Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 333 / 4 / iv

Put $X=x-x_0$, $Y=y-y_0$ and take $\beta ,r,H>0$. The constant-depth ocean transport streamfunction equation isIt 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 $\psi =e^{-aX}\varphi$. Thenwhere $K_0$ 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.

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 $\sqrt{\ell |X|}$. 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.