Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2024 / Paper 3 / 8 / b

Write ${\delta }_j=\sqrt{2K/{\omega }_j}$ for each component's thermal penetration depth. Its contribution at depth isThus its amplitude is multiplied by $e^{-x/{\delta }_j}$ and its wave phase lags the surface by $x/{\delta }_j$ radians. The corresponding unwrapped time lag is $x/({\delta }_j{\omega }_j)$; a phase measured from a single periodic signal is observable only modulo $2\pi$.

Higher-frequency variations have smaller penetration depths. In particular,The daily oscillation is therefore strongly attenuated at depths where a substantial annual oscillation survives. This is a thermal diffusion wave, whose phase lag is not the arrival time of a sharp thermal front. The displayed solution is the established periodic response; an additional transient would depend on initial subsurface data not needed for this interpretation.