Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2018 / 6 / d / Solution

Assuming the mean wind remains $10\mathrm{m}{\mathrm{s}}^{-1}$, the weakly stratified upper layer hasFor the inverse-width estimate $|k|\sim 10^{-4}{\mathrm{m}}^{-1}$, it is near the turning threshold $m=0$, with negligible upward group-velocity component. Components of larger $|k|$ are evanescent aloft and are substantially reflected; only horizontal wavelengths exceeding $2\pi U/N_{\mathrm{upper}}\simeq 62.8\mathrm{km}$ can propagate vertically there. For the alternative $10\mathrm{km}$ sinusoidal wavelength, $m_{\mathrm{upper}}^2<0$ and the evanescent decay length is about $1.6\mathrm{km}$.

The sharp inversion and change of stability can therefore cap the upward-radiating disturbance. Reflection between the ground and cap may trap energy in the lower layer and allow downstream lee-wave trains or resonant guided modes, with permitted wavelengths depending on the interface and lower-layer structure. Reflection, leakage and possible resonance are distinct: a specified $5\mathrm{km}$ height does not by itself fix a unique mode amplitude or wavelength.

Wave clouds may become confined below the cap or organized into persistent lee bands if there is enough moisture. Inversion-limited vertical mixing can also retain low cloud, haze or pollutants; sufficiently strong trapped motions can promote rotors or turbulence. These are possible changes, not inevitable weather outcomes: humidity, wave amplitude, detailed stability and wind shear are unspecified. The new structure suppresses upward energy transmission for the dominant short horizontal scales even though the lower atmosphere can still support waves.