Page 2 - Anatomy of a field-reversed configuration

Physics of Plasmas ARTICLE
respectable physics model on the other. The model should strike a bal- ance, neither too simplistic nor over-sophisticated. Overshadowing this is the goal of a fast tool, yielding results in nearly real time and certainly between experimental shots. Data inputs to Grushenka are from an array of magnetic-flux loops mounted at the wall of the con- finement vessel. This is distilled into a radius and half-length combina- tion {R/,Z/} to be defined shortly. The adopted physical model is an axisymmetric equilibrium of a rotating-fluid plasma with purely toroi- dal current. The model is constructed so as to account for kinetic effects in several respects.
The snapshot version of Grushenka reconstructs from data at a single time in an experiment. The timeline version uses a time sequence of data to create a sequence of snapshots of the evolving equilibrium and its properties. In the timeline, each successive snap- shot is treated as completely independent of its predecessor. No pre- sumptions are made about transport or relaxation, or sources that link successive snapshots. As such, Grushenka is not a transport code. However, the evolution of the equilibrium yields critical physical insights into transport, relaxation, and sources acting in the back- ground. Grushenka can also be operated as a hands-on analysis tool with operator-prescribed values {R/,Z/} to create a detailed snapshot of an equilibrium.
Several critical physical results follow from exercising Grushenka as an analysis tool. (a) FRC equilibria exist only within a limited range of inputs {R/,Z/} called the “shape domain.” (b) This domain is strongly affected by mirrors at the ends of the main confinement ves- sel. The limited computable-FRC domain reflects actual operational limitation in experiments. (c) Either an FRC (closed-field core) or a high-b mirror (no closed field core) is found, depending on the inputs. (d) Realistic plasma rotation as well as two-dimensionality strongly effect critical quantities such as the trapped flux and plasma energy, causing them to differ, sometimes markedly, from “standard” formu- las. (e) The plasma periphery plays an outsized role, accounting for the majority of the plasma energy and carrying a majority of the current in the confinement vessel.
The outline of the paper is as follows: The remainder of Sec. I reviews historical approaches to FRC equilibrium and reconstruction. Section II presents the method and details of “Grushenka.” Section III describes a variety of general equilibrium properties. Section IV presents timeline reconstructions of specific experiments and physical interpretations that follow. The paper concludes with a summary in Sec. V.
B. FRC equilibrium and reconstruction—Historical
Theoretical FRC equilibria have been computed since the early 1980s using the static-fluid model, the Grad–Shafranov (GS) equation. As reviewed in Sec. III of Ref. 1, these have been adapted to representa- tive snapshots of typical experimental FRCs by suitably adjusting the shape of the pressure function p(w) that appears in the GS equation. More recently, sophisticated FRC equilibrium models have been devel-
oped, e.g., multi-fluids3,4 as well as one with a fully kinetic treatment of
modern neutral-beam-sustained FRCs6 with lifetimes reaching tens of milliseconds, exceeding sac by three orders of magnitude. Any change on a time scale somewhat longer than sac allows the plasma to “keep up” as an evolving equilibrium. For example, the wall magnetic flux varies relatively slowly, whether the L/R time of the external circuit or a prescribed changing magnet current. Relaxation events may take longer than s ac but probably not greatly so since FRCs have no large toroidal field to retard instability growth. Beside “equilibrium,” other common approximations include (a) quasi-one-dimensionality, (b) force balance (radial and axial), and (c) thin SOL, see Ref. 7 (appendix) and Ref. 8. From these follow several standard approximation formu- las: (1) separatrix radius Rs 􏰂 R/; (2) separatrix half-length Zs 􏰂 Z/; (3) trapped flux /0 (magnitude of the poloidal flux at the O-point)
/0ðWbÞ 􏰂 BwðTÞR3/ðmÞ=RwðmÞ; (1) (4) plasma thermal energy
5 14
the ions.
Historically, “instant” reconstruction of FRCs from data has
relied on common approximations. Of these the most basic and the most reliable is “equilibrium.” Even in historical FRCs, plasma life- times <0.2 ms are 10–20 times the dynamical (acoustic) time scale, typically sac 􏰁 3–5 ls. Equilibrium is an even safer construct in
Phys. Plasmas 27, 112508 (2020); doi: 10.1063/5.0022663 Published under license by AIP Publishing
“EFIT” and other tools. In tokamaks, it has been advanced to a high level of sophistication, taking into account a range of diagnostic inputs. In the present case, the objective is less ambitious, focused on what can be learned from magnetic measurements alone, such data as are commonly available in FRC experiments. An advantage of the top- down approach is that it involves no guesswork about a transport
27, 112508-2
Ep0 􏰂 ð3=2ÞðB2w=2l0ÞV/; (2) and (e) core inventory N 􏰂 hn iV . Here, the flux-based separatrix vol-
Ð2 e/
ume is V/ 1⁄4 2p R/ dz (integral from mid-plane to z 1⁄4 Z/), the line-
averaged density hnei is from interferometric data, and l0 is the
permeability of free space. However, because of failures of these
assumptions, the formulas sometimes perform poorly, as pointed out
7,9
Historically, two reconstruction strategies have been pursued: “bottom-up” and “top-down.” The former adopts a particular plasma model including prescribed transport rates and computes forward in time to track the 2D evolution of the plasma. This approach is “bottom-up” because it builds on prescribed but adjustable transport rates posing as the underlying “physics.” The earliest such application to an FRC was the so-called “1–1/2D” construct,10,11 evolving a static- fluid equilibrium. Later Shumaker12 reconstructed the evolution of two experimental examples from the Field-Reversed-Experiment (FRX-C). A weakness of bottom-up is that it requires a hands-on guess of the transport model and an iterative process that may never con- verge. Further, it excludes relaxation phenomena (fundamentally 3D) despite the common view that FRCs are “relaxing” objects. Bottom-up methods are valuable for establishing points of understanding, such as demonstrating the failure of a particular transport mechanism13 but are impractical for fast reconstruction of experiment.
A top-down method is an alternate strategy: beginning with
observed data, work “downward” to reconstruct an equilibrium con-
sistent with the data. It relies on a plausible plasma model and uses an
automated procedure to solve for the equilibrium. Top-down recon-
struction has long been in common use in tokamaks, for example,
In particular, the actual separatrix radius Rs is only roughly R/: shorter elongation (E 1⁄4 Zs/Rs) causes Rs to rise above R/, while a thick peripheral plasma or a strong mirror can make it fall below. A more trustworthy method is a reasonable model with fully- 2D structure. This is the goal of the reconstruction tool developed here.
elsewhere.
scitation.org/journal/php