Rankine vortex ($u_{\theta }=\Omega r(r\le a),u_{\theta }=\Omega a^2/r(r>a)$)

A Rankine vortex models a finite rotating core joined continuously to an irrotational exterior. In cylindrical polar coordinates, its radial and vertical velocities vanish and its azimuthal velocity has the displayed form, where $a$ is the core radius and $\Omega$ is a signed angular velocity. The axial vorticity is $r^{-1}{\partial }_r(r{u}_{\theta })$, equal to $2\Omega$ inside and zero outside. The velocity is continuous at $r=a$, although its first radial derivative has different one-sided values. The circulation outside the core is $2\pi \Omega a^2$.

For a steady Rankine vortex in a fluid of constant density $\rho$ with gravity $-gk$, the Euler equations give $p_r=\rho u_{\theta }^2/r$ and $p_z=-\rho g$. Integrating on the two sides and matching pressure at $r=a$ givesSetting a constant surface pressure yieldsThus the free surface and its radial slope match at the core edge. The central depression is ${\Omega }^2{a}^2/g$, twice the depression at $r=a$. Bernoulli's theorem applies throughout the exterior; the rotational core instead uses the radial momentum balance above.