Page 3 - Anatomy of a field-reversed configuration

Physics of Plasmas ARTICLE
FIG. 1. Main plasma chamber.
model. Further, isolated relaxation events or benign instabilities pose no difficulty.
Such a tool was only recently developed in the FRC context,7
adopting the static-fluid model and a flexible (multi-parameter) pres-
sure function p(w). By manually adjusting the parameters in p(w) and
finding the equilibrium for each example, a database of nearly 500
equilibria relevant to the C-2 facility was constructed in advance.
Associated with each datum in this space are the shape dimensions
{R/,Z/} and other statistics. Given the target values of {R/,Z/} from
experiment, an interpolation extracts a host of statistics from the data-
base. This tool was used to reconstruct a timeline of a highly repeatable
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recently applied to FRCs was current tomography (CT).
system (r,h,z). These give rise to the familiar excluded-flux radius definition
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R/ðz; tÞ 1⁄4 R2wðzÞ 􏰃 /wðz; tÞ=pBzwðz; tÞ; (3)
where Rw(z) is the shaped radius of the confinement vessel. Two tangi- ble “shape dimensions” can be distilled from Eq. (3) as illustrated in Fig. 1: R/ is the maximum of the excluded flux radius; Z/ is the excluded-flux half-length, defined as the axial distance from the mid- plane to the point where R/(z,t) falls to 2/3 of its maximum.
In FRCs, the magnetics is poloidal-field dominated. Modest toroidal fields have been observed in translated FRCs but are small enough as not to compromise the high-b property (see Sec. IID of Ref. 2). Ampe`re’s law (h component) links the magnetic structure w to the plasma current jh,
􏰄
D w 1⁄4 􏰃l0rjhðw; rÞ;
condition on the C-2 experiment. Another top-down method
reconstructs the plasma current density jh(r,z) and the magnetic flux
w(r,z) from the magnetic diagnostics using Bayesian methods. This
has also been extended to include other diagnostics and to reconstruct
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II. RECONSTRUCTION METHOD A. Domain and magnetics
The present purpose is to reconstruct FRC equilibria beginning from tangible properties taken from routine magnetic diagnostics. These are the magnetic flux profile at the radial wall and three “point” values: the wall magnetic field at the mid-plane, and two principle dimensions (radius, half-length) of the plasma object. With a plasma model in hand one can then construct the intangibles: the fully-two- dimensional (2D) structure of the plasma object as well as many unmeasured quantities. Standard formulas, e.g., Eqs. (1) and (2), approximate some of these. A fully 2D reconstruction enables one to check the accuracy of these formulas.
The domain is the confinement vessel shown in Fig. 1, a shaped cylinder with a straight wall section of half-length Zstr and an attached conical end section. The plasma has mid-plane symmetry. The right boundary (z 1⁄4 Zm) is an open end-plane. The conical end section anticipates the application of a significant magnetic mirror at the ends of the main plasma chamber.
The inputs to Grushenka are readily available magnetic data: the wall flux /w(z,t) and the axial magnetic field at the wall Bzw(z,t). Here t is time and z is the axial coordinate in the cylindrical coordinate
Phys. Plasmas 27, 112508 (2020); doi: 10.1063/5.0022663 Published under license by AIP Publishing
(4) operator and
the electron density.
intensive, it is desirable to also have a fast reconstruction method built on a simple physics model.
Since Bayesian methods are computationally
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This method
D􏰄 1⁄4 r2$􏰀[(1/r2)$] is the Grad–Shafranov
w(r,z)1⁄4//2p is the magnetic flux per radian. The poloidal field is B1⁄4$w􏰅$h. The separatrix, illustrated by the dashed red line in Fig. 1, is the w 1⁄4 0 surface separating the closed-field core (w < 0) from the open-field periphery (w > 0). The radius Rs and half-length Zs are its semi-minor and semi-major axes. The solid red line is the excluded-flux radius profile Eq. (3). In the standard approximation (Sec. I B), {R/,Z/} are the approximations of the separatrix dimensions {Rs,Zs}.
Boundary conditions needed to complete the second-order equa- tion Eq. (4) are as follows. (1) A Dirichlet condition sets the “wall flux” at the radial boundary r 1⁄4 Rw(z), namely, w(Rw(z),z) 1⁄4 ww(z). In the end-cone proper (Fig. 1), the wall is treated as a magnetic surface (w 1⁄4 const). (2) Mid-plane symmetry requires a Neumann condition @w/@z 1⁄4 0 at z 1⁄4 0. (3) At the end-plane z 1⁄4 Zm, adopt a periodic boundary condition, @ w/@ z 1⁄4 0, a common artifice in equilibrium computations. Once a form for the current density jh is in hand, Eq. (4) can be solved by a relaxation algorithm. The methods for extracting from data a smooth ww(z) profile and the three point values Bw, R/, and Z/ are summarized in Appendix A. The right side of Eq. (4), specifically the current density jh, depends on the plasma model, which is described next.
B. Plasma model
Equation (4) links the magnetic structure w(r,z) to the plasma current density jh. Thus, finding the 2D structure calls for a physics-based flexible model for the current density, jh 1⁄4 F(r,w;{Ci}), where {Ci} is the set of adjustable parameters. The two size values {R/,Z/} are complemented by a commensurate pair of “adjustables” {Ci} i 1⁄4 1, 2.
1. Current density
The simplest plasma model is a static fluid for which the pressure is a “surface” function of w: p 1⁄4 p(w), i.e., the Grad–Shafranov system.
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jh 1⁄4 rP0ðwÞexpðmiX2r2=2kTtotÞ: (5)
27, 112508-3
where
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This is readily extended to accommodate azimuthal rotation. Accordingly, adopt the current density which is