Page 9 - Inference of field reversed configuration topology and dynamics during Alfvenic transients

NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03110-5
ARTICLE
1.2 1.0 0.8 0.6 0.4 0.2
Formation
Confinement
Formation
confinement vessel and 6 FC magnets, which can be used as passive FCs or be connected to power supplies.
The magnetic measurement system on C-2U32, 33 comprises a set of 19 magnetic pick-up probes placed inside the confining vessel and 8 external pick-ups located right underneath the 8 EQ magnets (Fig. 8). There are also Rogowski-based current measurements for all the FC magnets currents IEQ and also for some of the EQ magnets currents IEQ. For the rest of the magnets, only the set point used for its control is known.
The inference problem at hand requires finding the most likely solution for the elements of the total plasma current distribution arranged in a vector IP, along with the most likely solution for current induced in the confining vessel IV and all the magnets IM. A diagram illustrating the magnet location and grid used for the current distribution is shown in Fig. 8.
All the currents to be inferred are arranged into a single current vector
I 1⁄4 fIP; IV; IMg: ð29Þ
All current sources are modelled as GPs as described earlier. The information available to perform the inference comes from (i) set points for all IM, (ii) current measurements for IFC and a few IEQ, (iii) measurements of magnetic field at several locations outside the plasma region, both inside and outside the confining vessel, (iv) null boundary conditions for plasma current distribution, and (v) null boundary conditions for the flux change underneath the equilibrium magnets
0
–8 –6 –4 –2 0 2 4 6 8
Z (m)
Fig. 8 C-2U magnetostatic model. Toroidal plasma current is modelled using 734 discrete plasma current elements modelled as block coils (back grid inside the vessel contour in blue). The vessel is modelled using 31 flat block coil elements (not shown). Insulating quartz tubes (formation sections) are shown in red. All the magnets in confinement and formation sections are considered. Pulsed powered fast switching coils for plasma formation and acceleration (not shown) are located right outside the quartz tube. Magnetic probes inside and outside the confinement vessel are shown as red circles
The posterior for the hyper-parameters is p(θ|D), which from Bayes theorem is pðθjDÞ 1⁄4 pðDjθÞpðθÞ ð24Þ
pðDÞ
where p(D|θ) is the likelihood and p(θ) is the hyper-prior (prior for the hyper- parameters).
∂ψ ffi 0, which behave as perfect FCs on the timescale of the discharge.
In Bayesian model selection, the optimum set of hyper-parameters θopt is selected to maximize this probability.
θopt 1⁄4 argθ maxðpðθjDÞÞ
solutions where the plasma current distribution drops to zero at the domain, and the flux is conserved at the magnet locations (flux-conserving prior).
From the inferred currents in I is then straightforward to calculate the poloidal flux and magnetic field components on the domain grid using the matrix representations M,GR,GZ of the Biot–Savart operator34:
ψ 1⁄4 MI;
BR 1⁄4 GRI; ð30Þ BZ 1⁄4 GZI:
Main plasma shape and position variables of interest for control such as x- point, o-point and separatrix radius can then be obtained directly by searching for nulls on the magnetic field and flux along the axis and mid-plane. Low-order moments of the plasma current distribution of interest for control such as total plasma current or the axial position of current centroid can likewise be obtained from linear operations.
The prior over the hyper-parameters p(θ) in Eq. (24) is usually taken to be flat, since there is no indication of what are the best hyper-parameters before seeing the data. In this case, the optimal set of hyper-parameters that maximizes likelihood of the data with respect to the hyper-parameters is
θopt 1⁄4 argθmaxðpðθjDÞÞ 1⁄4 argθmaxðpðDjθÞÞ ð26Þ
Given a set of hyper-parameters θ, there is an infinite class of plasma profiles X (r) that can be generated by the corresponding prior covariance p(X|θ) through the corresponding GP. The quality of the data fit must be evaluated not just for one particular solution but for all the solutions that can be obtained for a given set of hyper-parameters. The likelihood should be integrated out (marginalized) with respect to all these possible profiles generated by a single set of hyper-parameters, so it becomes a marginal likelihood.
ð25Þ
ð31Þ
∂t
The boundary conditions (iv) and (v) are built directly into the prior, to obtain
IP 1⁄4 sumðIPÞ; z0IP 1⁄4 zTIP :
Data availability. All relevant data supporting the findings of this study are available from the authors on request.
Received: 12 October 2017 Accepted: 19 January 2018
References
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θopt 1⁄4 argθmaxðpðDjθÞÞ 1⁄4 argθmax
R
pðDjX; θÞpðXjθÞdX
ð27Þ
In the particular case at hand where p(X) is a GP, the likelihood is normal and the model linear, the marginal likelihood can be calculated analytically. The expression for its logarithm is16
1 1 1n
L1⁄4 log KΣXKT þΣD DT KΣXKT þΣD D logð2πÞ:
ð28Þ
222
For any given prior kernel, the maximum of the expression (28) with respect to
the hyper-parameters gives the optimal set of hyper-parameters that explain D.
Inference model for the C-2U device. The C-2U magnetic model used for the analysis comprises a total of 42 magnets, 31 vacuum vessel (passive) segments and a current distribution made of 734 discrete plasma current elements modelled as block coils (Fig. 8). Of special relevance are 8 equilibrium (EQ) magnets in the
NATURE COMMUNICATIONS | (2018)9:691 | DOI: 10.1038/s41467-018-03110-5 | www.nature.com/naturecommunications 9
R (m)