Large-Scale Gravity and Isostasy
9.5 The shape of the Earth .1 Seeing the ocean floor
in the ocean surface
The ocean surface is not perfectly level, even after allowance has been made for waves and tides, and Plate 3a shows its topography. Many readers will recognise a correspondence with many features of the ocean floors, particularly those associated with plate margins (compare with Plate 2c). Why does the ocean surface reveal the structure below? The link is through gravity anomalies.
Suppose a ball is resting on the Earth’s surface at B (Fig. 9.16a), to one side of a buried sphere.
9.5 The shape of the Earth ✦ 135
(a) (b)
x x
Figure 9.15 Vacancy diffusion.
The ball will tend to move to the left because the sphere pulls it sideways as well as downwards. It will tend to move until it is directly over the sphere, at A, where the force is straight down-wards. Though the force is too small to move a real ball, because of friction, it will move water, heaping it up over the sphere, until the water sur-face is everywhere perpendicular to the local pull of gravity (Fig. 9.16b), for then there is no longer a sideways pull to move it (such an equilibrium sur-face is called an gravity equipotential sursur-face).
Thus there is a small ‘hill’ where there is a positive gravity anomaly, a hollow for a negative one. A ship is unaffected by these slopes of the ocean sur-face, not because they are small, but because there is no sideways pull. The shape of the sea surface depends on the form of the gravity anomaly, but its calculation is beyond the scope of this book.
Because even the largest gravity anomalies (over oceanic trenches) are only a fraction of 1% of the average value of g, the ocean surface deflections are no more than a few metres.
The shape of the ocean surface is measured using satellites that measure the distance to the ocean surface below them. This is done by timing the reflection of radar waves, in a form of echo sounding; in turn, the positions of the satellites are found by observation from ground-based stations.
Great care is needed to eliminate the effects of tides and waves and other causes of error, but the aver-age height of sea level can be measured to a preci-sion of about 1 cm. The variations in sea level (Plate 3a) can be converted to provide a gravity map, accurate to about 1 mGal, which gives infor-mation about the structure of the ocean floors (Chapter 20).
9.5.2 The large-scale shape of the Earth The first approximation to the shape of the Earth is a sphere; one with a radius of 6371 km has the same volume as the Earth. But this ignores the equatorial bulge, so any cross-section of the Earth through the poles is an ellipse rather than a circle (Fig. 9.17), though the difference is only a few kilometres. Therefore, a better approximation to the Earth’s shape is an ellipsoid of revolution, the shape produced by rotating the ellipse about the Earth’s axis. The one that best approximates the sea-level surface is called the reference spheroid and it has a polar radius of 6357 km and an equatorial one of 6378 km, giving a flattening of 1/298.
B
water surface
sideways component of pull of sphere
pull of sphere
due to earth total force
A (a)
(b)
perpendicular
Figure 9.16 Distortion of a liquid surface by a buried mass.
ellipsoid sphere pole
pole
equator 6378.160 km
6371 km
6356.775 km
Figure 9.17 Shape of the Earth.
Even the reference spheroid is not an exact match to the sea surface, because of the effects of gravity anomalies, as described in the previous sec-tion. The actual mean sea level surface is called the geoid (for continental areas the geoid is the height that a sea level canal would have). The biggest dif-ferences between the reference spheroid and the geoid is no more than 80 m. Even so, they matter for exact surveying. Their cause is not well under-stood but must be due to extensive density differ-ences deep in the Earth.
Summary
1. On the large scale – lateral density differences extending for tens of kilometres or more – the Earth behaves as having a solid lithosphere floating on a yielding and denser asthenosphere.
2. In the simplest model of isostatic adjustment, the lithosphere is treated as a series of blocks that float up or down in the asthenosphere, according to Archimedes’ Principle, in response to changing loads. In the Airy model the blocks have the same density but different thicknesses; mountains are highest where the blocks are thickest and so go deepest, to form ‘roots’. In the Pratt model, the blocks extend down to the same depth but have different densities; highest parts are where the blocks are least dense. The Airy model is appro-priate for explaining major topographic varia-tions on the continents, but the Pratt model accounts better for the shapes of ocean ridges.
Continents are higher than ocean floors because of a combination of being lighter and thicker.
3. Isostatic adjustment of the simple block model follows Equations 9.1 and 9.2. [These are modi-fied for the Airy and Pratt models, see Section 9.1.4].
4. The regional compensation model of isostasy takes account of the lateral, flexural strength of the lithosphere, which results in an extensive load being supported by an area that extents beyond its limits. Loads up to a few kilometres across are supported by the strength of the lithosphere with negligible deflection, interme-diate-sized ones by both the strength of lithos-phere and buoyancy, while the most extensive loads – upwards of tens of kilometres across, depending on the type of lithosphere – are
sup-ported mainly by buoyancy and this approxi-mates to the simple isostasy model.
5. The mantle is able to behave as a liquid in isosta-tic adjustment, yet permits S-waves to propagate, because of solid-state creep. It has elastic rigidity when forces last only a short time, but deforms like a liquid under long-continued forces.
6. The boundary between the lithosphere and the asthenosphere is gradational and is where the temperature becomes high enough for the rate of solid-state creep to be significant. It also depends to some extent whether the rocks are acidic or basic. Its thickness varies from a few kilometres in young parts of the ocean floor to over 200 km in old continental areas (cratons).
7. Isostatic adjustment takes thousands of years because of the extremely large viscosity of the asthenosphere.
8. Isostatic compensation tends to eliminate grav-ity anomalies; therefore, gravgrav-ity surveys reveal whether isostatic equilibrium exists. Large and extensive gravity anomalies exist only if the loading – such as an ice sheet – has changed too recently – within the past few thousand years – for equilibrium yet to have been regained, or because an internal force is preventing equlibrium.
9. The water surface is up to a few metres higher than average where there is a positive gravity anomaly, lower where there is a negative one.
The ocean surface topography reveals major ocean bottom subsurface features, mostly reflecting plate tectonic structures.
10. The shape of the Earth is close to a reference spheroid with an elliptical polar cross-section.
The geoid is the actual mean sea level surface (on land: sea level canals). It differs from the reference spheroid by no more than 80 m.
11. You should understand these terms: lithosphere, asthenosphere; isostasy, isostatic compensation, isostatic rebound, isostatic equilibrium; isostatic anomaly, Airy and Pratt models, regional com-pensation; solid-state creep; reference spheroid, geoid.
Further reading
Isostasy is explained in Tsuboi (1983) and in Fowler (1990), which also discuss the shape of the Earth.
Further reading ✦ 137
Problems
1. Why does it seem a contradiction that the man-tle can both transmit S-waves and rebound iso-statically? Explain, on both an atomic and a bulk scale, how this apparent contradiction can be resolved.
2. How would you tell if an area is in isostatic equilibrium?
3. A large continental area covered with ice has a positive gravity anomaly. Which of the follow-ing might account for it?
(i) The thickness of ice increased recently.
(ii) The thickness of ice decreased recently.
(iii) The thickness of ice increased several tens of thousands of years ago.
(iv) The thickness of ice decreased several tens of thousands of years ago.
4. Explain how erosion of mountains can some-times result in uplift of the peaks.
5. Melting of the ice in the arctic region would cause the sea level to
(i) rise, (ii) fall, (iii) be unchanged.
6. If all the ice of Antarctica were to melt rapidly, would you expect (a) a thousand years later, (b) a million years later, that the shoreline around Antarctica would be, compared to the present, higher, lower, or the same?
7. If a continental area is in perfect isostatic equi-librium, which of the following are true?
(i) A Bouguer anomaly map would show no variations.
(ii) There are no lateral variations of density below the surface.
(iii) No uplift or subsidence is occurring.
8. A wide block of wood 100 cm high and with density 0.72 Mg/m3 is floating in a liquid with density 0.96 Mg/m3.
(a) Calculate how far the top of the block is above the surface.
(b) How far would it be if 12 cm were removed from the base of the block?
(c) How far would it be if 12 cm were removed from the top of the block?
9. A large area of continent consisting of 30 km of crust with average density 2.8 Mg/m3and over 90 km of material with density 3.1 Mg/m3 is covered with ice (density 0.9 Mg/m3) 1.6 km thick, and is in isostatic equilibrium. Then the ice melts. By how much has the rock surface of
the continent changed when equilibrium has been regained? (The density of the asthenos-phere is 3.2 Mg/m3.)
10. The crust of a continent contains a layer of salt, density 2.2 Mg/m3 and thickness 3 km, within sediments of density 2.4 Mg/m3. Over a period of time much of the salt is squeezed sideways, reduc-ing its thickness to 1 km. This causes the surface of the continent to lower by how much? (Assume the asthenosphere has a density of 3.2 Mg/m3.) 11. A extensive area is intruded by three basaltic
sills with uniform thicknesses of 30, 40, and 50 m. What is the change of the height of the sur-face after isostatic equilibrium has been restored? (Density of sill, 2.8 Mg/m3; density of asthenosphere, 3.2 Mg/m3.)
12. A sea, initially 2 km deep, over a long period of time fills to sea level with sediments. How deep are these sediments? (Use the following densities, in Mg/m3: water, 1; sediments, 2.4; asthenos-phere, 3.2.) If the sediments had been denser, would the thickness have been greater or less?
13. How would you expect the depth to the asthenosphere below a continental area with high geothermal gradient (i.e., the temperature increases with depth more rapidly than beneath most areas) to compare with that below an area with a low gradient?
14. Why cannot the depth to the base of the lithos-phere be measured using either seismic reflec-tion or refracreflec-tion surveys?
15. The uplift of former beaches around the Gulf of Bothnia is about 275 m. What thickness of ice would be needed to depress them back to sea level? (Density of ice, 0.9 Mg/m3; density of asthenosphere, 3.2 Mg/m3.)
16. The value of g at a place A is less than that at B.
Which of the following might be the explanation?
(i) A is at a higher latitude.
(ii) A is at a higher elevation.
(iii) A is underlain by lower-density rocks.
(iv) A but not B was covered by a thick ice sheet a million years ago.
17. The value of g at a place varies with time due to which of the following?
(i) Isostatic rebound.
(ii) The topography of the continents.
(iii) Lateral differences in the compositions of rocks.
(iv) The Earth’s rotation.
SUBPART I.4
Magnetism
Compasses point approximately north-wards because the Earth has a magnetic field which aligns them; it is as if the Earth has a powerful magnet at its centre. Many rocks in the past gained a magnetisation in the direction of this field and retain that direction today. These past directions act like fossil compasses and can show if the rock has moved since it was formed.
This has many applications, from following the movements of continents to the defor-mation of rocks in folding and even the angle through which a pebble has been rotated. The study and applications of fos-sil magnetism is called palaeomagnetism.
The direction of the magnetic field has var-ied over time; in particular, it has reversed (i.e., inverted) many times. The reversal record in rocks provides a way of ordering and even dating them, leading to magne-tostratigraphy.
Rocks become magnetised by various mechanisms, including cooling over a range of temperatures. This can be utilised to determine past temperatures, such as the heating caused by an intrusion. The mag-netic properties of surface rocks and soils partly depend on their history; mineral magnetism exploits this to investigate, for example, the sources of sediments, rates of erosion, and the presence of volcanic ashfalls.
The first of the two chapters of this subpart, Palaeomagnetism and mineral magnetism, describes the above topics. The second chapter, Magnetic surveying, describes how measuring variations in the present-day magnetic field at the sur-face can be used to investigate the sub-surface.