Flows that are multiples of vorticity are steady flows proof

Question

Define the flow $\vec{u}(\vec{x},t)=\alpha(\vec{x},t)\vec{\omega}(\vec{x},t)$ where $\alpha(\vec{x},t)$ is some scalar field. I am trying to show that $$ \frac{\partial\vec{u}}{\partial t}=\vec{0}. $$ From the incompressible vorticity equation (with zero body force) I know that $$ \begin{split} \frac{\partial\vec{\omega}}{\partial t} & = \nabla\times(\vec{u}\times\vec{\omega})\\ & = \nabla\times(\alpha\vec{\omega}\times\vec{\omega})=\vec{0}, \end{split} $$ so by the product rule I know also that $$ \frac{\partial \vec{u}}{\partial t} = \frac{\partial \alpha}{\partial t}\vec{\omega}, $$ and here I am stuck.