Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2025 / Paper 3 / 7 / g / Solution

Aliasing occurs because a sampled grid cannot distinguish certain different continuous Fourier modes. With spacing $h$, modes $k$ and $k+2\pi m/h$ agree at every sample point. Frequencies above the Nyquist frequency can therefore be misidentified as lower frequencies, changing the resolved spectrum and its subsequent diffusion rate.

For the corrected grid, $h=10/128\mathrm{cm}$ and the maximum resolved angular wavenumber is approximately $\pi /h\simeq 40.2\mathrm{c}{\mathrm{m}}^{-1}$. The initial Gaussian function has spectral envelope proportional to $e^{-a|k{|}^2/4}$. At that one-coordinate cutoff the factor is about $e^{-40.4}$, so the omitted high-frequency tail is tiny. Its continuation mismatch at the distant edges is also tiny. Significant aliasing is not expected for this resolved, smooth initial condition.

The linear constant-coefficient heat equation only damps existing modes; it does not generate higher frequencies through nonlinear products. An unresolved narrow peak, a sharp edge or a nonlinear source would require a separate resolution assessment. The duplicated-endpoint grid issue described above should not be diagnosed as high-frequency folding.