Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 321 / 3 / b / Solution

For the Keplerian shearing sheet background $\mathbf{U}=-3\Omega x{\mathbf{e}}_y/2$, the only spatial derivative is ${\partial }_x{U}_y=-3\Omega /2$. Since $U_x=0$ and all fields are independent of $y$, $\mathbf{U}\cdot \nabla \mathbf{U}=0$ and $\nabla \cdot \mathbf{U}=0$.

The Coriolis acceleration exactly balances the tidal term:With constant pressure and $\theta =0$, the buoyancy displacement variable equation also has zero advection, zero source $U_x$ and zero diffusion. Thereforeis a steady solution. Its absolute vorticity is ${\mathbf{\omega }}_0=(\Omega /2){\mathbf{e}}_z$, since the background relative vorticity is $-3\Omega {\mathbf{e}}_z/2$.