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 contracts and increases its column mass, causing isostatic deepening. Integrating a half-space cooling profile gives an excess mass proportional to $\sqrt{t}$. Converting this to seafloor depth includes the density contrast between mantle and seawater. Finite-plate cooling and basal heat supply limit indefinite square-root subsidence at old ages.