GS-DeepNet: mastering tokamak plasma equilibria with deep neural networks and the Grad-Shafranov equation

The force-balanced state of magnetically confined plasmas heated up to 100 million degrees Celsius must be sustained long enough to achieve a burning-plasma state, such as in the case of ITER, a fusion reactor that promises a net energy gain. This force balance between the Lorentz force and the pressure gradient force, known as a plasma equilibrium, can be theoretically portrayed together with Maxwell's equations as plasmas are collections of charged particles. Nevertheless, identifying the plasma equilibrium in real time is challenging owing to its free-boundary and ill-posed conditions, which conventionally involves iterative numerical approach with a certain degree of subjective human decisions such as including or excluding certain magnetic measurements to achieve numerical convergence on the solution as well as to avoid unphysical solutions. Here, we introduce GS-DeepNet, which learns plasma equilibria through solely unsupervised learning, without using traditional numerical algorithms. GS-DeepNet includes two neural networks and teaches itself. One neural network generates a possible candidate of an equilibrium following Maxwell's equations and is taught by the other network satisfying the force balance under the equilibrium. Measurements constrain both networks. Our GS-DeepNet achieves reliable equilibria with uncertainties in contrast with existing methods, leading to possible better control of fusion-grade plasmas.

Development of a filtered AXUV diode array for X-pinch soft x-ray spectra in the energy range of 1-10 keV

We develop a filtered absolute extreme ultraviolet (AXUV) diode array to measure the time evolution of the soft x-ray spectrum in the energy range of 1-10 keV. AXUV-HS5, the detector, has a fast rise time of 0.7 ns, a wide energy detection range, and high accessibility. We use Geant4 simulations to design an appropriate filter set for flat-and-sharp virtual channels (VCs), where a filter with no spectral edge removes large tails of the response curves. A Levenberg-Marquardt (LM) method, sensitive to the expected spectral function, is improved to reliably generate a continuous radiation spectrum, by utilizing spectral information from the least-squares (LS) method that reconstructs a discrete spectrum with low spectral resolution directly from the VCs. We test the filtered AXUV diode array on an X-pinch device with a peak current of 140 kA at Seoul National University; the array with ten channels is installed in a vacuum chamber. For a two-wire 40 μm stainless steel X-pinch, x-ray power, radiation temperature, and the reconstructed x-ray spectrum are obtained from the filtered AXUV diode array by the combined LS-LM method.

A modular X-pinch device for versatile X-pinch experiments at Seoul National University

This paper describes an X-pinch device recently developed at Seoul National University (SNU). The SNU X-pinch device is designed and fabricated to accommodate various diagnostics as well as conduct versatile experiments. It is easy to change the capacitance of the pulse generator because the capacitor bank has a modular design without insulation oil or gas. This allows us to perform a variety of experiments with a wide capacitance range from 80 to 800 nF. The operating voltage of the SNU X-pinch device is controlled from 20 to 100 kV by adjusting the gas pressure inside a triggered spark-gap switch. Triggering of the spark-gap switch is synchronized with the operation of a pulsed laser to diagnose the X-pinch plasma at the proper time. A large vacuum chamber precisely machined from an aluminum mono-block is attached to the top of the pulse generator. It is designed to accommodate not only various X-pinch loads but also various diagnostic apparatus such as optical components. Initial experiments with the SNU X-pinch device have successfully generated x rays with wires of various materials and sizes. The device will be used not only to explore the dynamics of X-pinch plasmas but also as a test stand for diagnostics of high-energy-density plasmas.

Data analysis scheme for correcting general misalignments of an optics configuration for a voltage measurement system based on the Pockels electro-optic effect

Having a sub-ns response time and not requiring physical contacts to the measurement points, a voltage measurement system based on the Pockels electro-optic effect, referred to as a PE (Pockels effect)-based voltmeter, is widely used for pulsed high voltage devices such as accelerators and X-pinch systems. To correct for the misalignment of a Pockels cell and the transmittance ratio of a beam splitter, a polar-coordinate-based data analysis scheme has been proposed. This scheme also overcomes a limitation on the measurable range of a PE-based voltmeter without ambiguity and can measure the half-wave voltage of a Pockels cell. We present an improved polar-coordinate-based data analysis scheme using an ellipse fitting method, which can correct for misalignments of all the optics components of a PE-based voltmeter while keeping the advantages of the previous scheme. We show the results of the improved data analysis scheme for measuring a slowly modulated voltage up to approximately 5 kV in about 30 s and a pulsed high voltage up to 7 kV with a rise time of less than 20 ns.

The new single crystal dispersion interferometer installed on KSTAR and its first measurement

Dispersion interferometers have been used to measure line integrated electron densities from many fusion devices. To optically suppress noise due to mechanical vibrations, a conventional dispersion interferometer typically uses two nonlinear crystals located before and after the plasma along the laser beam path. Due to the long beam path, it can be difficult to overlap the fundamental and second harmonic laser beams for a heterodyne dispersion interferometer and to focus the beams on the second nonlinear crystal located after the plasma, especially when the aperture of the nonlinear crystal is small, i.e., of the order of mm. To overcome such difficulties, a new concept of a heterodyne dispersion interferometer, a single crystal dispersion interferometer (SCDI), is developed and installed on KSTAR with the laser wavelength of 1064 nm. The concept and the optical setup of the KSTAR SCDI are discussed, as well as its first measurement during a shattered pellet injection that produces abrupt and large changes in the electron density. To demonstrate feasibility, the KSTAR SCDI measurements are also compared with those from the existing two-color interferometer.

Development of a forward model for Bayesian analysis of a single crystal dispersion interferometer

The Single Crystal Dispersion Interferometer (SCDI) is a newly developed dispersion interferometer (DI) system installed on KSTAR and has obtained the first data successfully in January 2020. Unlike conventional heterodyne DI systems, which use two nonlinear crystals, only one nonlinear crystal is used to eliminate the difficulty in overlapping the first and second harmonic beams, aligning and focusing the beams to a small aperture of the second nonlinear crystal, and resolving a problem of significant efforts to maintain the beam alignment to the second nonlinear crystal after a long beam transmission. The second nonlinear crystal is replaced by a frequency doubler, a simple electronic component. To infer a line integrated electron density with its associated uncertainty consistent with the measured data, we develop a forward model of the KSTAR SCDI that can be used as a likelihood within a Bayesian-based data analysis routine. The forward model consists of two main parts, which are an optical system and an electronics system, and it takes into account noises by modeling the mechanical vibrations and the electronic noises as Gaussian distributions, while the photon noise is modeled with a Poisson distribution. The developed forward model can be used for designing and improving the SCDI system.

Imputation of faulty magnetic sensors with coupled Bayesian and Gaussian processes to reconstruct the magnetic equilibrium in real time

A Bayesian with Gaussian process-based numerical method to impute a few missing magnetic signals caused by impaired magnetic probes during tokamak operations is developed such that the real-time reconstruction of magnetic equilibria, whose performance strongly depends on the measured magnetic signals and their intactness, is affected minimally. Likelihood of the Bayesian model constructed with Maxwell's equations, specifically Gauss's law for magnetism and Ampère's law, results in an infinite number of solutions if two or more magnetic signals are missing. This undesirable characteristic of the Bayesian model is remediated by coupling the model with the Gaussian process. Our proposed numerical method infers nine non-consecutive missing magnetic signals correctly in less than 1 ms suitable for the real-time reconstruction of magnetic equilibria during tokamak operations.

Bayesian modelling of the emission spectrum of the Joint European Torus Lithium Beam Emission Spectroscopy system

A Bayesian model of the emission spectrum of the JET lithium beam has been developed to infer the intensity of the Li I (2p-2s) line radiation and associated uncertainties. The detected spectrum for each channel of the lithium beam emission spectroscopy system is here modelled by a single Li line modified by an instrumental function, Bremsstrahlung background, instrumental offset, and interference filter curve. Both the instrumental function and the interference filter curve are modelled with non-parametric Gaussian processes. All free parameters of the model, the intensities of the Li line, Bremsstrahlung background, and instrumental offset, are inferred using Bayesian probability theory with a Gaussian likelihood for photon statistics and electronic background noise. The prior distributions of the free parameters are chosen as Gaussians. Given these assumptions, the intensity of the Li line and corresponding uncertainties are analytically available using a Bayesian linear inversion technique. The proposed approach makes it possible to extract the intensity of Li line without doing a separate background subtraction through modulation of the Li beam.

Experimental signatures of critically balanced turbulence in MAST

Beam emission spectroscopy (BES) measurements of ion-scale density fluctuations in the MAST tokamak are used to show that the turbulence correlation time, the drift time associated with ion temperature or density gradients, the particle (ion) streaming time along the magnetic field, and the magnetic drift time are consistently comparable, suggesting a "critically balanced" turbulence determined by the local equilibrium. The resulting scalings of the poloidal and radial correlation lengths are derived and tested. The nonlinear time inferred from the density fluctuations is longer than the other times; its ratio to the correlation time scales as ν(*i)(-0.8 ± 0.1), where ν(*i) = ion  collision rate/streaming rate. This is consistent with turbulent decorrelation being controlled by a zonal component, invisible to the BES, with an amplitude exceeding those of the drift waves by ∼ ν(*i)(-0.8).