For $u_x=0$ and $u_y=u_y(x)$, the material acceleration vanishes. The radial component of force balance is
$$
0=2\Omega u_y-c_s^2\frac{d\sigma}{dx}+3\Omega^2x.
$$
Therefore
$$
\boxed{u_y(x)=-\frac32\Omega x
+\frac{c_s^2}{2\Omega}\frac{d\sigma}{dx}}.
$$
The first term is the background <Keplerian shear>; the second is the <geostrophic balance> correction produced by the surface-density gradient.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 3 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-321/3/b/i/solution