Euler equation in a uniformly rotating frame (${\partial }_t\mathbf{u}+(\mathbf{u}\cdot \nabla )\mathbf{u}+2\mathbf{\Omega }\times \mathbf{u}=-\nabla \Pi$)

For constant mass density ${\rho }_0$, a frame with constant angular velocity $\mathbf{\Omega }$ absorbs centrifugal acceleration into reduced pressure $\Pi =p/{\rho }_0-|\mathbf{\Omega }\times \mathbf{r}{|}^2/2$. Indeed $\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})=-\nabla |\mathbf{\Omega }\times \mathbf{r}{|}^2/2$. Transforming the material acceleration in the Euler equations gives the displayed equation. The Coriolis acceleration remains velocity dependent and cannot be absorbed into a scalar potential. A conservative physical body force can be included by adding its potential to $\Pi$.