Numerical tests of formulae for volume enclosed by flux surfaces of integrable magnetic fields

TL;DR

The paper addresses efficiently computing the volume $V$ enclosed by flux surfaces for integrable magnetic fields by numerically testing and extending Mackay's 2024 volume formulas. It analyzes axisymmetric and toroidal-helical field examples, deriving 2D-reduction strategies via symmetry-based invariants, including a density-preserving extension (Theorem 3') and related density-averaging via Arnold-Liouville coordinates. Key findings show that Theorem 1 performs best for axisymmetric cases, while Theorem 3' provides a fast, accurate reduction for toroidal helices; Theorem 4, though valid, is typically slower, especially near separatrices. These results validate efficient volume computations in design-optimization contexts and guide method choice, with plan to extend to quasisymmetric and non-integrable scenarios and to implement vector-potential-based volume formulas; code is available for replication.

Abstract

Numerical tests of volume formulae are presented to compute efficiently the volume enclosed between flux surfaces for integrable 3D vector fields with various degrees of symmetry. In the process, a new case is proposed and tested.

Figures (13)

• Figure 1: Level sets of $\Psi$ on the poloidal section $\phi=0$ for $(m,n)=(2,1)$ and standard values given by (\ref{['eq:standard_values']}), in Cartesian (left) and symplectic (right) coordinates. The tori are coloured according to the region: inner (green), magnetic island (yellow), outer (magenta) and the separatrices (cyan).
• Figure 2: (Left) Volume enclosed by the flux-surface computed by (\ref{['V_general']}) (blue), (\ref{['eq:vol_lambda']}) (black), Theorem \ref{['thm1']} (red) and exact formula (\ref{['vol_axisym']}) (dashed grey) as a function of $\Psi$ for the field (\ref{['axisym_B']}) , with $C=1$. (Right) Example of the grid (top) and set of flux surfaces (bottom) used to compute (\ref{['V_general']}) and Theorems \ref{['thm1']}, respectively.
• Figure 3: Selection of the lattice generators $T_{1}$ and $T_{2}$ for a torus in the inner region. (Left): Generators $T_j$ in $(t_{u}, t_{v})$-coordinates, corresponding to the flow times under the $u$- and $v$-fields. (Right): Vectors $\tilde{T}_{j}$, the images of $T_{j}$ under $\varphi_T(0,0)$ in the $(\phi, \vartheta)$-plane. $\tilde{T}_1$ (in purple) is drawn over an orbit of $u$ (in green) through the initial point $(0,0)$; one translate of the $u$-orbit is also shown in green. The construction of $\tilde{T}_2$ requires the flow of $T$ units under $v$ (along the red curve) and then $-c$ units under $u$.
• Figure 4: Average $1/\hat{\rho}$ computed for subdivisions of sizes $q=4 - 12$: $\langle 1/\rho \rangle_q$, as a function of $\Psi$ in the three regions: (upper-left) inner, (upper-right) magnetic island, and (bottom) outer regions. The vertical lines indicate the value of $\Psi$ for the separatrix, and the horizontal dash--dot line indicates the numerical value of $\langle 1/\rho\rangle$ on the hyperbolic closed fieldline. Details of the plots near the separatrix are shown in dashed insets.
• Figure 5: Plot in the $(\phi,\vartheta)$ plane of the orbit (red) and the $u$-line (green) starting at the point $(\tilde{y},\tilde{z}) = (0.570, 0.211)$ within the magnetic island. The first intersection in this plot, between the orbit and the $u$-line, does not correspond to an actual crossing, but rather to the orbit passing through a different point $(\psi_1,\vartheta_1,\phi_1)$ on the flux surface that satisfies $\vartheta_0 = \vartheta_1-n\phi_1/m$ but not $\psi_1= \psi_0$.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• proof
• Theorem 4