Let $\boldsymbol\zeta=\nabla\times\boldsymbol u$ and $\boldsymbol\omega=\boldsymbol\zeta+2\Omega\boldsymbol e_z$ be the <relative vorticity> and <absolute vorticity>. The nonlinear acceleration identity rewrites the momentum equation as
$$
\partial_t\boldsymbol u-\boldsymbol u\times\boldsymbol\omega
=-\nabla\left(\frac P{\rho_0}+\frac12|\boldsymbol u|^2-\frac32\Omega^2x^2\right)-N^2\theta\boldsymbol e_x.
$$
Taking the <curl> removes the <pressure>, <kinetic energy> gradient and <shearing-sheet tidal potential>. Thus
$$
\partial_t\boldsymbol\omega-\nabla\times(\boldsymbol u\times\boldsymbol\omega)
=-N^2\nabla\theta\times\boldsymbol e_x.
$$
Both $\nabla\cdot\boldsymbol u$ and $\nabla\cdot\boldsymbol\omega$ vanish. The <vorticity equation> is consequently
$$
\partial_t\omega_j+u_i\partial_i\omega_j-\omega_i\partial_i u_j
=-N^2(\nabla\theta\times\boldsymbol e_x)_j.
$$
For the <absolute-vorticity flux tensor> $T_{ij}=u_i\omega_j-\omega_i u_j$, use the flux convention $(\nabla\cdot\boldsymbol T)_j=\partial_iT_{ij}$, with the first index denoting transport direction. This gives
$$
\boxed{\partial_t\boldsymbol\omega+\nabla\cdot\boldsymbol T=-N^2\nabla\theta\times\boldsymbol e_x.}
$$
The 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 form
$$
\boxed{\partial_t\omega_x+\partial_i(u_i\omega_x-\omega_i u_x)=0.}
$$
Its volume integral is constant when the net boundary flux vanishes. However $D_t\omega_x=\boldsymbol\omega\cdot\nabla u_x$ 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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 321 / 3 / a / Solution

{title2=$\partial_t\omega_x+\partial_iT_{ix}=0$}

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.

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 Incoming links: Radial vorticity conservation in a shearing sheet