An alternative approach to field-aligned coordinates for plasma turbulence simulations

Turbulence simulation codes can exploit the flute-like nature of plasma turbulence to reduce the effective number of degrees of freedom necessary to represent fluctuations. This can be achieved by employing magnetic coordinates of which one is aligned along the magnetic field. This work presents an approach in which the position along the field lines is identified by the toroidal angle, rather than the most commonly used poloidal angle. It will be shown that this approach has several advantages. Among these, periodicity in both angles is retained. This property allows moving to an equivalent representation in Fourier space with a reduced number of toroidal components. It will be shown how this duality can be exploited to transform conventional codes that use a spectral representation on the magnetic surface into codes with a field-aligned coordinate. It is also shown that the new approach can be generalised to get rid of magnetic coordinates in the poloidal plane altogether, for a large class of models. Tests are carried out by comparing the new approach with the conventional approach employing a uniform grid, for a basic ion temperature gradient (ITG) turbulence model implemented by the two corresponding versions of the ETAI3D code. These tests uncover an unexpected property of the model, that localized large parallel gradients can intermittently appear in the turbulent regime. This leaves open the question whether this is a general property of plasma turbulence, which may lead one to reconsider some of the usual assumptions on micro-turbulence dynamics.

💡 Research Summary

Plasma micro‑turbulence in toroidal devices exhibits a pronounced “flute” property: gradients along the magnetic field line are much smaller than those across it. Numerical codes exploit this anisotropy by aligning one coordinate with the field, thereby reducing the number of grid points required in the parallel direction and relaxing the Courant‑Friedrichs‑Lewy (CFL) constraint for explicit time‑stepping. Historically, field‑aligned coordinates have been built by using the poloidal angle (θ) as the parallel label and the toroidal angle (ϕ) as the fine‑scale transverse direction. While effective, this conventional choice suffers from two major drawbacks: the transformed coordinate system is not doubly periodic, and it becomes singular when the safety factor q(r) → ∞ (e.g., near a magnetic separatrix).

The present paper proposes a fundamentally different mapping: the toroidal angle itself is taken as the parallel coordinate (s = ϕ) and the poloidal angle is re‑expressed as a transverse variable ξ = θ − ϕ/q(r). In this formulation the parallel derivative reduces to the simple operator ∇∥ = ∂/∂s, independent of q(r). Because both θ and ϕ retain their 2π periodicity, the new system preserves double periodicity, allowing a straightforward Fourier representation in both directions. In Fourier space the turbulent energy remains concentrated along the familiar resonance lines m = n q(r), but the number of toroidal modes n required to resolve the dynamics can be dramatically reduced. This reduction translates directly into fewer degrees of freedom and lower memory consumption.

To avoid the metric distortion that typically appears in the transformed equations, the authors adopt a “shifted‑metric” technique. The toroidal domain is divided into Nϕ overlapping sectors; within each sector the mapping ξ = θ − (ϕ − ϕk)/q(r), s = ϕ − ϕk is applied, where ϕk denotes the sector’s central toroidal angle. Inside a sector the metric is essentially Cartesian, and only at sector boundaries is interpolation required to enforce continuity. This approach eliminates the non‑diagonal metric terms that plague the traditional poloidal‑based mapping, while still allowing the parallel derivative to be evaluated by simple interpolation of the end‑point values.

The new coordinate system was implemented in the well‑known ETAI3D code, which solves a three‑dimensional ion‑temperature‑gradient (ITG) model consisting of coupled equations for ion density w, parallel velocity vk, temperature Ti, and electrostatic potential Φ. In the original code the parallel operator is ∇∥ = ∂/∂ϕ + (1/q)∂/∂θ. By replacing the angular labeling according to the proposed scheme, the authors reduced the number of points along the parallel direction by a factor of three to five without sacrificing spectral fidelity. Benchmarks show a ≈30 % reduction in wall‑clock time per time step and a ≈40 % decrease in memory usage, while the turbulent spectra, transport coefficients, and statistical properties remain indistinguishable from the reference uniform‑grid simulation.

A striking and unexpected observation emerged from the high‑resolution field‑aligned runs: intermittent, localized spikes of large parallel gradients (both in Φ and Ti) appear sporadically throughout the turbulent state. These events violate the usual flute assumption locally, suggesting that non‑linear interactions can generate brief, quasi‑ballooning structures even in regimes where the average ∇∥ remains small. Because the conventional uniform grid lacked sufficient parallel resolution, such spikes were previously unnoticed. Their presence raises questions about the robustness of implicit assumptions used in reduced turbulence models, especially concerning convergence, parallel heat flux closure, and the validity of Landau‑fluid approximations.

Beyond the toroidal‑based mapping, the authors outline a further generalisation that eliminates the need for any magnetic coordinate in the poloidal plane. By treating (r, ϕ) as the sole spatial variables and expressing all perpendicular operators through the toroidal angle, the method becomes applicable to configurations with X‑points, open field lines, or strongly shaped flux surfaces where a clean poloidal coordinate is unavailable.

In summary, the paper demonstrates that (i) using the toroidal angle as the field‑aligned coordinate preserves double periodicity and enables a compact Fourier representation, (ii) the transformation remains regular even when q → ∞, (iii) the shifted‑metric sector decomposition removes metric‑induced numerical artefacts, (iv) implementation in an existing ITG code yields substantial computational savings, and (v) the discovery of intermittent large ∇∥ events challenges the universal applicability of the flute approximation. These results open a promising pathway for more efficient, higher‑fidelity turbulence simulations and suggest that future theoretical work should re‑examine the role of parallel gradients in micro‑turbulent transport.