The Hall dynamo effect and nonlinear mode coupling during sawtooth magnetic reconnection

Transcription

1 PHYSICS OF PLASMAS 13, The Hall dynamo effect and nonlinea mode coupling duing sawtooth magnetic econnection W. X. Ding, 1,2 D. L. Bowe, 1,2 B. H. Deng, 1 A. F. Almagi, 2,3 D. Caig, 2,3 G. Fiksel, 2,3 V. Minov, 2,3 S. C. Page, 2,3 J. S. Saff, 2,3 and V. Svidzinski 2,3 1 Electical Engineeing Depatment, Univesity of Califonia at Los Angeles, Los Angeles, Califonia Cente fo Magnetic Self-Oganization in Laboatoy and Astophysical Plasmas, Univesity of Wisconsin-Madison, Madison, Wisconsin Depatment of Physics, Univesity of Wisconsin-Madison, Madison, Wisconsin Received 5 June 2006; accepted 22 Septembe 2006; published online 29 Novembe 2006 Duing magnetic econnection associated with sawtooth activity in a evesed field pinch, we obseve a lage fluctuation-induced Hall electomotive foce, JB/n e e, which is capable of modifying the equilibium cuent. This Hall dynamo effect is detemined in the hot plasma coe by lase Faaday otation which measues equilibium and fluctuating magnetic field and cuent density. We find that the Hall dynamo is stongest when nonlinea mode coupling between thee spatial Fouie modes of the esistive teaing instability is pesent. Mode coupling altes the phase elation between magnetic and cuent density fluctuations fo individual Fouie modes leading to a finite Hall effect. Detailed measuements of the spatial and tempoal dynamics fo the dominant coe esonant mode unde vaious plasma configuations ae descibed poviding evidence egading the oigin of the Hall dynamo Ameican Institute of Physics. DOI: / I. INTRODUCTION Magnetic econnection is an impotant pocess in both laboatoy and natually occuing magnetized plasmas. Pehaps the best known example fo laboatoy plasmas is the tokamak sawtooth cycle fo the m=n=1 teaing mode, 1 whee m and n ae the poloidal and tooidal mode numbes, espectively. A pocess that esembles this sawtooth cycle also occus in some evesed field pinch RFP plasmas. In the RFP, the sawtooth cycle is connected to the magnetic dynamo, o self-geneation of mean cuent and magnetic flux, well known to undelie the self-oganization of RFP and spheomak plasmas. A standad model fo the RFP dynamo is povided by nonlinea esistive magnetohydodynamic MHD theoy. 2 A potion of this model has been expeimentally confimed, but the measuements ae incomplete, due to the complexity of coelating thee dimensional fluctuations of difficult to measue quantities. Magnetic econnection is govened by the genealized Ohm s law, i.e., the electon momentum equation 3 m e J e 2 n e t + E + v B 1 n e e J B + P e n e e = J, 1a whee n e and m e ae the electon density and mass, e is the electon chage, and P e is the electon pessue. J, E, v, B, and ae the plasma cuent density, electic field, ion velocity, magnetic field, and esistivity, espectively. This equation can be witten in the geneal fom 4 E + v B = R, 1b whee R is any nonideal tem due to collisions, paticle inetia, diamagnetic effects, o any fluctuation-induced foce. Some nonideal effects violate the fozen plasma condition and cause beaking of magnetic field lines leading to econnection. Electic field E is a measue of the change of magnetic flux and theefoe is sometimes efeed to as the econnection ate. 5 Detemination of the electic field and nonideal effects ae citical elements equied to undestand magnetic econnection physics. Possible dynamo mechanisms oiginate fom genealized Ohm s law by consideing spatial and possibly tempoal fluctuations in the vaious fields contained in the two-fluid desciption. The paallel mean-field Ohm s law can be constucted to isolate these dynamo mechanisms by decomposing each vaiable quantity in Eq. 1a into mean and fluctuating pats. Taking the ensemble aveage of the component along the mean magnetic field diection yields E + v B J B /n e e = J, whee denotes a fluctuating quantity and denotes a mean o ensemble-aveaged quantity. The inetial tem dj/dt is negligible in the expeiments descibed hee, and all quadatic tems diven by density and electon pessue fluctuations vanish upon ensemble aveaging. We note that mean cuent the ight hand side can be geneated by two fluctuation-induced tems: the MHD dynamo second tem on the left-hand side aising fom the coelation of magnetic and flow velocity fluctuations, and the Hall dynamo thid tem on the left-hand side aising fom the coheent inteaction of magnetic and cuent density fluctuations. It is well established expeimentally that E J in RFP plasmas sustained at constant cuent and flux. To date, the majoity of theoetical and expeimental investigation of the balancing of Ohm s law has focused on the MHD dynamo. It is also notewothy that a simple Ohm s law is also not satisfied in sawtoothing tokamak plasmas in the vicinity of the m=n=1 esonant suface. Wesson 1,6 obseved that the electic field geneated by the econnected helical flux nea the m=n=1 esonant laye is so lage that simple esistive X/2006/1311/112306/11/$ , Ameican Institute of Physics

2 Ding et al. Phys. Plasmas 13, dissipation is inadequate to descibe Joint Euopean Tous JET tokamak plasmas. He suggests that lage electon inetia is able to account fo the imbalance between the electic field and cuent in Ohm s law. Anothe model suggests that stochastization of field lines, esulting fom the inteaction of fundamental m=n=1 helical mode with modes of othe peiodicities, plays an impotant ole in the sawtooth collapse. 7 The possibility fo dynamo effects has not been investigated. In this pape we conside the dynamo pocess as obseved in Madison Symmetic Tous MST RFP plasmas. 8 In MST, the sawtooth cycle is vey distinct and obust, poviding a convenient make fo diagnosing magnetic elaxation effects. Plasmas in othe RFP expeiments exhibit less obust sawtooth behavio fo unknown easons. In the sawtooth cash phase, magnetic flux geneation is paticulaly stong, and thee is a fast global change in the equilibium cuent pofile. Thee ae othe connected effects as well, such as a degadation in enegy and paticle confinement, which ae not discussed hee. Multiple teaing instabilities ae obseved thoughout the sawtooth cycle, attaining thei lagest amplitude just befoe o duing the cash. In the nonlinea esistive MHD model, 9 these teaing modes ceate a vb dynamo which balances Ohm s law. Pobe measuements in the low-tempeatue plasma edge 10 /a0.90 eveal that Ohm s law is indeed balanced by the vb MHD dynamo. It is inteesting to note, howeve, that the MHD dynamo in these expeiments disappeas as the pobes ae moved towad the plasma inteio /a=0.85. It may be impotant that this is also the vicinity of the tooidal field evesal suface whee m=0 modes ae esonant. In the high-tempeatue plasma coe /a0.5, measuements using passive Dopple spectomety eveal significant MHD dynamo associated with m=1 modes, but the adial esolution was limited by the chod-integating natue of the measuement. 11 Active spectoscopy is cuently being developed to localize the coe MHD dynamo measuements. Recent analytical quasilinea calculations suggest that the Hall dynamo may be dominant fo the hot plasma inteio, at least nea a mode-ational suface. 12 Calculations of Hall dynamo fo astophysical applications 13,14 and laboatoy plasma measuements 15,16 also indicate that the Hall dynamo is significant. Theefoe, expeimental measuements of all impotant tems in the genealized Ohm s law with good adial esolution spanning the mino adius ae necessay to undestand a possible composite dynamo pocess that includes both MHD and Hall dynamo physics. In this pape, a fast lase-based Faaday otation diagnostic is employed to measue magnetic field and cuent density changes duing econnection events fo sawtoothing plasmas in MST. Both the time evolution of the pofiles of the mean axisymmetic quantities and the fluctuations of magnetic field and cuent density ae measued. We epot two expeimental esults. Fist, the two-fluid Hall dynamo effect is lage and balances the mean inductive electic field poduced duing the sawtooth cash in the hot plasma coe fo the dominant coe esonant mode. The cuent density pofile and its change duing a sawtooth cash is detemined by a combination of the Hall dynamo and inductive electic field. The Hall dynamo is geneated by teaing instabilities, which ae manifest as seveal spatial Fouie modes. Second, we obseve that the Hall dynamo is stongest when nonlinea coupling between spatial Fouie modes is pesent. Nonlinea coupling altes the phase elationship between cuent and magnetic field fluctuations so as to incease substantially the stength of the Hall dynamo. The two-fluid Hall electomotive foce acts to egulate the plasma electic field and impacts the magnetic econnection pocess. This pape is oganized as follows: in Sec. II, the expeimental technique and analysis methods ae descibed; in Sec. III, detailed expeimental measuements of magnetic fluctuations and Hall dynamo in the coe plasma ae pesented; Sec. IV contains discussion of Hall dynamo elative to MHD dynamo and the ole played by the Hall dynamo in magnetic econnection; and Sec. V contains a summay of expeimental esults and conclusions. II. EXPERIMENTAL TECHNIQUE AND MEASUREMENT METHOD The MST RFP is a tooidal confinement device, like the tokamak, but with tooidal magnetic field compaable to poloidal magnetic field. 8 It has majo adius R o =1.5 m, mino adius a=0.52 m. All data pesented in this pape ae fo deuteium plasmas with dischage cuent I p = ka, line-aveaged electon density n e m 3, evesal paamete F=B a/b 0.22, and electon tempeatue T e T i 300 ev. RFP plasmas have q1 eveywhee within the plasma coss section whee q=/rb /B is the safety facto and ae theefoe susceptible to lage amplitude magnetic fluctuations dominated by teaing instabilities. 2 We focus hee on standad RFP plasmas without cuent pofile contol so that magnetic fluctuations ae lage 1%. A set of 64 equally spaced magnetic coils with poloidal and tooidal sepaation is used to measue magnetic fluctuations at the plasma edge due to esistive teaing modes. The magnetic fluctuation amplitude fo a specified mode numbe m,n is obtained by spatial Fouie tansfomation fom the magnetic coil aay signals b m,n e im+n. Heein, we focus pimaily on the ole of the dominant, coe esonant, 1,6 esistive teaing mode with fequency in the ange khz. The intenal equilibium poloidal magnetic field, magnetic field fluctuations, and cuent density fluctuations ae measued by a fast time esponse 4 s Faaday otation diagnostic 17,18 whee 11 vetical chods sepaation 8 cm simultaneously pobe the plasma coss section. The diagnostic appoach employs two distinct but collinea fa-infaed FIR lase beams to pobe the plasma. The nonpetubing pobe beams ae fequency offset and have counte-otating cicula polaizations. Because of plasma biefingence, each beam expeiences a diffeent value of efactive index upon popagation though the plasma. The diffeence in efactive index is elated to the Faaday otation angle, accoding to the elation

3 The Hall dynamo effect and nonlinea mode coupling Phys. Plasmas 13, = 2 n R n L dz = n e B z dz = c F n e B z dz, 3 whee B z is the component of the magnetic field paallel to the FIR beams, n e is the electon density, 0 is the lase wavelength, z 2 = 2 x 2, x is the impact paamete of pobe beam x=0 coesponds to vacuum vessel cente, and is mino adius, all in MKS units. Refactive indices fo the ight-hand and left-hand ciculaly polaized waves ae denoted by n R and n L, espectively. A multichod intefeomete povides electon density along the same sightlines as the Faaday otation measuement with the measued phase, being given by = 2 n R + n L dz n e dz = c I n e dz. 4 The FIR lase beams ae at nominal wavelength 0 =432 m 694 GHz and opeated at a slight diffeence fequency of /2750 khz. The 11 chod polaimety and intefeomety data ae outinely sampled at 1 MHz with the diffeent fequency being aliased down to 250 khz. This esults in a 4 s time esponse nonaliased time esponse 1 s, which is sufficient to follow the equilibium and fluctuation dynamics of inteest. A digital phase compaato technique is employed to extact the phase infomation fo both the polaimete and intefeomete. 19 Typical polaimete system ms noise levels ae 1 mad 0.05 at 20 khz bandwidth. Combined Faaday otation and intefeomete measuements enable us to infe intenal magnetic field stuctue and its fluctuations. The equilibium tooidal cuent density distibution is obtained fom the measued poloidal field though use of Ampèe s law. Inductive tooidal electic field in the coe plasma, induced by the sawtooth collapse, can also be detemined using Faaday s law and is given by E = V L a 2R t B d, whee V L is the loop voltage measued at the plasma suface and the second tem is obtained by measuement of equilibium dynamics. 20 In the plasma coe, magnetic field diection is lagely in the tooidal diection; theefoe, E E. The electical esistivity has been measued to be m 1 in this egion FIG. 1. Time histoy of plasma dischage cuent and tooidal magnetic field fo a standad MST dischage. Each fast incease of tooidal magnetic field coesponds to a discete sawtooth cash o magnetic econnection event. Fo coe magnetic and cuent density fluctuation measuements, detailed analysis of the phase signals is equied. Since the measued Faaday otation angle depends on both the density and magnetic field, it is necessay to sepaate the two in ode to isolate the fluctuating magnetic field. 22 By ewiting Eq. 3 in tems of the equilibium and fluctuating quantities fo each vaiable e.g., = 0 +, B z =B 0z +B z, n e =n 0 +ñ, the fluctuating pat of the Faaday otation signal becomes = c F B 0z ñ dz + B zn 0 dz, whee the second-ode tem, c F B zñ dz, is negligible because both ñ and B z ae small. Fom this equation we see that the fluctuating pat of the polaimety signal is the sum of the fluctuating electon density weighted by equilibium magnetic field, and the fluctuating magnetic field weighted by equilibium density. Fo the six chods neaest the magnetic axis, the B 0z ñ dz tem is negligible and c F B zn 0 dz. This holds tue fo two easons: 1 the component of the equilibium poloidal field paallel to the cental chods B 0z is small, and 2 ñ dz 0 fo the measued fluctuations since m=1. 23 By using measued values fo equilibium poloidal magnetic field and electon density fluctuations, it can be shown that B 0z ñ dz0.05, which is less than the noise level. Finite contibutions fom the tooidal magnetic field to the Faaday signal esulting fom misalignment have also been consideed and found to be negligible. Futhe details on how the local magnetic and cuent density fluctuation infomation is obtained will be given in Sec. III C. III. EXPERIMENTAL RESULTS A. Equilibium cuent density pofile dynamics Many paametes in MST plasmas display a sawtooth cycle due to elaxation oscillations in the plasma coe. A typical MST dischage histoy is shown in Fig. 1. Sawtooth cashes, chaacteized by changes in tooidal flux, ae denoted by pompt inceases in the aveage tooidal field B. To detemine the time evolution of mean magnetic suface aveaged quantities, we ensemble aveage measued quanti- 6

4 Ding et al. Phys. Plasmas 13, FIG. 2. Ensemble aveaged sawtooth dynamics of a coe plasma soft-x-ay emission and b aveage tooidal magnetic field duing sawtooth cycle. Time t=0 denotes sawtooth cash. FIG. 3. Dynamics of a equilibium paallel cuent density pofile and b inductive electic field ove sawtooth cash. ties ove many epoducible sawtooth cycles, theeby eliminating the contibution of fluctuations. Fo example, ensemble aveaged soft-x-ay emission ove the sawtooth cash is plotted in Fig. 2a, which indicates an electon tempeatue dop in the plasma coe at the cash. Enegy is suddenly eleased within appoximately s afte the stable, slowly amping 2 3 ms heating phase. This cycle is vey simila to sawtooth phenomena obseved in tokamaks and othe tooidal fusion devices. At the cash, one sees an incease in aveage tooidal magnetic flux as shown in Fig. 2b whee t=0 denotes time of the cash. The pompt ise in B indicates that the RFP plasma itself must self-geneate tooidal magnetic flux dynamo effect since thee is no extenally applied poloidal electic field. It should be noted that the total magnetic enegy is educed afte a sawtooth cash even though the tooidal flux is inceased. Fast tempoal dynamics of the paallel cuent density pofile J,t ae detemined by measuing magnetic field using Faaday otation fo poloidal field, motional Stak effect fo tooidal field and extenal magnetic coils. 18,20 As shown in Fig. 3a, cuent density pofile flattening duing the sawtooth collapse equies vey fast s tanspot fom the plasma coe to the edge. This cuent tanspot timescale is much faste than a esistive time 100 ms fo MST. The induced electic field nea the magnetic axis associated with the sudden change of poloidal magnetic flux duing elaxation is shown in Fig. 3b. The foce associated with E,t40 V/m is fa geate than the collision foce J1 V/m. Moeove, although the enhanced electic field is in the same diection as the tooidal cuent density, J 0 actually dops about 20% at the cash. Thus anothe foce must be acting on the electons to balance the electic field. Phenomenologically, one could attibute this effect to enhanced anomalous esistivity in the coe. Howeve, in the plasma edge the inductive electic field is pependicula o even opposite to the cuent diection, making anomalous esistivity insufficient to balance the electic field. Theefoe, a physical mechanism fo the self-geneation of cuent is equied and needs to be expeimentally identified. A magnetic fluctuation-induced electomotive foce, eithe the MHD o Hall dynamo as shown in paallel Ohm s law see Eq. 2, is believed necessay to balance the electic field and edistibute the mean cuent duing a elaxation event. B. Paallel component of Hall dynamo Befoe poceeding to descibe measuements of the Hall dynamo effect in MST, it is necessay to fist expess the paallel component in a fom most suitable fo expeimental detemination. The poloidal and tooidal components in cylindical coodinates ae J B = j b j b, J B = j b j b, 7a 7b whee =1/ dd denotes a magnetic suface aveage. These two components can be simplified by using B=0, B= 0 J, whee ib l b l =1/ ib l b l dd=0, l=,,, / im,, / in fo a flux suface aveage. This yields

5 The Hall dynamo effect and nonlinea mode coupling Phys. Plasmas 13, J B = 0 j b j b = 1 Similaly, we obtain b 1 b b 1 1 =b b in R b ir 1 n 1 =b 0 J B = 0 j b j b = 1 b 1 b b. b 1 b b R b R m n b b 1 b b R 1 b R b = m ib ir n = R m 1 nb = R b R m n b 1 +b b + 1 m nb b R m ir n nb 1 b b b R m n b b b. 8a 8b The paallel component of Hall dynamo is defined as J B n e e J B = n e e B B = 1 B B n e ej B + J B B. B This expession can be futhe simplified in tems of b,b in the vicinity of the esonant suface = s by using the elation k B=m/ s B +n/rb =0, leading to J B = A 1 n e e n e eb b A 2 n e e 1 b b + B b b 1 B n e e A j b 1, 10 n e e whee A 1 = B B1+ mr n s A 2 = B B1 mr n s 2 = B 2 = B B1+ B B1 B B 2, B 2. B, B, and B ae the known equilibium tooidal, poloidal, and total magnetic field. The tem A 2 n e e 1 b b is found to be small in expeiments as will be shown late in Sec. III C whee b s 0 and b /b / fo teaing modes. In addition, 9 B b b 1 B n e e A 1 n e eb b nea the esonant suface see Eq. 10. To see this, we pefom a Taylo expansion nea esonant suface giving b b s = b b s b s b, whee s s 1 and b s =0. Theefoe, A 1 j b /n e e is the dominant tem in Eq. 10 and is used to detemine the Hall dynamo nea the esonant suface. C. Hall dynamo measuement To investigate the ole of the Hall dynamo J B /n e e on the mean cuent density pofile, we equie measuement of 1 the local cuent density fluctuation j ; 2 local magnetic field fluctuations b and b ; and 3 thei coheent inteaction o coelation. In the following, each of these measuements will be discussed. Fist, local cuent density fluctuations ae measued diectly by Faaday otation. It has peviously been established 22 that the cuent fluctuation between polaimete chods can be descibed by I x 1 B dl x 2 B dl = /c F n e 0, 11 whee =c F n e B dlc F n eb dl is the fluctuating Faaday otation signal, x 1 and x 2 ae impact paametes of the selected chod pai, and n e is the mean electon density.

6 Ding et al. Phys. Plasmas 13, FIG. 5. Faaday otation fluctuation amplitude fo diffeent chods. Cicles epesent measued Faaday otation fluctuation. Solid line is fitting esult. FIG. 4. Cuent density fluctuation spatial pofile fo 1,6 mode. Fluctuations peak at esonant suface and evese sign coss magnetic axis. Teating the density as constant and emoving it fom the integal is valid fo MST pofiles that ae athe flat. 23 Eos intoduced by this assumption ae 10%. The above equation then epesents a staightfowad application of Ampèe s law to the loop defined by adjacent Faaday otation chods and holds tue fo the cental six polaimete chods whee the density fluctuations ae negligible due to the small density gadient in the coe and the m=1 natue of the petubation, as discussed ealie. This line-aveaged measue of the cuent fluctuation I can then be inveted to obtain the local cuent density petubation j, though application of an asymmetic Abel invesion technique. 24 Since the measued helical cuent fluctuation is localized nea the esonant suface, the local cuent density fluctuation j can be diectly measued by the pai of chods neaest the esonant suface, as shown in Fig. 4. Itis detemined that the cuent density fluctuation 1 peaks at the esonant suface, 2 eaches an amplitude j /J 0 5% 6% at the sawtooth cash, and 3 petubation adial extension 8 cm. Results shown ae fo the dominant, coe esonant, 1,6 teaing mode. Cuent density fluctuation dynamics ae diectly measued, theeby pemitting one to coelate local cuent fluctuations to global magnetic fluctuations. Second, the local magnetic fluctuation pofile can be obtained by integating the cuent fluctuation distibution. Howeve, to detemine moe pecisely the magnetic and cuent density fluctuation pofiles, we have developed a simple fitting outine whee it is assumed that the esonant cuent density fluctuation pofile has the fom j = j a exp s /w 2, with j a amplitude, s suface location, and w width seving as fee paametes. Each paamete is quantified by making a best fit to the measued fluctuating Faaday otation pofiles. Once the fluctuating cuent density distibution is identified, the magnetic fluctuation spatial pofile can be obtained by using B= 0 J, B=0, and J=0 in a cylindical geomety. Detemination of the magnetic fluctuation spatial pofile fo a specific mode m,n, is accomplished by integating the above equations assuming j j and using the bounday condition b a=0. Theefoe, a modeled Faaday otation fluctuation fo a specified cuent petubation is descibed by M j a, s,w = c F n e b sin + b cos dz and can be constucted fo each chod by minimizing 6 2 = i i M j a, s,w b M a b a 2 i=1 i 2 with espect to the fee paametes j a, s, and w. Fo this expession, i and ae measuement eos, b a is measued by magnetic coils mounted inside the vessel, and b M a is a modeled value. In the minimization pocedue we specify j a, s, and w and obtain both the magnetic field and cuent density fluctuation pofiles. The measued Faaday otation fluctuation pofile and best fit esult ae shown in Fig. 5. Details of the analysis pocedue have been peviously descibed. 24 The esulting magnetic field and cuent density fluctuation spatial pofiles fo the coe esonant 1,6 mode ae shown in Fig. 6. Uncetainty due to the fitting pocedue and expeimental eo is also estimated. Radial magnetic field fluctuations ae obseved to extend continuously though the ational suface indicating thei esistive natue. Poloidal FIG. 6. a Radial and poloidal magnetic fluctuation spatial pofile fo dominant 1,6 mode. b Coesponding cuent density fluctuation spatial pofile.

7 The Hall dynamo effect and nonlinea mode coupling Phys. Plasmas 13, FIG. 8. Hall dynamo spatial pofile fo 1,6 mode showing a peak at the esonant suface. FIG. 7. a Dynamics of Hall dynamo solid line and inductive electic field dashed line duing magnetic elaxation event. Time t=0 denotes the sawtooth cash. Data have been ensemble aveaged ove 380 independent sawtooth events. b Phase diffeence between cuent and magnetic field fluctuations and c cuent density fluctuations dynamics ove sawtooth cash. magnetic fluctuations change sign acoss the esonant suface, in qualitative ageement with pevious measuements by pobes in OHTE and ZT and MHD computation. 2,9 A maximum in the cuent density fluctuation j /J 0 4.5% occus at s =17 cm whee the 1,6 mode esonant suface is located based on equilibium magnetic field measuements. The fluctuating cuent channel adial width is appoximately 8±3 cm. Amplitude, adial width, and location ae all consistent with the diect cuent density fluctuation measuement discussed ealie in this section. Nea the esonant suface, measuements show b s 0, b / /b /, and B B, justifying the simplification made in Eq. 10, whee the Hall dynamo paallel component was deived. The cuent sheet width is geate than the esistive MHD laye 0.2 cm and ion acoustic gyoadius s =2.0 cm. Howeve, it is found to be compaable to the magnetic island width, and ion skin depth, c/ pi 10 cm in MST. Although the econnection cuent laye width exceeds the simple linea MHD estimate, measued pofiles ae consistent with the teaing mode expectation that the cuent petubation is local and magnetic petubation is global. Finally, the phase between j and b can be obtained by ensemble aveaging. In MST, otation of the low-n magnetic modes tansfes thei spatial stuctue in the plasma fame into a tempoal evolution in the laboatoy fame. Since the magnetic modes ae global, fo convenience we coelate j to a specific helical magnetic mode obtained fom spatial Fouie decomposition of measuements fom 64 wall-mounted magnetic coils. Afte aveaging ove an ensemble of simila events, we can diectly detemine the phase between j s and b a fo the specified mode. Since 1 b a and b a have a fixed phase diffeence of /2 at the edge whee j =0 and 2 the adial magnetic petubation is expected to have a constant phase at all adii fo teaing modes 2,12 which has been veified by pobe measuements in lowe tempeatue plasmas, we ae able to detemine the phase between j s and b s. Using this infomation, along with the magnitude of b s fom Fig. 6, wenow evaluate j s b to detemine the Hall effect. The two-fluid Hall effect is measued to be nonzeo with stong tempoal dynamics. As shown in Fig. 7a, the Hall dynamo begins inceasing damatically immediately pio to the sawtooth cash, eaching 50±10 V/m, as shown by the solid line. This occus because the phase diffeence between the fluctuating cuent and magnetic field deviates fom 90 by 10 and the cuent density as well as magnetic field fluctuations incease, as shown in Figs. 7b and 7c. Away fom the sawtooth cash, the Hall electic field aveaged ove time window 2 to 1 ms is elatively small. At this time, the cuent density fluctuation has a nea 90 phase diffeence with adial magnetic field fluctuation, making the cosine of the phase small. The change in phase duing the sawtooth cash is pimaily ascibed to nonlinea mode coupling as will be discussed in Sec. III D. The fluctuating cuent j peaks at the esonant suface and hence the Hall dynamo has a maximum thee as well. As shown in Fig. 8, the measued width is 8 cm. This is estimated by using Eq. 10 and assuming the cuent fluctuation phase is spatially constant. The estimated width of the Hall dynamo is subject to faily lage uncetainties ±3 cm, since Eq. 10 is only valid nea the esonant suface. Nevetheless, the Hall dynamo is maximum at the esonant suface and an additional dynamo mechanism may be equied to balance the induced electic field elsewhee. This is especially tue in the coe whee the spacing between adjacent low ode atio-

8 Ding et al. Phys. Plasmas 13, nal sufaces is lagest. Such a pictue is qualitatively consistent with quasilinea theoy 12 which pedicts a localized Hall dynamo and a moe diffuse MHD dynamo away fom the esonant suface. Howeve, the measued Hall dynamo width is much geate than that pedicted, even when taking the measuement uncetainty into account. These esults imply that nonlinea dynamics play an impotant ole. The Hall dynamo effect is only measued in one location /a=0.35 coesponding to the 1,6 mode esonant suface. Contibutions fom othe modes i.e., 1,7, 1,8, to Hall dynamo at this location is negligible because cuent density fluctuations fom these modes ae measued to be small at /a=0.35. The Hall dynamo in the coe is found to oppose the mean cuent. Fom Fig. 7, it is obseved that the lage Hall electomotive foce is compaable to the induced electic field dashed line and acts to suppess equilibium cuent duing plasma elaxation. The esistive foce J is small compaed to both the induced electic field and the Hall dynamo duing the sawtooth cash. This diect measuement of a substantial Hall dynamo in a high-tempeatue plasma indicates that two-fluid effects ae necessay to undestand plasma elaxation. D. Nonlinea mode-mode inteaction Fo the Hall dynamo to be nonzeo, the phase between the cuent density and magnetic field fluctuations fo a contibuting mode must diffe fom /2. In MST, we obseve that the phase deviates significantly fom /2 only when thee-wave coupling is stong. That is, it appeas that the phase between cuent and magnetic field fo a specific mode is alteed if that mode is nonlinealy coupled to othe modes, and the alteation is such as to stengthen the Hall dynamo. We daw this conclusion by compaing the Hall dynamo in standad RFP plasmas to plasmas in which the dominant thee-wave inteactions ae eliminated. This effect of nonlinea mode coupling is simila to ecent esults fom MHD computation. 9 Expeimentally obseved phase deviation fom /2 see Fig. 7b duing a sawtooth cash implies that the Hall dynamo is enhanced. The phase change most likely oiginates fom nonlinea mode coupling in MST plasmas since magnetic fluctuations can inteact with the eddy cuents geneated by global magnetic field petubations, simila to the effect of an eo field 26 o imposed bounday. 27 The measued safety facto at the magnetic axis is q00.2, monotonically deceasing at lage mino adii. It passes though zeo at the evesal suface, being negative at the edge, as shown in Fig. 9. Fom the measued q pofile, esonant lown n=6 is the esonant mode closest to the magnetic axis, m=1 magnetic modes will dominate the coe magnetic fluctuation wave numbe spectum as is fequently obseved. In addition to the coe esonant modes, m=0 n=1,2,... modes ae esonant at the evesal suface. Both the m=1 and m=0 teaing modes have a global natue so that nonlinea mode coupling is common The thee-wave inteaction has to satisfy the sum ule FIG. 9. The safety facto q pofile 0.25 ms befoe sawtooth cash. The m=1 esonant sufaces and island width ae indicated by solid line and the m=0 suface by aow. Solid cicles epesent the measued q pofile. Mode esonant sufaces ae densely packed esulting in stong mode-mode inteaction. m 1 ± m 2 = m 3 and n 1 ± n 2 = n The coupling of two adjacent m=1 modes via inteaction with an m=0 mode has been shown to be vey impotant in both expeiments and MHD computation. A typical stong thee-wave inteaction obseved in MST plasmas is that between the 1,6, 1,7, and 0,1 modes. The suppession of one mode is expected to esult in a damatic eduction in the nonlinea mode coupling. In ode to identify the ole played by nonlinea coupling in the Hall dynamo duing the sawtooth cash, we compae standad RFP plasmas with those whee the evesal suface and the q=0 esonance suface has been emoved i.e., nonevesed MST plasmas whee the evesal suface is moved beyond o located at =a. Fo evesed plasmas, the m=1, n =5,6,7,... mode amplitudes duing the sawtooth cycle ae shown in Fig. 10a, and found to be compaable to the nonevesed case, as shown in Fig. 10b. This indicates that the m = 1 mode does not decease significantly fo non-evesed plasmas. Howeve, the m = 0 mode amplitude is significantly educed see Fig. 10c since its esonant suface is emoved. Fo nonevesed plasmas, measuements eveal the Hall dynamo effect is educed fivefold compaed to standad RFP plasmas and is obseved to peak at 10 V/m, as shown in Fig. 11a. The phase between the localized cuent density fluctuations and global magnetic fluctuation fo the 1,6 mode is also alteed as shown in Fig. 11b. Fom these esults it is appaent that to have the phase diffe significantly fom 90, substantial m = 0 activity is needed, implying that the change in 1,6 mode phase between j and b occus due to nonlinea coupling. IV. DISCUSSION The measued two-fluid Hall effect and the ole it plays in the plasma elaxation pocess povide expeimental evidence fo a new dynamo mechanism as well as insight into magnetic econnection in RFP plasmas. Howeve, the dynamo pictue fo the RFP emains incomplete. In this section,

9 The Hall dynamo effect and nonlinea mode coupling Phys. Plasmas 13, FIG. 11. a Hall dynamo and inducted electic field E and b phase between cuent and magnetic field fluctuations, fo nonevesed F=0, educed m=0 mode plasmas. FIG. 10. Colo Tempoal dynamics of coe esonant m=1 modes ove sawtooth cycle fo a standad plasma F= 0.22 and b nonevesed plasma F=0; andc m=0 mode amplitude fo standad F= 0.22 and nonevesed F=0 plasmas. Cash denoted by t=0. we biefly discuss a few issues elated to the Hall dynamo and MHD dynamo along with the ole of Hall dynamo in magnetic econnection. A. Hall and MHD dynamos In MST, the imbalance between inductive electic field and cuent exists ove nealy the entie plasma column. 31 It is pedicted that vaious dynamo mechanisms may be playing a ole in poviding the equied balance. Both Hall dynamo and MHD dynamo have been identified expeimentally in MST with the elative weighting of these two effects thoughout the plasma coss section being a topic of geat inteest. As mentioned in the Intoduction, pobes inseted at the plasma edge have measued a significant MHD dynamo fo /a0.90. The MHD dynamo is balanced by local electic field, consistent with RFP MHD computation. 9 Howeve, at the innemost pobe position /a=0.85, the effect is no longe obseved. 10 Pevious Hall dynamo measuements by magnetic pobes show it to be elatively small in the edge egion /a=0.92, contibuting 25% to total electic field, but becoming significant futhe inwad at /a= These measuements eveal a spatial tend opposite that of the MHD dynamo. In the plasma coe, it is obseved that the Hall dynamo dominates nea the esonant suface fo the 1,6 mode /a=0.35, as shown in Fig. 7. Fo this egion the Hall dynamo alone is sufficient to balance the electic field, theeby implying the MHD dynamo is small. To date, Hall dynamo measuements have not been made in the egion 0.5/a0.8, due to lage density fluctuations which complicate extaction of magnetic field fluctuation infomation fom the Faaday otation measuements. Wok in this aea is actively being pusued. Based on measuements at the 1,6 mode esonant suface, it seems easonable to expect that a simila Hall dynamo should occu at the 1,7, 1,8, 1,9, etc., esonant sufaces as well. Since these esonant sufaces become moe closely spaced with inceasing tooidal mode numbe see Fig. 9, the Hall dynamo may essentially become quasicontinuous in this egion. This may also hold tue in the egion 0.35/a0.5, whee spacing between the 1,6 and 1,7 mode esonant sufaces is 5 cm and compaable to the Hall dynamo width shown in Fig. 8. Howeve, magnetic stochasticity may also be playing a ole theeby pointing out the need fo a diect measuement at othe esonant sufaces. Pevious line-aveaged MHD dynamo measuements 11 by Dopple spectomety indicate a nonzeo MHD dynamo in the coe but the spatial distibution emains unesolved. An inteesting question is whethe the MHD dynamo plays any ole at the esonant suface location whee the Hall dynamo effect is maximum. An active MHD dynamo at the esonant suface equies a finite coelation between adial velocity fluctuations and magnetic fluctuations. To see this, we expess the MHD dynamo paallel component at the esonant suface in cylindical coodinates: B v B =v B B + v B, 13 B whee vb =v b v b, vb =v b v b.ifv does not coelate with magnetic fluctuations, Eq. 13 can be ewitten as B

10 Ding et al. Phys. Plasmas 13, v B = v b B B v b B B = 1 m B k Bv b, 14 whee v=0 is used. Howeve, at the esonant suface, k B= 0 making the MHD dynamo vanish. Finite coelation between adial velocity fluctuations and magnetic fluctuations is theefoe equied fo the MHD dynamo to be active at the esonant suface. To date, measuements have not obseved a finite coelation between adial velocity fluctuations and magnetic fluctuations 11 implying vb vanishes at the esonant suface. By combining measuements of Hall dynamo and MHD dynamo, both edge and coe, a pictue emeges whee the Hall effect dominates in the vicinity of the esonant suface wheeas the MHD dynamo may be active elsewhee. This expeimental pictue is simila to quasilinea theoy 12 which shows the Hall dynamo dominating at the esonant suface whee ions and electon ae decoupled, but MHD dynamo active elsewhee. New measuements of local ion velocity fluctuations by chage exchange ecombination spectoscopy ae being developed on MST to exploe the detailed spatial pofile of the MHD dynamo. 32 Meanwhile, electon velocity measuement by the elativistic Fizeau effect is also being developed on MST and potentially povides an electon dynamo measuement combination of MHD and Hall dynamo in the plasma coe. 33 B. Reconnection time in MST plasmas The magnetic econnection time fo typical MST plasmas is about 100 s. Sawtooth econnection in the RFP shaes many simila featues with the tokamak. Fast econnection occus nea a laye fomed at the esonant suface and poloidal flux is emoved duing the econnection pocess. Kadomtsev intoduced a magnetic econnection model fo the tokamak sawtooth cash by consideing finite esistivity effects duing econnection. 34,35 This is simila to the Sweet-Pake model, 4 whee a collision tem in paallel Ohm s law is included, giving E + v B * = J, 15 whee B * is the helical component of the magnetic field. Late, Wesson 4 intoduced an electon inetial tem into Eq. 15 to explain fast econnection fo lage tokamaks. Accoding to Ohm s law, the Kadomtsev econnection time is K R A, 16 whee R the is esistive time and A is Alfvén time. Fo typical MST paametes, we find A 1 s, and R 0 s 2 /200 ms. The computed K 450 s fo MST plasmas is much slowe than expeimental obsevations 100 s. Based on esults povided in this pape, this diffeence is not supising since paallel Ohm s law nea the esonant suface on MST is essentially balanced by the twofluid Hall dynamo effect associated with nonlinea modemode coupling athe than a esistive collision tem. We can heuistically teat the Hall dynamo as an anomalous esistivity in Ohm s law Eq. 15 given by * = J B ne J The time-aveaged Hall dynamo amplitude duing econnection is about 20 V/m, and the equilibium cuent density J 2.2 MA/m 2 leading to an anomalous esistive time * R / * 4.4 ms, and an estimated econnection time * R A 66 s. This esult is faste than the Kadomtsev econnection time and much close to expeimental obsevation. It should be pointed out that magnetic econnection in MST is much moe complicated in eality. Magnetic econnection simultaneously occus at diffeent locations i.e., diffeent esonant sufaces and the global inteaction of magnetic econnection events egulates the magnetic econnection time. Pope teatment may equie a fully nonlinea two-fluid MHD computation. V. CONCLUSION In summay, time-esolved obsevations of cuent density fluctuations and magnetic field fluctuations in a hightempeatue collisionless plasma have been made by using a fast, lase-based, Faaday otation diagnostic. The amplitude, tempoal dynamics, spatial distibution, and phase elation between cuent density and magnetic fluctuations fo the dominant, coe esonant, 1,6 mode have been measued. These electomagnetic fluctuations have chaacteistics consistent with expectations fo esistive teaing modes. The Hall electomotive foce along the mean magnetic field diection, JB /n e e, is also detemined and found to be significant nea the mode esonant suface. The Hall dynamo acts to educe mean cuent density in the coe and balances the electic field induced at a sawtooth cash. Expeiments demonstate that nonlinea mode-mode coupling is an essential ingedient to geneate a significant Hall dynamo. Nonlinea mode coupling altes the phase elation between the cuent density and magnetic field fluctuations with significant phase deviation fom /2 occuing only when thee-wave coupling is stong. The Hall effect dominates in the vicinity of the esonant suface wheeas the MHD dynamo may be active elsewhee. ACKNOWLEDGMENTS The authos acknowledge contibutions fom all membes of the MST goup. In paticula, we acknowledge Pofesso C. Hegna fo helpful discussion of Hall dynamo paallel component, and Pofesso C. Foest and D. J. Andeson who gave suppot fo MST equilibium econstuction. This wok is suppoted by the U.S. Depatment of Enegy and the National Science Foundation. 1 J. Wesson, Tokamaks Oxfod Univesity Pess, New Yok, S. Otolani and D. D. Schnack, Magnetohydodynamics of Plasma Relaxation Wold Scientific, Singapoe, L. Spitze, J., Physics of Fully Ionized Gases, 2nd ed. Intescience, New Yok, 1962, p E. Piest and T. Fobes, Magnetic Reconnection Cambidge Univesity Pess, New Yok, 2000.

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