Ambipolar magnetic fluctuation-induced heat transport in toroidal devices*

Transcription

1 Ambipolar magnetic flctation-indced heat transport in toroidal devices* P. W. Terry, G. Fisel, H. Ji, a) A. F. Almagri, M. Ceic, D. J. Den Hartog, P. H. Diamond, b) S. C. Prager, J. S. Sarff, W. Shen, M. Stoneing, and A. S. Ware Department of Physics, University of Wisconsin Madison, Madison, Wisconsin Received 10 November 1995; accepted 4 Janary 1996 The total magnetic flctation-indced electron thermal flx has been determined in the Madison Symmetric Tors MST reversed-field pinch Fsion Technol. 19, from the measred correlation of the heat flx along pertrbed fields with the radial component of the pertrbed field. In the edge region the total flx is convective and intrinsically ambipolar constrained, as evidenced by the magnitde of the thermal diffsivity, which is well approximated by the prodct of ion thermal velocity and the magnetic diffsivity. A self-consistent theory is formlated and shown to reprodce the experimental reslts, provided nonlinear charge aggregation in streaming electrons is acconted for in the theory. For general toroidal configrations, it is shown that ambipolar constrained transport applies when remote magnetic flctations i.e., global modes resonant at distant rational srfaces dominate the flx. Near locations where the dominant modes are resonant, the transport is nonambipolar. This agrees with the radial variation of diffsivity in MST. Expectations for the toama are also discssed American Institte of Physics. S X I. INTRODUCTION The motion of charged particles along the stochastic fields of magnetic trblence is generally regarded as an important transport mechanism in fsion and astrophysical plasmas. Becase the rates of streaming along magnetic fields are greatly different for electrons and ions of comparable temperatre, electron and ion loss rates are not necessarily ambipolar i.e., eqal as they are for electrostatic trblence. For particle transport, the flxes are constrained by Ampère s law to be ambipolar for localized, internally generated flctations, independent of the vale of the spatial mean electric field. 1 The same self-consistency constraint, however, does not lead to intrinsic ambipolarity in the heat flx, either nder qasilinear theory, 1 or nder more realistic renormalized theories of the trblent response.,3 This difference is attribted to the fact that the electromagnetic potentials of ambipolarity are created in response to, and ltimately control, charge densities. They cannot distingish between events that interchange two particles of disparate energies from those that interchange particles of eqal energy. 3 Sch argments imply that the electron heat loss is tied to the electron thermal velocity, whereas the particle loss is tied to the slower ion thermal velocity. This leads to a mared disparity in the rates of convective and condctive heat loss. 4 We describe herein experimental observations and an analytical theory that directly contradict the above assertions. Measrement of the magnetic flctation-indced electron heat flx 5 in the edge of the Madison Symmetric Tors 6 MST shows that it is convective, despite the presence of a * Paper 7Q10, Bll. Am. Phys. Soc. 40, Invited speaer. a Present address: Princeton Plasma Physics Laboratory, Princeton, New Jersey b Department of Physics, University of California at San Diego, La Jolla, California temperatre gradient, and ambipolar constrained. The ambipolar constraint is nderscored by a simple modeling exercise that shows that the flx is well described by a Rechester Rosenblth diffsivity, 7 bt with ion thermal velocity as the streaming factor. In the theory, ambipolar constraints are shown to arise throgh nonlinear bnching of electrons and the effect of this bnching on the collective plasma dielectric response throgh resonant momentm and energy exchanges. When the ions of the dielectric response are adiabatic i.e., their thermal velocity exceeds the flctation phase velocity, the edge electron heat flx is convective, manifestly ambipolar constrained, and agrees well with the measrement in terms of its magnitde and scaling. The electron bnching crcial in present considerations is not acconted for in conventional theories of trblence. 8 This bnching arises from correlations among streaming electrons, whose relative separations lie within a correlation length of the scattering fields. Sch electrons remain correlated for sfficiently long to act as discrete clmps of ballistically propagating electrons. The moving clmps indce a shielding response in the plasma dielectric, casing a transfer of energy and momentm to the fields throgh resonant emission. The excited fields, in trn, transfer energy bac to the distribtions via Landa damping, a process governed by the identical resonance condition. The self-consistent momentm and energy exchange between electrons and electrons and ions, as mediated by the trblent fields, is ths elastic. Conseqently, the trblent collisions behave lie Colomb collisions, with scattering between isothermal particles of the same species prodcing no transport. In a selfconsistent treatment of collisionless magnetic trblence involving no temperatre gradients, this constraint was fond to eliminate transport involving trblent electron electron scattering, the type described by qasilinear theory. 9 In that treatment, it was also assmed that the residal ambipolar constrained losses de to trblent electron ion collisions are negligibly small. This led to the conclsion that electro- Phys. Plasmas 3 (5), May X/96/3(5)/1999/7/$ American Institte of Physics 1999 Downloaded 07 Jn 006 to Redistribtion sbject to AIP license or copyright, see

2 static flctations alone reglate transport. Here, in contrast, the temperatre gradient allows the existence of a qasilinear-lie condctive flx associated with trblent electron electron transfer. This flx arises from the perpendiclar energy moment. As with field-aligned momentm, 10 there is no transport of parallel energy via trblent electron electron collisions. The ambipolar-constrained flx de to trblent electron ion collisions has both convective and condctive components and is not necessarily small. For MST parameters in the edge, the convective flx dominates both its condctive conterpart and the nonambipolar electron electron flx. In this case, magnetic trblence reglates transport, bt in a way that is ambipolar constrained. II. EXPERIMENT The magnetic flctation-indced electron thermal flx was measred in MST, 6 a large reversed-field pinch RFP with a minor radis of a0.5 m, a major radis of R1.5 m, and a plasma crrent of I0.7 MA. Plasmas in MST typically have a line-average density and temperatre of cm 3 and 300 ev, respectively. Limited profile diagnostics for-chord far infrared FIR and Thompson scattering indicate that density and temperatre profiles are broad bt not flat. The density profile tends to pea more than the temperatre profile, particlarly at lower densities. The plasma core typically rotates. The imprity ion rotation speed is 10 4 m/s; the main plasma ion rotation speed is not measred. However, correlation between imprity ion slowing down and mode rotation rate decrease before and dring sawtooth crashes sggests that imprity and main plasma ion rotation rates are similar. 11 Althogh the rotation profile is not nown in MST, a decrease of rotation rate with radis is implied by the lac of rotation beyond the reversal layer. MST discharges are characterized by a broad spectrm of magnetic trblence. Magnetic flctations are measred both by insertable probes and by extensive wall-monted probe arrays. Power is concentrated in the low freqencies 0 Hz, where flctations are of the order of a percent B /B and helicities correspond to m1, n5 8. At higher freqencies the flctation spectrm obeys a power law decaying as 5/ and encompasses higher helicities. 1 In the high-freqency range, the toroidal wave nmber varies linearly with freqency, with freqencies from Hz corresponding to toroidal mode nmbers from These measrements indicate that even at the edge, magnetic trblence is dominated by flctations whose resonant srface lies well in the core. Extensive nmerical modeling stdies 13 have established that the low-freqency spectrm component corresponds to internally resonant nonlinearly satrated global tearing modes. These modes are responsible for sstaining the reversal of the mean toroidal magnetic field, bt in so doing they stochasticize the magnetic field within the reversal radis. 14 Simlations also indicate that they interact qasicoherently, giving rise to a spectrm that is nonstationary at low wave nmber. The temporal evoltion of the magnetic field for the dominant modes n5 8 indicates a broad spectrm with signatres de to sawtoothing at 1 Hz, a 10 Hz signal associated with rotation of the n6 modes at the FIG. 1. Cross-power spectrm for the correlation of the flctating parallel heat flx with the flctating radial magnetic field for r/a0.75. plasma rotation speed, and a high-freqency tail. 15 At least some of the high-freqency signal is believed to reslt from aliasing effects with shorter-wavelength flctations. For a fixed wave nmber in the range of the dominant tearing modes, the freqency spectrm is dominated by rotation and qite coherent. The width of the freqency spectrm halfwidth at half-maximm is no greater than one-qarter the vale of the pea. A pea at very low freqency de to sawtoothing appears as a distinct featre. Magnetic probe array measrements indicate that the electron mean-free path is many times larger than the parallel correlation length. Measrements of the magnetic flctation-indced heat and particle flxes have been reported previosly. 5,1 The particle flx e was obtained sing an electrostatic energy analyzer in conjnction with a magnetic probe to correlate the parallel electron crrent flctation with B r. The heat flx Q e was obtained with a fast pyrobolometer and magnetic probe to correlate the flctating parallel heat flx with B r. Both measrements covered a region from r/a0.8 to the wall. The toroidal field reverses sign at r/a The cross power of parallel heat flx and B r is shown in Fig. 1. The cross power peas at 10 Hz with virtally all of the power below 0 Hz. From the dispersion crve for the toroidal mode nmber, this freqency range is identified with the global tearing modes n5 8. The absence of any significant contribtion from higher-freqency flctations will be an important element of later analysis. A comparison of 3 the convective heat flx, e T e, calclated from the measred magnetic flctation-indced particle flx, and the measred magnetic flctation-indced total heat flx Q e is shown in Fig. as a fnction of radis in the oter region of the plasma. Bonds on an estimate of the total electron heat flx inferred from a global power balance are indicated by the hatched region. Figre shows that Q e acconts for all thermal losses at r/a0.8. Closer to the wall, however, Q e drops sharply, becoming mch smaller than the power balance-inferred heat flx. This is consistent with stochasticity ending at the reversal layer in the region r/a and some other mechanism e.g., electrostatic flctations driving the heat flx in the edge. Figre indicates that Q e is convective over the entire range sampled. This observation invalidates the ineqality e Q e / e T e 5 as a signatre that transport is dominated by magnetic braiding. 4 A vale of e greater than nity wold occr if the total heat flx were not ambipolar constrained while the convective flx were. 000 Phys. Plasmas, Vol. 3, No. 5, May 1996 Terry et al. Downloaded 07 Jn 006 to Redistribtion sbject to AIP license or copyright, see

3 h e t v A b 0 h e v B h e 0 c B 0 b 0 h e ef e T e i * T, v c A, expiti x, FIG.. Total magnetic flctation-indced electron thermal flx and convective thermal flx as a fnction of radis from r/a0.8 otward. The shaded region indicates the magnitde of the total heat flx inferred from a power balance. The inference from Fig. is that ambipolar constraints apply to the total heat flx Q e. This is frther illstrated by comparing the reslts of Fig. with the Rechester Rosenblth model, Q er (B /B 0 ) L. Using the spectrmaveraged magnetic flctation power B /B and a parallel correlation length L 1mnown approximately from magnetic probe array measrements to within a factor of 3, this expression agrees with the reslts of Fig., bt only if the electron thermal velocity is replaced by the ion thermal velocity. Otherwise it is off by over an order of magnitde. where T * e(ct e /eb 0 )b 0 rˆl 1 n 1 e (v /v e 3 is the diamagnetic freqency, e L ne /L Te is the ratio of density gradient scale length to temperatre gradient scale length, is the electron thermal velocity, and f e is the eqilibrim distribtion fnction. The soltion of the DKE has the form h e,r,, R A,A, h e,, 3 where the first two terms are referred to as the coherent response, representing the response of the distribtion to potential distrbances at the same wave nmber and freqency, and the third term is the incoherent distribtion. In a moderate to wea trblence regime where resonance broadening effects are of second order, the coherent responses are given by T R,i * v ef e/t e 4 and III. THEORY R A, v c R,. 5 In light of these measrements, we calclate the Lenard Balesc trblent collision integral LBTCI, 9 in order to evalate the electron heat flx moment of the flctating electron distribtion for an eqilibrim with temperatre and density gradients. The electron heat flx is Q e Re d 3 v m ev, ic B 0 b 0 rˆ v c A h, 1 e, where and A are the electrostatic and magnetic potentials, h e is the nonadiabatic part of the electron distribtion, b 0 is the nit vector along the eqilibrim magnetic field, and rˆ is the radial nit vector. This expression describes EB advection of heat, and heat loss via electron streaming along the pertrbed magnetic field B A b 0. The component of Q e proportional to A yields the heat flx measred by experiment. The electron distribtion appearing in Eq. 1 is obtained from the electron drift inetic eqation DKE, The neglect of resonance broadening effects is consistent with the narrow spectrm at low wave nmber, as described in the previos section. The incoherent distribtion h e, arises from the nonlinear copling. In Forier space, the nonlinearities of Eq. are convoltions, allowing h e (,) to be driven by potential amplitdes at different and. The strctre and dynamical properties of the incoherent distribtion are obtained from the soltion of the evoltion eqation 8 for the two-point phase space density correlation h e x 1,v 1,th e x,v,t. The incoherent distribtion is identified with correlated electron clmps that propagate ballistically along the eqilibrim magnetic field. The spatial and temporal properties of the incoherent distribtion mae it distinct from the plasma dielectric. Specifically, qasinetrality and Amp`ere s law describe the shielding of the clmps by the dielectric. The shielding relationship is implicit in the strctre of Eq. 3: the response fnctions R A and R are electron ssceptibilities that contribte to the dielectric in the sal way. The incoherent distribtion, however, is not proportional to or A, and does not contribte to the dielectric. The shielding of the particle-lie clmp is analogos to the shielding of moving test particles by a plasma dielectric. Sbstitting the electron distribtion, Eq. 3, into the heat flx, Eq. 1, leads to the LBTCI, Phys. Plasmas, Vol. 3, No. 5, May 1996 Terry et al. 001 Downloaded 07 Jn 006 to Redistribtion sbject to AIP license or copyright, see

4 Q e Re d 3 v m ev, ic B 0 b 0 rˆ v c R A, A A, v c A h e, R,, h e, R A,A, v c R, A,. In this expression, the coherent and incoherent components of the distribtion play roles consistent with those played in the constititive relations. The coherent responses, throgh the gradient dependence of R A and R, prodce diffsive terms when the divergence of the flx is taen. If the coherent response to the magnetic potential is linearized, the diffsivity is that of qasilinear theory with a Rechester Rosenblth-lie scaling. Similar scaling also follows when a nonlinear resonance broadened coherent response is sed. 3 The incoherent distribtion leads to a drag-lie term in the LBTCI. The ballistically propagating electron correlations are shielded by the plasma dielectric, indcing emission into the collective modes of the dielectric in the form of a resonant exchange of momentm and energy between macroparticle and modes. This emission process sbjects the clmp to a drag. The shielding and drag processes enter Eq. 6 when the transport description is made self-consistent by relating the potentials to the charge distribtions throgh Amp`ere s law and qasinetrality. These constraints are given, respectively, by 4e d 3 vv c F i R,, R A,A, h e,] A, and 4e d 3 vf i R,, R A,A, h e,]0, where F i f i e/t i h i is the fll ion distribtion, h i is the nonadiabatic ion distribtion, satisfying ( v )h i () ( T * i) f i J 0 ( v / i )(e/t i )(, v c 1 A, ), T * i T * e(e i), J 0 is the zeroth-order Bessel fnction, and i is the ion gyrofreqency. Note that the ion and coherent electron distribtions are compatible with shielding of tearing mode crrent layers by oter tearing eigenmode strctres on neighboring rational srfaces in a finite inetic description. However, parts of these distribtions also * go beyond simple tearing mode physics. The nonadiabatic ions, for example, provide the dissipative electron ion copling that ltimately governs the ambipolar-constrained parts of the flx. Ampère s law and qasinetrality are imposed on Eq. 6 by solving Eqs. 7 and 8 for A, and, and sbstitting into Eq. 6. In the reslting expression, the magnetic drag term, A h e,, becomes a linear combination of the correlations ñ e h e, and J h e,, where J ed 3 vv h e and ñ e d 3 v h e are the incoherent crrent and density terms of Ampère s law and qasinetrality. The magnetic diffsion term, A A,, becomes a linear combination of the d 3 v and d 3 vv moments of the correlations J h e, and ñ e h e,. Soltions of the two-point evoltion eqation indicate 8 that the correlations J h e, and ñ e h e, are ballistic in character, allowing the twotime two-point correlations to be written in terms of one-time two-point correlations, as K h e, v K h e, where K is either ñ e or J. These constraints impose on the emission process and reslting drag force the same resonant condition that characterizes the damping of wave energy and ltimately leads to diffsion. As a conseqence, these constraints prodce a partial cancellation of the diffsion with the drag, considerably simplifying the LBTCI. Ampère s law and qasinetrality are then sed to rewrite the correlations of incoherent flctations in terms of power densities A, and,. The algebraic procedre described above is straightforward bt tedios. Becase it has been detailed elsewhere 9 we do not repeat the steps here. Following the above steps, the LBTCI can be written as e e e i e i Q e D T v L id n D Te L T ni L 0 T e, Tin 10 where, for sae of comparison with experiment, only the magnetic components have been displayed. In this expression, D e) T is the diffsivity of the condctive heat flx of (e (e i) (e i) qasilinear theory and D n and D T describe transport prodced by the momentm and energy exchange between electron clmps and ions as mediated by collective modes. The diffsivity of the condctive heat flx comes from the temperatre gradient-dependent parts of the coherent electron response, and is given by e D e T D, 11 where D b B 0 1 1/, exp 9 1 / and b b 0 ra. The convective part of the sal qasilinear heat flx does not appear in the LBTCI de to the cancellation referred to above. As sggested in Ref., the condctive part of the energy moment srvives the cancellation. However, this is tre only for the perpendiclar energy. Both the condctive and convective parts of the parallel energy moment are canceled by the drag. This otcome contradicts the assertions of Ref.. As noted, the presence of a condctive flx and absence of a convective flx mimics collisional transport, which, for lie particles, yields a zero diffsivity and a nonzero thermal condctivity. 00 Phys. Plasmas, Vol. 3, No. 5, May 1996 Terry et al. Downloaded 07 Jn 006 to Redistribtion sbject to AIP license or copyright, see

5 Assming adiabatic ions ( ), the ion components of Eq. are given by 1 * e1 i e D i n ṽ D exp, e D i T ṽ D exp, v 1 i 1 4 i where ṽ (v dv h e) 1 v dv v h e is the perpendiclar energy of electrons in the incoherent distribtion and the lowest-order finite ion gyroradis corrections have been inclded. The ion components arise from the ion integrals d 3 v F i and d 3 vv F i in qasinetrality and Ampère s law and are ths ambipolar constrained, as indicated by the ion velocity factor in Eq.. The ion terms describe wave-moderated electron ion scattering. Thogh these expressions are nominally convective and condctive, the driving gradients are in the ion distribtion. Therefore, these terms are not convective and condctive in the sal sense electron gradient drives electron heat loss; rather, they are more ain to off-diagonal terms involving gradients of the opposite charge species. For MST, it is possible that is as large as maing the hydrodynamic regime of interest. For i and wea collisions provide the necessary dissipation to prodce transport. In this regime the diffsivities are e D i n e D i T 1/ D 1/ D i ṽ 1 i ṽ * e1 i, i. 16 The condctive heat flx Eq. 14 yields a heat pinch in the deeply adiabatic regime / 1. For in the electron diamagnetic direction, this pinch is smaller than the otward convective flx Eq. 13, maing the net flx otward. In the wealy adiabatic and hydrodynamic regimes / 1, the condctive flx changes sign and becomes otward. The ion components of the electron heat flx are not completely general. In particlar, magnetic drifts grad B and crvatre have not been considered. Evalation of the heat flx Eq. 10 ltimately reqires an assessment of the relative magnitdes of the electron and ion components. This, in trn, reqires a consideration of the spectrm sms in the diffsivities. Becase magnetic flctation-indced transport is prodced by particles streaming along trblent fields, the flx is critically sensitive to spectral variation in. In prior treatments, spectra that are peaed abot 0 with some width have generally been assmed. 16 Sch an assmption is valid near the rational srfaces of the modes that locally dominate the spectrm. It is not valid if the modes that locally dominate the spectrm are resonant at a distant rational srface. For a centrally peaed spectrm, the Rechester Rosenblth heat flx expression is recovered from Eq. 10. This is most easily seen for a spectrm that is flat ot to and zero thereafter. To facilitate evalation, we tae the continos limit and convert the sm over to an integral. From dimensional considerations, b b /, for, where b is the spectrm-averaged magnetic flctation level. With 1/, (e the expression for D e) T becomes e D e T 1/ b /B 0 1 d 4 1/ exp b /B /, 17 when /. Following the same procedre for the sms in Eqs. 13 and 14 then leads to the conclsion that the electron term dominates Eq. 10 by a factor proportional to the ratio ( / ). The resltant transport satisfies a Rechester Rosenblth expression. Consider now a peaed spectrm that is shifted off 0 by an amont 0 /. This spectrm has virtally no power in the locally resonant modes for which 0, bt is dominated by modes resonant at remote rational srfaces. Sch a spectrm cold apply for a limited range of freqencies, one that dominates the heat flx, with flctations otside the range having 0, as reqired to prodce a stochastic field locally. As shown below, this is precisely the sitation in MST. For a shifted spectrm that is flat, with width, the sms in Eqs. 11 and can again be (e carried ot withot difficlty. Evalating D e) T yields e D e 0 / b /B 0 1 T d 0 / 4 1/ exp 1 4 0, 18 b /B / 0 where it has been assmed that all power lies in the adiabatic regime, /( 0 /). The factor ( / 0 ) (1 /4 0 ) arises from the difference in the nmber of resonant particles at the extremes of the spectrm. For this spectrm the magnetic diffsivity for electron motion is smaller than the Rechester Rosenblth expression by the factor / 0. Becase the corresponding factor / 0 also appears in the ion diffsivities, the ion terms are larger than the electron term. These factors reflect the nmber of particles of species j that are resonant with the collective mode. With both species adiabatic, the phase velocity / 0 falls in the bl of both distribtions. Becase the ion distribtion is narrower, there are more resonant ions, and trblent electron ion scattering dominates the electron heat flx. Note that in this case, the heat loss is manifestly ambipolar. IV. DISCUSSION We now comment on Eqs as they pertain to the electron magnetic flctation-indced heat loss in MST. 1 In the edge of MST, where the measrements reported in Sec. II were made, the magnetic flctation spec- Phys. Plasmas, Vol. 3, No. 5, May 1996 Terry et al. 003 Downloaded 07 Jn 006 to Redistribtion sbject to AIP license or copyright, see

6 trm as a fnction of is peaed well away from 0, with little power at 0. Since b is not directly measred, this spectrm strctre mst be inferred from b m,n where m and n are the poloidal and toroidal mode nmbers and from the expression for in the edge of a RFP. Expanding B 0 (r) B 0 in the region of the reversal layer, r,m,n n R sm,n rr 1 db 0, 19 B 0 dr rr s m,n where r s (m,n) is the resonant srface for the m,n helicity. For the global tearing modes that dominate the spectrm, is not close to zero becase rr s is large we are interested primarily in the region r0.8a, while r s m1, n5 8 is deep in the core. The parallel wave nmber does, or corse, approach zero when rr s 0.8a. Using a Bessel fnction model for B 0 (r) and B 0 (r), this reqires n In this range the flctation power is down by two orders of magnitde. More critically, measrements of toroidal mode nmber verss freqency place flctations in this range at freqencies above 100 Hz. The cross-power spectrm Fig. 1 indicates that flctations with freqencies above 0 Hz mae virtally no contribtion to transport. For r0.8a, and with the power concentrated at m1, n6, the spectrm peas in the range 1 m 1 and falls to zero in a narrow band arond this range. Under this circmstance, the spectrm is lie that assmed in deriving Eq. 18. Conseqently, the ion contribtion to the flx dominates the electron component and the heat loss is ambipolar. Assming that the freqencies of the dominant magnetic flctations in the plasma frame are in the range of measred freqencies 10 0 Hz, is somewhat larger than the diamagnetic freqency * e. The convective ion component is therefore modestly larger by a factor of 6 than the condctive component, for comparable density and temperatre gradient scale lengths. Of corse, this conclsion mst be regarded with cation becase the freqency spectrm has rotation-indced Doppler shifts. However, it is qite possible that the freqency of the dominant modes, as measred in the plasma frame at r0.8a where there is little rotation, is given primarily by the rotation rate at the resonant srfaces of the dominant modes. It also appears that the temperatre is weaer than the density gradient, frther favoring the convective component over the condctive component. 3 For the freqency and wave nmber ranges presented above, the phase velocity / 0 is comparable to the ion thermal velocity, maing the factor / 0 close to nity. The ion diffsivity is ths close in magnitde to the Rechester Rosenblth diffsivity, while the flx carries an ion streaming factor. 4 On the basis of inferred spectrm shapes, it is possible to predict the radial dependence of the heat flx in a RFP. Moving inward from the edge, the spectrm shifts toward 0 as the distance to the resonant srfaces of the global tearing modes decreases. Approaching these srfaces, the spectrm first overlaps 0 and then peas at 0. In this process, the heat flx goes from being ambipolar, with the loss rate fixed by an ion thermal velocity, to nonambipolar, with the rate governed by the electron thermal velocity. The magnitde of the heat flx therefore rises dramatically in moving from the edge into the core. This is consistent with observations, which indicate a sharp rise in heat flx as radis decreases. It is also consistent with the rather flat temperatre profiles of the core region and confinement controlled at the edge. These qalitative statements are validated by a more qantitative analysis based on the central temperatre response to sawtooth events, as seen by Thompson scattering. Using a Rechester Rosenblth expression with central vales for the magnetic flctation amplitde, pressre, and gradient scale length, the temperatre response is well fit sing the electron thermal velocity. 17 This discssion may also have some bearing on magnetic flctation-indced transport in toamas. In toamas, it is generally assmed that the spectrm either peas abot zero or has a significant 0 component. However, if the flctation energy is concentrated at low-order srfaces and these are sfficiently separated radially, it is possible that the heat flx maes a partial transition from being dominated by the electron term to being ambipolar constrained. This wold reslt in regions of better confinement interspersed with regions of poor confinement, despite the absence of good flx srfaces. Recent observations on Continos Crrent Toama 18 are consistent with sch an effect. Two points relating to the freqency dependence of the heat flx are also worth maing. The first concerns the Lndqist nmber scaling of the flx. Normally it is assmed that the flx scaling is governed by the scaling of the power spectrm. However, when the spectrm peas away from 0, the flx depends on the freqency. It is conceivable that the freqency has some Lndqist nmber dependence. The second point is to note the isotope scaling of the heat flx. The thermal velocity dependence of Eqs. 15 and 16, in conjnction with velocity mltiplier in Eq. 11, is somewhat nfavorable in its isotope scaling. However, the freqency liely has a favorable isotope scaling based on observed decreases of sawtooth freqencies in going from hydrogen to deterim. The points discssed above represent significant areas of plasible agreement between theory and experiment. This agreement cannot be expected for theories that ignore the nonlinear bnching of electrons. Clearly, more information is needed to determine the freqency of the tearing mode flctations in the plasma frame. This necessarily reqires sorting ot the freqency spectra of the dominant low n modes in relation to sawtoothing, rotation, diamagnetic effects, and aliasing from high n modes. In smmary, measrements of the magnetic flctationindced electron heat flx in MST indicate that it is predominantly convective and its rate is governed by the ion thermal velocity. This reslt does not agree with common theoretical expectations that hold that electron heat loss is not sbject to ambipolar constraints, allowing the loss rate to be governed by the electron thermal velocity and favoring condctive heat loss over convective heat loss. In this paper, we have proposed an explanation for the observations based on a proper acconting of the effects of nonlinear electron bnching. This leads in a natral way to ambipolar constraints. 004 Phys. Plasmas, Vol. 3, No. 5, May 1996 Terry et al. Downloaded 07 Jn 006 to Redistribtion sbject to AIP license or copyright, see

7 Magnetic trblence in MST appears to offer the first experimental evidence for sch constraints in three-dimensional 3-D magnetized plasmas. Theoretical expressions for the heat flx that inclde bnching are consistent with the experimental observations, within experimental ncertainties. ACKNOWLEDGMENT This wor was spported by the U.S. Department of Energy. 1 R. E. Waltz, Phys. Flids 5, A. A. Thol, P. L. Similon, and R. N. Sdan, Phys. Rev. Lett. 59, A. A. Thol, P. L. Similon, and R. N. Sdan, Phys. Plasmas 1, F. W. Perins, in New Ideas in Toama Confinement, edited by M. N. Rosenblth American Institte of Physics, New Yor, 1994, p G. Fisel, S. C. Prager, W. Shen, and M. Stoneing, Phys. Rev. Lett. 7, R. N. Dexter, D. W. Kerst, T. W. Lovell, S. C. Prager, and J. C. Sprott, Fsion Technol. 19, A. B. Rechester and M. N. Rosenblth, Phys. Rev. Lett. 40, T. Botros-Ghali and T. H. Dpree, Phys. Flids 4, P. W. Terry, P. H. Diamond, and T. S. Hahm, Phys. Rev. Lett. 57, ; P. W. Terry, P. H. Diamond, and T. S. Hahm, Phys. Rev. Lett. 60, P. W. Terry and P. H. Diamond, Phys. Flids B, D. J. Den Hartog, A. F. Almagri, J. T. Chapman, H. Ji, S. C. Prager, J. S. Sarff, R. J. Fonc, and C. C. Hegna, Phys. Plasmas, M. R. Stoneing, S. A. Hoin, S. C. Prager, G. Fisel, H. Ji, and K. J. Den Hartog, Phys. Rev. Lett. 73, S. Ortolani and D. D. Schnac, Magnetohydrodynamics of Plasma Relaxation World Scientific, Singapore, Y. L. Ho, Ncl. Fsion 31, J. S. Sarff, S. A. Hoin, H. Ji, S. C. Prager, and C. R. Sovinec, Phys. Rev. Lett. 7, J. A. Krommes, C. Oberman, and R. G. Kleva, J. Plasma Phys. 30, M. Ceic, D. J. Den Hartog, M. R. Stoneing, J. S. Sarff, S. C. Prager, and P. W. Terry, Bll. Am. Phys. Soc. 40, G. Fisel, S. C. Prager, P. Pribyl, R. J. Taylor, and G. R. Tynan, Phys. Rev. Lett. 75, Phys. Plasmas, Vol. 3, No. 5, May 1996 Terry et al. 005 Downloaded 07 Jn 006 to Redistribtion sbject to AIP license or copyright, see