= Solution
Because the velocity is independent of $z$, the vortex-stretching term is $(\boldsymbol\omega\cdot\nabla)\mathbf u=\omega_z\partial_z\mathbf u=0$. Substituting $\omega_z=-\nabla^2\psi$, $u=\psi_y$ and $v=-\psi_x$ into the <vorticity equation> and cancelling a minus sign gives
$$
\boxed{\partial_t(\nabla^2\psi)+\psi_y\partial_x(\nabla^2\psi)
-\psi_x\partial_y(\nabla^2\psi)=0.}
$$
The planar <vorticity> is transported by the flow without vortex stretching under these inviscid assumptions.
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