= Free surface of a Rankine vortex
{c}
{title2=$z=C-\Omega^2(2a^2-r^2)/(2g)\text{ in the core}$}
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 $r=a$ gives
$$
p=\begin{cases}B-\rho gz-\frac12\rho\Omega^2(2a^2-r^2),&r\leq a,\\B-\rho gz-\rho\Omega^2a^4/(2r^2),&r>a.\end{cases}
$$
Setting a constant surface pressure yields
$$
z=\begin{cases}C-\Omega^2(2a^2-r^2)/(2g),&r\leq a,\\C-\Omega^2a^4/(2gr^2),&r>a.\end{cases}
$$
Thus the <free surface> and its radial slope match at the core edge. The central depression is $\Omega^2a^2/g$, twice the depression at $r=a$. <Bernoulli's theorem> applies throughout the exterior; the rotational core instead uses the radial momentum balance above.
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