Isostasy (Greek ísos 'equal', stásis 'standstill') or isostatic equilibrium is the state of gravitational equilibrium between Earth's crust (or lithosphere) and mantle such that the crust "floats" at an elevation that depends on its thickness and density. This concept is invoked to explain how different topographic heights can exist at Earth's surface. Although originally defined in terms of continental crust and mantle,^[1] it has subsequently been interpreted in terms of lithosphere and asthenosphere, particularly with respect to oceanic island volcanoes,^[2] such as the Hawaiian Islands.

Although Earth is a dynamic system that responds to loads in many different ways,^[3] isostasy describes the important limiting case in which crust and mantle are in static equilibrium. Certain areas (such as the Himalayas and other convergent margins) are not in isostatic equilibrium and are not well described by isostatic models.

The general term isostasy was coined in 1882 by the American geologist Clarence Dutton.^[4][5][6]

History of the concept

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.^[7]

The American geologist Clarence Dutton use the word 'isostasy' in 1889 to describe this general phenomenon.^[4][5][6] 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.^[8] The Airy hypothesis was later refined by the Finnish geodesist Veikko Aleksanteri Heiskanen and the Pratt hypothesis by the American geodesist John Fillmore Hayford.^[3]

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.^[9] 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.^[10] The concept was further developed in the 1950s by the Dutch geodesist Vening Meinesz.^[3]

Models

Three principal models of isostasy are used:^[3][11]

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.

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.^[12] Models such as deep dynamic isostasy (DDI) include such viscous forces and are applicable to a dynamic mantle and lithosphere.^[13] 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.^[14]

Airy

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:

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

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

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

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

Pratt

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]

Vening Meinesz / flexural

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

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

As the rigid layer becomes weaker, ${\displaystyle \kappa }$ approaches infinity, and the behavior approaches the pure hydrostatic balance of the Airy-Heiskanen hypothesis.^[14] Zdroj:https://en.wikipedia.org?pojem=Isostasy
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