= Solution
Taking the curl of the <Euler equation> gives the <vorticity equation> for an inviscid <incompressible flow> without body forces:
$$
\boxed{\partial_t\boldsymbol\omega+(u\cdot\nabla)\boldsymbol\omega
=(\boldsymbol\omega\cdot\nabla)u,\qquad\boldsymbol\omega=\nabla\times u.}
$$
The given velocity has divergence $-1+0+1=0$ and curl $(0,0,\omega(t))$. Its vorticity is spatially uniform, so its advective derivative vanishes. The stretching term is $\omega\partial_zu=(0,0,\omega)$. Thus
$$
\omega'=\omega,\qquad\boxed{\omega(t)=\omega_0e^t.}
$$
The signed vertical component is $\omega_0e^t$; its magnitude is $|\omega_0|e^t$, with the source's magnitude convention taking $\omega_0\geq0$. As an additional consistency check, after imposing this equation the material acceleration is $(x,0,z-1)$, so a pressure $p=-\rho[x^2+(z-1)^2]/2+p_0(t)$ satisfies the full <Euler equation>.
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