 Solution

ID: past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2022/paper-1/10/b/ii/solution

Choose

t_{1} = 1/ (κ k^{2})

to make the sketch explicit. The three profiles are

T (0, y) = A cos k y, T (t_{1}, y) = A e^{- 1} cos k y, T (2 t_{1}, y) = A e^{- 2} cos k y .

They have the same wavelength, zero crossings and phase, with amplitudes

A

0.3679 A

and

0.1353 A

. Diffusion reduces the temperature gradient and amplitude; it does not advect or change this individual mode’s wavelength.

Original diffusion solutions: one Fourier mode loses amplitude, and a two-mode profile loses its short-wavelength structure faster

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