Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 331 / 2 / a / Solution

Use the Boussinesq approximation with a constant inertial reference mass density ${\rho }_0$, keeping the background mass density gradient in buoyancy. Write $V(z)=\overline{U}(z)-c$ and $P=\hat{p}/{\rho }_0$. For $k\ne 0$, the linearized normal mode equations areThe reference mass density is chosen close to the background values, whose relative variation must be small. Hydrostatic background pressure satisfies ${\overline{p}}^{\prime }=-g\overline{\rho }$.

From incompressibility, $\hat{u}=i{\hat{w}}^{\prime }/k$. The horizontal momentum equation and the density equation then giveSubstitute these into the vertical momentum equation and multiply by $ik$:For $V\ne 0$, rearrangement gives the Taylor–Goldstein equationThe buoyancy frequency $N$ is real for stable stratification, where ${\overline{\rho }}^{\prime }<0$. Points where $\overline{U}=c$ are critical levels of an internal gravity wave; the derivation there is understood through a limiting or piecewise formulation. The printed density gradient is ${\overline{\rho }}^{\prime }$, not the pressure gradient accidentally substituted in the local TeX.