 Solution

Taking the curl of the Euler equation gives the vorticity equation for an inviscid incompressible flow without body forces: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 $\omega {\partial }_{z}u=(0,0,\omega )$. ThusThe signed vertical component is ${\omega }_0{e}^t$; its magnitude is $|{\omega }_0|e^t$, with the source's magnitude convention taking ${\omega }_0\ge 0$. 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.