Thermal buoyancy support of topography ($\Delta H\simeq ({\rho }_s/{\rho }_{\mathrm{ref}})\alpha \Delta TL$)

Suppose a compensated vertical column contains thickness $L$ of material warmed by $\Delta T$. For a small anomaly, the volumetric thermal expansion coefficient gives $\delta \rho \simeq -{\rho }_s\alpha \Delta T$. Its column mass change is $\delta M\simeq -{\rho }_s\alpha \Delta TL$. Define ${\rho }_{\mathrm{ref}}\Delta H$ to be the additional column mass associated with the permitted uplift and interface motion in the chosen isostasy model. The mass balance ${\rho }_{\mathrm{ref}}\Delta H+\delta M=0$ proves the displayed scaling.

The model must specify ${\rho }_{\mathrm{ref}}$: translating an upper column of fixed internal thickness above mantle compensation gives a mantle-density scale, while fixing an internal interface and changing only surface-layer thickness gives that surface layer's density scale. If their ratio is near one, $L=7$ km, $\Delta T=350$ K and $\alpha =10^{-5}{\mathrm{K}}^{-1}$ imply only about 25 m of relief. Extending this to a whole lithosphere requires its depth-dependent temperature anomaly, not only the mean temperature anomaly of the crust. Dynamic topography and flexural stresses lie outside a simple local-column calculation.