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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.

An oceanic plate cooling model solves the heat equation in a layer of fixed thickness $a$, with fixed cold surface and hot basal temperatures. A hot initial interior cools towards a linear steady profile. Unlike a half-space cooling model, it has a finite limiting thermal thickness and a nonzero old-age surface heat flux. Its transient sine coefficients must reproduce the chosen initial profile as well as the boundary values.

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.

Take positive heat flux upwards, along increasing $z$, and denote thermal conductivity by $k$. Fourier's law gives $q_s=-k{T}_z{|}_{z=a}$. Since $\cos (n\pi )=(-1{)}^n$, differentiation of the literal printed expression givesIts negative young-age flux is another sign of the unphysical initial profile. For the explicitly corrected hot initial plate, the surface heat flux of a cooling plate is insteadThe old-age limit is $k\Delta T/a$. For $t\ll {\tau }_0$, replacing the Gaussian sum by its leading integral gives $q_s\sim k\Delta T/\sqrt{\pi \kappa t}$, recovering the half-space cooling model. Neither $k$ nor $\kappa$ is numerically specified, so the age-dependent flux remains symbolic as requested.

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 isWith volumetric thermal expansion coefficient $\alpha$, the excess column mass relative to hot mantle isCompensation with seawater density ${\rho }_w$ givesThus 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.^[ref]

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: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 $(D{k}^4+\Delta \rho g)w_k=q_k$ 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.

New oceanic lithosphere is hot and mechanically weak. As it cools, the depth over which rock can sustain elastic stress on the loading timescale generally increases, so effective elastic thickness tends to increase with plate age at the time of loading. It is this mechanical state when the load is emplaced, together with subsequent relaxation, that matters; the present age alone is insufficient for an ancient load.

The half-space cooling model illustrates the thermal dependence. With age $\tau$, surface temperature $T_s$, initial interior temperature $T_m$ and constant thermal diffusivity $\kappa$,A chosen temperature $T_{\ast }$ lies atIf the base of the mechanically strong layer approximately follows such an isotherm, $T_e$ initially follows a similar age trend. Cooling models constrained by ocean-floor depth and heat flow provide the thermal motivation; their fitted thermal thickness must not be mistaken for measured elastic thickness.^[ref]

In practice $T_e$ is an effective bending thickness rather than the depth of one universal temperature. Stress-dependent yielding, grain size, water content, composition, loading duration and reheating can alter it. Finite-plate cooling and basal heat input also limit indefinite square-root growth at old ages. Nor should $T_e$ be identified with either the thickness of the crust or the full seismically defined lithosphere.