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).


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].


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.


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.


[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).