Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 3 / a

For an inviscid Boussinesq approximation fluid in a nonrotating frame, write buoyancy as $\sigma$ and kinematic pressure as $p$. The governing equations areThey require density variations to be small compared with a constant reference density, while retaining those variations in buoyancy; the flow scale must be small compared with the background density scale height. Ideal flow additionally neglects viscosity and scalar diffusion.

Linearize aboutwhere $N$ is the buoyancy frequency, and defineFor two-dimensional disturbances, the linear equations areThe condition $N^2>0$ expresses stable density stratification.