= Solution
Put $X=x-x_0$, $Y=y-y_0$ and take $\beta,r,H>0$. The constant-depth <ocean transport streamfunction> equation is
$$
r\nabla_h^2\psi+\beta\psi_x=H\delta(X)\delta(Y).
$$
It is a steady <advection-diffusion equation> for the transport response, with westward <pseudovelocity>. For an unbounded domain the <point-forced Sverdrup–drag Green function> can make its contours precise. Set $a=\beta/(2r)$ and $\psi=e^{-aX}\phi$. Then
$$
(\nabla_h^2-a^2)\phi=\frac Hr\delta(X)\delta(Y),\qquad \boxed{\psi=-\frac{H}{2\pi r}e^{-aX}K_0\left(a\sqrt{X^2+Y^2}\right),}
$$
where $K_0$ is the <Modified Bessel function of the second kind>. This optional explicit expression is used only to produce the requested sketch; no weak-drag assumption is imposed.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-333-point-forcing.png]
{title=Streamfunction contours around a point wind-curl source with finite bottom drag, showing a long western wake and a short eastern response}
{height=400}
Near the source the logarithmic singularity produces almost circular contours. At distances large compared with $\ell=r/\beta$, $K_0(aR)\sim\sqrt{\pi/(2aR)}e^{-aR}$, so the response is proportional to $e^{-a(R+X)}/\sqrt R$. It decays exponentially eastward, but only algebraically on the western axis; its broad western wake has transverse width of order $\sqrt{\ell|X|}$. \b[Contours are stretched westward], with closed finite-level contours around the source and sharper eastern gradients. Larger drag broadens the scale $\ell$ and makes any fixed nearby view more nearly circular. For $r=0$ the elliptic smoothing disappears; for $\beta=0$ the infinite-plane response is logarithmic up to a gauge and cannot be fixed by a zero-at-infinity condition.
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