ANULUM / SCPN Reactor Systems / Closed magnetic confinement / Physics

The physics of closed magnetic confinement

A closed-field device holds its plasma on nested toroidal flux surfaces, so a charged particle streams along a field line that never leaves the vessel. The five configurations of this family differ in who makes the field: external coils, the plasma's own current, or a relaxed mixture of both. This page collects the closed-form relations the family's device cores evaluate, from the externally dominated tokamak to the self-organised compact toroids.

What these relations are. Every relation on this page is one a device core of this family implements as a closed-form evaluation on a declared operating point at computational_prototype maturity: exact geometry, a vacuum field, empirical consistency instruments, and the fully relaxed states of an ideal cylinder. No equilibrium, stability, transport or current-drive equation is solved; the relaxed states are identities of a model that real devices depart from; no confinement, fusion power, gain or breakeven statement is made; no value describes or validates any real machine.

1 · Geometry and the vacuum field of a tokamak

A tokamak plasma is a torus of major radius $R_0$, minor radius $a$, elongation $\kappa$ and triangularity $\delta$. Its volume, exact for an elliptic cross-section by Pappus's theorem, and the vacuum toroidal field, which falls as one over the major radius, are the first two closed forms the tokamak core owns:

$$V = 2\pi^2 R_0\, a^2\, \kappa, \qquad B_t(R) = B_0\,\frac{R_0}{R}, \qquad \frac{B_{\mathrm{in}}}{B_{\mathrm{out}}} = \frac{R_0 + a}{R_0 - a}$$The volume ignores the triangularity, and a test asserts that it does; the field is the vacuum field, the plasma's paramagnetism and the discreteness of the coils are not modelled. The ordering inboard > axis > outboard is asserted, not assumed. (SCPN-TOKAMAK-CORE, level-0 device physics.)

Operational instruments, composed rather than restated

Two empirical consistency instruments live on the core's operational limits and are only read by the level-0 record: the Greenwald density limit in the form and units the source prints, and an elongation-corrected cylindrical safety factor. The normalised current sits beside them with the ceiling the spherical-torus source prints:

$$n_G = \frac{I_p}{\pi a^2}\ [10^{20}\,\mathrm{m^{-3}},\ I_p\ \mathrm{in\ MA}], \qquad q_{\mathrm{cyl}} = \frac{5\, a^2 B_t\,(1+\kappa^2)}{2 R_0 I_p}, \qquad I_N = \frac{I_p}{a B_t} \lesssim 7\ \mathrm{MA\,m^{-1}\,T^{-1}}$$$n_G$ is equation 1.3 of Greenwald 2002; the ceiling on $I_N$ and the pairing of aspect ratio with natural elongation are what Peng & Strickler 1985 print. These are consistency instruments with documented applicability, not predictions of disruption or stability, and the safety-factor floor is reported, never enforced.

2 · Rotational transform without a plasma current: the stellarator

A stellarator, heliotron or torsatron makes the rotational transform entirely with three-dimensional external coils, so it needs no net plasma current and no current drive. The family's stellarator core declares that geometry — major radius, average minor radius, number of field periods, rotational transform, coil realisation and count, field on axis, volume-averaged beta, flat-top duration — and the diagnostic and clock semantics of a three-dimensional device. It implements no level-0 physics yet: the three-dimensional equilibrium and stellarator solver work sits, explicitly, with SCPN-Fusion-Core. The stellarator page says exactly what is declared and what is not.

3 · The relaxed state: reversed-field pinch

In a reversed-field pinch the plasma current makes most of the confining field, and the plasma relaxes toward the Taylor state $\nabla\times\mathbf{B} = \mu\mathbf{B}$ with constant $\mu$. In a cylinder of minor radius $a$ that state is the Bessel-function model, written with the pinch parameter $\Theta$ and the reversal parameter $F$ as the source defines them:

$$B_\phi(r) = B_0 J_0(\mu r), \qquad B_\theta(r) = B_0 J_1(\mu r), \qquad \Theta = \frac{B_\theta(a)}{\langle B_\phi\rangle}, \qquad F = \frac{B_\phi(a)}{\langle B_\phi\rangle}$$ $$\mu = \frac{2\Theta}{a}, \qquad B_0 = \frac{\langle B_\phi\rangle\,\Theta}{J_1(2\Theta)}, \qquad F_{\mathrm{bfm}}(\Theta) = \frac{\Theta\, J_0(2\Theta)}{J_1(2\Theta)}, \qquad \Theta_{\mathrm{rev}} = \frac{j_{0,1}}{2} = 1.2024\ldots$$The toroidal field reverses inside the plasma at $r_{\mathrm{rev}} = j_{0,1}/\mu$ once $\Theta > \Theta_{\mathrm{rev}}$; the model is defined for $0 < \Theta < j_{1,1}/2 = 1.9158\ldots$, where $J_1(2\Theta)$ vanishes and $F_{\mathrm{bfm}}$ has its pole. The safety factor of the cylinder is $q(r) = (r/R_0)\,J_0(\mu r)/J_1(\mu r)$, with $q(0) = a/(\Theta R_0)$ by the series limit and $q(a) = F_{\mathrm{bfm}}\,a/(\Theta R_0)$. Paccagnella 2015, eqs. 4–5, single-region limit; the Bessel functions and their zeros from the shared kernel library. (SCPN-RFP-CORE.)

The source itself records that real reversed-field pinches depart from the fully relaxed state — its $F$–$\Theta$ curve is steeper than the operational range — so the model's reversal parameter is reported against the declared one as an advisory, never as a prediction.

4 · The relaxed state in a flux conserver: spheromak

A spheromak is a simply connected compact toroid: both field components are generated by internal plasma currents near a Taylor minimum-energy state, with no central column and no toroidal-field circuit linking the plasma. In a right circular flux conserver of radius $R$ and length $L$ with a conducting wall and end plates, the lowest axisymmetric solution of $\nabla\times\mathbf{B} = \lambda\mathbf{B}$ separates:

$$k_r = \frac{j_{1,1}}{R}, \qquad k_z = \frac{\pi}{L}, \qquad \lambda_{\mathrm{fc}} = \sqrt{k_r^2 + k_z^2}$$ $$B_z = B_0\,J_0(k_r r)\sin(k_z z), \qquad B_\theta = B_0\,\frac{\lambda}{k_r}\,J_1(k_r r)\sin(k_z z), \qquad B_r = -B_0\,\frac{k_z}{k_r}\,J_1(k_r r)\cos(k_z z)$$Divergence-free, satisfying the curl identity component by component, meeting the wall because $J_1(j_{1,1}) = 0$ and the end plates because the sine vanishes there. For the SSPX conserver, 1 m in diameter by 0.5 m high, $\lambda_{\mathrm{fc}} = 9.91\ \mathrm{m^{-1}}$ against the printed 9.9 (Wood 2005), within the declared one per cent. Doubling both extents halves the eigenvalue: the inverse-radius scaling PPPL-2257 prints. (SCPN-SPHEROMAK-CORE.)

Formation by helicity injection

A coaxial magnetised gun injects helicity with the figure of merit $\lambda_{\mathrm{gun}} = \mu_0 I_{\mathrm{gun}}/\psi_{\mathrm{gun}}$. The operating rule Wood 2005 prints is a disposition: hollow current profiles when $\lambda_{\mathrm{gun}} > \lambda_{\mathrm{fc}}$, peaked when below, relaxed when the ratio sits within a declared band around one. The core reports the ratio and the disposition; it predicts nothing about any discharge.

5 · The compact toroid without a toroidal field: FRC

A field-reversed configuration is a prolate compact toroid held by a poloidal field alone. The FRC core's level-0 physics is radial pressure balance across the separatrix and an empirical bound on kinetic scale:

$$p_{\max} = \frac{B_e^2}{2\mu_0}, \qquad \langle\beta\rangle = 1 - \frac{x_s^2}{2}, \qquad S^* = \frac{r_s}{\delta_i},\; \delta_i = \frac{c}{\omega_{pi}}, \qquad E = \frac{l_s}{2 r_s}, \qquad \frac{S^*}{E} < 3.5, \qquad v_A = \frac{B_e}{\sqrt{\mu_0 n m_i}}$$The plasma pressure at the null carries the whole external magnetic pressure; averaging over the separatrix cross-section gives the average-beta relation. The bound 3.5 is printed by Bala et al. (arXiv:2204.07978, eq. 14); a configuration inside it is ordered, not claimed stable. The split between electron and ion temperature is not modelled. (SCPN-FRC-CORE.) The pulsed merge-compression of two FRCs belongs to the magneto-inertial family.

6 · What the closed forms leave to the solver laboratory

Equilibrium. The Grad–Shafranov equilibrium, free-boundary and predictive, and its three-dimensional stellarator counterpart are SCPN-Fusion-Core surfaces; a device core detects an accepted surface and never copies it.
Stability and disruption. MHD stability criteria, tilt of compact toroids, tearing and relaxation dynamics are solver work; the cores carry empirical instruments and advisory bands (RFP tearing and relaxation fluctuations at 1–500 kHz after Ortolani & Schnack 1993; spheromak tilt, shift and relaxation activity at 1–200 kHz after Bellan 2000), reported and never clamped.
Transport, current drive, helicity balance, resistive decay. Not evaluated by any device core in this family.

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Explorer

Tokamak instruments, RFP profiles across the reversal, spheromak eigenvalue and formation, FRC balance — live.

The five configurations

What each core owns, excludes and anchors on.

Sources

Every printed value and where it comes from.