Isostasy (Greek
''ísos'' "equal",
''stásis'' "standstill") or isostatic equilibrium is the state of
gravitational
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
equilibrium between
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
'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
Continental crust is the layer of igneous, sedimentary, and metamorphic rocks that forms the geological continents and the areas of shallow seabed close to their shores, known as continental shelves. This layer is sometimes called '' sial'' be ...
and mantle, it has subsequently been interpreted in terms of
lithosphere and
asthenosphere, particularly with respect to oceanic island
volcanoes
A volcano is a rupture in the crust of a planetary-mass object, such as Earth, that allows hot lava, volcanic ash, and gases to escape from a magma chamber below the surface.
On Earth, volcanoes are most often found where tectonic plates a ...
, such as the
Hawaiian Islands.
Although Earth is a dynamic system that responds to loads in many different ways,
isostasy describes the important limiting case in which crust and mantle are in static equilibrium. Certain areas (such as the
Himalayas
The Himalayas, or Himalaya (; ; ), is a mountain range in Asia, separating the plains of the Indian subcontinent from the Tibetan Plateau. The range has some of the planet's highest peaks, including the very highest, Mount Everest. Over 10 ...
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
Clarence Edward Dutton (May 15, 1841 – January 4, 1912) was an American geologist and US Army officer. Dutton was born in Wallingford, Connecticut on May 15, 1841. He graduated from Yale College in 1860 and took postgraduate courses there until ...
.
History of the concept
In the 18th century, French
geodesist
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
s attempted to determine the shape of the Earth (the
geoid
The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
) by measuring the
length of a degree of latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
at different latitudes (
arc measurement
Arc measurement, sometimes degree measurement (german: Gradmessung), is the astrogeodetic technique of determining of the radius of Earth – more specifically, the local Earth radius of curvature of the figure of the Earth – by relating the la ...
). A party working in
Ecuador
Ecuador ( ; ; Quechua: ''Ikwayur''; Shuar: ''Ecuador'' or ''Ekuatur''), officially the Republic of Ecuador ( es, República del Ecuador, which literally translates as "Republic of the Equator"; Quechua: ''Ikwadur Ripuwlika''; Shuar: ' ...
was aware that its
plumb line
A plumb bob, plumb bob level, or plummet, is a weight, usually with a pointed tip on the bottom, suspended from a string and used as a vertical reference line, or plumb-line. It is a precursor to the spirit level and used to establish a vertic ...
s, 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
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
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 In geodesy and geophysics, the Bouguer anomaly (named after Pierre Bouguer) is a gravity anomaly, corrected for the height at which it is measured and the attraction of terrain. The height correction alone gives a free-air gravity anomaly.
Defini ...
) 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.
The American geologist
Clarence Dutton
Clarence Edward Dutton (May 15, 1841 – January 4, 1912) was an American geologist and US Army officer. Dutton was born in Wallingford, Connecticut on May 15, 1841. He graduated from Yale College in 1860 and took postgraduate courses there until ...
coined the term 'isostasy' in 1882 to describe this general phenomenon.
[ However, two hypotheses to explain the phenomenon had by then already been proposed, in 1855, one by ]George Airy
Sir George Biddell Airy (; 27 July 18012 January 1892) was an English mathematician and astronomer, and the seventh Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the E ...
and the other by John Henry Pratt
John Henry Pratt FRS (4 June 1809 – 28 December 1871) was a British clergyman, astronomer and mathematician. A Cambridge Apostle, he joined the British East India Company in 1838 as a chaplain and later became Archdeacon of Calcutta. Although ...
. The Airy hypothesis was later refined by the Finnish geodesist Veikko Aleksanteri Heiskanen
Veikko Aleksanteri Heiskanen (23 July 1895, in Kangaslampi – 23 October 1971, in Helsinki) was a famous Finnish geodesist.
He is mostly known for his refinement of the theory of isostasy by George Airy and for his studies of the global geoid.
...
and the Pratt hypothesis by the American geodesist John Fillmore Hayford
John Fillmore Hayford (May 19, 1868 – March 10, 1925) was an eminent United States geodesist. His work involved the study of isostasy and the construction of a reference ellipsoid for approximating the figure of the Earth. The crater Hayford on ...
.[
Both the Airy-Heiskanen and Pratt-Hayford hypotheses assume that isostacy reflects a local hydrostatic balance. A third hypothesis, ]lithospheric flexure
Lithospheric flexure (also called regional isostasy) is the process by which the lithosphere (rigid, thin outer layer of the Earth) bends under the action of forces such as the weight of a growing orogeny or changes in ice thickness related to glac ...
, takes into account the rigidity of the Earth's outer shell, the lithosphere. Lithospheric flexure was first invoked in the late 19th century to explain the shorelines uplifted in Scandinavia following the melting of continental glacier
In glaciology, an ice sheet, also known as a continental glacier, is a mass of glacial ice that covers surrounding terrain and is greater than . The only current ice sheets are in Antarctica and Greenland; during the Last Glacial Period at La ...
s 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
Lake Bonneville was the largest Late Pleistocene paleolake in the Great Basin of western North America. It was a pluvial lake that formed in response to an increase in precipitation and a decrease in evaporation as a result of cooler temperature ...
. The concept was further developed in the 1950s by the Dutch geodesist Vening Meinesz.[
]
Models
Three principal models of isostasy are used:[
# The Airy–Heiskanen model – where different topographic heights are accommodated by changes in crustal thickness, in which the crust has a constant density
# The Pratt–Hayford model – where different topographic heights are accommodated by lateral changes in ]rock
Rock most often refers to:
* Rock (geology), a naturally occurring solid aggregate of minerals or mineraloids
* Rock music, a genre of popular music
Rock or Rocks may also refer to:
Places United Kingdom
* Rock, Caerphilly, a location in Wales ...
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
.
# The Vening Meinesz, or flexural isostasy model – where the lithosphere acts as an elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
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. Models such as deep dynamic isostasy (DDI) include such viscous forces and are applicable to a dynamic mantle and lithosphere. Measurements of the rate of isostatic rebound
Post-glacial rebound (also called isostatic rebound or crustal rebound) is the rise of land masses after the removal of the huge weight of ice sheets during the last glacial period, which had caused isostatic depression. Post-glacial rebound ...
(the return to isostatic equilibrium following a change in crust loading) provide information on the viscosity of the upper mantle.
Airy
The basis of the model is Pascal's law
Pascal's law (also Pascal's principle or the principle of transmission of fluid-pressure) is a principle in fluid mechanics given by Blaise Pascal that states that a pressure change at any point in a confined incompressible fluid is transmitted ...
, 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):[
h1⋅ρ1 = h2⋅ρ2 = h3⋅ρ3 = ... hn⋅ρn
For the simplified picture shown, the depth of the mountain belt roots (b1) is calculated as follows:
:
:
:
where is the density of the mantle (ca. 3,300 kg m−3) and is the density of the crust (ca. 2,750 kg m−3). Thus, generally:
]
:''b''1 ≅ 5⋅''h''1
In the case of negative topography (a marine basin), the balancing of lithospheric columns gives:
:
:
:
where is the density of the mantle (ca. 3,300 kg m−3), is the density of the crust (ca. 2,750 kg m−3) and is the density of the water (ca. 1,000 kg m−3). Thus, generally:
:''b''2 ≅ 3.2⋅''h''2
Pratt
For the simplified model shown the new density is given by: , where is the height of the mountain and c the thickness of the crust.[
]
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
Lithospheric flexure (also called regional isostasy) is the process by which the lithosphere (rigid, thin outer layer of the Earth) bends under the action of forces such as the weight of a growing orogeny or changes in ice thickness related to glac ...
, 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.[
For example, the vertical displacement ''z'' of a region of ocean crust would be described by the ]differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
:
where and are the densities of the aesthenosphere and ocean water, ''g'' is the acceleration due to gravity, and is the load on the ocean crust. The parameter ''D'' is the ''flexural rigidity'', defined as
:
where ''E'' is Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...
, is Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Po ...
, and is the thickness of the lithosphere. Solutions to this equation have a characteristic wave number
: