Connection Length¶

Connection length is one of the basic geometric quantities in scrape-off-layer (SOL) physics. The SOL is the open-field-line region outside the closed confinement volume: particles and energy leave the confined plasma mostly by rapid transport along magnetic field lines and are intercepted by a limiter, divertor plate, first wall, or another plasma-facing component. This makes the distance measured along the magnetic field, not the straight-line Euclidean distance, the relevant length scale for parallel heat conduction, sheath closure, neutral recycling, and divertor heat-load interpretation. Standard SOL reviews emphasize that open field lines connect the edge plasma to material surfaces and set the exhaust channel; see Fundamenski's JET SOL review and the IPP overview of SOL heat-exhaust physics: Fundamenski 2007, IPP SOL overview. A concise definition is also given in the CIEMAT FusionWiki entry: Connection length.

Definitions¶

Let \(\mathbf{x}(\ell)\) be a magnetic field line parameterized by arc length \(\ell\), with

For a starting point \(\mathbf{x}_0\), the one-sided forward and backward connection lengths are

where \(\ell_+\) and \(\ell_-\) are the first positive arc lengths at which the forward and backward field-line integrations intersect a material boundary or leave the computational SOL domain. A target-to-target connection length is the distance along the same field line between two material intersections,

Many reduced SOL closures use \(L_\parallel\) for the effective parallel distance between the modeled perpendicular plane and a sheath/target. Depending on the model, \(L_\parallel\) may be one-sided, half of a target-to-target length, or a prescribed constant. For example, the sheath-connected blob example in the reduced fluid-model literature uses \(\nabla\cdot\mathbf{j}_{sh}=n_e\phi/L_{||}\) with a configurable connection_length. In axisymmetric tokamak estimates near a poloidal limiter or divertor, \(L_\parallel\) often scales like the distance traveled along a helical field line, approximately \(qR\) times an order-one poloidal-angle factor. In shaped, diverted, or non-axisymmetric geometry, this approximation is replaced by field line tracing.

When toroidal angle \(\phi\) is used as the integration coordinate, a field line in cylindrical coordinates satisfies

In flux coordinates, the same idea is written with contravariant components. For a VMEC-like fixed-flux-surface map with \(s\) held constant,

and the arc length over a toroidal step is computed from the metric along the mapped curve. The important point is that \(L_c\) is a property of field-line topology and wall intersection, while \(L_\parallel\) in a reduced model is the particular effective length inserted into a closure.

Why It Matters¶

Connection length controls parallel transport time scales. A short connection length tends to couple a point strongly to a nearby sheath or wall, while a long connection length gives more distance for parallel conduction, radiation, ionization, charge exchange, and cross-field diffusion to modify the plasma before it reaches a target. W7-X connection-length studies explicitly connect \(L_c\) to SOL temperature profiles, high-recycling access, and heat-flux spreading: W7-X connection-length study. Recent stellarator divertor optimization work uses target-to-target connection length as a heat-load design metric and notes that long island-divertor connection lengths help spread heat flux: stellarator island-divertor optimization.

Historically, tokamak SOL modeling often reduced the parallel direction to a one-dimensional or two-point model along a representative field line. That is still useful for closures and scaling arguments. Stellarator and resonant magnetic perturbation geometries made the full three-dimensional connection map unavoidable: islands, stochastic layers, and localized wall intersections produce large spatial variation in \(L_c\). This is why modern 3D boundary workflows use field-line tracing, magnetic meshes, and repeated map reconstruction. FLARE and magnetic-mesh work for EMC3-EIRENE describe this field-line reconstruction viewpoint for stellarator SOL/divertor modeling: FLARE field-line analysis, magnetic mesh generation for stellarators.

Calculation In DRBX¶

DRBX keeps the connection-length calculation separate from the drift-reduced Braginskii RHS. The RHS consumes an array called connection_length; geometry adapters decide how that array was produced and document whether it is a true wall-hit length, an adjacent-plane arc length, or a proxy.

The synthetic self-contained stellarator lane uses a controlled proxy in src/drbx/geometry/stellarator.py. _estimate_connection_length(...) repeatedly applies the forward FCI map, counts how many toroidal-plane steps remain inside the radial domain, and multiplies the bounded step count by an average metric step length. This is not a wall-hit trace. It is a deterministic non-axisymmetric proxy used for clean-clone examples, metric/operator tests, and qualitative turbulence weighting.

The imported-field lane in src/drbx/geometry/essos_import.py has three map-source semantics:

• coil: field lines are traced with the external coil-field runtime. DRBX interpolates forward/backward endpoints at adjacent toroidal planes and uses the traced exit length to the boundary when that exit length is finite. Otherwise it falls back to the bidirectional adjacent-plane arc length.
• vmec: DRBX integrates a VMEC-coordinate map using \(d\theta/d\phi=B^\theta/B^\phi\). Because this map stays on closed VMEC surfaces, it stores a bidirectional adjacent-plane arc length and has no sheath endpoint masks.
• hybrid: DRBX uses the smooth VMEC map coordinates but keeps the coil-derived endpoint masks, \(|B|\), and connection-length array. This is the current bridge for open-field SOL closure tests on a smooth non-axisymmetric interpolation map.

The imported validation campaign in src/drbx/validation/essos_imported_fci_campaign.py checks the connection-length arrays before they are used for physics claims. build_essos_imported_fci_map_diagnostics(...) verifies finite and nonnegative values, records radial means, and computes single-grid neighbor jump diagnostics. build_essos_imported_connection_length_refinement_diagnostics(...) then compares nested grids by restricting each fine grid to the coarser grid and reporting normalized RMS, 95th-percentile, and \(L_\infty\) errors plus an observed order when three or more levels are available. Manufactured examples use conservative block restriction. Live imported levels can instead use coordinate-aware restriction, which interpolates the fine grid at the coarse \((\rho,\phi,\theta)\) points before forming the difference. For the closed VMEC adjacent-plane map, the grid-invariant refinement quantity is \(L_{\parallel,\Delta\phi}/\Delta\phi\), the parallel step length per toroidal radian, because the raw adjacent-plane step length changes when the toroidal spacing is refined. The importer now keeps three quantities separate: raw_connection_length for backward-compatible mixed reports, adjacent_step_length for the consumed FCI map step, and target_exit_length for wall-hit or target-exit diagnostics. Open-field target lengths still need a separate gate because target-exit lengths and adjacent-plane fallback lengths have different grid-scaling behavior. The clean-clone example examples/geometry-3D/essos-field-lines/imported_connection_length_refinement.py exercises that exact refinement path on manufactured non-axisymmetric data and sets require_observed_order=True so a two-level report cannot be promoted as convergence evidence. The same example writes a lightweight sweep summary JSON that aggregates promotion_ready, evidence_role, finest errors, observed order, and finite-overlap thresholds across all resolved map sources.

Current Promotion Rule¶

Connection-length arrays are accepted for publication-grade imported-field turbulence only when all of the following are true:

• Values are finite and nonnegative on the consumed FCI grid.
• Endpoint masks used by sheath/recycling exactly match the imported forward/backward map boundary masks.
• Directional target labels distinguish forward, backward, and bidirectional exits and exactly reconstruct the endpoint counts consumed by sheath and recycling closures.
• Open-field endpoint diagnostics report finite, nonnegative wall-hit target_exit_length values on endpoint cells and finite, nonnegative adjacent_step_length values where adjacent-map interpolation exists.
• Direction-specific wall-hit lengths must be finite and nonnegative on all forward and backward boundary masks used by the imported maps; aggregate target-exit coverage alone is not sufficient for promotion.
• Single-grid roughness diagnostics do not show unresolved grid-scale jumps.
• Promotion campaigns set require_connection_resolution=True, so the single-grid roughness diagnostic is a hard imported-map gate rather than advisory metadata.
• A live multi-grid control passes with the correct quantity: VMEC adjacent-step maps use parallel step length per toroidal radian, while open-field coil/hybrid maps must separate target-to-target exit length from adjacent-step fallback length.
• Hybrid VMEC/coil maps use the VMEC adjacent step for FCI-map refinement and the coil target-exit length only for endpoint/sheath-source diagnostics.
• The multi-grid control has at least three nested levels when it is used to support an observed-order convergence claim.
• The refinement report has promotion_ready=true; this requires finite pairwise comparisons, the configured RMS and \(L_\infty\) thresholds, monotonic error reduction, and an explicitly required and available observed-order check.
• Open-field adjacent-step and target-exit live refinement reports may use a documented minimum_finite_pair_fraction below one, because endpoint cells are expected to lack adjacent-plane lengths and are covered by the separate endpoint and target-label diagnostics.
• Reports with advisory_only=true or an evidence_role beginning with negative_ may be kept as diagnostics and regression evidence, but they do not support publication, README, or movie promotion claims.
• The turbulence figure or movie reports which connection-length definition was used: wall-hit/exit length, adjacent-plane arc length, or synthetic proxy.

Until the live multi-grid imported run passes, synthetic and manufactured connection-length figures are validation evidence for the algorithms, not a claim that a particular stellarator wall/target design has been resolved.