isostasy

In the 17th and 18th centuries, French geodesists (for example, Jean Picard) attempted to determine the shape of the Earth (the geoid) by measuring the length of a degree of latitude at different latitudes (arc measurement). A party working in Ecuador was aware that its plumb lines, used to determine the vertical direction, would be deflected by the gravitational attraction of the nearby Andes Mountains. However, the deflection was less than expected, which was attributed to the mountains having low-density roots that compensated for the mass of the mountains. In other words, the low-density mountain roots provided the buoyancy to support the weight of the mountains above the surrounding terrain. Similar observations in the 19th century by British surveyors in India showed that this was a widespread phenomenon in mountainous areas. It was later found that the difference between the measured local gravitational field and what was expected for the altitude and local terrain (the Bouguer anomaly) is positive over ocean basins and negative over high continental areas. This shows that the low elevation of ocean basins and high elevation of continents is also compensated at depth.{{cite book |last1=Kearey |first1=P. |last2=Klepeis |first2=K.A. |last3=Vine |first3=F.J. |title=Global tectonics. |date=2009 |publisher=Wiley-Blackwell |location=Oxford |isbn=9781405107778 |edition=3rd |page=42}}

The American geologist Clarence Dutton use the word 'isostasy' in 1889 to describe this general phenomenon. However, two hypotheses to explain the phenomenon had by then already been proposed, in 1855, one by George Airy and the other by John Henry Pratt.{{sfn|Kearey|Klepeis|Vine|2009|p=43}} The Airy hypothesis was later refined by the Finnish geodesist Veikko Aleksanteri Heiskanen and the Pratt hypothesis by the American geodesist John Fillmore Hayford.

Both the Airy-Heiskanen and Pratt-Hayford hypotheses assume that isostacy reflects a local hydrostatic balance. A third hypothesis, lithospheric flexure, takes into account the rigidity of the Earth's outer shell, the lithosphere.{{sfn|Kearey|Klepeis|Vine|2009|pp=44-45}} Lithospheric flexure was first invoked in the late 19th century to explain the shorelines uplifted in Scandinavia following the melting of continental glaciers at the end of the last glaciation. It was likewise used by American geologist G. K. Gilbert to explain the uplifted shorelines of Lake Bonneville.{{cite journal |last1=Gilber |first1=G.K. |title=Lake Bonneville |journal=U.S. Geological Survey Monograph |date=1890 |volume=1 |doi=10.3133/m1}} The concept was further developed in the 1950s by the Dutch geodesist Vening Meinesz.

Three principal models of isostasy are used:{{sfn|Kearey|Klepeis|Vine|2009|pp=42-45}}

1. The Airy–Heiskanen model – where different topographic heights are accommodated by changes in crustal thickness, in which the crust has a constant density
2. The Pratt–Hayford model – where different topographic heights are accommodated by lateral changes in rock density.
3. The Vening Meinesz, or flexural isostasy model – where the lithosphere acts as an elastic plate and its inherent rigidity distributes local topographic loads over a broad region by bending.

Airy and Pratt isostasy are statements of buoyancy, but flexural isostasy is a statement of buoyancy when deflecting a sheet of finite elastic strength. In other words, the Airy and Pratt models are purely hydrostatic, taking no account of material strength, while flexural isostacy takes into account elastic forces from the deformation of the rigid crust. These elastic forces can transmit buoyant forces across a large region of deformation to a more concentrated load.

{{anchor|Isostatic anomaly}}

Perfect isostatic equilibrium is possible only if mantle material is in rest. However, thermal convection is present in the mantle. This introduces viscous forces that are not accounted for the static theory of isostacy. The isostatic anomaly or IA is defined as the Bouger anomaly minus the gravity anomaly due to the subsurface compensation, and is a measure of the local departure from isostatic equilibrium.

At the center of a level plateau, it is approximately equal to the free air anomaly.{{sfn|Kearey|Klepeis|Vine|2009|pp=45-48}} Models such as deep dynamic isostasy (DDI) include such viscous forces and are applicable to a dynamic mantle and lithosphere.{{cite journal | title=Mantle Flow and Determining Position of LAB Assuming Isostasy | last=Czechowski | first=L. | journal=Pure and Applied Geophysics | year=2019 | volume=176 | issue=6 | pages=2451–2463 | doi=10.1007/s00024-019-02093-8| bibcode=2019PApGe.176.2451C | doi-access=free }} Measurements of the rate of isostatic rebound (the return to isostatic equilibrium following a change in crust loading) provide information on the viscosity of the upper mantle.{{sfn|Kearey|Klepeis|Vine|2009|p=45}}

File:Airy Isostasy.jpg

File:Backstripping and eustasy correction.jpg

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):{{sfn|Kearey|Klepeis|Vine|2009|p=43}}

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:

:$(h\_1+c+b\_1)\backslash rho\_c\; =\; (c\backslash rho\_c)+(b\_1\backslash rho\_m)$

:$\{b\_1(\backslash rho\_m-\backslash rho\_c)\}\; =\; h\_1\backslash rho\_c$

:$b\_1\; =\; \backslash frac\{h\_1\backslash rho\_c\}\{\backslash rho\_m-\backslash rho\_c\}$

where $\backslash rho\_m$ is the density of the mantle (ca. 3,300 kg m⁻³) and $\backslash rho\_c$ 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:

:$c\backslash rho\_c\; =\; (h\_2\backslash rho\_w)+(b\_2\backslash rho\_m)+[(c-h\_2-b\_2)\backslash rho\_c]$

:$\{b\_2(\backslash rho\_m-\backslash rho\_c)\}\; =\; \{h\_2(\backslash rho\_c-\backslash rho\_w)\}$

:$b\_2\; =\; (\backslash frac\{\backslash rho\_c-\backslash rho\_w\}\{\backslash rho\_m-\backslash rho\_c\})\{h\_2\}$

where $\backslash rho\_m$ is the density of the mantle (ca. 3,300 kg m⁻³), $\backslash rho\_c$ is the density of the crust (ca. 2,750 kg m⁻³) and $\backslash rho\_w$ 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: $\backslash rho\_1\; =\; \backslash rho\_c\; \backslash frac\{c\}\{h\_1+c\}$, where $h\_1$ is the height of the mountain and c the thickness of the crust.{{sfn|Kearey|Klepeis|Vine|2009|pp=43-44}}

File:Local-regional isostasy - flexure, elastic thickness.jpg

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.{{sfn|Kearey|Klepeis|Vine|2009|pp=44-45}}

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

:$D\backslash frac\{d^4z\}\{dx^4\}+(\backslash rho\_m-\backslash rho\_w)zg\; =\; P(x)$

where $\backslash rho\_m$ and $\backslash rho\_w$ are the densities of the aesthenosphere and ocean water, g is the acceleration due to gravity, and $P(x)$ is the load on the ocean crust. The parameter D is the flexural rigidity, defined as

:$D=ET^3\_c/12(1-\backslash sigma^2)$

where E is Young's modulus, $\backslash sigma$ is Poisson's ratio, and $T\_c$ is the thickness of the lithosphere. Solutions to this equation have a characteristic wave number

:$\backslash kappa=\backslash sqrt[4]\{(\backslash rho\_m-\backslash rho\_w)g/4D\}$

As the rigid layer becomes weaker, $\backslash kappa$ approaches infinity, and the behavior approaches the pure hydrostatic balance of the Airy-Heiskanen hypothesis.{{sfn|Kearey|Klepeis|Vine|2009|p=45}}

==Depth of compensation==

The depth of compensation (also known as the compensation level, compensation depth, or level of compensation) is the depth below which the pressure is identical across any horizontal surface. In stable regions, it lies in the deep crust, but in active regions, it may lie below the base of the lithosphere.{{cite book |editor1-last=Jackson |editor1-first=Julia A. |title=Glossary of geology. |date=1997 |publisher=American Geological Institute |location=Alexandria, Virginia |isbn=0922152349 |edition=Fourth |chapter=depth of compensation}} In the Pratt model, it is the depth below which all rock has the same density; above this depth, density is lower where topographic elevation is greater.{{cite book |last1=Allaby |first1=Michael |title=A dictionary of geology and earth sciences |date=2013 |publisher=Oxford University Press |location=Oxford |isbn=9780199653065 |edition=Fourth |chapter=Pratt model}}

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.{{sfn|Kearey|Klepeis|Vine|2009|pp=45-46}}

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.{{sfn|Kearey|Klepeis|Vine|2009|p=43}}{{cite book |last1=Monroe |first1=James S. |title=Physical geology : exploring the Earth |date=1992 |publisher=West Pub. Co |location=St. Paul |isbn=0314921958 |page=305}}

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 {{convert|80|km||sp=us}} thick,{{sfn|Kearey|Klepeis|Vine|2009|p=322}} versus {{convert|40|km||sp=us}} for average continental crust.{{sfn|Kearey|Klepeis|Vine|2009|p=19}} 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.{{sfn|Kearey|Klepeis|Vine|2009|p=48}}

Mid-ocean ridges are explained by the Pratt hypothesis as overlying regions of unusually low density in the upper mantle.{{sfn|Kearey|Klepeis|Vine|2009|p=48}} This reflects thermal expansion from the higher temperatures present under the ridges.{{cite book |last1=Philpotts |first1=Anthony R. |last2=Ague |first2=Jay J. |title=Principles of igneous and metamorphic petrology |date=2009 |publisher=Cambridge University Press |location=Cambridge, UK |isbn=9780521880060 |edition=2nd |pages=6–10}}

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.{{sfn|Kearey|Klepeis|Vine|2009|p=48}}

{{Main|Post-glacial rebound}}

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{{cite journal |last1=Eronen |first1=Matti |last2=Gluckert |first2=Gunnar |last3=Hatakka |first3=Lassi |last4=Van de Plassche |first4=Orson |last5=Van der Plicht |first5=Johannes |last6=Rantala |first6=Pasi |date=28 June 2008 |title=Rates of Holocene isostatic uplift and relative sea-level lowering of the Baltic in SW Finland based on studies of isolation contacts |url=https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1502-3885.2001.tb00985.x |journal=Boreas |volume=30 |issue=1 |pages=17–30 |doi=10.1111/j.1502-3885.2001.tb00985.x |s2cid=54582233 |access-date=15 November 2022}} and Hudson Bay.{{cite journal |last1=Balestra |first1=Barbara |last2=Bertini |first2=Adele |last3=De Vernal |first3=Anne |last4=Monechi |first4=Simonetta |last5=Reale |first5=Viviana |date=1 October 2013 |title=Late Quaternary sea surface conditions in the Laurentian Fan: Evidence from coccolith and dinocyst assemblages |url=https://www.sciencedirect.com/science/article/abs/pii/S0031018213003234#! |journal=Palaeogeography, Palaeoclimatology, Palaeoecology |volume=387 |pages=200–210 |doi=10.1016/j.palaeo.2013.07.002 |bibcode=2013PPP...387..200B |access-date=15 November 2022|url-access=subscription }} 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.{{sfn|Kearey|Klepeis|Vine|2009|pp=45-46}}

In addition to the vertical movement of the land and sea, isostatic adjustment of the Earth also involves horizontal movements.{{cite journal |last1=James |first1=Thomas S. |last2=Morgan |first2=W. Jason |title=Horizontal motions due to post-glacial rebound |journal=Geophysical Research Letters |date=June 1990 |volume=17 |issue=7 |pages=957–960 |doi=10.1029/GL017i007p00957|bibcode=1990GeoRL..17..957J }} It can cause changes in Earth's gravitational field{{cite journal |last1=Alexander |first1=J. C. |title=Higher harmonic effects of the Earth's gravitational field from post-glacial rebound as observed by Lageos |journal=Geophysical Research Letters |date=November 1983 |volume=10 |issue=11 |pages=1085–1087 |doi=10.1029/GL010i011p01085|bibcode=1983GeoRL..10.1085A }} and rotation rate, polar wander,{{cite journal |last1=Wahr |first1=John |author-link=John M. Wahr |last2=Dazhong |first2=Han |last3=Trupin |first3=Andrew |last4=Lindqvist |first4=Varna |title=Secular changes in rotation and gravity: Evidence of post-glacial rebound or of changes in polar ice? |journal=Advances in Space Research |date=November 1993 |volume=13 |issue=11 |pages=257–269 |doi=10.1016/0273-1177(93)90228-4|bibcode=1993AdSpR..13k.257W }} and earthquakes.{{cite book |last1=Davenport |first1=Colin A. |last2=Ringrose |first2=Philip S. |last3=Becker |first3=Amfried |last4=Hancock |first4=Paul |last5=Fenton |first5=Clark |title=Earthquakes at North-Atlantic Passive Margins: Neotectonics and Postglacial Rebound |chapter=Geological Investigations of Late and Post Glacial Earthquake Activity in Scotland |date=1989 |pages=175–194 |doi=10.1007/978-94-009-2311-9_11|isbn=978-94-010-7538-1 }}

The hypothesis of isostasy is often used to determine the position of the lithosphere-asthenosphere boundary (LAB).{{cite journal |last1=Grinč |first1=M. |last2=Zeyen |first2=H. |last3=Bielik |first3=M. |year=2014 |title=Automatic 1D integrated geophysical modelling of lithospheric discontinuities: a case study from Carpathian-Pannonian Basin region |journal=Contributions to Geophysics and Geodesy |volume=44 |number=2 |pages=115–131 |doi=10.2478/congeo-2014-0007 |bibcode=2014CoGG...44..115G |s2cid=129497623 |url=https://journal.geo.sav.sk/cgg/article/download/109/104 |access-date=13 December 2021|doi-access=free }}

• {{annotated link|Archimedes' principle}}
• {{annotated link|William Bowie (engineer)}}
• {{annotated link|Lau, Gotland}}
• {{annotated link|Marine terrace}}
• {{annotated link|Gravity anomaly}}
• Gravity map
• Timeline of the development of tectonophysics (before 1954)

{{Reflist}}

• {{cite book |last1=Lisitzin |first1=Eugenie |author-link1=Eugenie Lisitzin |title=Sea-level changes |date=1974 |publisher=Elsevier Scientific Pub. Co |location=Amsterdam |isbn=9780080870441 |url=https://books.google.com/books?id=oohairZYezMC&dq=Lisitzin,+E.+(1974)++Sea+level+changes+.+Elsevier+Oceanography+Series,+8&pg=PP1 |access-date=23 March 2022}}

• {{Cite EB1922 |last=Oldham |first=Richard Dixon |author-link=Richard Dixon Oldham |wstitle=Isostasy |short=x}}

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Category:Geodynamics

Category:Geomorphology

Category:Buoyancy

Category:Earth's crust