= Oceanic thermal subsidence
{title2=$d(t)-d_r\propto\sqrt t$}
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 model>
$$
T(z,t)=T_s+\Delta T\,\operatorname{erf}\left(\frac{z}{2\sqrt{\kappa t}}\right),\qquad \Delta T=T_m-T_s>0.
$$
For 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> is
$$
\Delta\Sigma=\rho_m\alpha\Delta T\int_0^\infty\operatorname{erfc}\left(\frac{z}{2\sqrt{\kappa t}}\right)dz
=\frac{2\rho_m\alpha\Delta T}{\sqrt\pi}\sqrt{\kappa t}.
$$
This uses $\int_0^\infty\operatorname{erfc}u\,du=1/\sqrt\pi$.
At a common compensation depth, additional water depth replaces the dense upper column by seawater of <mass density> $\rho_w$. Balancing column masses gives $(\rho_m-\rho_w)[d(t)-d_r]=\Delta\Sigma$, hence
$$
d(t)-d_r=\frac{2\rho_m\alpha\Delta T}{(\rho_m-\rho_w)\sqrt\pi}\sqrt{\kappa t}.
$$
The 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.
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