Flexure of the lithosphere

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Lithosphere

Figure 2.30 Flexural downbending of the lithosphere as a result of a two-dimensional load of half-width a, height h, and density ps.

between the lithosphere and the behavior of an elastic sheet under load. Figure 2.30 illustrates the elastic response to loading; the region beneath the load subsides over a relatively wide area by displacing asthenospheric material, and is complemented by the development of peripheral bulges. Over long periods of time, however, the lithosphere may act in a viscoelastic manner and undergo some permanent deformation by creep (Section 2.10.3).

For example, the vertical displacement z of the oceanic lithosphere under loading can be calculated by modeling it as an elastic sheet by solving the fourth order differential equation:

d4 z

where P(x) is the load as a function of horizontal distance x, g the acceleration due to gravity, and pm, pw the densities of asthenosphere and sea water, respectively. D is a parameter termed the flexural rigidity, which is defined by:

where E is Young's modulus, a Poisson's ratio, and Te the thickness of the elastic layer of the lithosphere.

The specific relationship between the displacement z and load for the two-dimensional load of half-width a, height h, and density p5 shown in Fig. 2.30 is:

where

More realistic models of isostasy involve regional compensation. A common approach is to make the analogy

and pw, pm the densities of water and the mantle, respectively.

Note that as the elastic layer becomes more rigid, D approaches infinity, X approaches zero, and the depression due to loading becomes small. Conversely, as the layer becomes weaker, D approaches zero, X approaches infinity, and the depression approaches h(ps — pw)/ (Pm — Ps) (Watts & Ryan, 1976). This is equivalent to Airy-type isostatic equilibrium and indicates that for this mechanism to operate the elastic layer and fluid substrate must both be very weak.

It can be shown that, for oceanic lithosphere away from mid-ocean ridges, loads with a half-width of less than about 50 km are supported by the finite strength of the lithosphere. Loads with half-widths in excess of about 500 km are in approximate isostatic equilibrium. Figure 2.31 illustrates the equilibrium attained by the oceanic lithosphere when loaded by a sea-mount (Watts et al., 1975). Thus, as a result of its flexural rigidity, the lithosphere has sufficient internal strength to support relatively small loads without sub-

Observed

Observed

100 200 Distance (km)

Figure 2.31 Interpretation of the free air anomaly of the Great Meteor Seamount, northeast Atlantic Ocean, in terms of flexural downbending of the crust. A model with the flexural rigidity (D) of6 x 1022 Nm appears best to simulate the observed anomaly. Densities in Mg m~3. Arrow marks the position 30°N, 28°W (redrawn from Watts et al., 1975, by permission of the American Geophysical Union. Copyright © 1975 American Geophysical Union).

100 200 Distance (km)

Figure 2.31 Interpretation of the free air anomaly of the Great Meteor Seamount, northeast Atlantic Ocean, in terms of flexural downbending of the crust. A model with the flexural rigidity (D) of6 x 1022 Nm appears best to simulate the observed anomaly. Densities in Mg m~3. Arrow marks the position 30°N, 28°W (redrawn from Watts et al., 1975, by permission of the American Geophysical Union. Copyright © 1975 American Geophysical Union).

surface compensation. Such loads include small topographic features and variations in crustal density due, for example, to small granitic or mafic bodies within the crust. This more realistic model of isostatic compensation, that takes into account the flexural rigidity of the lithosphere, is referred to as flexural isostasy (Watts, 2001).

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