Radiative decay of a localized temperature perturbation in an idealized atmosphere

We use Fourier analysis and Spiegel's results (1957) to calculate the radiative decay of a small, smooth, localized, temperature perturbation in a homogeneous, isothermal atmosphere, assuming the grey case. We find approximate, analytic expressions for the decay in both the optically thin and optically thick limits. We also derive a method for checking the accuracy of the solution(s) using the conservation of energy of the perturbation. -Authors