= Solution
For constant density, divide the incompressible <Euler equations> by $\rho$. The vector identity $(\mathbf u\cdot\nabla)\mathbf u=\nabla(|\mathbf u|^2/2)-\mathbf u\times\boldsymbol\omega$ converts them into $\partial_t\mathbf u=\mathbf u\times\boldsymbol\omega-\nabla(p/\rho+|\mathbf u|^2/2)$. Taking the <curl> eliminates the gradient. Expand the curl of the cross product, use $\nabla\cdot\mathbf u=0$ and $\nabla\cdot\boldsymbol\omega=0$, and obtain
$$
\partial_t\boldsymbol\omega=(\boldsymbol\omega\cdot\nabla)\mathbf u-(\mathbf u\cdot\nabla)\boldsymbol\omega,\qquad\boxed{\frac{D\boldsymbol\omega}{Dt}=(\boldsymbol\omega\cdot\nabla)\mathbf u.}
$$
This is the <vorticity equation>, including <vortex stretching>. Without constant density there would in general be an additional baroclinic term; the displayed source equations use the constant-density interpretation.
For the specified velocity, direct differentiation gives $\boldsymbol\omega=2\Omega(t)\mathbf k$. It is spatially uniform, while $(\boldsymbol\omega\cdot\nabla)\mathbf u=2\beta\boldsymbol\omega$. Thus $\Omega'=2\beta\Omega$ and
$$
\boxed{\boldsymbol\omega=2\Omega(0)e^{2\beta t}\mathbf k.}
$$
If the positive $z$ direction agrees with the initial vorticity, $2\Omega(0)=\omega_0$ as written in the question. For opposite rotation the initial component is signed and the vector has the corresponding minus sign.
For a material particle, $d(x^2+y^2)/dt=2x(-\beta x-\Omega y)+2y(-\beta y+\Omega x)=-2\beta(x^2+y^2)$, while $dz/dt=2\beta z$. The initial material circle therefore evolves into
$$
\boxed{a(t)=e^{-\beta t},\qquad x^2+y^2=a(t)^2,\qquad z=e^{2\beta t}=a(t)^{-2}.}
$$
The azimuthal rotation changes the labels of points on the circle but not the circle as a set. Its counterclockwise <circulation>, viewed from positive $z$, is
$$
\Gamma=\oint\mathbf u\cdot d\mathbf r=2\pi\Omega(t)a(t)^2=2\pi\Omega(0)=\pi\omega_0
$$
for the positive-orientation convention. The strain contributes no tangential velocity. Thus \b[circulation stays constant although vorticity grows and the enclosed material area shrinks], illustrating <Kelvin's circulation theorem> for a material loop in constant-density inviscid flow with no nonconservative body force.
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