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

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Non-Axisymmetric Equilibrium Reconstruction for Stellarators, Reversed Field Pinches and Tokamaks 1 TH/7-2 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: [email protected] 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 2 TH/7-2 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 o m 2 # S (d)" S (p)& (p) i i ! ! )% S ( i $ " i ' 2 o where ! is the function to be minimized, Si (d) are the observed signals which depend on m the experimental data d, Si (p) are the model-computed signals which depend on the S reconstruction parameters p , and ! i 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. 3 TH/7-2 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 Figure 1: Edge iota, plasma current, and reconstructed discharge with a vacuum current-profile parameter a vs. time for the CTH plasma. The (stellarator) transform of 0.13 at inset plot shows the current profile at time 1.635 s. The gray the plasma edge. The bars around reconstructed a values and dashed lines in the reconstructed parameters are the inset indicate posterior parameter uncertainties. 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. Figure 2: CTH flux surfaces reconstructed with the V3FIT code, at times 1.60 s (above) and at On the Helically Symmetric Experiment 1.64 s (below). The upper surfaces are the (HSX) stellarator [7], V3FIT was used to vacuum surfaces. perform equilibrium reconstructions of the plasma pressure and current profiles using measurements from magnetic diagnostics. Profiles of the electron density and temperature 4 TH/7-2 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 Pa 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
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