1, Eq. (19) does not have a solution when ft = 1, whichindicates that a flute-like mode would not be unstable. For a marginally stable solution toexist the right hand sideof Eq. (19) must be < 1 and therefore *T > 1. From Eq. (18) weobserve that when 2 *T > 1 marginal stability is obtained for r7 > 2/3. This is a necessarycondition for the existence of unstable solutions. Therefore when r7 > 2/3, the conditionn*T < 1 is a sufficient condition for stability. From the solutions that appear in Ref. 10we observe that typically 0.5 < Qo < 2. Rewriting the stability condition we obtain:Wd > (%T (20)with wT = 7Z'*. Normally r7 > 1 and this condition is less restrictive than the interchangecondition: IdI > C% (1 + 7).9VI. Dissipative Trapped Ion ModeIn the simplest approximation the dissipative trapped ion mode can be derived byassuming collisional electrons and collisionless ions, i.e. by taking the limit ve -+ oo andVi -+ 0. In this limit Eq. (17) would be replaced byE//T + Q.(3/2 - 1/7)/(1 - ) (21)E/T - Q ,(1with ( = (2/ft - 1)/( 2 *T - 1). The marginally stable solution to Eq. (21) is Q = o=(3/2 - 1/7)/(Q*T - 1) and C = 1, i.e. Q*T = 2/ft (as pointed out in Ref. "). Notice thatthis solution of Eq. (21) is possible when ft = 1 which corresponds to a flute-like mode.This indicates that the dissipative mode is driven by the difference in the (collisional)electron and the (collisionless) ion response and instability does not require a localizationof the mode in the trapped particle region.If we take Q = no + 6 + i-y and= 1 + A with A > 0 the imaginary part yields aconstraint= 0, (22)Q2 + y2while the real part gives:Q02 +,y2i.e. A < 0. This indicates that there is no unstable solution when ( > 1, i.e. whenf*T < 2/ft. Thus a sufficient condition for stability is C2 d > W*T/2. This condition is lessrestrictive than the stability condition for either the collisionless mode or the interchangemode.10VII. DiscussionThe results shown above are generally applicable to a magnetic confinement devicethat that is stabilized by compressibility, i.e. one that satisfy's the inequality C~2d > w..For a tokamak w*/Cd ~ A with A the aspect ratio, so that the compressibility is usuallyconsidered to be a small correction. For a low aspect ratio tokamak, however, the stabilizingcompressibility term can become important.The dipole reactor concept is a radical departure from the tokamak or from othersimilar toroidal magnetic fusion reactor concepts. The magnetic field lines are closed, thefield is poloidal and the flux surfaces are defined by the toroidal drifts. There are no driftsoff the flux surfaces and the dipole is not subject to neoclassical effects. In addition thehigh degree of axisymmetry inherent in the coil set insures the absence of non-axisymmetrydriven "ripple" losses.Hasegawa 2 considered a collisionless plasma confined in a dipole that is characterizedby an equilibrium distribution function which is non-Maxwellian. He showed that when theequilibrium distribution function can be characterized by FO = Fo(p, J), i.e. OFo/8& = 0,drift frequency modes are not unstable.We have shown that in the more fusion relevant case, when FO is Maxwellian andtherefore 9Fo/O49 < 0, the dipole may still not be subject to drift frequency fluctuations.Therefore a magnetic dipole based fusion reactor may exhibit classical transport.AcknowledgementsThe author would like to thank Professor M. Mauel (Columbia University) for manystimulating discussions about dipole confinement and Dr. Barton Lane (Plasma Dynamics,Belmont, Mass) for insights about the gyro-kinetics. This research was supported by theUS Department of Energy.11References:1 A. Hasegawa, Comments on Plasma Physics and Controlled Fusion, 1, (1987) 147.2 A. Hasegawa, L. Chen and M. Mauel, Nuclear Fus. 30, (1990) 2405.3 A. Hasegawa, L. Chen, M. Mauel, H. Warren, S. Murakami, Fusion Technology 22,(1992) 27.4 E. Teller, A. Glass, T.K. Fowler, A. Hasegawa, J. Santarius, Fusion Technology 22,(1992) 82.' N.F. Ness Ness, M.H. Acuna, R.P. Lepping, L.F. Burlanga, K.W. Behannon, F.M.Neubauer, Science 206, (1979) 966.6 S.M. Krimigis T.P. Armstrong, C.Y. Fan, E.P. Keath, W.I. Axford, G.G. Gloeckder, R.D.Zwid, C.O. Bostrom, L.J. Lanzerotti, J.F. Carbary, D.C. Hamilton, Science 206, (1979)977.7 M. Rosenbluth, Phys. Fluids 11, 869 (1968).8 T. Antonsen and B. Lane, Phys Fluids 23, (1980) 1205; T. Antonsen, B. Lane and J.J.Ramos, Phys Fluids 24, (1981) 1465.9 H.L. Berk, M.N. Rosenbluth, V. Wong, T. Antonsen, D.E. Baldwin, Fiz. Plasmy 9,(1983) 176, [Sov. J. Plasma Phys. 9 (1983) 108].10 M. Tagger, G. Lavel and R. Pellet, Nuc. Fus 17, 109 (1977)." W. Tang, J. Adam and D. Ross, Phys Fluids 20, 430 (1978)12

Stability of electrostatic modes in a levitated dipole

http://dspace.mit.edu/bitstream/1721.1/95296/1/96ja028_full.pdf

Stability of electrostatic modes in a levitated dipole

Abstract

Similar works

Full text

Available Versions

DSpace@MIT