= Solution
The <Free surface of a Rankine vortex> follows from setting its pressure constant. At a <free surface> with constant <pressure> $p_s$, solve the two <pressure> expressions from part (b) for $z$. With $C=(B-p_s)/(\rho g)$, this gives
$$
\boxed{z(r)=\begin{cases}
C-\dfrac{\Gamma^2(2a^2-r^2)}{2ga^4},&r\leq a,\\
C-\dfrac{\Gamma^2}{2gr^2},&r>a.
\end{cases}}
$$
Substituting $r^2=x^2+y^2$ recovers the required Cartesian form. The two values coincide at $r=a$, and both radial slopes there equal $\Gamma^2/(ga^3)$. The surface is parabolic in the core and approaches height $C$ as $r\to\infty$; at the centre its depression below $C$ is $\Gamma^2/(ga^2)$.
\Image[/past-exam-of-the-options-i-course-of-the-university-of-oxford/2012-vortex-surface.png]
{title=Free surface through the vortex core and irrotational exterior}
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