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

The vorticity-related conservation theorems for ideal flows, such as Helmholtz's theorem, break down in non-ideal flows which can be viscous or with non-conservative body forces. On the other hand, the vorticity-related quantities within a carrier convected by a virtual circulation-preserving velocity can still be conserved in some non-ideal flows. We provide the conditions for the existence and uniqueness of a globally smooth virtual velocity with several explicit examples. By incorporating the virtual velocity into the vortex-surface field (VSF), we can track vortex surfaces in some non-ideal flows. If a flow has a viscous-like diffusion term which is orthogonal to the vorticity without singularity, 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 is still useful. We use the approximate virtual velocity to track vortex surfaces in a magnetohydrodynamic flow. Compared with the VSF evolution convected by the physical velocity, the conservation of vorticity flux is significantly improved, and the spurious vortex deformation induced by the Lorentz force is eliminated.