Absolute-vorticity flux tensor ($T_{ij}=u_i{\omega }_j-{\omega }_i{u}_j$)

For a flow satisfying incompressibility, this antisymmetric tensor writes advection and vortex stretching in conservative form. With its first index as transport direction, $(\nabla \cdot T{)}_j={\partial }_i{T}_{ij}$.

When the only nonpotential force points radially, its curl has no radial component. The radial absolute vorticity therefore obeys a local flux conservation law, although vortex stretching can change it along a fluid trajectory.