Page 1 - Global particle simulation of field reversed configuration with field aligned mesh in cylindrical coordinates

Bayesian Inference of an FRC
J.A.Romero, S. Dettrick, M. Onofri and the TAE team
TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610
Motivations
FRC physics understanding benefits from advanced scientific analysis tools such as Bayesian analysis, which allow comparison of various physics models according to their data evidence.
When plasma variables are interlinked, Bayesian analysis combining the information of several diagnostics renders solutions with less uncertainty than any of the solutions obtained by independent analysis of each of the diagnostics.
Magneto-Kinetic control of an FRC requires real time inference of a reduced set of plasma variables, such as FRC magnetic flux, density, etc. When relationships between plasma variables and sensors are linear, inference using conjugate priors requires only linear operations and is then suitable for deterministic real time computations.
Bayesian methods
Forward Model
D=HX+ε D ( )
Inference model for Magnetics
Inference model for density profile X(ρ)
Sustained FRC discharges in C-2U
Current drive sustains the FRC against resistive losses. FRC lasts as long as the characteristic time of the external flux conserver (~5 ms)
Field reversal on axis found
with very large confidence, particularly early in the discharge.
Conclusions
Bayesian methods can extract useful physics even when the number of available diagnostics is small.
Inference using conjugate priors is good enough to get very reasonable solutions with little computational time, making it suitable for inter-shot analysis and future real time applications.
Bayesian analysis of field reversed configurations reveals strong field reversal on axis as well as “double hump” features on the radial density profiles. The later feature is only observed in global transport simulations in cases where significant fast ion pressure and current drive are present. Hence the result is indicative of strong fast ion current drive in the experiment.
References
E. Rasmussen and C. K. I. Williams (2006). Gaussian Processes for Machine Learning. The MIT Press. ISBN 0-262-18253-X
J. Svensson, A. Werner (2008). Current tomography for axisymmetric plasmas. Plasma Physics and Controlled Fusion, 50, N 8
J.A. Romero, J. Svensson (2013). Optimization of out-vessel magnetic diagnostics for plasma boundary reconstruction in tokamaks. Nuclear Fusion,Volume 53, Number 3.
E. Granstedt et al. Fast imaging and modeling of the C-2U outflow jet and pre-ionization plasmas. On this conference BP11.00047.
B. H. Deng. J. S. Kinley, and J. Schroeder. Electron density and temperature profile diagnostics for C-2 field reversed configuration plasmas. Review of Scientific Instruments 83, 10E339 (2012)
M. Onofri, S. Dettrick, D. Barnes, T. Tajima. Simulations of the C-2 and C-2U Field Reversed Configurations with the Q2D code. APS 2015.
Model is linear ( Biot –Savart) Inference of plasma, vessel and
flux conserving current in all magnets using magnetic probes.
Forward model schematics
Modelislinear D=∫Xdl→D=K⋅N SE type prior kernel Σ X
Asingleacousticmodulatorisusedfor
all lasers. As a result the sensor noise is correlated across all channels. ΣD Is then
non diagonal. Obtained from interferometer noise traces before the shot.
CO2 Laser LoS
Boundary conditions for flux at
the magnet location. ∂ψ ! 0 ∂t
Boundary conditions for plasma current j=0
Squared Exponential prior kernel
(Z−Z)2⎞ XX⎜σ2σ2⎟
⎛ (R−R)2 Σ(i,j)=σ2exp⎜− i j − i j ⎟
⎝R Z⎠
Data covariance kernel is diagonal ( independent sensor noise)
Typical interferometer noise for all the channels
Inferred density profile
Inferred results depend strongly on the interferometer noise model
When correlations of the interferometer noise are properly modeled
in ΣD a “double hump” structure appears.
Profile variable X (ρ ) Discretization X
Inference of Alfvenic transients
Flux | Probes (External/internal/Predicted)
D p(D|X) ∝exp(−εT ∑−1ε)
∂2ψ
k =∫j ext drdz
Measurements
(If noise ε is normal)
zφ2 ∂z
Likelihood
Using correlated noise model
Using uncorrelated noise model
Prior
(If X is Gaussian process)
()
∝exp−X−μ ∑ X−μ (( )T −1( )T
)
pX
X
PriorKernelexample(SE) ∑ Evidence
(
)
⎛⎞ (ρ − ρ' )2
ρ,ρ'
p(D)= ∫ p(D | X)p(X)dX
X
=σ2 exp⎜− ⎟ ⎜ 2λ2 ⎟
⎝⎠
Hooke’s and Newton law are recovered
Conjugatepriorssolutionwhen D=KX
∝ exp (X − μ) ∑−1 (X − μ) ( T )
SOL emission vs. inferred flux
Evidence of fast ion current
“Double hump” structure is consistent with Q2D transport simulations when a significant amount of fast ion pressure and current drive exists.
p(D | X)p(X) p(D)
p(X | D)=
Σ= K Σ K+Σ −1 μ=ΣKTΣ−1D
Emission from the 3d 3p transition 4+
(T) DID
Priorhyper-parameters θ={σ,λ,..} determinedfrom
⎧11⎫ θopt =argmaxθ ⎨⎩−2logKΣI (θ)KT +ΣD −2DT (KΣI (θ)KT +ΣD)D⎬⎭
(at 650.0 nm) of O compared
with the inferred flux surfaces.
Emission peaks in the SOL where
Te and Ne are low. Minimal emission from this spectral line is found in the core , where Te and Ne are sufficient to ionize O4+