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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.

In the oceanic plate cooling model, $T$ is temperature, $z$ height above the plate base, $a$ plate thickness, and $t$ age since formation at the spreading axis. With half-spreading rate $v$, a parcel at distance $x$ has $t=x/v$. The surface and basal temperatures are $T_0$ and $T_1$ respectively. The positive integer $n$ labels a sine diffusion mode. For constant thermal diffusivity $\kappa$,Substituting one sine mode into $T_t=\kappa T_{zz}$ fixes this normalization; $a^2/\kappa$ itself is the diffusion timescale before the factor ${\pi }^2$.

Source qualification. The printed transient coefficients have the wrong sign for the intended initially hot plate. Indeed, in the interior,Thus the literal zero-age profile is $T_0+\Delta T(1-2z/a)$, where $\Delta T=T_1-T_0$, rather than $T_1$. It would even approach $T_0-\Delta T$ from inside the upper boundary. The intended hot-ridge cooling solution reverses the transient sign, using $2(-1{)}^{n+1}/(n\pi )$. Both versions have the same boundary temperatures and old-age limit. Part (b) gives both surface-flux expressions; the physical ridge-elevation and ridge-push calculations explicitly use the hot initial plate. This sign is present in the original PDF as well as the held transcription.