Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2 / a / Solution

Let $D/Dt={\partial }_t-Sx{\partial }_y$ be the material derivative along the background linear shear flow ${\mathbf{u}}_0=-Sx{\mathbf{e}}_y$. Linearizing about constant $\Sigma$ givesThe coefficient $2\Omega -S$ includes the contribution $\mathbf{v}\cdot \nabla {\mathbf{u}}_0=-S{v}_x{\mathbf{e}}_y$ from perturbation advection of the background shear.