= Solution
\b[<Oceanic thermal subsidence>.] Newly formed oceanic <lithosphere> is hot and buoyant. Cooling increases its column <density>, causing subsidence by <isostasy> as it moves away from a ridge. The <half-space cooling model> has profile is
$$
T(z,t)=T_s+(T_m-T_s)\operatorname{erf}\left(\frac{z}{2\sqrt{\kappa t}}\right).
$$
With volumetric <thermal expansion> coefficient $\alpha$, the excess column <mass> relative to hot <mantle> is
$$
\Delta\Sigma\simeq\rho_m\alpha\int_0^\infty(T_m-T)\,dz
=2\rho_m\alpha(T_m-T_s)\sqrt{\frac{\kappa t}{\pi}}.
$$
Compensation with seawater <density> $\rho_w$ gives
$$
\boxed{d(t)=d_r+
\frac{2\rho_m\alpha(T_m-T_s)}{\rho_m-\rho_w}
\sqrt{\frac{\kappa t}{\pi}}.}
$$
Thus young-to-intermediate-age seafloor generally deepens approximately as the square root of age. At old ages, finite thermal-plate thickness and basal <heat> supply flatten this trend; indefinite half-space deepening is inappropriate. Sediments, variable <crust>, volcanic swells and <dynamic topography> also perturb bathymetry. Ocean-floor depth and <heat> flow jointly constrain cooling models.<An analysis of the variation of ocean floor bathymetry and heat flow with age by Barry Parsons and John G. Sclater>[1977, half-space cooling, bathymetric age trend and finite-plate alternative]{ref}
\b[<Effective elastic thickness>.] This is the thickness of a homogeneous elastic plate with the observed effective <flexural rigidity>, not necessarily the crustal or thermal lithospheric thickness:
$$
\boxed{D=\frac{ET_e^3}{12(1-\nu^2)}.}
$$
Infer it by modelling deflection around known loads, such as seamounts or trenches, or fitting <gravity-topography admittance> and <gravity-topography coherence> across wavelengths. A simple Fourier balance is $(Dk^4+\Delta\rho g)w_k=q_k$ for a signed transverse load. The transition from buoyancy-dominated to bending-dominated behaviour constrains $D$; then assumed <Young's modulus> $E$ and <Poisson's ratio> $\nu$ give $T_e$.
Good inference includes load geometry and <density>, buried interfaces, sediment and internal loading, <boundary conditions> and data resolution. These can trade off with rigidity. Plastic yielding, faulting, mechanical layering and creep make $T_e$ an effective, timescale- and stress-dependent quantity. Older, colder <lithosphere> is often stronger, but age at loading and subsequent relaxation matter; a single universal age-to-$T_e$ conversion is unjustified.
\Image[/past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2014/paper-1-pep-thermal-flexure.png]
{title=Square-root cooling versus finite-plate saturation, and stiffness sensitivity of the flexural response}
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