Numerical investigation of the 3D regularized Biot-Savart model towards vortex reconnection*

As the inverse of the curl operator, the Biot-Savart law calculates the velocity field produced by a vortex line, and plays a central role in the analysis of vortex motion.  As is well-known that the Biot-Savart integral has a logarithmic divergence if the denominator of the integrand is close to zero. To avert this logarithmic singularity, Rosenhead (1930) introduced an artificial positive parameter μ in the denominator and regularized the singularity. In this paper, we investigate numerically the motion of two tilted circular vortex rings,  which has been proposed as a model for the finite time singularity problem for the Navier-Stokes equation [1][2],  using the regularized Biot-Savart model by Rosenhead. We observe that μ particularly moderates the development of curvature at the tipping points, the closest approach on the vortex rings. The moderation grows in a small circular region around a tipping point the extent of which is a function of μ. We demonstrate that a large value of μ distorts the development of curvature significantly to produce a spurious bubble around the tipping points. 

[1] H.K.Moffatt & Y.Kimura,  JFM (2019) 861 930—967. [2] H.K.Moffatt & Y.Kimura,  JFM (2019) 870, R1, and JFM (2020) 887, E2 (corrigendum).