Reduced pressure in a rotating fluid ($P=p+\rho \Phi -\rho |\mathbf{\Omega }\times \mathbf{x}{|}^2/2$)

For constant density and a conservative force per mass $-\nabla \Phi$, gravity and centrifugal acceleration can be incorporated into the displayed reduced pressure. The resulting Euler equations in rotating axes are ${\mathbf{u}}_t+(\mathbf{u}\cdot \nabla )\mathbf{u}+2\mathbf{\Omega }\times \mathbf{u}=-\nabla P/\rho$. The sign of the centrifugal contribution follows from $\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{x})=-\nabla |\mathbf{\Omega }\times \mathbf{x}{|}^2/2$. Constant density is what permits combination into one pressure gradient without extra density-gradient terms.