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

Assume inviscid, unforced evolution, so shallow-water potential vorticity is conserved. If a column has initial depth $H_1$ and relative vorticity ${\zeta }_1$, moving to $H_2>H_1$ at fixed Coriolis parameter givesIts absolute vorticity is amplified by vortex stretching. A column initially without relative vorticity develops cyclonic spin: positive in the Northern Hemisphere and negative in the Southern Hemisphere. More generally the increase is in the direction of its original absolute vorticity, not necessarily the direction of $f$ if the initial spin is unusually strong.

Conservation of shallow-water potential vorticity with fixed relative vorticity givesFor ordinary planetary-vorticity-dominated columns, moving towards the equator decreases the magnitude of absolute vorticity, so the column must become shallower and spread horizontally. This is the compression counterpart of vortex stretching. For $\zeta =0$, the depth ratio is $f_2/f_1$, in either hemisphere before reaching the equator. The displayed general formula is essential if the fixed relative vorticity is large: the sign of the depth change then depends on the original absolute vorticity, and conservation cannot continue through zero absolute vorticity with positive finite depth and nonzero conserved potential vorticity.

Let $\mathbf{\zeta }=\nabla \times \mathbf{u}$ and $\mathbf{\omega }=\mathbf{\zeta }+2\Omega {\mathbf{e}}_z$ be the relative vorticity and absolute vorticity. The nonlinear acceleration identity rewrites the momentum equation asTaking the curl removes the pressure, kinetic energy gradient and shearing-sheet tidal potential. ThusBoth $\nabla \cdot \mathbf{u}$ and $\nabla \cdot \mathbf{\omega }$ vanish. The vorticity equation is consequentlyFor the absolute-vorticity flux tensor $T_{ij}=u_i{\omega }_j-{\omega }_i{u}_j$, use the flux convention $(\nabla \cdot \mathbf{T}{)}_j={\partial }_i{T}_{ij}$, with the first index denoting transport direction. This givesThe scalar $\theta$ is a buoyancy displacement variable, so dimensional consistency gives it units of length; the printed potential-temperature terminology presupposes this normalization. Its curl source directly forces only the $y,z$ components of absolute vorticity. Hence radial vorticity conservation in a shearing sheet takes the local flux formIts volume integral is constant when the net boundary flux vanishes. However $D_t{\omega }_x=\mathbf{\omega }\cdot \nabla u_x$ still permits vortex stretching; the radial component is not generally a materially conserved scalar in three dimensions. Using divergence on the second index of the stated tensor would reverse the transport sign, so the tensor convention matters.

Potential vorticity combines fluid rotation with stretching and stratification. For an inviscid barotropic shallow-water layer,where $\zeta$ is relative vorticity and $h$ is layer thickness.

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.