Barotropic cylindrical rotation theorem (${\partial }_z{\Omega }^2=0$)

In a steady axisymmetric barotropic fluid with purely azimuthal velocity $r\Omega {\mathbf{e}}_{\varphi }$ and a conservative body force, the barotropic enthalpy function obeys $\nabla (w+\Phi )=r{\Omega }^2{\mathbf{e}}_r$. Taking its curl yields the displayed cylindrical rotation law. The first integral is $w+\Phi -\int r{\Omega }^{2}dr=C$ in a connected fluid region. Baroclinic pressure and density need not satisfy this conclusion.