Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 2

Let $p=K{\rho }^{1+1/m}$. Hydrostatic equilibrium in the uniform gravitational field givesSubstitution and integration with $\rho =0$ at $z=0$ giveswhere $H$ fixes the normalization. Integrating the hydrostatic equation from $z$ to the free surface gives the pressure directly:This indeed has $p\propto {\rho }^{1+1/m}$ and $p(0)=0$, as required for a polytropic atmosphere.

Linearize the inviscid momentum equation in the uniformly rotating frame. The Coriolis acceleration is $2\mathbf{\Omega }\times \delta \mathbf{u}$, and the equilibrium pressure gradient cancels gravity. For perturbations proportional to $e^{i{k}_{x}x-i\omega t}$, the horizontal components areThe vertical component is

Linearizing mass conservation,givesFinally, linearizing the adiabatic pressure equation givesThese are the stated five equations. Self-gravity contributes no perturbation because it is neglected, and the equilibrium centrifugal term has already been absorbed or omitted.

Set $\delta \mathbf{u}=-i\omega \mathbf{\xi }$. The continuity and adiabatic equations give the Lagrangian perturbationsBecause the equilibrium is a neutrally stratified polytropic atmosphere, $p\propto {\rho }^{\gamma }$, so the displacement terms cancel andUsing $dp/dz=-\rho g$ then gives

Eliminating ${\xi }_y$ from the two horizontal momentum equations and combining gravity with the vertical pressure force yieldsThusand the requested coupled first-order system is

For an incompressible perturbation, $\Delta =0$, so $\Psi =g{\xi }_z$. The vertical equation givesIncompressibility together with the horizontal equation also givesEquating the two expressions yieldsThe positive-frequency branch is the surface gravito-inertial waveIts vertical displacement decays as $e^{{\omega }^{2}z/g}$ into the atmosphere $z<0$.

Eliminate ${\xi }_z$ from the coupled system to obtainWriting $\Psi =\psi e^{kz}$ givesFor the neutral polytrope found in part (a),If $\psi$ is a polynomial of degree $n$ with leading term $z^n$, the coefficient of the highest power $z^n$ in the ODE isA polynomial solution therefore requiresThe factor $e^{kz}$ traps every mode below the free surface. The $n=0$ solution is the incompressible surface gravito-inertial or $f$ mode of part (d). The $n\ge 1$ solutions are vertically structured acoustic p modes; neutral stratification leaves no buoyancy-driven $g$-mode family.