UC Davis

We study vortex motion using a Hopf flow on a 3-sphere, in place of the standard 3-torus defined by periodic boundary conditions. Although both of these underlying manifolds look locally like R3, the vortex behavior differs significantly. This raises the question of how well calculations on the 3-torus correspond to behavior in R3, for this problem or others. We find that, on both the 3-sphere and 3-torus, vortex tangles appear, with their line density related to the strength of the driving velocity field. However, tangles on the 3-sphere are highly anisotropic, with vortices tending to align along the flow direction. Standard power-law dependences change accordingly from their values in more isotropic tangles. The line length density (L) is proportional to vns1.28, where vns is the drive velocity, and the reconnection rate depends roughly on (L)2. These compare to vns2 and (L)2.5 for the more isotropic tangles on the 3-torus. In addition, in simulations using the local induction approximation rather than the full Biot-Savart law, the tangle collapses so that all vortices are nearly aligned with a single flow line. By contrast, vortices on the 3-torus become perpendicular to the driving velocity.