The sensitivity of the vortex filament method to different reconnection models

We present a detailed analysis on the effect of using different algorithms to model the reconnection of vortices in quantum turbulence, using the thin-filament approach. We examine differences between four main algorithms for the case of turbulence driven by a counterflow. In calculating the velocity field we use both the local induction approximation (LIA) and the full Biot-Savart integral. We show that results of Biot-Savart simulations are not sensitive to the particular reconnection method used, but LIA results are.

💡 Research Summary

The paper investigates how the choice of vortex‑reconnection algorithm influences the outcomes of vortex‑filament simulations of quantum turbulence. Quantum turbulence in superfluid helium consists of quantized vortex lines that constantly reconnect, and the vortex‑filament method (VFM) models these lines as thin space curves whose dynamics are governed by the Schwarz equation. The velocity field induced by the vortices can be computed either by the full Biot–Savart (BS) integral, which accounts for all non‑local interactions, or by the Local Induction Approximation (LIA), which retains only the local curvature contribution and is computationally cheaper.

Four reconnection schemes are examined. Type I reconnects any pair of points whose separation falls below a critical distance Δ = δ/2, with no further constraints. Type II adds a physical constraint: after reconnection the total vortex‑line length must decrease, reflecting the dissipative nature of reconnections. Type I II is an “ultra‑dissipative” variant of Type I that removes the two points that triggered the reconnection, thereby maximising line‑length loss. Type IV, introduced by Kondaurova and Nemirovskii, predicts whether two line segments will intersect during the next time step by solving a set of linear equations for the interpolation parameters (φ, ψ); if a solution exists within the segment bounds, a reconnection is performed. This algorithm explicitly checks for an imminent geometric collision.

The numerical experiments are performed in a periodic cube of side D = 0.1 cm, populated initially with a random tangle of vortex loops of radius 0.0095 cm. The spatial resolution is δ = 1.6 × 10⁻³ cm and the time step Δt = 10⁻⁴ s. Counter‑flow turbulence is driven by imposing opposite normal‑fluid and superfluid velocities along the x‑axis, with a temperature of 1.6 K (mutual‑friction coefficients α = 0.98, α′ = 0.016). The normal‑fluid–superfluid velocity difference v_ns is varied (starting from 0.55 cm s⁻¹) to test the experimentally verified steady‑state law L = γ v_ns², where L is the vortex‑line density and γ is a temperature‑dependent parameter.

Two sets of simulations are carried out: one using the full Biot–Savart law, the other using LIA, each combined with the four reconnection algorithms. The primary diagnostics are the vortex‑line density L(t) and the reconnection rate per unit line length ζ/Λ (ζ is the number of reconnections per unit time, Λ is the total line length). Results are presented in Figures 5–8 of the paper.

Biot–Savart results: All four reconnection schemes produce virtually identical L(t) curves. After an initial rapid increase, the line density settles into a statistically steady state whose value scales as v_ns², confirming the γ v_ns² law. The reconnection rate ζ/Λ is also essentially the same for all algorithms, with only a modestly higher rate for the unconstrained Type I scheme. The insensitivity to reconnection details is attributed to the accurate representation of non‑local vortex interactions in the BS integral; small differences in the timing or geometry of reconnections do not propagate into the large‑scale dynamics.

LIA results: The picture changes dramatically. Because LIA neglects non‑local contributions, the geometry of the vortex tangle after a reconnection strongly influences subsequent motion. Some algorithms (notably Type IV with tolerance ε = 10⁻⁴–10⁻³) lead to a “degenerate” steady state in which the vortices become nearly straight, aligned in planes parallel to the counter‑flow direction. In this configuration the line density stops evolving and the reconnection rate collapses to near zero. Other algorithms (Type I, Type II, Type I II) either keep the tangle active, allowing L to grow continuously, or produce oscillatory behaviour, but never reach the same robust steady state observed with BS. The unconstrained Type I algorithm yields the highest reconnection rate, as expected, but this does not translate into a markedly different L(t) evolution compared with the other LIA runs.

The authors also verify that, for the BS simulations, varying v_ns yields consistent values of the macroscopic parameter γ, independent of the reconnection algorithm. In contrast, LIA simulations produce γ values that depend on the reconnection rule, highlighting the algorithmic sensitivity of the approximation.

Conclusions: The study demonstrates that the vortex‑filament method, when coupled with the full Biot–Savart law, is robust against the choice of reconnection algorithm; the macroscopic statistics of quantum turbulence (line density, reconnection rate, γ) are essentially algorithm‑independent. However, the computationally cheaper LIA is highly sensitive to reconnection details; inappropriate reconnection criteria can drive the system into unphysical, degenerate configurations. Therefore, researchers employing LIA must carefully test the impact of their reconnection scheme, possibly calibrating against BS results or experimental data. The work provides the first systematic quantitative assessment of reconnection‑algorithm sensitivity in VFM and offers practical guidance for future quantum‑turbulence simulations.