 Solution

Because the velocity is independent of $z$, the vortex-stretching term is $(\mathbf{\omega }\cdot \nabla )\mathbf{u}={\omega }_z{\partial }_z\mathbf{u}=0$. Substituting ${\omega }_z=-{\nabla }^2\psi$, $u={\psi }_y$ and $v=-{\psi }_x$ into the vorticity equation and cancelling a minus sign givesThe planar vorticity is transported by the flow without vortex stretching under these inviscid assumptions.