Inertial-wave vorticity equation (${\partial }_t(\nabla \times \mathbf{u})=2(\mathbf{\Omega }\cdot \nabla )\mathbf{u}$)

Linearize the Euler equation in a uniformly rotating frame about rest and impose incompressible flow, $\nabla \cdot \mathbf{u}=0$. Taking the curl eliminates the reduced pressure. For constant $\mathbf{\Omega }$, $\nabla \times (\mathbf{\Omega }\times \mathbf{u})=-(\mathbf{\Omega }\cdot \nabla )\mathbf{u}$, giving the displayed vorticity equation. Taking another curl and time derivative yields ${\partial }_t^2{\nabla }^2\mathbf{u}+4(\mathbf{\Omega }\cdot \nabla {)}^2\mathbf{u}=0$. These equations describe the restoring influence of background rotation on small disturbances without buoyancy or viscosity.