ISOSTASY
INTRODUCTION
The word "isostasy" is derived from Greek roots meaning "equal" and "standing". Isostasy
represents the theory that the outer parts of the earth are in a state of dynamic
equilibrium such that the surface elevation depends on the mean density of the
underlying rock, moreso than its rigidity or flexural strength. A consequence of this
hypothesis is that the surface of the earth moves down and up as surface (or subsurface)
loads are added or subtracted. In a broad sense it means that the Principle of
Archimedes' applies to the earth. And that loads are compensated immediately below
where they occur so the earth behaves like a fluid, without much long distance flexural
strength, similar to ice or putty. Given the loads, dimensions and geological time scales
involved, this is a pretty fair approximation and is well supported by geodetic survey data
on crustal motions.
HISTORY
The first hint of isostasy was provided in the middle of the eighteenth century when Piere
Bouguer participated in a survey to measure the length of a degree of latitude at the
equator. This survey, in turn, was part of a French project to determine the shape of the
earth: is it a prolate or oblate spheroid? In the former case a degree of latitude would be
shorter at the poles than at the equator. For an oblate spheroid the opposite would be
true. To make the test Maupertius led an expedition to Lapland and La Condamine, to
Peru (now Equador). The results indicated that the earth is oblate.
During this work Bouguer noted that there was a difference between the vertical direction
as based on the position of stars and that based on a spirit level or plumb bob. At first this
"deflection of the vertical" was explained as an effect of Newtonian gravitational attraction
between the plumb bob and the Andes Mountains that lay parallel to the survey route; the
mountains pulled the plumb bob sideways away from the astronomical vertical direction.
On closer examination this explanation didn't quite work; the calculated deflection of the
vertical was too great. In other words the calculated Newtonian attraction of the
mountains was too large. The most logical explanation for the excessive attraction is that
the deep rocks of an underlying "mountain root" are less dense than the surrounding
rocks at those depths.
A similar deficit of horizontal attraction was noted for the Himalayas during geodetic
surveys in India during the nineteenth century. Analyses of these data led to the two
classical views of isostasy.
THE AIRY HYPOTHESIS
George Airy was the royal astronomer. In his view isostacy is realized by means of a
crust of low and uniform density "floating" on a mantle of high and uniform density. Under
mountains the crust is thick and elevations correspondingly high. Under the sea the crust
is thin and the water correspondingly deep. The base of the crust undulates as an
exaggerated mirror of the topography. The depth of uniform pressure lies below the base
of the crust (the "Moho"). Recall that this hypothesis was proposed decades before
seismic waves were used to probe the interior of the earth and the words "crust" and
"mantle" did not have their current restricted meanings. This model was supported by
gross similarity of rock densities from mountains, cratons and ocean islands.
Today the Airy model is thought to provide a good explanation for the elevation difference
between the continents and the oceans. It is a fair picture of intra-continental elevations
but a very poor one for ocean depths. Nowadays the Airy-Woollard model, which allows
variable densites in both the crust and mantle, does a better job within the continents.
THE PRATT HYPOTHESIS
John Pratt was the archdeacon of Calcutta and a mathematician of great power. His
isostatic model features a horizontal crustal base. The crustal density varies from high
under the seas to low in high mountains. Thus the pressure is everywhere equal at the
base of the crust.
To a first approximation crustal rocks do not show the hypothesized density differences.
Even worse, seismic refraction measurements show that the base of the crust roughly
mirrors the topography and is not level. Thus the Pratt model doesn't explain either the
elevation differences in elevation between oceans and continents or within continents. In
a modified form it does explain variations in depth of the ocean floor.
When recast in modern terminology, "Thermal" or "Pratt-Thermal" isostasy accounts very
well for the systematic age-related changes in ocean depth. Nowadays we think that the
lithosphere (not crust) is hot and light at the ridge axes and becomes colder and denser
as the plate ages. The sea is shallow where the lithosphere is light and deep where it is
heavy. Wegener apparently understood this decades before continental drift and seafloor spreading were accepted. However, there were reasons why isostasy in general and
these models in particular were not accepted right away.
TWO OBJECTIONS TO ISOSTASY IN GENERAL
At the same time that these models of isostasy were being developed, other scientists
were learning to probe the deep interior of the earth. Two strong arguments for a solid
earth were advanced. If the earth were "solid", in our everyday use of the term, rock
could not flow and isostatic equilibrium would not occur.
The older argument for a mainly solid earth was based on analysis of ocean tides. Were
the earth fluid, like the sea, it would respond to tidal forces as does the ocean water and
only small tides would be seen. The earth and ocean water would move together and
there would be little relative motion (that is, no observable tide).
At about the beginning of the twentieth century seismologists tracked shear waves
throughout all the earth down to a small liquid core (the solid inner core was discovered
much later). These studies provided convincing evidence for a mainly solid earth because
shear waves will not propagate through liquids.
If the earth is solid how can rocks flow to gain isostatic balance?
ANOTHER OBJECTION TO AIRY AND PRATT ISOSTASY
In both these models isostasy is local. That is, any adjacent rock columns can move up
and down independently in response to changing loads. You'll recall lab exercises with
floating wooden blocks. The problem, of course, is that this freedom of motion implies
that rock has no shear strength. On the other hand the mere existence of steep
topographic slopes demonstrates that rocks do have long-term shear strength; mountains
don't seem to ooze away and flatten out (or is there really post-orogenic collapse
involving detachment faults?). How can we reconcile these contradictions?
VENING-MEINESZ OR FLEXURAL ISOSTASY
F. A. Vening-Meinesz was a Dutch geophysicist active in the middle years of the
twentieth century. He is famous for his early studies of gravity anomalies at sea,
especially over deep-sea trenches. These measurements were made with special
pendulums in a submerged and stationary submarine.
Along with other scientists he realized that the key to our puzzle lay in the rheological
behavior of rocks. When cold, rocks are strong and can sustain loads for millions or
billions of years without deforming. Near the melting point, however, rocks are weak and
deform readily over geologic time scales. "Hot" and "cold" refer to closeness to melting
temperatures on the Kelvin (absolute) temperature scale.
Applied to the earth these observations indicate that there is a "cold", strong outer
rheological layer. This may or may not coincide with the crust or lithosphere or any other
named layer. Similarly there is a "hot", weak, underlying layer that can flow rapidly over
intervals as short as a few thousand years (post-glacial rebound shows this). This
deformable layer is solid with respect to short-term loading (earthquake waves) but flows
for loads of long duration. Even deeper lies stronger rock: it's very hot but not so close to
melting temperatures as melting is inhibited by high pressures.
In the Vening-Meinesz or flexural model surface loads such as ice-sheets are largely
supported by bending of a cold, outer, elastic layer with a thickness in the range of about
10 to 100 km. We can model this kind of support by using engineering formulae for bent
beams. At the earth's surface the rocks are strong and can sustain steep slopes. At depth
the rocks are weak, but still solid, and flow in response to differential loading. In this
model the mass of unit columns down to the depth of flow is not necessarily constant; it
can vary from place to place.
Using our engineering formulae we can estimate the elastic thickness. Where it is thin,
the vertical displacements are large for a given load and the are concentrated in a narrow
region under and around the load. Where the thickness of the elastic layer is large the
vertical displacements are smaller but occur to great distances from the load.
In the ocean such studies show that the elastic thickness increases as the plate ages.
The base of the elastic plate seems to follow roughly the 450°C isotherm, not the base of
the lithosphere plate (about 1300°C). On the continents the picture is less clear because
of often complex rheological layering (for example, weak quartz-rich plutonic rock).
Clearly, given the dominant control be temperature, the elastic thickness can change with
time. How can we harmonize this local isostasy with the regional isostasy based on a
thickening lithosphere?
Studies of flexural isostasy have been made around ice sheets, ice caps, deltas, deepsea fans, volcanic islands, foreland basins and even subduction zones. The loading may
be external, such as an ice cap, or internal such as a mafic pluton intruded into lighter
rock.
TESTS OF ISOSTASY
We can test isostasy with both static and dynamic observations.
Perhaps the most obvious static test is to estimate underground densities and thus find
the depth at which lithostatic pressure should be uniform (we'll discuss temperature
estimates later). Underground density differences influence the acceleration of gravity.
Analysis of "gravity anomalies" (ES 734/834) gives constraints on density.
It is well known the seismic P-wave velocities are proportional to densities for common
rocks. Thus we can convert measured seismic velocities to densities provided we make
appropriate adjustments for in situ pressure and temperature conditions.
Dynamic tests involve observing or estimating the effect of changing loads. The most
famous studies involve the rapid uplift of Scandinavia after the Wisconsinan ice-sheet
vanished. The rate and extent of the uplift indicates that more-or-less complete balance
can be achieved in a matter of a few thousand years.
REFERENCES
Heiskanen, W. A. and F. A. Vening Meinesz, 1958, "The Earth and Its Gravity
Field", McGraw-Hill, 470 pp. This is the classical pre-plate tectonic text on
gravity and isostasy. Besides its historical interest this book is a treasury of
relevant mathematics. ES 734 students will be interested in the diagrams of the
Worden gravimeter.
"Lillie", "Sleep and Fujita", "Kearey and Vine" and “Turcotte and Schubert" all discuss
isostasy in modern terms.
Kearey, P. and Vine, F. J. (1996) Global tectonics. Blackwell Science, Oxford
Robert J. Lillie, R.J. (1999) Introduction to Geophysics- (Ch2)
Sleep, N.H. & Fujita, K., 1997. Principles of Geophysics, Blackwell Science, Malden,
Mass., 586 pp.
Turcotte, D.L. and Schubert, G. (2002) Geodynamics 472p, (Ch3).