= Thermal buoyancy support of topography
{title2=$\Delta H\simeq(\rho_s/\rho_{\rm ref})\alpha\Delta T L$}
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 $\delta M\simeq-\rho_s\alpha\Delta T L$. Define $\rho_{\rm ref}\Delta H$ to be the additional column mass associated with the permitted uplift and interface motion in the chosen <isostasy> model. The mass balance $\rho_{\rm ref}\Delta H+\delta M=0$ proves the displayed scaling.
The model must specify $\rho_{\rm ref}$: 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.
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