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Comparison of Plasma Flows and Currents in HSX to Neoclassical Theory

D.T. Anderson1, A.R. Briesemeister1, J.C. Schmitt2, F.S.B. Anderson1, K.M. Likin1,

J.N. Talmadge1, G.M. Weir1, K. Zhai1

1University of Wisconsin, Madison, WI, USA 2Princeton Plasma Physics Laboratory, Princeton, NJ, USA

E-mail contact of main author: dtanders@wisc.edu

Abstract. The Helically Symmetric Experiment (HSX) was designed to have an axis of symmetry in the helical direction, reduced neoclassical transport and small equilibrium currents due to the high effective transform. Unlike other stellarators in which |B| varies in all directions on a flux surface, plasmas in HSX are free to rotate in the direction of quasihelical symmetry. In this paper we will present measurements with Charge Exchange Recombination Spectroscopy (CXRS) that demonstrate that intrinsic plasma flows are predominantly in the direction of symmetry. Whereas previous neoclassical calculations did not conserve momentum, we show that the experimental results agree better with recent modifications to neoclassical theory that do conserve momentum. Also, we present a 3-D equilibrium reconstruction of the plasma pressure and current profile utilizing a set of magnetic diagnostics and demonstrate that these results are in reasonable agreement with expectations from the same neoclassical model.

1. Introduction

Measuring and modeling the radial electric field and plasma flow is important for understanding plasma transport and stability. Sheared flows driven by steep radial electric field profiles have been linked to reduced turbulent transport in both “H-mode” plasmas and plasmas with internal transport barriers [1]. The radial electric field can also reduce neoclassical particle transport, which is especially important in stellarators and other devices with significant non-symmetric magnetic field components. Non-axisymmetric fields exist in tokamaks because of the finite coil effects and have been intentionally introduced primarily for ELM suppression. Non-resonant magnetic perturbations in the DIII-D tokamak have been shown to cause a non-zero flow offset that can cause the plasma to spin without momentum injection from neutral beams [2].

A common method to calculate the flows and electric field in stellarators is to use the DKES [3] (Drift Kinetic Equation Solver) code. DKES uses the incompressible flow

approximation: B

BE

B

BE22

, where denotes flux surface average. This

approximation eliminates the changes in kinetic energy which would occur as a result of radial drift in the presence of Er, making the drift kinetic equation mono-energetic and allowing the use of a collision operator that only includes pitch angle scattering. This form of the drift kinetic equation has a non-physical singularity at the resonant value of the radial

electric field: θTaresr Bv

mι

nmE

[4], where θB is the poloidal magnetic field, Tav is the

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thermal velocity for species a, n and m are respectively the toroidal and poloidal mode numbers of the dominant component(s) of the magnetic field spectrum. For HSX, the dominant helical mode has n=4 and m=1. It has been shown that near a resonance in the radial electric field DKES does not correctly calculate the particle diffusion coefficient [4]. This paper will focus on the outer half of the HSX plasma where Er is predicted to be much smaller than the resonant value for the hydrogen ions.

The pitch angle scattering collision operator does not conserve momentum. Three techniques [5,6,7] developed to correct the mono-energetic transport coefficients calculated by DKES to account for momentum exchange have been implemented in the current version of the PENTA [8] code. All three techniques use the diffusion coefficients calculated by DKES, and are therefore not accurate near resonant values of Er. The Sugama-Nishimura method was used for the calculations including momentum conservation shown in this paper.

The plasma in HSX for these studies and the reconstruction efforts detailed in the next section are created and heated by up to 100 kW of 1st harmonic O-mode ECRH with a 28 GHz gyrotron at a magnetic field strength of B=1.0 T. The launch angle is perpendicular to the magnetic axis, introducing negligible toroidal current drive. Electron temperature and density profiles are measured by a 10 channel Thomson scattering system. Electron temperatures up to nearly 3 keV are measured in the core. With a central density ~ 4 x 1018 m-3, ECE measurements show a minimal supra-thermal component to the plasma, and good agreement is seen in stored energy between integrated profiles and diamagnetic measurements. Doppler spectroscopy indicates that, by comparison with the electrons, the ions are cold with temperatures in the range of 30–60 eV.

2. CXRS Measurements and Plasma Flow

The CXRS system is used to measure the velocity, temperature, and density of C+6 ions in HSX. A 30 keV, 4 Amp, 3 ms diagnostic neutral beam [9] is fired vertically through the plasma. The neutral beam does not cause any measurable perturbations to the electron temperature and density profiles. The Doppler shift, broadening and strength of the 529 nm emission line from C+5 ions is measured using two 0.75 m Czerny-Turner spectrometers. Electron multiplying ccd’s capture a series of images of the spectra during each discharge. Each image is integrated for 5 ms and read out in 1 ms. Images captured before and after the beam is fired are used to remove the background light. The spectral position of the emission line is measured during a series of shots with the magnetic field in the counter clockwise direction and compared to the position measured with the field in the clockwise direction. All components of the flow velocity should reverse when the field is reversed. Using the difference in the measured emission line position between the two cases effectively doubles the Doppler shift caused by the plasma velocity. This technique also eliminates systematic measurement error caused by the uncertainty in the relative strengths of the fine structure

Figure 1 The electron temperature and density profiles were measured using Thomson scattering and used for the PENTA and DKES calculations. The C+6

temperature and density were measured using the CHERS system. nH+ is found by subtracting the C+6 charge density from ne.

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components of the emission line. A spectral calibration is performed every shot using a neon calibration lamp to account for instrumental drift throughout the day.

The CXRS system measures C+6 density. Coronal equilibrium calculations and passive spectroscopic measurements indicate that significant populations of lower ionizations states of carbon are present, especially in the outer half of the plasma. Since no quantitative measurement of the density of the other ion species exists, an upper bound on the H+ density is set by the difference between the electron density and the C+6 charge density, and used as

an input for the neoclassical calculations. The flow velocity is measured at ten radial locations from two different directions as shown on the diagram in Figure 2. One direction is a measurement of the flow in an approximately poloidal direction while the other is in an approximately toroidal direction. The views are not orthogonal. The two measurements provide constraints on the local flow velocity. A third constraint on the flow velocity vector is provided by the assumption that there will be no net flow in the Ψ direction. The velocity measured by each view, shown in Figure 3(a), is a beam density weighted average of the local flow velocity along the viewing direction throughout each beam/view intersection volume. These measurements can be used to determine the flows parallel and

perpendicular to the direction of symmetry within a magnetic surface. Outside of r/a = 0.4, the errors due to the large beam width compared to the magnetic surface size are small. Figure 3(b) shows that the measured flows are predominantly in the direction of symmetry as one would expect (equivalent to the toroidal direction in a tokamak), with only small flow velocities in the perpendicular direction within the magnetic surface.

Figure 2 Drawing of the HSX CHERS system showing the neutral beam along with the viewing chords, the flux surfaces and a portion of the vacuum vessel

Fig. 3 (a) Measured flow velocity in the “toroidal” view (green) and the “poloidal” view (purple); (b) Projection of flow velocity in the quasihelically symmetric direction and its normal.

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The radial electric field, along with the pressure gradient, drive flow perpendicular to the magnetic field direction. A flow parallel to the magnetic field can arise as a result of the plasma viscosity. HSX was optimized for quasi-helical symmetry. This helical direction of

quasi-symmetry reduces flow damping along the helical direction [10], allowing significant parallel flows to arise. Parallel flows have been calculated both with the DKES code and with the PENTA code using the momentum conservation correction techniques. Momentum conservation has little effect on the particle fluxes and the ambipolar electric field, but a large effect is observed on the parallel flow as shown in Figure 4. The PENTA calculation is in reasonable agreement with the measurements over a large portion of the plasma, while DKES predicts virtually no parallel flow.

3. Equilibrium Currents

In the absence of an induced ohmic current or current drive, the parallel current consists of the Pfirsch-Schluter current (which integrates to zero on a flux surface) and the bootstrap current. The bootstrap current can be significant at higher beta values and can modify the rotational transform and the magnetic configuration. Accurately modeling this current is crucial in predicting performance in optimized stellarator devices. We have seen above that including momentum conservation in modeling quasisymmetric devices leads to considerably different results in the parallel flow compared to previous models, and in much better agreement with measurements. It could be expected that momentum conservation might have a large effect on the bootstrap current as well. Calculations with PENTA do show a strong dependence of the bootstrap current on the radial electric field. Experiments have been performed to examine the equilibrium currents in HSX. For the studies reported we have utilized a set of magnetic flux loops to measure signals external to the plasma and used reconstruction techniques (V3FIT [11]) to infer the current and pressure profiles which give good agreement with the magnetic diagnostic signals. The reproducibility of HSX discharges and low MHD activity provide reasonable signals for this analysis. Analysis of these signals has been used to demonstrate that the Pfirsch-Schluter current in HSX rotates helically in going around the torus and has a magnitude reduced by a factor of (N-m) (3 for HSX) from an equivalent tokamak. These results were reported at the IAEA in 2008 [12]. This factor algebraically goes through all of neoclassical transport for quasi-helical symmetry. Early in the discharge, the magnetic signals are dominated by the Pfirsch-Schluter currents which are set up rapidly on the equilibrium timescale. The unidirectional bootstrap current then evolves over a timescale longer than the discharge time in HSX.

Fig. 4 Measured parallel flow and neoclassical calculation with (red) and without (blue) momentum conservation.

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The total toroidal current in HSX is measured by a Rogowski coil encircling the plasma. Discharges were analyzed with both 100 kW and 50 kW ECH heating power, but only the 50

kW are reported here. The current evolution is even slower at 100 kW due to the increased electron temperature. Figure 5 shows the time evolution of the total current as measured by the Rogowski coil. To be sure that we were not driving current in a particular direction due to misalignment of the ECH launching structure, the field direction was reversed. The solid red curve is the current in this case (the dashed red curve has the sign reversed for comparison). This clearly shows no significant current driven in a specific direction through some external momentum input, and also illustrates the reproducibility of the discharges.

Figure 6(a) shows the profiles measured by the Thomson scattering and CXRS system used for analysis by PENTA, while 6(b) shows the electric field as calculated by PENTA. Near the core where we have a steep gradient in Te we can have either ion or electron root solutions. The bootstrap current density depends critically on the sign of the electric field in the multiple root region as shown in Figure 6(c).

(a) (b) (c) While the profiles and stored energy are stationary, the bootstrap current continues to evolve in time during the discharge. Fitting this evolution to an exponential L/R rise gives a steady-state value of ~380A with a rise time of ~50 ms. As seen in Figure 6(c), the bootstrap current in the core actually changes sign based on whether the plasma is in the electron or ion root in this region. If the core plasma is assumed to be in the electron root, then the integrated current is <10 A. If the ion root solution is assumed, the integrated current is of the order 260A, much closer to the experimental value. A 1-D diffusion equation for the rotational transform by Strand and Houlberg [13] is used to model the current evolution on the way to the steady-state conditions. This is the profile that is then used as input into the equilibrium reconstruction for analysis of the measured flux loop signals.

0 0.01 0.02 0.03 0.04 0.05 0.06-300

-200

-100

0

100

200

300

Am

ps

sec

50 kW ECH

Figure 5 Total toroidal current measured by the Rogowski coil for BT clockwise (red) and counter-clockwise

Figure 6 Penta input profiles, electric field, and bootstrap current density predictions

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To perform the reconstruction, an initial guess of the pressure and current profile is made. The initial pressure profile corresponds to the experimental measurements of the density and temperature. The initial current profile estimate has just been described. The reconstruction is performed by the V3FIT code which minimizes the mismatch between the measured signals and a model pressure and current profile. There are a total of 5 plasma parameters to be

optimized: two free parameters for the pressure profile, two for the current profile and one for the net toroidal flux within the last closed flux surface. Figure 7 shows how the initial pressure profile (shown in blue) based on the experimental data gives a good fit to the poloidal magnetic field measured by a B-dot coil array internal to the vacuum chamber. The black dashed line is the local signal predicted by V3FIT based solely on the Pfirsch-Schluter component of the current. The DC offset and slight variations seen in the blue trace are the modification of those signals introduced by the modeled bootstrap current, With the reconstruction (red trace), the χ2 value is reduced by 1/3, providing an even better fit to the coil data and slightly modifying the pressure profile as shown in Figure 8. Good agreement of the reconstructed current profile with the calculated profile has also been observed, however a wide range of profiles also show reasonable agreement with the data, indicating that the measured edge magnetic signals are not that sensitive to the small bootstrap current, especially given its localization to the core.

4. Summary

A direction of symmetry in quasihelically symmetric stellarators offers the possibility for large flows with positive impacts on confinement as in the tokamak. Large flows in the direction of quasisymmetry have been observed in HSX without any external momentum input. Agreement between the level of the parallel flow between calculations and experiment are only achieved if momentum conservation is included in the model through use of the PENTA code. Using the PENTA formulation allows calculations to be performed with a single code spanning cases from pure axisymmetry, to rippled/perturbed tokamaks, to quasisymmetric configurations, to full 3D systems.

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

Pa

/LCFS

Figure 7 Predicted and measured flux loop signals from the Pfirsch-Schluter and bootstrap currents (blue) and improved agreement with V3FIT reconstruction.

Figure 8 Initial pressure profile (blue) and reconstructed profile by V3FIT (red)

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Understanding the bootstrap current and its evolution is critical for future devices where the current will evolve over minutes or more. The current can have significant impact on the configuration and device performance. The bootstrap and Pfirsch-Schluter currents in HSX have been estimated from external flux loops signals modelled with PENTA and the V3FIT codes. An evolution equation has been applied to estimate the bootstrap current profile during its approach to steady-state. V3FIT has been used in a forward direction to predict signals from the modelled currents and in a reconstruction mode to achieve better agreement between the measured and predicted signals with slight profile adjustments. Results also point to the need for additional constraints and data (in addition to the present external flux loops) to definitively determine the current profile with high confidence. The results confirm the unique helical nature of the Pfirsch-Schluter current in HSX and the reduction in magnitudes of both currents by the predicted factor of three due to the quasihelical symmetry.

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