Page 6 - Inference of field reversed configuration topology and dynamics during Alfvenic transients

ARTICLE
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03110-5
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1000
5b 4.5
When GP priors are used, and linear relationships among current sources and measurements can be established, the CT solution involves non-iterative matrix operations and is then ideally suited for deterministic real-time applications. Because no equilibrium assumptions are used in this case, inference of plasma topology and dynamics up to Alfvenic frequencies then becomes possible. The FRC topology and dynamics have been determined during Alfvenic oscillations, with excellent self-consistency of results.
Methods
FRC approximations. The inference results of experimental data presented have been compared with first-order approximations for FRC parameters, which are summarized below. These are valid for an elongated FRC inside a FC of constant radius Rw with negligible field line curvature at the mid-plane, termed the long FRC approximation.
The radial pressure balance condition relates the magnetic field component in the axial direction right beneath the inner vessel walls at the o-point plane Bw with the average kinetic pressure of the plasma:
4 3.5 3 2000
c
1000 0 –1000
Fz
ma –kz Z0
0.4
PðrÞ ffi
B 2w B 2 ð r Þ
2μ : ð5Þ
0
a
0.1 0.2
0.3
t (ms)
0.5
So the average kinetic pressure of the plasma at the o-point (where the magnetic pressure is null) must necessarily be equal to the magnetic pressure at the confinement vessel walls
Pð0 pointÞ 1⁄4 B2w : ð6Þ 2μ0
When the plasma pressure is a flux function, and Eq. (5) is fulfilled, then the o- point radius is proportional to the separatrix radius 1.
R0 ffi Rs :
p2 ð7Þ
In addition to the former, the plasma is in axial force balance, and the maximum beta achievable by an ideal FRC surrounded by a perfect FC of constant radius Rw is given by the Barnes’ average β condition29,
2
βffi1 0:5 Rs ð8Þ
Rw
which depends solely on the ratio of separatrix radius Rs to FC wall radius Rw. When both axial and radial pressure balance are fulfilled, the flux at the o-point
Fig. 5 Inference of Alfvenic oscillations. a Hooke’s constant as obtained from the inferred magnetic configuration. b Estimated particle inventory as stored in the C-2U database. c Electromagnetic force (solid green), plasma mass times acceleration (red) and Hooke’s constant times axial plasma displacement (blue). The shaded area corresponds to one standard deviation
with hybrid and Grad–Shafranov equilibrium codes27, 28. (c) The magnitude of field reversal on axis is very significant and consistent with radial force balance (Eq. (10)) predictions. (d) The approximation (14) does not reproduce well the inferred FRC length. (e) There is a correlation between FRC length and plasma current, as expected from Eq. (11). However, the approximation (11) does not reproduce well the inferred results, partly because this approximation does not consider a current distribution flowing outside the separatrix. (f) Vessel current decays in about ~ 5 ms, comparable with the characteristic time over which the FRC is passively stabilized.
Discussion
We have used the CT method to provide a direct inference of the internal FRC magnetic topology, both during steady state and fast Alfvenic transients. The viability of the approach has been ver- ified in a number of ways, including comparisons with approx- imate results from a long FRC approximation, recovering of a force balance dynamic equation, and comparison with imaging of visible plasma emission.
All current sources have been modelled as GPs and inferred from external magnetic measurements using Bayesian analysis. Smoothing priors (for plasma current and vessel current dis- tributions) and a flux-conserving prior derived from Lenz’s law (for the magnet currents) have been used in the inference. From all the inferred current sources, FRC topology and dynamic properties have been obtained. This includes the separatrix geo- metry and the axial stability properties of the magnetic config- uration, among others.
(trapped flux) is given by
ψ0 ffi
B w R 3s Rw
: ð9Þ
The plane perpendicular to the machine axis that contains the o-point is termed the o-point plane. The intersection of the o-point plane with the machine axis determines the point were the axial component of the magnetic field is minimum. From Eqs. (5) and (9), the magnitude of the field reversal Bax at this point is
Bax ffi Rs Bw ð10Þ Rw
From Ampere’s law and Eq. (10), the total plasma current in a very elongated FRC of length L can be approximated by

IffiðBw BaxÞLffiBw 1þRs ð11Þ μ0 μ0 Rw
The plasma elongation is defined as
E1⁄4 L : ð12Þ
2Rs
A common approximation to separatrix radius and length comes from the excluded flux radius axial profile, which can be derived directly from magnetic sensors30, as explained below.
6 NATURE COMMUNICATIONS | (2018)9:691
| DOI: 10.1038/s41467-018-03110-5 | www.nature.com/naturecommunications
F (N) Nd (× 1019) Kz (N/m)