Non-Perturbative Measurements of Low and Null Magnetic Field in High Temperature Plasmas Using the Hanle Effect
P. 1

  High-fidelity Bayesian inference of transient FRC plasma perturbations in C-2W
P.C. Norgaard1, T. Baltz1, M. Dikovsky1, S. Geraedts1, I. Langmore1, T. Madams1, N. Neibauer1, J. Platt1, R. von Behren1,
J. Romero2, S. Dettrick2, M. Thompson2, E. Trask2, H. Gota2, R. Mendoza2, N. Bolte2, T. Roche2, and the Google/TAE Team1,2 (1) Google LLC, 1600 Amphitheatre Parkway, Mountain View, CA 94043
  (2) TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610
61st Annual Meeting of the APS Division of Plasma Physics, October 21-25, 2019 • Fort Lauderdale, Florida
Analysis via Bayesian inference is initiated after the C-2W instrument data and machine settings for an experimental shot are uploaded to Google Cloud. The analysis can be run for an individual shot or for hundreds of shots by distributing the work using the Borg cluster-management system[1]. Each shot is divided into intervals of about 60 times, and each interval is processed independently on a separate GPU. Computation is substantially accelerated by running on GPUs using TensorFlow software implementation[2]. The current system can analyze about 20 time frames of data from C-2W per GPU hour. The results are collected into an hdf5 file, a summary html page (example shown below), and a more detailed html for debugging and detailed inspection, which is sent back to TAE servers. [1] [2]
 Summary html of the electron density reconstruction for a C-2W shot. The top row shows the relative amplitude of each azimuthal modes vs time. When the amplitude is small compared to the statistical uncertainty, the mode is not plotted. The second row plots the corresponding statistical samples for rotational frequency of each mode using the same time axis as the first plot.
The third and fourth row correspond to three time values indicated by dotted lines in the first two plots. On top is a sample of reconstructed midplane electron density. Below the radial density profile from the plasma center is plotted for the axisymmetric component and azimuthal modes.
The authors wish to thank the many team members at TAE and Google who contributed significant work, including the
images here, to make the poster publication possible.
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The midplane of C-2W has the most instruments and measurements of any section of the confinement vessel, making it optimal for reconstruction of the plasma profile. An electron density profile and azimuthal mode structure is reconstructed by combining data from FIR interferometry, secondary electron emission from the neutral beams (SEE), and Mirnov probes in the two azimuthal arrays closest to the midplane.
Seven FIR interferometer beams measure chord-integrated electron density in the midplane, while seven more are tilted 15 degrees. When projected onto the midplane this provides 14 chords with 8.1 cm spacing. Interferometry is the most accurate measure of electron density, but since it is chord-integrated on nearly parallel paths it cannot distinguish certain features of translation and azimuthal modes, so additional instruments are used to improve modeled density resolution.
Shine through from the eight neutral beams is measured by secondary electron emission detectors located on four diagnostic ports at the midplane. The out-of-plane projection of these beams are incorporated into the model to bound expected model error from plasma translation and 3d shape.
Mirnov Probes
Our present analysis uses the two arrays of eight Mirnov probes closest to the midplane. The Bz and Bθ components help the model differentiate the rotational direction and frequency of azimuthal perturbation modes.
  and Mirnov probe locations (blue), as located in the C-2W confinement vessel.
A holistic, high-fidelity plasma reconstruction based on Bayesian inference is applied to study the advanced beam-driven Field Reversed Configuration (FRC) in the C-2W machine[1] at TAE Technologies, Inc. The method employs a statistical distribution of possible plasma states, including variables that persist across time points to enhance resolution of plasma dynamics. This implementation of “time-linked” Bayesian inference is used to evaluate electron density fluctuations at the C-2W midplane. Likely values and statistical confidence intervals are generated for density perturbation mode amplitudes, frequencies, and radial profiles. Synthesis of multiple high-frequency diagnostics provides significantly increased reconstruction fidelity compared to single instrument analysis. These include FIR interferometry, Mirnov magnetic field probes, and neutral beam induced secondary electron emission detection. This analysis is performed at
2 us resolution for the entire 10-30 ms plasma lifetime via distributed computing. Results include evaluation of the midplane density profile evolution, and observation of low-level intermittent azimuthal modes that rotate about the machine axis with frequencies in the range of 10-100 kHz. [1] H. Gota et al., Nucl. Fusion 59, 112009 (2019).
Bayesian Inference
We employ Bayesian inference to obtain a statistical description of the electron density profile that is consistent with the plasma diagnostic measurements. The resolution of the method is significantly improved by "time-linking", in which the change of a random variable between time steps is also modeled by a statistical process.
A parameterized model is used to represent the plasma state. Our midplane profile model consists of three parts:
- The axisymmetric part of the density
- A translational shift in the midplane
- Azimuthal perturbation modes
The prior specifies conditions on the random variables for each parameter. Samples drawn from the prior represent possible plasma states. Some samples corresponding to the parameterization above are plotted below.
Forward Model
The forward model transforms a set of random variables to a corresponding set of measurements. For interferometry, the density is integrated along the laser path. Noise terms are needed to provide for discrepancy between the parameterized representation and real physical system.
The posterior is obtained from the defined prior and forward model, and allows for statistical sampling of the parameterization (z) given a set of instrument measurements (mtrue). We use Hamiltonian Monte Carlo (HMC) to sample from the posterior.
With sufficient samples of the posterior, the reconstructed profile and credibility bounds are obtained from the mean value and quantiles.
The predicted measurement interval is computed by running posterior samples through the forward model. This allows for comparison to the plasma diagnostic measurements.
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Blue points are the measurements from the interferometry chords. Blue shading is the predicted measurement interval, which is computed by running posterior samples through the forward model.
Red line is the mean of the posterior samples. Red shading is the confidence interval, similar to one standard deviation error bounds.
FIR interferometry chords (pink), neutral beam paths for SEE (purple),