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Non-Axisymmetric Equilibrium Reconstruction for Stellarators, Reversed Field Pinches and Tokamaks

J.D. Hanson1, S.A. Lazerson2, D.T. Anderson3, M. Cianciosa1, P. Franz4, D.A. Gates2, J.H. Harris5, S.P. Hirshman5, K. Ida6, S.F. Knowlton1, L.L. Lao7, E.A. Lazarus5, L. Marrelli4, D. A. Maurer1, N. Pablant2, S. Sakakibara6, J.C. Schmitt2, A.C. Sontag5, B.A. Stevenson1, Y. Suzuki6, D. Terranova4

1Auburn University, Auburn, AL, USA 2Princeton Plasma Physics Laboratory, Princeton, NJ, USA 3University of Wisconsin, Madison, WI, USA 4Consorzio RFX, EURATOM-ENEA Association, Padova, Italy 5Oak Ridge National Laboratory, Oak Ridge, TN, USA 6National Institute for Fusion Science, Toki, Gifu, Japan 7General Atomics, San Diego, CA, USA E-mail contact of main author: jdhanson@auburn.edu Abstract. Axisymmetric equilibrium reconstruction using MHD equilibrium solutions to the Grad-Shafranov equation has long been an important tool for interpreting tokamak experiments. This paper describes recent results in applying non-axisymmetric (3D) equilibrium reconstruction codes to nominally axisymmetric plasmas (tokamaks and reversed field pinches), and fully non-axisymmetric plasmas (stellarators). Results from applying the V3FIT code to CTH and HSX stellarator plasmas, RFX-mod reversed field pinch plasmas and DIII-D tokamak plasmas and the STELLOPT code to LHD stellarator plasmas and DIII-D tokamak plasmas are presented.

1. Introduction

Determination of the experimental magnetohydrodynamic (MHD) equilibrium configuration of toroidal plasmas has become essential for understanding and optimizing stability and confinement in fusion research devices. Reconstruction of the experimental equilibrium from a combined set of magnetic, radiant and kinetic measurements is effected by minimizing the mismatch between modeled and observed signals by changing the parameters that specify the model equilibrium. Equilibria of idealized axisymmetric plasma in tokamaks and reversed field pinches (RFPs) are routinely reconstructed with procedures employing the Grad-Shafranov equation[1,2]. Stellarator equilibria are inherently non-axisymmetric, and cannot be reconstructed using traditional axisymmetric methods. Moreover, identification and creation of significant non-axisymmetric, or three-dimensional (3D) effects have become increasingly important to improving high-performance reversed field pinch and tokamak plasmas. 3D reconstruction tools with the functionality of proven two-dimensional reconstruction tools are needed to fully explore the exciting possibilities of utilizing 3D shaping to enhance performance of axisymmetric systems as well as 3D stellarators. This paper describes recent activities to develop and field two reliable, accurate 3D reconstruction codes, V3FIT and STELLOPT, for experimental use.

The V3FIT [3] and STELLOPT [4] codes both employ the VMEC [5] 3D equilibrium solver. In VMEC, a non-axisymmetric but field-period symmetric equilibrium is modeled with closed nested flux surfaces, without magnetic islands or regions of chaotic field lines. The pressure and either rotational transform or toroidal current profiles, as functions of either the toroidal

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or poloidal flux, must be specified. The radial profiles are specified with adjustable coefficients. These coefficients are used as reconstruction parameters. VMEC can treat both axisymmetric and non-axisymmetric configurations, both free- and fixed-boundary equilibria, and both stellarator-symmetric and non-stellarator-symmetric equilibria. For fixed-boundary equilibria, the shape of the outermost flux surface must be specified, while for free-boundary equilibria, the magnetic field due to current in external coils, and the total toroidal magnetic flux contained in the plasma must be specified.

Both V3FIT and STELLOPT can utilize signals from magnetic diagnostics, soft X-rays (SXR), Thomson scattering, and geometrical information from plasma limiters. STELLOPT can also utilize Motional Stark Effect (MSE) signals. V3FIT can also utilize chordal interferometry signals. Generally speaking, both codes reconstruct the equilibrium by numerically minimizing a function

! 2 (p) ! Si

o(d)" Sim (p)

" iS

#

$%

&

'(

2

i)

where ! 2 is the function to be minimized,

!

Sio(d) are the observed signals which depend on

the experimental data

!

d ,

!

Sim (p) are the model-computed signals which depend on the

reconstruction parameters p , and ! iS is the square root of the variance of the observed

signal. The reconstruction parameters are changed until a minimum in ! 2 is reached. Both codes calculate a finite difference approximation to the Jacobian matrix (partial derivatives of modeled signals with respect to reconstruction parameters) for the ! 2 minimization, and both use the approximate Jacobian to compute posterior parameter covariances. V3FIT and STELLOPT differ in the details of their minimization algorithms, their utilization of auxiliary profiles (like electron density, electron temperature, and soft x-ray emissivity), and in their computation of model signals.

For given equilibria, the model signals from machine-specific magnetic diagnostics are computed as a four-dimensional Biot-Savart integral over the plasma volume and along the diagnostic coil in V3FIT. The STELLOPT code utilizes a virtual casing principle whereby a three-dimensional Biot-Savart integral over the plasma surface and along the diagnostic coil is performed. A single soft X-ray (SXR) signal is computed as a one-dimensional integral of the SXR emissivity along the straight-line chord ending at the X-ray detector. Thomson scattering signals are modeled as point measurements of the electron temperature. Geometrical information about the plasma limiter or boundary is incorporated into the reconstruction primarily through the enclosed magnetic flux of the reconstructed equilibrium in V3FIT. The STELLOPT code includes limiter shapes, but relies on Thomson data to determine the plasma edge.

V3FIT is currently in use on stellarators (HSX, CTH), reversed field pinches (RFX-mod) and tokamaks (DIII-D) for a wide variety of studies including: interpretation of Pfirsch-Schlüter and bootstrap currents, vertical instabilities and density-limit disruption activity, design of new magnetic diagnostics, conformance of multiple data sources to a single set of flux surfaces, quasi-single helicity states in reversed field pinches, and error-field effects on nominally axisymmetric tokamak plasmas. STELLOPT is currently in use on stellarators (LHD) and tokamaks (DIII-D) providing detailed profile reconstructions for transport calculations and diagnostic inversions.

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2. Stellarators

As a first example, current-driven equilibria of discharges in the CTH stellarator/tokamak hybrid [6] are calculated with V3FIT. In CTH, ohmic currents of up to Ip = 70 kA are driven

to significantly modify the original vacuum rotational transform profile in order to study MHD stability and disruption susceptibility of hybrid discharges. The experimental rotational transform profile is obtained by V3FIT reconstruction using external magnetic diagnostics. The model current profile is parameterized in the form (1! s! )5 , where s is the fractional toroidal flux enclosed by a flux surface and ! is a reconstruction parameter characterizing the width of the current profile. A continuous reconstruction (~600 time slices) is shown in Figure 1 for a discharge with a vacuum (stellarator) transform of 0.13 at the plasma edge. The reconstructed parameters are the maximum enclosed toroidal flux

and the current profile shape parameter ! . The time-dependent reconstruction shows the peaking of the current profile during the current ramp-up as ! decreases with time. Similar

to the current rise phase of tokamak discharges in which surface kink modes are observed when the edge transform is at or somewhat below a rational value, the profile undergoes rapid narrowing as the net edge rotational transform passes through the rational values of 1/3 and 1 /2, when a large island is likely to define the edge of the plasma. Figure 2 shows the computed shape of the CTH plasmas at two different times, demonstrating the strong 3D shaping of both the vacuum and hybrid current-driven equilibria. On the Helically Symmetric Experiment (HSX) stellarator [7], V3FIT was used to perform equilibrium reconstructions of the plasma pressure and current profiles using

measurements from magnetic diagnostics. Profiles of the electron density and temperature

Figure 1: Edge iota, plasma current, and reconstructed current-profile parameter a vs. time for the CTH plasma. The inset plot shows the current profile at time 1.635 s. The gray bars around reconstructed a values and dashed lines in the inset indicate posterior parameter uncertainties.

Figure 2: CTH flux surfaces reconstructed with the V3FIT code, at times 1.60 s (above) and at 1.64 s (below). The upper surfaces are the vacuum surfaces.

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have been measured by the Thomson scattering system and ion temperature profiles with a CXRS system. The PENTA [8] code, which includes momentum conservation, was used to calculate the bootstrap current from these profiles for inclusion in the model. A poloidal array of coils outside of the separatrix measured radial and poloidal magnetic field components. The reconstruction of the plasma pressure profile and stored energy based on this poloidal array agrees reasonably well with that measured by Thomson scattering. Figure 3 shows the reconstructed plasma pressure profile compared to the measured pressure profile. The measured profile is based on Thomson scattering and Doppler spectroscopy. While the uncertainty in the current profile in the core is large, the total bootstrap current measured by the Rogowski coil was in agreement with the modeled total current. Several discharges on the LHD heliotron at the National Institute for Fusion Science have

been reconstructed with STELLOPT. Experimental data for the pressure and current profiles are provided by high resolution Thomson scattering, MSE, and magnetic diagnostics [9-11]. The Thomson scattering system provides approximately 136 electron temperature and density data points across the horizontally elongated cross section of the plasma. The total enclosed toroidal flux is varied during optimization in order to place the edge of the VMEC equilibrium at the points where the electron temperature goes to zero. This allows the location

of the plasma edge to be determined without the necessity of a limiter. The electron density profile is assumed to be a function of toroidal flux and is optimized to fit the Thomson measurements. No restriction on the edge electron density is enforced. The electron pressure is scaled by a multiplicative factor to allow the optimizer to match the measured diamagnetic diagnostic signal. The reconstructed electron temperature and Thomson data are shown together in Figure 4. The initial (dotted) and final (solid) profiles indicate that the reconstruction has found a more accurate value for the total enclosed toroidal flux and hence size of the plasma cross section. The reconstructed toroidal current profile in LHD is determined using measurements of the total toroidal

!

!

Pa

!/!LCFS Figure 3: Plasma pressure vs. flux surface label. The dashed black curve is the profile reconstructed with V3FIT, the solid red curve is measured, using Thomson scattering and Doppler spectroscopy data.

Figure 4: Electron temperature reconstruction for LHD shot 85384 at time 0.90 s. The dotted blue line indicates the initial profile and plasma edge. The solid line depicts the reconstructed pressure profile. A set of ten Akima spline knots equally spaced in toroidal flux parameterize the equilibrium Te profile.

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current and MSE polarimetry. The MSE signal provides a measurement of the change in magnetic field pitch angles from the vacuum pitch angle. In a high-beta conventional stellarator, the pressure driven currents are of the same order or larger than the toroidal

current drive. As a result, the MSE measurement reflects the effects of both net toroidal current and finite plasma beta. A goal of the reconstruction is to discriminate these two sources of the magnetic field through the inclusion of additional diagnostics. A set of Rogowski coils around the device provides a robust measurement of the net toroidal current. The MSE data helps determine the model current profile, which is parameterized by 10 spline knots equally spaced in toroidal flux. The experimental and reconstructed MSE polarization angles are shown in Figure 5. The accurately reconstructed equilibrium also allows for the inversion of ion temperature data from the X-Ray Imaging Crystal Spectrometer installed on the device [12].

3. Reversed Field Pinches

Reversed Field Pinches (RFPs) are nominally axisymmetric devices. With the recent observation of spontaneously generated helical states in all RFP [13] experiments, the shortcomings of equilibrium reconstructions based on the assumption of axisymmetry have become apparent. On the RFX-mod reversed field pinch in Padova, Italy, 3D equilibrium reconstructions are now routinely obtained with V3FIT. Magnetic diagnostic, Thomson scattering, and SXR emissivity data are being used to assess the conformance of the data to a single set of flux surfaces for plasmas in helical states.

RFX-mod is equipped with an extensive system of external magnetic measurements to provide local measurements of the three components of the magnetic field as well as toroidal and poloidal fluxes. If only magnetic diagnostic information is used for reconstructions, it is often the case that two distinct local minima of ! 2 can be found, one corresponding to a nearly axisymmetric equilibrium, and the other to a more helical equilibrium. The more helical equilibrium characteristically has a non-monotonic q-profile, and often has

Figure 2: Motional Stark Effect (MSE) reconstructions of LHD shot 82716 at 3.00 s. Blue open circles indicate reconstructed MSE signal, error bars indicate measured signal. Red open circles indicate equilibrium MSE signals that were not used in the equilibrium reconstruction.

Figure 6: Electron temperature measurements (left) and their remapping onto flux surfaces (right) for two different equilibria: one without pressure (middle) and one with pressure (right). Full and empty circles refer to measurement points located on the two sides of the helical magnetic axis.

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significantly reduced ! 2 compared to the axisymmetric equilibrium. The use of electron temperature measurements from the Thomson scattering diagnostic can often remove the ambiguity in the reconstructed equilibria [14]. The electron temperature information is included in the force balance (through the pressure profile) using measured electron densities and an assumption about the fraction of the total pressure contributed by the electrons. The combined use of magnetic and thermal measurements allows for a more accurately determined non-monotonic q-profile.

The Thomson scattering electron temperature observations can also be included in V3FIT reconstructions without a direct contribution to the pressure profile – only an assumption that the electron temperature is a flux-surface function is sufficient to improve equilibrium reconstructions. Soft X-ray (SXR) measurements can be similarly included in equilibrium reconstruction, using only the assumption that the emissivity is a flux surface function. The reason that an assumption of constancy on a flux surface can affect the equilibrium reconstruction is that observed data from more than one location on a given flux surface must have the same value. Figure 6 shows a clear example of this. In the left plot, the electron temperature, as measured by Thomson scattering, is plotted versus the major radius. Empty (full) circles indicate on the inboard (outboard) side of the helical magnetic axis. The middle plot and right plots show the same Thomson scattering data, now plotted versus the radial flux coordinate (flux surface label) s. In the middle plot, the flux surfaces are determined by a reconstruction with the assumption of zero pressure. In the right plot, the model pressure has an assumed radial profile, with a pressure scaling factor determined by the equilibrium reconstruction. Note that the Thomson scattering data for inboard and outboard sides of the same flux surface do not align in the middle plot, but do align in the right plot. The constant on a flux surface assumption of the electron temperature observations allows the pressure scaling factor to be determined.

4. Tokamaks

Recent work on DIII-D [15] shows that there is a discrepancy on the order of two centimeters between the vertical position of the separatrix as determined by axisymmetric equilibrium

reconstruction and that measured by Thomson scattering for some discharges with an externally applied n=1 magnetic perturbation. VMEC non-axisymmetric equilibrium calculations for some of these discharges show non-axisymmetric perturbations to the magnetic field structure that are on the order of half a millimeter. Figure 7 shows the magnitude of the spatial deviation of the outermost flux surface from the axisymmetric value. The current and pressure profiles used for this non-axisymmetric equilibrium calculation were taken from an axisymmetric kinetic EFIT reconstruction. The computed

Figure 7: Spatial deviation from axisymmetry of the last closed flux surface for poloidal slices at several toroidal angles of a DIII-D plasma with an externally imposed n=1 magnetic perturbation. Kinetic EFIT profiles were used for VMEC free-boundary run. Poloidal angle 0 (180) corresponds to the outboard (inboard).

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deviations are significantly smaller than those observed by Lao et al. Preliminary results of non-axisymmetric V3FIT reconstructions, where the pressure and current profiles are allowed to vary, indicate that the surface deviations from axisymmetry are larger than those of Figure 7, but still substantially smaller than those experimentally observed.

The 3D MHD equilibrium of DIII-D shot 142603 has been reconstructed by STELLOPT. The equilibrium was constrained by flux loops, Rogowski coils, Thomson scattering data, and MSE polarimetry. The reconstructed plasma shape indicates only slight boundary deviations from axi-symmetry with the largest deviations being approximately 9 mm. The bulk parameters for the plasma remained largely unchanged from their values as calculated by the 2D equilibrium fitting code EFIT [10]. The STELLOPT reconstruction finds a plasma current of 1.39 MA, plasma

beta of 1.78%, q95 of 3.4, and plasma volume of 19.66 m3. This up-down symmetric shot did not experience ELM suppression despite the application of an n=3 RMP field. Thomson scattering reveals penetration of applied 3D fields and flattening of the Thomson data around rational surfaces.

5. Conclusion

The V3FIT and STELLOPT codes have both been successfully used on multiple experiments for non-axisymmetric equilibrium reconstruction using experimental data. As the non-axisymmetric equilibrium reconstruction capabilities mature and increase in sophistication, and as our experience with 3D reconstruction grows, we expect to further improve our understanding of the effects of 3D shaping on axisymmetric and intrinsically non-axisymmetric plasmas, in preparation for ITER.

Appendix 1: References [1] LAO, L.L., et al., “Reconstruction of current profile parameters and plasma shapes in

tokamaks”, Nucl. Fusion 25 (1985) 1611. [2] ANDERSON, J.K., et al., “Equilibrium reconstruction in the Madison Symmetric Torus

reversed field pinch”, Nucl. Fusion 44, (2004) 162. [3] HANSON, J.D., et al., “V3FIT: a code for three-dimensional equilibrium

reconstruction”, Nucl. Fusion 49 (2009) 075031. [4] SPONG, D.A., et al., “Physics issues of compact drift optimized stellarators”, Nucl.

Fusion 41 (2001) 711.

Figure 8: Reconstructed DIII-D pressure profile from Thomson scattering (shot 142603). Flat regions correlate with rational surfaces.

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[5] HIRSHMAN, S.P. and WHITSON, J.C., “Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria”, Phys. Fluids 26 (1983) 3553.

[6] PETERSON, J., et al., “Vacuum magnetic field mapping experiments for validated determination of the helical coil location in stellarators”, Phys. Plasmas 17 (2010) 03205.

[7] ANDERSON, F.S.B., et al., “The Helically Symmetric Experiment, (HSX) goals, design and status”, Fusion Technology 27 (1995) 273.

[8] SPONG, D.A., “Generation and damping of neoclassical plasma flows in stellarators”, Phys. Plasmas 49 (2004) 056114.

[9] NARIHARA. K., et al., “Development of Thomson scattering diagnostics for the large helical device”, Fusion Engineering and Design 34-35 (1997) 67.

[10] IDA, K., et al., “Measurements of rotational transform with motional stark effect spectroscopy”, Fusion Science and Technology 58 (2010) 383.

[11] SAKAKIBARA, S. et al., “Magnetic measurements in LHD” Fusion Science and Technology 58, (2010) 471.

[12] PABLANT, N. et al., “Layout and results from the initial operation of the high-resolution x-ray imaging crystal spectrometer on the Large Helical Device”, Rev. Sci. Inst. 83 (2012) 083506.

[13] LORENZINI, R., et al., “Self-organized helical equilibria as a new paradigm for ohmically heated fusion plasmas”, Nature Phys. 5 (2009) 570.

[14] TERRANOVA, D., et al., “RFP helical equilibria reconstruction with V3FIT-VMEC” 38th EPS Conference on Controlled Fusion and Plasma Physics, Strasbourg, France, 27 June-1 July 2011, P5.084.

[15] LAO, L.L., et al., “Plasma Equilibrium Response to slowly rotating 3D Magnetic Perturbations in DIII-D RMP Experiments”, Bull. Am. Phys. Soc., 56 (2011) BAPS.2011.DPP.TP9.12 295.