Oceanic plate cooling model (${\tau }_0=a^2/({\pi }^2\kappa )$)

An oceanic plate cooling model solves the heat equation in a layer of fixed thickness $a$, with fixed cold surface and hot basal temperatures. A hot initial interior cools towards a linear steady profile. Unlike a half-space cooling model, it has a finite limiting thermal thickness and a nonzero old-age surface heat flux. Its transient sine coefficients must reproduce the chosen initial profile as well as the boundary values.

For a hot initial plate cooled from above, Fourier's law gives the displayed upward heat flux. At old ages only the steady value $k\Delta T/a$ remains. At young ages the sum approaches a Gaussian integral, giving $q_s\sim k\Delta T/\sqrt{\pi \kappa t}$. This divergence is an idealized sudden surface cooling, not an infinite observed ridge heat flux.