Isostasy

Approximate gravitational compensation of loads by subsurface mass redistribution. Airy and Pratt models use local column balances; flexural support spreads the load laterally. Real surface topography can also be maintained by dynamic stresses.

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.

Let a deposited layer have thickness $H$ and density ${\rho }_b$, replacing a fluid of density ${\rho }_f$, while the original crust translates downward by $w$ above mantle of density ${\rho }_m$. Assume all deposited material stays in the same fluid environment, unchanged pre-existing layer thicknesses, an external material source, and complete local isostatic adjustment without bending resistance.

At a fixed compensation depth, added column mass is $({\rho }_b-{\rho }_f)H$ per unit area, whereas depression replaces a column contrast $({\rho }_m-{\rho }_f)w$. Equal hydrostatic pressure therefore requiresIt follows thatThe pre-existing ground subsides by $w$, while the top of the new layer rises by $\Delta z$ if ${\rho }_b<{\rho }_m$. For a subaerial deposit, air density is negligible. For a fully submerged deposit, use seawater density. A deposit that crosses the fluid surface requires separate treatment of its submerged and emerged portions. Specifying only volume $V$ leaves the answer dependent on loaded area through $H=V/A$.

Addition of erupted or intruded rock as a gravitational load on a planetary surface or interior. Deposited thickness, density, geometry and emplacement history control isostatic adjustment and lithospheric flexure. Thermal and compositional changes in the source can supply additional effects distinct from the weight of the deposited material.

A common reference depth at which lateral pressure differences are assumed relaxed in an isostatic model. It must lie below relevant lateral density anomalies; it need not be a sharp material interface.

The addition of sediments as a mechanical load, contributing to subsidence through isostasy and lithospheric flexure.

Deformation and redistribution of material tending to restore gravitational support after a load or density distribution changes.

A cold linear plate profile has excess column mass ${\rho }_m\alpha \Delta Ta/2$ relative to hot mantle. Equal hydrostatic pressure below the plates is restored by replacing part of the rock column with water. The displayed elevation follows to first order in thermal expansion; its denominator is the rock-minus-water density difference, not the rock density alone.

A local compensation model in which columns of different density reach a common compensation depth. Higher surface topography is associated with a less dense column; density variation replaces the variable-root mechanism of an Airy model.

A local compensation model with approximately fixed crustal density and variable crustal thickness. A height $h$ is supported by a crustal root $b={\rho }_{c}h/({\rho }_m-{\rho }_c)$ in a simple constant-density, air-loaded column model.