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Measurement and Simulation of Electron Thermal Transport in the MST Reversed-Field Pinch

D. J. Den Hartog1,2, J. A. Reusch1, J. K. Anderson1, F. Ebrahimi1,2,*, C. B. Forest1,2

D. D. Schnack1, and H. D. Stephens1,2,† 1Department of Physics, University of Wisconsin–Madison, Madison, Wisconsin, 53706 U.S.A. 2The Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas, University of Wisconsin–Madison, Madison, Wisconsin, 53706 U.S.A.

E-mail contact of main author: djdenhar@wisc.edu Abstract. Comparison of measurements made in the MST reversed-field pinch (RFP) to the results from extensive single-fluid nonlinear resistive MHD simulations provides two key observations. First, thermal diffusion from parallel streaming in a stochastic magnetic field is reduced by particle trapping in the magnetic mirror associated with the toroidal equilibrium. Second, the structure and evolution of long-wavelength temperature fluctuations measured in MST shows remarkable qualitative similarity to fluctuations appearing in a finite-pressure nonlinear MHD simulation.

1. Introduction

The reversed-field pinch (RFP) is a toroidal magnetic confinement configuration with low externally applied toroidal magnetic field. The plasma β is relatively high and the magnetic field at the external coils is relatively low, both desirable characteristics for a fusion reactor configuration. In addition, the effective plasma resistance is large enough that it may be possible to ohmically heat an RFP reactor to ignition, mitigating the need for expensive and complicated auxiliary heating. These advantages have motivated a careful study of thermal transport in the RFP, both of the mechanisms governing transport and the quantitative level of transport. In standard multi-helicity RFP plasmas, the low magnetic safety factor (q-profile) allows a large number of resistive magnetic tearing modes to be resonant. The resulting magnetic islands typically overlap, leading to stochastic wandering (diffusion) of magnetic field lines in the plasma, and enhanced thermal transport. This enhanced thermal transport can be dramatically reduced either by reducing the free energy source that drives the tearing modes [1], or by entering a single-helicity regime in which one tearing mode dominates the spectrum [2]. But understanding and predicting thermal transport in a stochastic magnetic field is the focus of a great deal of research in both laboratory and astrophysical plasma, so the MST RFP has been operated in a multi-helicity regime in which stochastic transport dominates. Detailed measurements of electron temperature and thermal transport have been made in this operational regime, and exhibit quantitative agreement with the results of extensive MHD simulation when the effect of particle trapping is included. * Currently at University of New Hampshire, Durham, New Hampshire, 03824 U.S.A. † Currently at Pierce College, Lakewood, Washington, 98498 U.S.A.

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The MST RFP is circular cross-section torus with major radius R = 1.5 m and minor radius a = 0.5 m. For the measurements described in this paper, standard RFP discharges were run at a plasma current of 400 kA, with electron density approximately 1 × 1019 m-3 and electron temperature varying between 200 and 400 eV. These standard multi-helicity discharges in MST typically exhibit a limit-cycle behavior often referred to as “RFP sawteeth [ 3 ].” The sawtooth limit cycle has three phases: a slow rise of magnetic mode amplitudes, a crash in which mode amplitudes spike and stochasticity peaks, and a quiet period following the crash in which mode amplitudes are relatively small. This sawtooth sequence is illustrated in the time evolution of the magnetic mode amplitude shown in Fig. 1.

2. Measurements

New high-time-resolution measurements of the evolution of the electron temperature profile [Te(r,t)] through a sawtooth event in high-current RFP discharges have been made using the enhanced capabilities of the upgraded multi-point, multi-pulse Thomson scattering diagnostic on MST. Thomson-scattered light is collected across the plasma minor radius and analyzed by a set of 21 filter polychromators [4]. This system is based on a pair of commercial flashlamp-pumped Nd:YAG lasers operated at 1064 nm, producing Q-switched pulses of ~10

ns duration. Each laser has been upgraded to “pulse-burst” capability, enabling it to produce a burst of up to fifteen 2 J pulses at repetition rates of 1–12.5 kHz. Variable pulse width drive of the flashlamps and direct control of the Pockels cell are the key components of the laser upgrade [5].

This system provides radial profiles of Te at repetition rates of 1–25 kHz, resulting in precise ensemble measurements of rapid Te evolution (Fig. 2). The quality of this data enables the determination of magnetic equilibrium and thermal transport at levels of precision and time resolution that were previously unobtainable (Fig. 3). Thermal diffusion is calculated by performing a low resolution fit of the χe

FIG. 1. Sawtooth cycle evolution of the RMS amplitude of the core m = 1 modes in MST. The red vertical line demarcates the sawtooth crash at time t = 0. This plot is an ensemble of measurements taken from hundreds of highly reproducible MST plasma shots.

FIG. 2. Electron temperature versus time for a point near the core (black) and a point near the edge (green) of the plasma. Time t = 0 is when the sawtooth crash occurs. This plot is an ensemble of measurements taken from hundreds of highly reproducible MST plasma shots.

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profile to the electron temperature data via the energy conservation equation which, assuming Fourier’s law qe = -ne χe∇Te (where qe is the electron heat flux and ne is the electron density), can be written as ∂Ee/∂t = div(qe) + ηJ||

2 – sinks (Ee is the energy stored in the electrons and ηJ||

2 is ohmic input power). Note that measured χe varies by over two orders of magnitude in the core region of the plasma over the course of the sawtooth cycle.

3. Simulations

The measured evolution of the magetic equilibrium and χe are compared directly to simulations using the nonlinear, single-fluid MHD code DEBS [6], run at parameters matching the RFP discharges in MST [ 7 ]. In particular, the simulation Lundquist number S = τR/τA matches the ~4 × 106 recorded in the experiment, and the measured Te(r) is used for calculation of the simulation resistivity profile. These simulations display MHD activity and sawtooth behavior similar to that seen in MST (Fig. 4). Both the slow and fast evolution of the magnetic equilibrium during a sawtooth is reproduced well (Fig. 5). The simulated mode amplitudes are, on average, about a factor of two larger than those measured in MST, though the spectral distribution of mode amplitudes is similar

FIG. 3. Measured electron thermal diffusion (a) and temperature (b) versus radius at three times relative to the sawtooth crash. The plasma is divided into four regions: core, midradius, reversal surface, and edge. Thermal diffusion is calculated for each of these regions at each of the three times.

FIG. 4. Comparison of the simulated and measured m = 1 mode amplitudes. Sawtooth crashes are marked by spikes in the mode amplitude. The simulation can be allowed to run ad infinitum, producing regular periodic sawtooth crashes, whereas each MST plasma shot has a finite duration.

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to that seen in experiment. The mode amplitude discrepancy may be due to the lack of two-fluid effects in DEBS, particularly ion gyroviscosity [8]. The simulated mode amplitudes are scaled to match experiment in order to realistically calculate magnetic field line diffusion Dmag using a magnetic field line tracing code and the simulated mode eigenfunctions.

In a zero β simulation, the measured χe is compared to the thermal diffusion due to parallel losses along diffusing magnetic field lines, v||Dmag, where Dmag is determined from the simulation by tracing magnetic field lines. Agreement within uncertainties is only found if the reduction in thermal diffusion due to electron trapping is taken into account [ 9 , 10 ]. The resulting effective electron thermal diffusion is χMD = fcv||Dmag ≈ fcvth,eDmag (Fig. 6), where fc is the circulating fraction of the electron population. In the core, the measured χe is greater than χMD prior to and during the sawtooth crash, suggesting that other transport mechanisms such as temperature profile flattening due to magnetic islands may be important at these times. The circulating fraction fc in MST is generally not much more than 50% over much of the minor

radius [11]. Mirroring of trapped particles is a result of the RFP equilibrium field configuration; |B| variation over a flux surface is not much smaller than a typical tokamak. In a second simulation, the pressure field was evolved self-consistently assuming Ohmic heating and anisotropic thermal conduction. Critically, these simulations allow pressure fluctuations to develop, allowing temperature structures, which have been observed in MST, to be investigated. Although these pressure-evolved simulation results need further confirmation, the fluctuations in the simulated temperature are very similar in character and time evolution to temperature fluctuations measured in MST (Fig. 7) [12]. This includes m = 1, n = 6 fluctuations that flatten the temperature profile as well as m = 1, n = 5 fluctuations that generate hot island structures near the core shortly after sawtooth crashes.

FIG. 5. Comparison of the ensemble-averaged simulated and measured on-axis safety factor q(0) through a sawtooth crash.

FIG. 6. Comparison of the average electron thermal diffusion from measurement (χe) and simulation (χMD) for (a) the core region (0 ≤ r/a ≤ 0.45) and (b) the midradius region (0.45 ≤ r/a ≤ 0.7).

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Acknowledgements

This work was supported by the U. S. Department of Energy and the National Science Foundation.

References

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FIG. 7. The electron temperature fluctuation amplitude correlated with the m = 1, n = 6 magnetic tearing mode from (a) a finite-pressure DEBS simulation and (b) Thomson scattering measurement. The dashed and solid vertical lines indicate the approximate location of the mode resonant rational surface.

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[10] REUSCH, J.A., et al., “Experimental Evidence for a Reduction in Electron Thermal Diffusion due to Trapped Particles,” Phys. Rev. Lett. 107 (2011) 155002.

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