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

Diffusion smooths and broadens the hot spot while conserving its integral. For an isolated two-dimensional initial profile $u_0=e^{-r^2/a}$, the Gaussian hot spot under diffusion isHere $a=0.1\mathrm{c}{\mathrm{m}}^2$ and $t=10\mathrm{s}$, giving a width parameter of $0.5\mathrm{c}{\mathrm{m}}^2$ and a peak of $0.20$ in the initial temperature units. The periodic-grid calculation agrees to displayed precision because the periodic copies are far away. Running the endpoint-inclusive listing as written instead gives a sampled initial peak near $0.9695$ and final peak near $0.1963$; the difference is consistent with the coordinate-grid issue, rather than a different diffusion law.

Increasing positive thermal diffusivity at fixed time produces a wider, lower peak. Each nonconstant Fourier mode decays faster, while the spatial mean of a periodic heat equation remains fixed. The profiles approach a uniform mean at long times under these periodic boundary conditions.