Bondeson, A.

Hubbard, A.E.

Hutchinson, I.H.

In, Y.

Porkolab, M.

Ramos, J.J.

Snipes, J.

Wolfe, S.

DSpace@MIT

October 10, 2000n=1 RESISTIVE MODESin REVERSED MAGNETIC SHEARALCATOR C-MOD PLASMASY. IN, J.J.RAMOS, A.E.HUBBARD, I.H.HUTCHINSON,M. PORKOLAB, J. SNIPES, S. WOLFE, A. BONDESON†MIT Plasma Science and Fusion CenterCambridge, Massachusetts 02139United States of America†Chalmers University, Go¨teborg, SwedenDuring current ramp-up discharges with early auxiliary RF heating, re-versed magnetic shear proﬁles have been obtained on Alcator C-Mod. MHDoscillations are often observed on ECE diagnostics, as well as magnetics.Although no direct q-proﬁle measurement was available, a hollow currentdensity proﬁle with qmin=2.5 was reconstructed by EFIT. The location ofthe biggest electron temperature ﬂuctuations was close to the outer q = 3rational surface. Based on a resistive linear stability code (MARS), the MHDoscillations were identiﬁed as n=1 resistive “multiple” tearing modes. Sincethe pressure proﬁle was hollow and q > 1 in the core, a resistive interchangemode was also predicted to be unstable at the inner q = 4 rational surface,despite the low βN .1I INTRODUCTION 2I. IntroductionReversed magnetic shear proﬁles are predicted to give better perfor-mance1 in terms of conﬁnement, stability and self-sustaining bootstrap cur-rents. Many experiments have since veriﬁed such predictions in various toka-maks.2–4 In particular, in the negative central magnetic shear (NCS) region,the plasma shows excellent stability to n = ∞ ideal magnetohydrodynamic(MHD) ballooning modes.5 However, when reversed magnetic shear plasmasare accompanied by MHD ﬂuctuations, plasma performance is reduced. Un-derstanding the characteristics of the MHD activity associated with reversedmagnetic shear plasmas is required prior to adopting reversed magnetic shearoperation as an advanced tokamak operation scenario.6,7A resistive MHD mode speciﬁc to reversed magnetic shear plasmas is a resis-tive double tearing mode, which can be destabilized when a pair of rationalsurfaces with the same q are close to each other. Since it is a global mode, itcan be observed in experiments rather easily. For a single tearing mode,8 ina classical tearing theory, ∆′> 0, where ∆′ ≡ (ψ′/ψ)|rs+w/2rs−w/2, is an unstablecondition with magnetic island width (w) centered at r=rs. In a neoclassicaltheory, even negative ∆′can be unstable in that transport and polarizationterms should be also considered .9 On the other hand, for a double tearingmode,10–14 we do not have a similarly simple analytic criterion.In addition, a pressure gradient driven resistive interchange mode might beobserved near a rational surface in the NCS region. Since the resistive inter-change mode is localized close to a resonant layer, it is not easy to observe inexperiments unless other MHD modes are suppressed. Although resistive in-terchange modes in a tokamak were predicted in theory15 as early as resistivetearing modes, the experimental observation was only recently reported.16A reversed magnetic shear conﬁguration is generally created during currentII EXPERIMENTS 3ramp discharges because the current diﬀusion time (τR = µ0a2/η where µ0:vacuum permeability [4π×10−7 H/m], a: scale length, η: resistivity) is muchlonger than the plasma current rise time. If additional heating is launchedat the early current ramp stage, it retards the current penetration further asτR is proportional to T3/2e , which helps to keep the reversed magnetic shearproﬁles. To achieve high performance reversed magnetic shear plasmas inAlcator C-Mod,17 ICRF heating is added to early ohmic plasmas as its auxil-iary power, unlike most other tokamaks whose main power source is neutralbeam injection (NBI).II. ExperimentsA. DiagnosticsThe Alcator C-Mod is a high ﬁeld (BT = 5∼8 T), compact (R ∼ 0.67m, a ∼ 0.22 m), diverted tokamak. Poloidal magnetic ﬁeld pickup loops inAlcator C-Mod are used to measure MHD ﬂuctuations, which are usuallydigitized at a rate of 1 MHz. For these studies, we use four sets of toroidallydisplaced coils located on outboard limiters. Two sets are composed of 7 coilsand the others of 4 coils. In addition, there are some inboard coils whichare located closer to the plasma, which are useful for determining poloidalmode numbers. For electron temperature measurements, Thomson scatteringand ECE (electron cyclotron emission) diagnostics, including a Michelsoninterferometer, radiometer and grating polychromator (GPC), are installedon Alcator C-Mod. The Michelson interferometer is absolutely calibrated;all the other ECE temperature measurement diagnostics are cross-calibratedon the basis of the Michelson calibration. For the purpose of observing MHDactivity on electron temperature measurements, the GPC was chosen as amain diagnostic because it provides good spatial resolution (∼ 1 cm) andII EXPERIMENTS 4temporal response (up to 2 µsec). The GPC is based on the extraordinarymode (X-mode) of emission at the second harmonic of the electron cyclotronfrequency.B. MHD observationsFor Alcator C-Mod, a typical current ramp up rate is 5.0 MA/sec up to0.1 sec. During the experiment discussed here, the current ramp up ratewas initially 5.1 MA/sec and then was reduced to 1.5 MA/sec until thetarget current Ip=0.8 MA was reached. As shown in Fig. 1, there were shorttrips during the very early ICRF heating phase due to poor coupling tothe plasma. The power was up to 1.5 MW during the period of the MHDactivity, which was observed not only on magnetics, but also on GPC withthe same coherent frequencies. The toroidal mode numbers (n) of all theMHD ﬂuctuations on magnetics were identiﬁed as n = 1, while their poloidalmode numbers (m) decreased from 7 to 5 until the big MHD ﬂuctuationswere observed on Channel 6 of GPC (R ∼ 0.8 m) around t = 0.11 sec (SeeFig. 2). Using the EFIT18 program, the equilibrium associated with theMHD activity was reconstructed. An internal pressure proﬁle was used as aconstraint for EFIT, in addition to external magnetics data. Fig. 3 shows theevolution of the temperature ﬂuctuations (∆T ≡ (Tmax − Tmin)/2) aroundt=108 msec, while Fig. 4 shows the q-proﬁle reconstructed by EFIT at t=108msec.For the pressure proﬁle, Ti(e), ni(e) (i(e): ion (electron)) are needed. Theelectron temperatures (Te) were given by GPC, while the ion temperature(Ti) is inferred based on a simple equilibration model which assumes that theelectrons and ions are thermally equilibrated and that the ion conﬁnementtime is constant over the proﬁle. The ion density proﬁle (ni(R)) was assumedIII INTERPRETATION 5to be the same as ne(R). The densities (ne) were inferred from Abel-invertedvisible Bremsstrahlung from a multichord diagnostic whose view is horizon-tal and near the midplane.19,20 The brightness of the visible Bremsstrahlungdiagnostic is a function of Zeff , ne, and Te. When the electron temperatures(Te) are provided from GPC and Zeff is assumed to be a constant, ne can beinferred. As a note, the inverted density proﬁle obtained from far-infrared(FIR) interferometry is regarded as less reliable for this analysis. This is dueprimarily to the vertical view of the diagnostic, which passes cold, relativelydense plasma in the vacuum chamber above and below the main plasma.Since the contributions of the edge plasmas to the FIR line integrated den-sity measurements are not separated from those of the main plasma, theinversion process is sensitive to rather arbitrarily assumed edge plasma den-sities. For this reason, the horizontal visible Bremsstrahlung diagnostic waspreferred for density measurement. Therefore, the pressure proﬁle is givenby P = neTe+∑niTi ∼ ne(Te+Ti) ≤ 2neTe, where ne is inferred from visibleBremsstrahlung brightness, while Te is measured by GPC.The dashed curve of Fig. 5 is the pressure proﬁle calculated in the afore-mentioned way. Based on this internal pressure proﬁle, as well as magnetics,a most likely reversed magnetic shear conﬁguration has been reconstructedduring the current ramp stage. Based on the reconstructed equilibrium, theMHD stability at t=108 msec will be discussed.III. InterpretationFor stability analysis, the MARS code21 has been used. This is a linearresistive MHD stability code with a given toroidal mode number and we con-sider n = 1, which has been ascertained in magnetics unambiguously. Forthe input parameters, the Lundquist number (or magnetic Reynolds numberIII INTERPRETATION 6S0) and the shape of the temperature proﬁle are needed. Unless otherwisespeciﬁed, in the following case S0 ≡ (τR/τA)0 = 1.67× 107 is used, where τAis the Alfve´n time. The shape parameter (ηi ≡ d(lnT )/d(lnn)) has been de-termined from the relation between density (n(ψ)/n0 = (P (ψ)/P0)1−α) andtemperature (T (ψ)/T0 = (P (ψ)/P0)α), so ηi = α/(1− α). Since the densityproﬁle was rather ﬂat, a best ﬁt for α was found to be 67, giving ηi= 6.0.The ﬁrst distinguishable mode has resistive double tearing-like behavior, asshown in Fig. 6. When the magnetic Reynolds number is increased (in otherwords, η, which is inversely proportional to S0, decreases), such an instabilitydisappears, which indicates that it is a non-ideal (resistive) mode. Around theq = 3, 4 rational surfaces, there were big pressure ﬂuctuations, whose modecomponents are represented by diﬀerent symbols (e.g. •(m = 3), (m = 4)).Also, near each resonant surface, each mode component changes its phase.When all of the components are summed up, the pressure ﬂuctuation (δp)theoretically predicted in the MARS code is peaked near the outer q = 3rational surface, whose radial location is 83.5 cm (ρ ∼0.73) (See the reddash-dot curve and solid thick black curve of Fig. 6). Since other resonantlayers are seen at various rational surfaces, the instability can be called aresistive “multiple” tearing mode. In comparison with the experimental ob-servations shown in Fig. 3, the location of the peaked ﬂuctuations predictedin the MARS code is less than 2-3 cm away from the Channel 6 (R = 80-81cm (ρ ∼0.62)), whose ∆T is the biggest. In addition, a range of possibleq proﬁles is shown by the solid thin black curves, while other thick shortand long dashed curves show other reconstructed q-proﬁles based on diﬀer-ent constraints. From these proﬁles, the outer q=3 surfaces (ρ ∼ 0.7) aremuch closer to the location of the peak ∆Te channel. Considering that thepotential errors in determining the q-proﬁle via EFIT reconstruction are notinsigniﬁcant, the theoretical result based on the solid thick q-proﬁle seems toIII INTERPRETATION 7match the experimental observation quite well. Since this “multiple” tearingmode is a global mode, it is not surprising that most of the GPC channelsshowed large or small electron temperature ﬂuctuations in the experiment.The perturbed magnetic ﬂux (∆b)21 linear eigenfunction is shown in Figure 7.According to the MARS code results, γτA ∼ 1.8×10−4, where γ is the growthrate [sec−1] and τA ∼ 8× 10−8 sec. Thus, the growth time is approximately0.44 msec, which is within a typical range of tearing mode evolution.Interestingly, the MARS code also predicted a non-global mode, whoseﬂuctuation is highly localized near one of the q=4 rational surfaces in theNCS region. It was predicted as a resistive interchange mode. This wouldbe diﬃcult to observe without suppressing all the other global MHD insta-bilities, apart from the capability of spatially well-resolved diagnostics. Thisresistive interchange mode is contrasted with those found in DIII-D experi-ments.16 According to the DIII-D results, the resistive interchange instabilitycan occur beyond a threshold βN1. However, the βN of the equilibrium dis-cussed here was 0.27[%mT/MA], which is low. Basically, this instabilitywas predicted to occur due to a hollow pressure proﬁle, not from high βN .Since the elongation (κ ∼ 1.14) during the current ramp was close to one,we could assume a circular plasma. The ideal (Mercier) stability criterion(DI < 0)2 was satisﬁed. In this case, the H-factor3 in the NCS region wasno more than -0.02. Since |H| = 0.02 << 1/2 and (q2 − 1) was positive, the1βN ≡ β/(Ip/aB0) [% m T/MA]; β = 2µ0 < p > /B20 , Ip: plasma current [MA], a:minor radius [m], B0: magnetic field at magnetic axis [T], and < p >: volume averagedpressure [Pa]2According to Glasser, Greene and Johnson,15 DI = −1/4 + 2ap′/(B2sS2)(q2 − 1)3H =µ0p′V ′f2πq′[< 1|∇ψ|2 > −<B2/|∇ψ|2><B2>], where p′ = dp/dψ, V ′ = dV/dψ, q′ = dq/dψ, < · >: flux surface average quantity andB = f∇φ+∇φ ×∇ψ.IV CONCLUSIONS 8resistive interchange mode stability criterion (DR < 0)4 was violated by thepositive sign of p′. In more recent discharges, there were some indications ofhighly localized Te ﬂuctuations that we believe have occurred due to such aresistive (or possibly ideal) interchange mode. The temperature ﬂuctuationsappeared on only one of the GPC core channels in a discharge with a hollowTe proﬁle. Detailed analysis is under way and will be reported separately.IV. ConclusionsDuring current ramp discharges of reversed magnetic shear experiments,MHD oscillations have been observed on ECE, as well as on magnetics. Thetoroidal mode number was identiﬁed as n = 1 unambiguously, while thepoloidal mode number decreased from 7 to 5 until the biggest Te ﬂuctuationwas observed near the outer q = 3 rational surface. The equilibrium of thereversed magnetic shear conﬁguration was reconstructed by EFIT. Using alinear resistive MHD code (MARS), its stability has been analysed. TheMHD activity of the Te measurements was identiﬁed as an n = 1 resistive“multiple” tearing mode. This code gave γτA ∼ 1.8 × 10−4 and the growthtime was predicted to be approximately 0.44 msec.Another n=1 resistive mode has been predicted in NCS region of the re-versed magnetic shear plasma. Its driving force comes from inverted pressuregradient (p′ >0) in the core which violates the resistive interchange mode sta-bility criterion (DR < 0). Although the associated magnetic island size ischallenging to Alcator C-Mod diagnostic capabilities, there were some indi-cations of highly localized ﬂuctuations on GPC in other recent dischargeswhich seemed to have been caused by the resistive interchange mode.4DR = DI + (H − 1/2)2. In particular, if |H | << 1/2, DR  DI + 1/4 2ap′/(B2sS2)(q2 − 1)IV CONCLUSIONS 9In summary, the global MHD mode during reversed magnetic shear ex-periments of Alcator C-Mod has been ascertained as n=1 resistive “multiple”tearing mode. In addition, a resistive interchange mode in NCS region of lowβN has been predicted due to a hollow pressure proﬁle.AcknowledgementsThe authors would like to thank E. Marmar for providing density proﬁlesusing visible Bremsstrahlung diagnostic and M. Greenwald for helpful dis-cussion regarding pressure proﬁles. This work has been supported at MITby USDOE Contract No. DE-FC02-99ER54512.REFERENCES 10References1KESSEL, C., MANICKAM, J., REWOLDT, G., and TANG, W. M., Phys.Rev. Lett. 72 (1994) 12122LEVINTON, F. M. et al., Phys. Rev. Lett. 75 (1995) 44173STRAIT, E. J. et al., Phys. Rev. Lett. 75 (1995) 44214The JET Team, Plasma Phys. and Controlled Fusion 39 (1997) B3535GREENE, J.M. and CHANCE, M.S., Nucl. Fusion 21 (1981) 4536TURNBULL, A.D., TAYLOR, T.S., LIN-LIU, Y.R., and ST. JOHN, H.,Phys. Rev. Lett. 74 (1995) 7187TAYLOR, T.S., Plasma Phys. and Controlled Fusion 39 (1997) B478FURTH, H.P., KILLEEN, J. and ROSENBLUTH, M.N., Phys. Fluids 6(1963) 4599LA HAYE, R.J., SAUTER, O., Nucl. Fusion 38 (1998) 98710WHITE, R.B., MONTICELLO, D.A., ROSENBLUTH, M.N. and WAD-DELL, B.V., Plasma Physics and Controlled Fusion Research 1977(IAEA,Vienna, 1977), Vol.1, 56911CARRERAS, B., HICKS, H.R. and WADDELL, B.V., Nucl. Fusion 19(1979) 58312PRITCHETT, P.L., LEE, Y.C. and DRAKE, J.F., Phys. Fluids 23 (1980)136813CHANG, Z. et al., Phys. Rev. Lett. 77 (1996) 3553REFERENCES 1114GU¨NTER, S. et al., Nucl. Fusion 39, special issue, (1999) 179315GLASSER, A.H., GREENE, J.M. and JOHNSON, J.L., Phys. Fluids 19(1976) 56716CHU, M.S. et al., Phys. Rev. Lett. 77 (1996) 271017PORKOLAB, M. et al., Proceedings of the 24th EPS Conf on Contr. Fusionand Plas. Phys. Part II, (1997) 56918LAO, L.L. et al., Nucl. Fusion 30 (1990) 103519FOORD, M.E., MARMAR, E.S. and TERRY, J.L., Rev. Sci. Instru. 53(1982) 140720CHRISTENSEN, C., PhD thesis, MIT, 199921BONDESON, A., VLAD, G. and LU¨TJENS, H., Phys. Fluids B4 (1992)1889REFERENCES 12   Ch3 (0.731m)Ch4 (0.757m)Ch2 (0.704m)Ch1 (0.675m)Ch5 (0.781m)Ch6 (0.804m)t=0.108 sMagnetic pickup coil  at outer limiter [T/s]DαPlasma  current [MA] Line integrated density [x1020 m-2] ICRF Power [MW]Te [keV]Figure 1: MHD activity during reversed magnetic shear plasma. Note thatthe electron temperatures of the two innermost channels are lower than thethird channel (i.e. a hollow temperature proﬁle).REFERENCES 13   0.60.811.21.4 Te 1 [keV] R=0.675 m0.811.21.4 Te 2 [keV] R=0.704 m0.811.21.4 Te 3 [keV] R=0.731 m11.2 Te 4 [keV] R=0.757 m0.80.9 Te 5 [keV] R=0.781 m0.50.60.7 Te 6 [keV] R=0.804 m0.40.5 Te 7 [keV] R=0.827 m0.20.30.4 Te 8 [keV] R=0.848 m0.10.2 Te 9 [keV] R=0.868 m-20020 Poloidal pick up coil at outer limiter 105 110 115 120time [msec]Figure 2: Enlarged view of the MHD activity observed on GPC and magnet-ics of Fig. 1 near t=108 msec. In particular, observe that the Te ﬂuctuationson Channel 6 are the biggest and that the oscillation frequency (3 ∼ 4 kHz)is the same as that of the magnetics.REFERENCES 14Figure 3: At t=0.108 sec considered here, Channel 6, which is located atR=80 ∼ 81 cm (ρ ∼ 0.62), shows the biggest ﬂuctuations. Also, note thatthe ﬂuctuations of Channel 6 evolve, peak near t=0.11 sec and then thebiggest ﬂuctuations move outward into Channel 7 (R ∼ 83 cm) at t=0.120sec.REFERENCES 15   MagneticAxis, R=0.668 mSeparatrixR=0.889 mR~0.835 mFigure 4: q-proﬁle at t=108 msec. Note that the q=3 outer rational surfaceis located at R∼ 83.5 cm (ρ ∼ 0.73), which is close to (∆T )max.REFERENCES 16   MagneticAxis, R=0.668 mSeparatrixR=0.889 mFigure 5: Pressure proﬁles: Solid curve - from EFIT reconstruction;Dashed curve - calculated from Te of GPC and ne inferred from visibleBremsstrahlung; Dash-dot curve - 2 ×(neTe), based on Te of GPC and nefrom TCI-inversion using routinely given EFIT reconstruction.REFERENCES 17   Figure 6: Perturbed pressure eigenfunction for the n=1 resistive “multiple”tearing mode. When all the components are summed up, the total predictedpressure ﬂuctuations (δp) are delineated with the red dash-dot curve. Notethat resonant layers are observed not only at the outer q = 3 surface, butalso at the inner q = 3 and at the q = 4 rational surfaces, which characterizesthe “multiple” tearing mode. This can be compared with the temperatureﬂuctuations (∆Te) of t=0.108 sec in Fig. 3. Theory and experiment resultsmatch each other. The growth rate for this mode is γτA ∼ 1.8 × 10−4 forS0 = 1.67× 107, where τA: Alfve´n time ∼ 8× 10−8 sec. The local Lundquistnumber S at the outer q = 3 surface is 6.15 ×106. The other two cases, whoseq-proﬁles are shown in thick short and long dashed black curves respectively,also predicted almost the same ﬂuctuations.REFERENCES 18Figure 7: Perturbed radial magnetic ﬂux deﬁned as ∆b ≡ dΦdθdφ= [∇ψ · (∇θ×∇φ)]−1B˜ · ∇ψ. The sum of the Fourier components giving the perturbedmagnetic ﬂux at θ =0 is delineated with the dash-dot curve.

n=1 Resistive Modes in Reversed Magnetic Shear Alcator C-Mod Plasmas

http://dspace.mit.edu/bitstream/1721.1/93952/1/00ja030_full.pdf

n=1 Resistive Modes in Reversed Magnetic Shear Alcator C-Mod Plasmas

Abstract

Similar works

Full text

Available Versions

DSpace@MIT