Tracking vortex surfaces frozen in the virtual velocity in non-ideal flows

Abstract

We demonstrate that, if a globally smooth virtual circulation-preserving velocity exists, Kelvin’s and Helmholtz’s theorems can be extended to some non-ideal flows which are viscous, baroclinic or with non-conservative body forces. Then we track vortex surfaces frozen in the virtual velocity in the non-ideal flows, based on the evolution of a vortex-surface field (VSF). For a flow with a viscous-like diffusion term normal to the vorticity, we obtain an explicit virtual velocity to accurately track vortex surfaces in time. This modified flow is dissipative but prohibits reconnection of vortex lines. If a globally smooth virtual velocity does not exist, an approximate virtual velocity can still facilitate the tracking of vortex surfaces in non-ideal flows. In a magnetohydrodynamic Taylor–Green flow, we find that the conservation of vorticity flux is significantly improved in the VSF evolution convected by the approximate virtual velocity instead of the physical velocity, and the spurious vortex deformation induced by the Lorentz force is eliminated.