Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 331 / 3 / b / Solution

Forboth material derivatives vanish because the fields are independent of $x$. Since $\hat{\mathbf{z}}\times \mathbf{U}=z\hat{\mathbf{y}}$, steady momentum balance requiresThese derivatives are compatible and integrate toThe cross-stream temperature gradient and vertical stratification are in thermal-wind balance with the vertical shear.