Half-space cooling model ($T(z,\tau )=T_s+(T_m-T_s)\mathrm{erf}[z/(2\sqrt{\kappa \tau })]$)

The half-space cooling model solves the one-dimensional heat equation for initially hot material cooled at a fixed-temperature surface. For constant thermal diffusivity $\kappa$, a fixed interior-temperature isotherm deepens proportional to $\sqrt{\tau }$. Applied to an oceanic lithosphere, it explains the leading age dependence of cooling; finite plate thickness and basal heat supply limit its extrapolation to old ages.

Cooling oceanic lithosphere becomes denser and subsides under isostasy. A simple calculation explains the square-root age law. Let depth $z$ be measured below a fixed cooling surface, take constant thermal diffusivity $\kappa$, and write the half-space cooling modelFor mantle reference mass density ${\rho }_m$ and small constant volumetric thermal expansion coefficient $\alpha$, the cold-column density excess relative to hot mantle is $\delta \rho ={\rho }_m\alpha (T_m-T)$. Its integral per unit horizontal area isThis uses ${\int }_0^{\infty }\mathrm{erfc}udu=1/\sqrt{\pi }$.

At a common compensation depth, additional water depth replaces the dense upper column by seawater of mass density ${\rho }_w$. Balancing column masses gives $({\rho }_m-{\rho }_w)[d(t)-d_r]=\Delta \Sigma$, henceThe relevant geophysical density contrast is therefore mantle versus seawater, and $t$ is seafloor age measured from the ridge reference depth $d_r$. The model neglects sediment loading, dynamic support and changes of material properties. A finite oceanic plate cooling model and basal heating limit indefinite square-root deepening at old ages.