Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 61 / 3 / i

Let ${\Sigma }_0$ be the uniform background surface density, ${\mathbf{u}}_0=-Sx{\mathbf{e}}_y$, and write $\Sigma ={\Sigma }_0+{\Sigma }^{\prime }$, $\mathbf{u}={\mathbf{u}}_0+\mathbf{v}$, ${\Phi }_d={\Phi }_{d0}+{\Phi }_d^{\prime }$. The background sheet has no horizontal gravitational gradient; its vertical field is irrelevant to the stated planar equations. Put $D_0={\partial }_t-Sx{\partial }_y$. Keeping first-order terms givesThe $-S{v}_x$ contribution comes from $\mathbf{v}\cdot \nabla {\mathbf{u}}_0$, while the Coriolis acceleration supplies the remaining coupling. The prescribed tidal potential has no perturbation, and its background force already balances the background Coriolis term. The isothermal pressure linearizes to $c_s^2\nabla {\Sigma }^{\prime }/{\Sigma }_0$.