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Flexural Isostasy Hypothesis Statement

Isostasy (Greek ísos "equal", stásis "standstill") is the state of gravitational equilibrium between Earth's crust 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. When a certain area of Earth's crust reaches the state of isostasy, it is said to be in isostatic equilibrium. Isostasy does not upset equilibrium but instead restores it (a negative feedback). It is generally accepted [1] that Earth is a dynamic system that responds to loads in many different ways. However, isostasy provides an important 'view' of the processes that are happening in areas that are experiencing vertical movement. Certain areas (such as the Himalayas) are not in isostatic equilibrium, which has forced researchers to identify other reasons to explain their topographic heights (in the case of the Himalayas, which are still rising, by proposing that their elevation is being supported by the force of the impacting Indian plate; the Basin and Range Province of the Western US is another example of a region not in isostatic equilibrium.)

Although originally defined in terms of continental crust and mantle, it has subsequently been interpreted in terms of lithosphere and asthenosphere, particularly with respect to oceanic island volcanoes such as the Hawaiian Islands.

In the simplest example, isostasy is the principle of buoyancy wherein an object immersed in a fluid is buoyed with a force equal to the weight of the displaced fluid. On a geological scale, isostasy can be observed where Earth's strong crust or lithosphere exerts stress on the weaker mantle or asthenosphere, which, over geological time, flows laterally such that the load is accommodated by height adjustments.

The general term 'isostasy' was coined in the year 1889 by the American geologist Clarence Dutton.[2]

Isostatic models[edit]

Three principal models of isostasy are used:

  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 rockdensity.
  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, whereas flexural isostasy is a statement of buoyancy when deflecting a sheet of finite elastic strength.

Airy[edit]

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). In other words:

h1⋅ρ1 = h2⋅ρ2 = h3⋅ρ3 = ... hn⋅ρn

For the simplified picture shown the depth of the mountain belt roots (b1) are:




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, we may generally consider:


b1 ≅ 5⋅h1

In the case of negative topography (i.e., 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, we may generally consider:


b2 ≅ 3.2⋅h2

Pratt[edit]

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[edit]

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.

Isostatic effects of deposition and erosion[edit]

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.

An analogy may be made with an iceberg—it always floats with a certain proportion of its mass below the surface of the water. If more ice is added to the top of the iceberg, the iceberg will sink lower in the water. If a layer of ice is somehow sliced off the top of the iceberg, the remaining iceberg will rise. Similarly, Earth's lithosphere "floats" in the asthenosphere.

Isostatic effects of plate tectonics[edit]

When continents collide, the continental crust may thicken at their edges in the collision. If this happens, much of the thickened crust may move downwards rather than up as with the iceberg analogy. The idea of continental collisions building mountains "up" is therefore rather a simplification. Instead, the crust thickens and the upper part of the thickened crust may become a mountain range.[citation needed]

However, some continental collisions are far more complex than this, and the region may not be in isostatic equilibrium, so this subject has to be treated with caution.[citation needed]

Isostatic effects of ice sheets[edit]

Main article: 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 and Hudson Bay. 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.

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

Eustasy and relative sea level change[edit]

Main article: Eustasy

Eustasy is another cause of relative sea level change quite different from isostatic causes. The term eustasy or eustatic refers to changes in the volume of water in the oceans, usually due to global climate change. When Earth's climate cools, a greater proportion of water is stored on land masses in the form of glaciers, snow, etc. This results in falling global sea levels (relative to a stable land mass). The refilling of ocean basins by glacial meltwater at the end of ice ages is an example of eustatic sea level rise.

A second significant cause of eustatic sea level rise is thermal expansion of sea water when Earth's mean temperature increases. Current estimates of global eustatic rise from tide gauge records and satellite altimetry is about +3 mm/a (see 2007 IPCC report). Global sea level is also affected by vertical crustal movements, changes in Earth's rotation rate, large-scale changes in continental margins and changes in the spreading rate of the ocean floor.

When the term relative is used in context with sea level change, the implication is that both eustasy and isostasy are at work, or that the author does not know which cause to invoke.

Post-glacial rebound can also be a cause of rising sea levels. When the sea floor rises, which it continues to do in parts of the northern hemisphere, water is displaced and has to go elsewhere.

See also[edit]

References[edit]

Further reading[edit]

External links[edit]

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
Cartoon showing the isostatic vertical motions of the lithosphere (grey) in response to a vertical load (in green)
  1. ^A.B. Watts, Isostasy and flexure of the lithosphere,Cambridge Univ. Press., 2001
  2. ^"Clarence Edward Dutton"(PDF). 1958. Retrieved 7 October 2014. 

The Theory of Isostasy

Shih-Arng Pan
December 8, 2007

(Submitted as coursework for Physics 210, Stanford University, Fall 2007)

Fig. 1: Pratt Theory (left) and Airy's Theory (right).

Introduction

Isostasy is a fundamental concept in the Geology. It is the idea that the lighter crust must be floating on the denser underlying mantle. It is invoked to explain how different topographic heights can exists on the Earth's surface. Isostatic equilibrium is an ideal state where the crust and mantle would settle into in absence of disturbing forces. The waxing and waning of ice sheets, erosion, sedimentation, and extrusive volcanism are examples of processes that perturb isostasy. The physical properties of the lithosphere (the rocky shell that forms Earth's exterior) are affected by the way the mantle and crust respond to these perturbations. Therefore, understanding the dynamics of isostasy helps us figure out more complex phenomena such as mountain building, sedimentary basin formation, the break-up of continents and the formation of new ocean basins [1].

Theory

There are two main ideas, developed in the mid-19th century, on the way isostasy acts to support mountain masses. In Pratt's theory, there are lateral changes in rock density across the lithosphere. Assuming that the mantle below is uniformly dense, the less dense crustal blocks float higher to become mountains, whereas the more dense blocks form basins and lowlands. On the other hand, Airy's theory assumes that across the lithosphere, the rock density is approximately the same, but the crustal blocks have different thicknesses. Therefore, mountains that shoot up higher also extend deeper roots into the denser material below. Both theories rely on the presumed existence of a denser fluid or plastic layer on which the rocky lithosphere floats. This layer is now called the asthenosphere, and was verified in the mid-20th century to be present everywhere on Earth due to analysis of earthquakes - seismic waves, whose speed decrease with the softness of the medium, pass relatively slowly through the asthenosphere.

Both theories predict a relative deficiency of mass under high mountains, but Airy's theory is now known to be a better explanation of mountains within continental regions, whereas Pratt's theory essentially explains the difference between continents and oceans, since the continent crust is largely of granitic compostion which is less dense than the basaltic ocean basin.

Evidence

Since isostasy predicts deficiencies of mass under higher topological regions, one way to test isostasy on the planetary scale is to measure the variation of the local gravitational field. A simple pendulum can be utilized to measure the local strength of gravity; indeed, this was how the first gravity measurements in the U.S. were performed. Nowadays, physical geodesy, the study of physical properties of the gravity field of Earth, utilizes relative and absolute gravimeters for gravity surveys. Modern absolute gravimeters work by measuring the acceleration rate of a free-falling mass in vacuum - the mass includes a retroflector which acts as one arm of a Michelson interferometer, thus the velocity of the mass can be inferred from the interference fringes. Modern relative gravimeters mostly use quartz zero-length springs, and are calibrated to absolute gravimeters. A portable spring-based gravimeter can now measure the earth's gravitational field up to accuracies of nanometer per second squared.

Due to self-rotation, the Earth bulges at its own equator, roughly forming an ellipsoid, hence at sea level the value of gravity is dependent on the latitude, and is less at latitudes near the equator than at latitudes near the poles. This value of gravity at a particular point on the ellipsoid is called the theoretical value for that point. Subtracting the theoretical value of gravity from the observed value of gravity at a point gives a difference called the "gravity anomaly." After correcting both for elevation and for the gravitational attraction of the rocks between the instrument and the ellipsoid, the measured value of gravity minus the theoretical value is called the "Bouguer gravity anomaly."

Quantitatively, correcting for the flattening of the Earth and its rotation, the gravitational strength at a given latitude φ is

where ge is the gravitational strength at the equator, ω is the angular velocity of the Earth's rotation, and a and b are the semi-major and semi-minor axes, respectively, that describe the ellipsoid that best describes Earth's flattening. The above formula is based on Claurant's first-order theorem and is derived by solving Laplace's equation to a spheroidal boundary.

Next, we correct for elevation by Taylor-expanding the gravitational force with respect to r, and obtain the free-air correction (FAC), which assumes no mass between the measuring instrument and sea level:

where ggeoid is the gravitational strength at the geoid (the hypothetical local sea level), h is the elevation, and re is the radius of the earth.

Correcting for the effect of mass distributions between the instrument and the geoid is more involved, and will depend on local topography and density distributions. Assuming that the topography extends at similar heights in all directions around the instrument, and L is the thickness of the crust above sea level, within which the local rock density ρ is constant, the Bouguer correction (BC) then becomes

Assuming that the geoid and reference ellipsoid are sufficiently similar, the Bouguer gravity anomaly (BA) is then

As expected, measurements of Earth's gravitational field indicate that Bouguer gravity anomalies are generally very negative over mountains and plateaus, and zero or positive over oceans. Gravity is indeed weaker than expected over much of the Alps, the Himalayas, and many other mountain ranges. In regions that have had the time to reach isostatic equilibrium without being disturbed by other geological effects, such as the south-western United States, very good correlation exists between the local elevation and Bouguer gravity anomalies, providing compelling evidence for isostasy [4].

Resulting Geological Processes from Isostasy

The laws of buoyancy act on continents just as they would on icebergs and rafts. An iceberg will rise further out of the water when the top is melted, and a raft will sink deeper when loads are added. However, the adjustment time for continents is much slower, due to the viscosity of the asthenosphere. This results in many dynamic geological processes that are observed today. The following paragraphs illustrate some of these examples.

The formation of ice sheets could cause the Earth's surface to sink. In areas which had ice sheets in the last ice age, the land is now "rebounding" upwards since the heavy ice has melted and the load on the lithosphere is reduced. Evidence from geological features include former sea-cliffs and associated wave-cut platforms that are found hundreds of meters above the sea level today. In the Baltic and in Canada, the amount and rate of uplift can be measure. In fact, due to the slowness of rebound, much of the land is still rising.

Isostatic uplift also compensates for the erosion of mountains. When large amounts of material are carried away from a region, the land will rebound upwards to be eroded further. Due to drainage patterns, the erosion and removal of material is more prominent at plateau edges. Isostatic uplift may raise the edge higher than it used to be, so the ridge tops can be at an elevation considerably higher than the plateau itself. This mechanism is especially probable in mountain ranges bounding plateaus, such as the Himalayas and Kunlun Mountains bounding the Tibetan Plateau [3].

Interestingly, given enough time and reaction kinetics, due to chemical transformations, the thick crustal root underneath mountains can become denser and founder into the mantle. The removal of the dense root can happen by the convection of the underlying asthenosphere or by delamination. After the root has detached, the asthenosphere rises and isostatic equilibrium leads to more mountain building at that region. For instance, this is thought to be the reason behind the late Cenozoic uplift of the Sierra Nevada in California. In fact, seismic data provide images of crust-mantle interactions during the supposed active foundering of the dense root beneath the southern Sierra Nevada. It appears that dense matter flowed asymmetrically into a mantle drip beneath the adjacent Great Valley. At the top of this drip, a V-shaped cone of crust is being dragged down tens of kilometers into the center of the mantle drip, leading to the disappearance of the Mohorovicic discontinuity (the boundary between crust and mantle) in seismic images [5]. Likewise, at the northern Sierra Nevada, there is also a seismic "hole" known as the Redding anomaly, lending to the assumption that lithospheric foundering occurred there as well. Moreover, beneath the southern Sierra Nevada, Boyd et al.. imaged what may be the foundering lithosphere itself when they generated a density map of the region via seismic tomography [2].

In conclusion, isostasy is yet another example of a deceptively simple idea in physics that provides crucial and sweeping explanatory power for other sciences.

(c) 2007 Shih-Arng Pan. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] A. B. Watts, Isostasy and Flexure of the Lithosphere (Cambridge, 2001).

[2] O. S. Boyd, C. H. Jones and A. F. Sheehan, "Foundering Lithosphere Imaged Beneath the Southern Sierra Nevada, California, USA," Science 305, 660 (2004).

[3] C. Ollier and C. Pain, The Origin of Mountains (Routledge, 2000).

[4] J. Gilluly, "Crustal Deformation in the Western United States," in The Magatectonics of Continents and Oceans, ed. by H. Johnson and B. L. Smith (Rutgers, 1970), p. 47.

[5] G. Zandt et al., "Active Foundering of a Continental arc Root Beneath the Southern Sierra Nevada in California," Nature 431, 41 (2000).

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