Isostasy

Airy isostasy, in which a constant-density crust floats on a higher-density mantle, and topography is determined by the thickness of the crust.

Airy isostasy applied to a real-case basin scenario, where the total load on the mantle is composed by a crustal basement, lower-density sediments and overlying marine water

The basis of the model is Pascal's law, and particularly its consequence that, within a fluid in static equilibrium, the hydrostatic pressure is the same on every point at the same elevation (surface of hydrostatic compensation):[3][8]

h₁⋅ρ₁ = h₂⋅ρ₂ = h₃⋅ρ₃ = ... h_n⋅ρ_n

For the simplified picture shown, the depth of the mountain belt roots (b₁) is calculated as follows:

${\displaystyle (h_{1}+c+b_{1})\rho _{c}=(c\rho _{c})+(b_{1}\rho _{m})}$

${\displaystyle {b_{1}(\rho _{m}-\rho _{c})}=h_{1}\rho _{c}}$

${\displaystyle b_{1}={\frac {h_{1}\rho _{c}}{\rho _{m}-\rho _{c}}}}$

where ${\displaystyle \rho _{m}}$ is the density of the mantle (ca. 3,300 kg m⁻³) and ${\displaystyle \rho _{c}}$ is the density of the crust (ca. 2,750 kg m⁻³). Thus, generally:

b₁ ≅ 5⋅h₁

In the case of negative topography (a marine basin), the balancing of lithospheric columns gives:

${\displaystyle c\rho _{c}=(h_{2}\rho _{w})+(b_{2}\rho _{m})+[(c-h_{2}-b_{2})\rho _{c}]}$

${\displaystyle {b_{2}(\rho _{m}-\rho _{c})}={h_{2}(\rho _{c}-\rho _{w})}}$

${\displaystyle b_{2}=({\frac {\rho _{c}-\rho _{w}}{\rho _{m}-\rho _{c}}}){h_{2}}}$

where ${\displaystyle \rho _{m}}$ is the density of the mantle (ca. 3,300 kg m⁻³), ${\displaystyle \rho _{c}}$ is the density of the crust (ca. 2,750 kg m⁻³) and ${\displaystyle \rho _{w}}$ is the density of the water (ca. 1,000 kg m⁻³). Thus, generally:

b₂ ≅ 3.2⋅h₂

For the simplified model shown the new density is given by: ${\displaystyle \rho _{1}=\rho _{c}{\frac {c}{h_{1}+c}}}$, where ${\displaystyle h_{1}}$ is the height of the mountain and c the thickness of the crust.[3][15]

Cartoon showing the isostatic vertical motions of the lithosphere (grey) in response to a vertical load (in green)

This hypothesis was suggested to explain how large topographic loads such as seamounts (e.g. Hawaiian Islands) could be compensated by regional rather than local displacement of the lithosphere. This is the more general solution for lithospheric flexure, as it approaches the locally compensated models above as the load becomes much larger than a flexural wavelength or the flexural rigidity of the lithosphere approaches zero.[3][9]

For example, the vertical displacement z of a region of ocean crust would be described by the differential equation

${\displaystyle D{\frac {d^{4}z}{dx^{4}}}+(\rho _{m}-\rho _{w})zg=P(x)}$

where ${\displaystyle \rho _{m}}$ and ${\displaystyle \rho _{w}}$ are the densities of the aesthenosphere and ocean water, g is the acceleration due to gravity, and ${\displaystyle P(x)}$ is the load on the ocean crust. The parameter D is the flexural rigidity, defined as

${\displaystyle D=ET_{c}^{3}/12(1-\sigma ^{2})}$

where E is Young's modulus, ${\displaystyle \sigma }$ is Poisson's ratio, and ${\displaystyle T_{c}}$ is the thickness of the lithosphere. Solutions to this equation have a characteristic wave number

${\displaystyle \kappa ={\sqrt[{4}]{(\rho _{m}-\rho _{w})g/4D}}}$

As the rigid layer becomes weaker, ${\displaystyle \kappa }$ approaches infinity, and the behavior approaches the pure hydrostatic balance of the Airy-Heiskanen hypothesis.[14]

When large amounts of sediment are deposited on a particular region, the immense weight of the new sediment may cause the crust below to sink. Similarly, when large amounts of material are eroded away from a region, the land may rise to compensate. Therefore, as a mountain range is eroded, the (reduced) range rebounds upwards (to a certain extent) to be eroded further. Some of the rock strata now visible at the ground surface may have spent much of their history at great depths below the surface buried under other strata, to be eventually exposed as those other strata eroded away and the lower layers rebounded upwards.[18]

An analogy may be made with an iceberg, which always floats with a certain proportion of its mass below the surface of the water. If snow falls to the top of the iceberg, the iceberg will sink lower in the water. If a layer of ice melts off the top of the iceberg, the remaining iceberg will rise. Similarly, Earth's lithosphere "floats" in the asthenosphere.[8][19]

When continents collide, the continental crust may thicken at their edges in the collision. It is also very common for one of the plates to be underthrust beneath the other plate. The result is that the crust in the collision zone becomes as much as 80 kilometers (50 mi) thick, [20] versus 40 kilometers (25 mi) for average continental crust.[21] As noted above, the Airy hypothesis predicts that the resulting mountain roots will be about five times deeper than the height of the mountains, or 32 km versus 8 km. In other words, most of the thickened crust moves downwards rather than up, just as most of an iceberg is below the surface of the water.

However, convergent plate margins are tectonically highly active, and their surface features are partially supported by dynamic horizontal stresses, so that they are not in complete isostatic equilibrium. These regions show the highest isostatic anomalies on the Earth's surface.[22]

Mid-ocean ridges are explained by the Pratt hypothesis as overlying regions of unusually low density in the upper mantle.[22] This reflects thermal expansion from the higher temperatures present under the ridges.[23]

In the Basin and Range Province of western North America, the isostatic anomaly is small except near the Pacific coast, indicating that the region is generally near isostatic equilibrium. However, the depth to the base of the crust does not strongly correlate with the height of the terrain. This provides evidence (via the Pratt hypothesis) that the upper mantle in this region is inhomogeneous, with significant lateral variations in density.[22]

The formation of ice sheets can cause Earth's surface to sink. Conversely, isostatic post-glacial rebound is observed in areas once covered by ice sheets that have now melted, such as around the Baltic Sea[24] and Hudson Bay.[25] As the ice retreats, the load on the lithosphere and asthenosphere is reduced and they rebound back towards their equilibrium levels. In this way, it is possible to find former sea cliffs and associated wave-cut platforms hundreds of metres above present-day sea level. The rebound movements are so slow that the uplift caused by the ending of the last glacial period is still continuing.[18]

In addition to the vertical movement of the land and sea, isostatic adjustment of the Earth also involves horizontal movements.[26] It can cause changes in Earth's gravitational field[27] and rotation rate, polar wander,[28] and earthquakes.[29]

The hypothesis of isostasy is often used to determine the position of the lithosphere-asthenosphere boundary (LAB).[30]