 Solution

ID: past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2025/paper-3/7/b/solution

Apply the Fourier transform of a derivative twice in each spatial coordinate, and commute the time derivative with the spatial Fourier transform. For the Laplacian,

F [Δ u] = - (k_{x}^{2} + k_{y}^{2}) u .

Thus

\partial_{t} u = - α (k_{x}^{2} + k_{y}^{2}) u .

The minus sign is essential: positive thermal diffusivity damps short spatial scales rather than amplifying them. The zero Fourier mode is unchanged in the absence of a source.

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