Functional perturbation theory under axisymmetry: Simplified formulae and their uses for tokamaks

TL;DR

This work shows that Functional Perturbation Theory (FPT) simplifies dramatically under axisymmetry in tokamaks, allowing closed-form, near-instantaneous predictions of how magnetic-topology features shift under perturbations. By deriving axisymmetric forms that replace all $\phi$-integrals with analytical results, the paper presents concise formulas for shifts of orbits and X/O-cycles, shifts of stable/unstable manifolds, and flux-surface deformation, leveraging matrix exponentials such as $\mathcal{D}\mathcal{P}^{m=±1}=e^{2\pi \mathbf{A}}$ and simple cycle-shift expressions like $\delta \mathbf{x}_{cyc} = - \mathbf{A}^{-1} \cdot \delta ( \frac{R B_{pol}}{B_{\phi}} )$. The axisymmetric framework also provides practical pathways to control detachment and maximize vacuum-vessel space by tuning $f_x$ and the eigenvalues of $\mathcal{D}P^m$, as well as robustly computing flux-surface deformations via $(\theta,r)$ flux coordinates. Overall, the approach enables real-time, topology-aware plasma control in tokamaks without requiring full 3D-field responses, while outlining higher-order considerations when $\mathcal{D}P^m$ nears unity and the potential for snowflake divertor configurations under degraded DP dynamics.

Abstract

In strictly axisymmetric configurations of tokamaks, field-line tracing reduces from a three-dimensional ODE system to a two-dimensional one, where Poincaré-Bendixson theorem applies and guarantees the nonexistence of chaos. The formulae of functional perturbation theory (FPT) mostly simplify to compact closed-form expressions to allow the computation to finish instantly, which could improve and accelerate the existing plasma control systems by detangling the plasma dynamics from the magnetic topology change. FPT can conveniently calculate how the key geometric objects of magnetic topology: 1. the divertor X-point(s) and the magnetic axis, 2. the last closed flux surface (LCFS) 3. flux surfaces change under perturbation. For example, when the divertor X-point shifts outwards, the LCFS there must expand accordingly, but not necessarily for other places of the LCFS, which could also contract, depending on the perturbation. FPT can not only facilitate adaptive control of plasma, but also enable utilizing as much as possible space in the vacuum vessel by weakening the plasma-wall interaction (PWI) via tuning the eigenvalues of $\mathcal{DP}^m$ of the divertor X-point(s), such that the field line connection lengths in the scrape-off layer (SOL) are long enough to achieve detachment. Increasing flux expansion $f_x$ is another option for detachment and can also be facilitated by FPT. Apart from the edge, FPT can also benefit the understanding of the plasma core. Since the magnetic axis O-point would also shift under perturbation and the shift is known by FPT, the O-point can be controlled without full knowledge of the plasma response, which shall not significantly change the tendency.

Figures (2)

• Figure 1: EAST PF coil vacuum fields per 1kA. The #103950 LCFS at $t=3.0s$ and the first wall are drawn for reference. The magnetic field calculation is based on formulae given by [labinac_magnetic_2006].
• Figure 2: Shifts $\delta \mathbf{\boldsymbol{X}}^{\text{u/s}}_{\text{pol}} (s)$ of the stable and unstable manifolds grown from the divertor X-point under the perturbation of 1kA PF1 coil vacuum field. The streamline in (a) represents the EFIT poloidal field while that in (b) represents the vacuum field of PF1 coil with 1kA current. The shift $\delta \mathbf{\boldsymbol{X}}_{\text{pol}} ^\text{u}(s)$ progressed on the unstable branch shall have the same normal component as that of $\delta \mathbf{\boldsymbol{X}}_{\text{pol}} ^\text{s}(s)$ progressed on the stable branch, which is verified by the arrows in (a). The shift in the normal direction is an intrinsic property in terms of differential geometry, unlike the shift in the tangent direction (which can be affected by the parameterization). The first variations $\delta \mathbf{\boldsymbol{X}}^{\text{u/s}}_{\text{pol}} (s)$ here are progressed by Eq. \ref{['eq:deltaXuspol_progression']}.