Straining effect in the process of vortex reconnection

During the process of vortex reconnection, the vortices are subject to the local velocity gradient, and the finite vortex cores are inevitably deformed by the straining action. The nature of the deformation is determined by the eigenvalues of the rate-of-strain tensor. Using a tent model for vortex reconnection which consists of vortex filaments in the form of two anti-parallel tilted hyperbolae, we investigated the structure of the eigenvalues and eigenvectors of the rate-of-strain tensor near the points of the closest approach (tipping points). We observed the following things: (1) The second largest eigenvalue λ₂ is positive and its eigenvector e₂ is in the axial direction; each vortex is therefore persistently stretched at the tip. (2) The eigenvectors e₁ and e₃ are in the plane perpendicular to the axial direction and are mutually orthogonal as expected for a real symmetric matrix. (3) At the tipping point on one vortex, the rate-of-strain produced by the other vortex alone is dominant, and the magnitudes of the components in the perpendicular plane of e₁ and e₃ are nearly equal; these vectors are therefore close to the directions along a line inclined at an angle of ±π/4. These observations can be verified asymptotically using a model of two tilted vortex rings.