Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2013 / Paper 1 / 1 / a

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.

At the top, $T(a,t)=T_0$. Ocean water provides a cold thermal reservoir relative to mantle temperatures, so a fixed surface temperature is a useful idealization. The model does not resolve near-axis hydrothermal circulation or spatially variable bottom-water temperatures.

At the ridge axis, $T(z,0)=T_1$ for $0<z<a$ in the intended cooling model. Upwelling and magmatic heat supply continually bring hot material into the newly formed plate. The abrupt meeting of this hot interior with the cold upper boundary is an idealized corner discontinuity, resolved diffusively for $t>0$. The printed series fails this initial condition, as shown in the parent Solution.

At the bottom, $T(0,t)=T_1$. Convecting mantle is treated as maintaining the basal temperature and supplying heat to the plate. This is a prescribed thermal boundary at fixed thickness, not a claim that the real mechanical base is an impermeable material interface.

Far from the ridge, $T(z,\infty )=T_0+\Delta T(1-z/a)$. Transients have decayed and the heat equation reduces to $T_{zz}=0$. A finite basal heat supply sustains this gradient. Thermal thickness, seafloor depth and surface heat flux therefore approach finite limits rather than continuing the laws of the half-space cooling model indefinitely.