Edlund, E.M.

Kramer, G.J.

Lin, L.

Lin, Y.

Porkolab, M.

Wukitch, S.J.

DSpace@MIT

        PSFC/JA-08-48               Observation of reversed shear Alfvén eigenmodes during sawteeth in Alcator C-Mod  Edlund, E.M., Porkolab, M., Kramer*, G.J., Lin, L., Lin, Y., Wukitch, S.J.   * Princeton Plasma Physics Laboratory                   Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge  MA  02139  USA               This work was supported by the U.S. Department of Energy, Grant Nos. DE-FC02-99-ER54512 and DE-FC02-04ER54698  Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.  Observation of reversed shear Alfve´n eigenmodes during sawteethin Alcator C-ModE.M. Edlund1, M. Porkolab1, G.J. Kramer2, L. Lin1, Y. Lin1, and S.J. Wukitch11MIT Plasma Science and Fusion Center,Cambridge, Massachusetts 02139, USA2Princeton Plasma Physics Laboratory,Princeton, New Jersey 08543, USA(Dated: March 15, 2009)AbstractGroups of frequency chirping modes observed between sawtooth crashes in the Alcator C-Modtokamak are interpreted as reversed shear Alfve´n eigenmodes near the q=1 surface. These modesindicate that a reversed shear q profile is generated during the relaxation phase of the sawtoothcycle. Two important parameters, qmin and its radial position, are deduced from comparisons ofmeasured density fluctuations with calculations from the ideal MHD code NOVA. These studiesprovide valuable constraints for further modeling of the sawtooth cycle.1The sawtooth cycle has long been recognized as a robust feature of tokamak plasmas[1]. The cycle is composed of an abrupt crash in the central electron temperature and arelaxation, or reheat, phase (see Fig. 1, t ≥ 0.25 s). The sawtooth crash is a process wherebythe magnetic field lines interior to the q=1 surface (where q = rBφ/R0Bθ is the magneticsafety factor, r is the minor radial coordinate, Bφ is the toroidal magnetic field, R0 is thetokamak major radius and Bθ is the poloidal magnetic field) reconnect to lines outside of theq=1 surface. In this process the magnetic field ”frozen-in” laws are broken leading to rapidradial transport wherein the temperature, density, and current profiles are flattened [2–4].Following the cessation of magnetic reconnection the plasma relaxation phase is characterizedby a nearly linear increase in the central electron temperature and an inward diffusion ofparallel current formerly displaced by the reconnection. Understanding the evolution of thecurrent profile and its relation to sawtooth phenomena is important, especially for futureburning plasma experiments such as ITER, where an accurate model of the physics ofthe crash event and the subsequent relaxation process is crucial for developing predictivecapability for sawtooth control. In this Letter we report observations of frequency chirpingAlfve´nic modes excited during the sawtooth cycle in Alcator C-Mod tokamak plasmas, hereinidentified as reversed shear Alfve´n eigenmodes (RSAEs, also Alfve´n cascades) located nearthe q=1 surface. RSAEs should be of great interest for future burning plasma experimentsboth for their use as a form of MHD spectroscopy and because Alfve´nic modes may negativelyimpact the confinement of the energetic ions in the plasma core [5]. Importantly, the presenceof RSAEs near the q=1 surface is interpreted as evidence of a hollow current profile duringthe sawtooth cycle.The phase contrast imaging (PCI) diagnostic [6], a type of internal reference beam inter-ferometer, is an outstanding tool for the study of core localized Alfve´nic activity. The out-put signal of the PCI system is a 1-d image, decomposed into 32 elements of approximately4.5mm chord separation in the direction of the major radius, which is linearly proportionalto the line integral of the electron density perturbations along the beam path. The reportingof q=1 RSAEs in Alcator C-Mod, and also in JET [7, 8] (see also comment [9]), is due tothe availability of diagnostics sensitive to small perturbations in the plasma core. In a fewAlcator C-Mod experiments the amplitude of the q=1 RSAEs was large enough allow formode number identification with the magnetic pickup (Mirnov) coils.Typical plasma parameters in Alcator C-Mod (R0 = 0.68 m, a = 0.22 m, κ ≤ 1.85) in2which these modes are seen are Bφ = 5.4 T, Ip = 600-800 kA and ne0 <∼ 1.5 × 1020m−3.Up to 5 MW of auxiliary heating is provided by ion-cyclotron resonance heating (ICRH) atthe fundamental cyclotron frequency of the H+ minority species in a deuterium majorityplasma (nH/nD ≈ 0.05), creating central electron temperatures in the range of 3-6 keV [6].Analysis of many experiments shows that q=1 RSAE activity is seen in about a third of allsawtoothing L-mode shots with central electron densities below about 1.5×1020m−3 and on-axis ICRH, with a strong positive correlation with electron temperature. The experimentspresented here were optimized for the investigation of RSAEs during the current-ramp phaseof the discharge (t < 0.25 sec), where a maximum in the current density exists transientlyoff-axis as the ohmic current diffuses toward the plasma core [10]. Application of early ICRF(turned on at 0.08 seconds after plasma creation) was used to heat the plasma to retardthe current penetration and create a fast ion population to drive the RSAEs at the q=2and higher surfaces. By extending the ICRH pulse to the current flattop phase, multiplegroups of frequency chirping modes similar to the RSAEs excited during the current rampwere observed following the second sawtooth crash and continued, at varied amplitudes, formany cycles thereafter.RSAEs are known to exist near the minimum of the q profile (qmin) in a reversed shearequilibrium and are driven unstable by the spatial gradient of the fast ion density fromneutral beams, fusion born alpha particles, or in the case of Alcator C-Mod, an ICRHdriven minority ion species [11, 12]. RSAEs and other Alfve´nic modes have been used for”MHD spectroscopy”, that is, the inference of equilibrium parameters such as qmin or q onaxis (q0) from frequency spectra [12, 13]. The characteristic frequency sweep of the RSAEshas been well qualified in previous experimental [11, 12], computational [14], and theoretical[15, 16] studies. An approximate dispersion relation for the RSAEs can be found in Ref.[16],ω = 2TeR20Mi(1 +74TiTe)+V 2AR20(mqmin− n)21/2 + ω∆, (1)where R0 is the tokamak major radius, Mi the majority ion mass, VA = B/√µ0niMi isthe Alfve´n velocity, B is the modulus of the magnetic field, ni(≈ ne) is the ion density,m is the poloidal mode number, n the toroidal mode number, and the ω∆ term includescorrections for fast ion pressure and finite pressure gradients and can be neglected in the3current experiments because these terms are small near the q=1 surface. It should be notedthat though Eq. 1 was derived in the limit of high q and high n, comparison with theideal MHD code NOVA [17] has shown that it is accurate within 5%, and for the analysispresented here Eq. 1 is sufficient to determine mode numbers from the rate of the frequencychirping. The first term in brackets in Eq. 1 derives from the geodesic deformation of theAlfve´n continuum and provides the minimum frequency offset (fmin). The Alfve´nic nature ofthe RSAEs is represented by the second term in brackets, and being inversely proportionalto qmin it is largely responsible for the frequency sweep as the equilibrium evolves andqmin decreases. The maximum frequency of the RSAEs is set by the toroidicity-inducedAlfve´n eigenmode (TAE) frequency, fTAE = VA/4piqR0. The minimum frequency and TAEfrequency (taking q = qmin) are plotted in Fig. 1 for reference. Toroidal mode numbers of2-5 in the ion-diamagnetic direction have been directly measured by magnetic pick-up coils,shown in Fig. 2. The n=6-9 mode numbers are inferred through their frequency spectra andrate of frequency chirp, which is expected to scale approximately proportional to n whenthe Alfve´nic contribution is dominant (see Eq. 1).The ideal MHD code NOVA, employing a non-variational method for solution of theeigenmode equations in full toroidal geometry, is used to model RSAEs with qmin ≈ 1. NOVAhas been used formerly to explain a range of observations, including TAE behavior duringsawteeth in JT-60 [18], the anti-balooning structure of RSAEs in TFTR [19], and modecoupling in DIII-D [20]. The analysis presented in this paper is an extension of the MHDspectroscopy technique which shows how both the value of qmin, its radial position (rmin),and to some extent the value of q0 (q at the magnetic axis), can be deduced from the spatialstructures of the q=1 RSAEs. When possible, experimental data are used in the NOVAanalysis. The electron temperature profiles are taken from the electron cyclotron emission(ECE) diagnostic, the deuterium majority ion temperature is calculated from the neutronflux (Ti/Te ≈ 0.8), and electron density profiles from the Thomson scattering system. Theq profile is derived from a combination of edge constraints using the equilibrium code EFIT[21], which is based on edge magnetic measurements, determination of the q=1 surface bythe sawtooth inversion radius in the soft x-ray data, and a core solution determined throughanalysis of the PCI data. Within the ideal MHD framework NOVA has produced q=1RSAE solutions for toroidal mode numbers from 2 through 9 whose spectra are in goodagreement with the PCI observations, shown in Fig. 3 as the dashed overlay. NOVA shows4that qmin decreases to approximately 0.92 prior to the third sawtooth crash, and a linearextrapolation to the beginning of the sawtooth period estimates q to be 1.02, suggesting acomplete reconnection process has occurred. In the present case the RSAEs are not observedin the first half of the sawtooth cycle and hence we have no quantitative information on theq profile during this time (t = 0.262 to 0.270 sec in Fig. 3). The stability of these modesis calculated with the kinetic extension of the NOVA code, NOVA-K [22], which shows thatthe n=3 RSAE solution can be unstable for fast ion distributions with tail temperaturesabove 100 keV with a central βH of 0.5%, βH/β0 ≈ 0.2 (where β0 = 2µ0p0/B2 is the ratioof the kinetic pressure to magnetic pressure and the subscript ”H” refers to the ”hot” ioncomponent), all reasonable values for the heating scheme used [23]. An example of a possiblecore q profile based on this analysis and the associated n=3 eigenmode are presented in Fig.4.A synthetic PCI signal can be created from the two dimensional density perturbationcalculated by NOVA by integrating the perturbations along a PCI viewing chord and con-volving it with the system response function. A complex pattern of peaks and nodes is seenin the PCI signal and is attributed to integration through regions of positive and negativedensity fluctuations which tend to cancel. The spacing of the peaks has been studied in thesynthetic PCI signal and varies linearly with the value of rmin used in the simulation. Itis the dominance of the value of rmin in determining the spatial structure of the syntheticdiagnostic signal that allows this parameter to be determined. The dependence of the valueof q0 has also been explored using parameter scans in NOVA. Our studies suggest that verylow shear, represented here by the quantity δq ≡ q0− qmin, is required to generate the rangeof observed RSAEs. The n=4,5,6 RSAEs appear to be especially sensitive to this parameterand cease to exist when the δq exceeds 0.02. At this time there is no additional experimen-tal measurement of the value of q0 in Alcator C-Mod. Two free parameters are used in thecomparison of the experimental and synthetic PCI signals. Though the channel to channelspacing of the PCI signal is known very accurately the position of the array in major radiusis known only to ±1cm. To account for this the experimental data are shifted in majorradius when comparing to the synthetic diagnostic signal to minimize the RMS differenceand find the best fit. The second free parameter is the absolute value of the synthetic di-agnostic signal which accounts for the fact that NOVA calculates relative fluctuation levels.The best-fit solution is found by scanning the synthetic diagnostic signal amplitude until5the RMS difference with the experimental data is minimized.A comparison of the synthetic and experimental PCI signals is presented in Fig. 5 foran n=3 RSAE, where the synthetic PCI signal represents the best-fit solution. Similarcalculations have been performed for n = 4, 5, 6 and together determine rmin/a to be near0.20. The magnitude of the central shear (δq) was scanned over the range δq = 0.01 to 0.05for each of the modes and at each value of rmin. The conclusion from this study is that thatRSAE solutions are found only for δq ≤ 0.02 at rmin/a = 0.20 and exist up to δq = 0.04for larger rmin/a. Larger values of shear force the RSAEs into the continuum which causesthem to become heavily damped. As a final note we would like to mention that the presenceof q > 1 following the sawtooth crash may allow for down-chirping RSAEs to exist. Thispossibility has been explored with NOVA with the result that relatively undamped andunstable solutions have been discovered [24]. The existence of the down-chirping solutionsseems to require a local maximum in q near the core, equivalent to a peak in the Alfve´ncontinuum, around which the RSAEs form.In summary, we have shown that the frequency chirping modes excited during the saw-tooth cycle are consistent with the behavior of q=1 RSAEs. Calculations performed withthe code NOVA, using a weakly reversed shear q profile with q0 − qmin <∼ 0.02 are ableto produce RSAE solutions for the range of mode numbers identified in the experiments.Calculations of the growth rates show that the RSAEs can be unstable in the presence of anenergetic hydrogen ion tail produced with minority ion cyclotron resonance heating. Usingthe measured frequency spectra of the RSAEs as a diagnostic for q, the NOVA calculationsshow that q is likely greater than unity immediately following the crash and may decrease tovalues in the range of 0.92 to 0.95. The constraints derived from the analysis of the RSAEsshow remarkable agreement with with recent results using equilibrium reconstructions fromMSE constrained EFIT for the sawtoothing phase of plasmas in the DIII-D tokamak [25].Future experiments would benefit from multiple core perturbation diagnostics and energeticion loss detectors to study the relation between the energetic ions, Alfve´n eigenmodes andsawteeth.We thank the Alcator C-Mod team for their support of these experiments. We alsoacknowledge the contributions of Jesus Ramos, Peter Catto, Per Helander, and Chuck Kesselfor the valuable discussions of our modeling and neoclassical transport theory. This worksupported by the US DOE under contracts DE-FC02-99-ER54512 and DE-FC02-04ER54698.6[1] S. von Goeler, W. Stodiek, and N. Sauthoff, Phys. Rev. Lett. 33, 1201 (1974).[2] B.B. Kadomtsev, Fiz. Plazmy 1, 710 (1975) [Sov. J. Plasma Phys. 1, 389 (1975)].[3] F. Porcelli, D. Boucher, and M.N. Rosenbluth, Plasma Phys. Cont. Fus. 38, 2163 (1996).[4] D. Biskamp, Astrophys. Space Sci. 242, 165 (1998).[5] W.W. Heidbrink et al., Phys. Rev. Lett. 99, 245002 (2007).[6] M. Porkolab et al., IEEE Trans. Plasma Sci. 34, 229 (2006).[7] P. Sandquist, S.E. Sharapov, M. Lisak, and T. Johnson, Phys. Plasmas 14, 122506 (2007).[8] S.E. Sharapov et al., Proc. 21st IAEA Conf. IAEA-CN-149, EX/P6-16 (2006).[9] Sandquist et al. concluded that a reversed shear q profile is incompatible with the observedtornado modes. However, they did not consider the possibility that a reversed shear q profilemay exist transiently in the early part of the giant sawtooth phase before the tornado modesdevelop. Our studies indicate that a reversed shear q profile is necessary to excite RSAEs.[10] M. Porkolab et al., Proc. 21st IAEA Conf. IAEA-CN-149, EX/P6-19 (2006).[11] H. Kimura et al., Nucl. Fus. 38, 1303 (1998).[12] S.E. Sharapov et al., Phys. Plasmas 9, 2027 (2002).[13] G.J. Kramer et al., Nucl. Fus. 41, 1135 (2001).[14] G.J. Kramer, N.N. Gorelenkov, R. Nazikian, and C.Z. Cheng, Plasma Phys. Cont. Fus. 46,L23 (2004).[15] N.N. Gorelenkov, G.J. Kramer, and R. Nazikian, Plasma Phys. Cont. Fus. 48, 1255 (2006).[16] B.N. Breizman, M.S. Pekker, and S.E. Sharapov, Phys. Plasmas 12, 1 (2005).[17] C.Z. Cheng and M.S. Chance, J. Comput. Phys. 71, 124 (1987).[18] G.J Kramer et al., Nucl. Fus. 40, 1383 (2000).[19] R. Nazikian et al., Phys. Rev. Lett. 91, 125003 (2003).[20] M.A. Van Zeeland et al., Phys. Plasmas 14, 056102 (2007).[21] L.L. Lao et al., Nucl. Fus. 25, 1611 (1985).[22] C.Z. Cheng, Phys. Rep. 211, 1 (1992).[23] V. Tang et al., Plasma Phys. Cont. Fus. 49, 873 (2007).[24] E.M Edlund, et al., Phys. Plasmas 16, (2009) accepted for publication.[25] E.A. Lazarus et al., Phys. Plasmas 14, 055701 (2007).7100300500700900frequency [kHz]100103  log scale [a.u.]0.100 0.150 0.200 0.250 0.300time [sec]T e0 [keV]P ICRH [MW]024024TAEsRSAEs    q=1RSAEscurrent ramp phase sawtooth   phase2 3 41fminfTAEFIG. 1: (color online) PCI spectrogram showing the range of modes observed in the current-rampand early sawtoothing phase. The numbers below the Te0 trace denote the sawtooth crash numberrelative to the beginning of the shot. Ti/Te = 0.8 has been used for the calculation of fmin.271 273 275 277 279 281time [msec]200300400500600271 273 275 277 279 281time [msec]+pi−pi0+pi−pi00 90 180 270 360coil position [deg]+pi−pi0frequency [kHz]+pi−pi0n = 5n = 4n = 3n = 2n = 8n = 9n = 7(a) (b) (c)n = 5n = 6n = 3n = 4n = 2fminn = 5n = 4n = 3n = 2FIG. 2: (color online) Experimental data for RSAEs following the second sawtooth crash showing(a) a composite PCI spectrogram (b) Mirnov coil spectrogram and (c) measured mode phases as afunction of toroidal angle. The mode numbers in panel (a) are derived from comparison with Eqn.1, shown here as the dashed lines.80.255 0.265 0.275 0.285time [sec]1.000.980.960.940.90qmin0.92no RSAE    data100300500700frequency [kHz]n=2n=3n=4n=5n=6n=7n=8n=92nd sawtooth     crash3rd sawtooth     crashfminFIG. 3: (color online) PCI spectrogram showing the modes following the second sawtooth crash.NOVA results of the frequency spectra are overlaid in dashed line and have been used to infer theevolution of qmin from 0.99 to 0.92 over the period of RSAE activity.0.0 0.2 0.4 0.6 0.8 1.0q1.001.051.100.950.90displacement,χ [a.u.]n = 3 RSAE   f = 321 kHzfrequency [kHz]120016008004000n = 3RSAETAEq=1qmin q=1qminr/a r/a0.0 0.2 0.4 0.6 0.8 1.0m=3m=4m=2FIG. 4: (color online) On the left, the n=3 Alfve´n continuum calculated for the q profile shown atbottom right, with q0 = 0.96 and qmin = 0.95 at r/a = 0.25 corresponding to the conditions neart=0.275 s in Fig. 3. On the top right, an n=3 RSAE solution with the typical single dominantpoloidal mode. The q profile is shown on the bottom right for reference.9(b)Exp.NOVAmajor radius [m] major radius [m]0.5 0.6 0.7 0.8 0.9z [m]0.0+0.1-0.1-0.2-0.3+0.2+0.3amplitude [a.u.]+1- 10.6 0.7 0.80.00.20.40.60.81.0(a)FIG. 5: (color online) (a) Real part of the 2-D density perturbation structure for an n=3 RSAEcalculated by NOVA with the PCI beam path marked by the grey background, and (b) the com-parison of the measured radial structure from PCI (blue circles) to the synthetic diagnostic resultusing the NOVA solution (red line).10

Observation of reversed shear Alfvén eigenmodes during sawteeth in Alcator C-Mod

http://dspace.mit.edu/bitstream/1721.1/94272/1/08ja048_full.pdf

Observation of reversed shear Alfvén eigenmodes during sawteeth in Alcator C-Mod

Abstract

Similar works

Full text

Available Versions

DSpace@MIT