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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

    BE 22

     

      , 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: θTa res r Bvm

    ι nm

    E 

     [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 l