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Gravity III: Isostasy
The deflection of plumb bob
near mountain chains is less
than expected. Calculations
show that the actual deflection
may be explained if the excess
mass is canceled by an equal
mass deficiency at greater
depth.
A plumb-bib
Picture from wikipedia
Isostasy: the Airy hypothesis (application of Archimedes’ principal)
• Two densities, that of the rigid
upper layer, u, and that of the
substratum, s.
• Mountains therefore have deep
roots. A mountain height h1 is
underlain by a root of thickness:
h1u
r1 
.
s  u
h1
d
u
s
r3
r1
• Ocean basin depth, h2, is underlain by an anti-root of
thickness:

d(u  w )
r3 
.
s  u
Isostasy: the Pratt’s hypothesis
• The depth to the base of the
upper layer is constant.
• The density of rocks beneath
mountains is less than that
beneath valleys.
• A mountain whose height is h1
is underlain by a root whose
density 1 is:
D
1  u
h1  D
.
• Ocean basin whose depth is h2 is underlain by a high
density material, 2, that is given by:

d 
u D  w d
D d
.
Isostasy
Questions:
• Which is the correct hypothesis?
• Does isostatic equilibrium apply everywhere?
Isostasy
Is the person resting on top of a spring-matress in a state of
isostatic equilibrium?
Isostasy: elastic flexure
Like the springs inside the mattress, the elastic lithosphere can
also support excess mass.
Thick plates can support more excess mass than thin plates.
Isostasy: elastic flexure
The response of the lithosphere to a vertical load depends on the
lithosphere elastic properties as follows:
d4w
D 4  V (x) ,
dx
where D is the flexural rigidity, that is given by:

Eh 3
D
,
12(1  )
with:
E being Young Modulus
h being the
 plate thickness
 being Poisson’s ratio
Isostasy: elastic flexure
The figure below show the solution for the simplest case for:
V (x)  0 for x  0
and
V (x)  0 for x  0 .

Figure from Fowler
Note the flexural bulge on either side of the depression.
Of course in reality things are more complex…
Isostasy: example from the Hawaii chain
free-air
bathymetry
Two effects:
• Elastic flexure due to
island load.
• A swell due to mantle
upwelling.
Figure from Fowler
Isostasy: example from the Mariana subduction zone
deflection [km]
Fluxural bulge
Figure from Fowler
distance [km]
• The accretionary wedge loads the plate edge causing it to bend.
• A flexural bulge is often observed adjacent to the trench.
• Topography of Mariana bulge implies a 28 km thick plate.
deflection [km]
Isostasy example from the Tonga subduction zone
Figure from Fowler
distance [km]
• The Tonga slab bends more steeply than can be explained by an
elastic model.
• It turned out that an elastic-plastic model for the lithosphere can
explain the bathymetry data.
Isostasy: local versus regional isostatic equilibrium
According to Pratt and Airy hypotheses, excess mass is perfectly
compensated everywhere. This situation is referred to as local
isostasy.
Isostasy: local versus regional isostatic equilibrium
The situation where some of the load is supported by the strength
of the lithosphere is referred to as regional isostasy. In this case,
isostatic equilibrium occurs on a larger scale, but not at any point.
Isostasy
Questions:
1. Isostatic equilibrium means no excess mass. Does this mean
no gravitational anomaly.
2. Can we distinguish compensated from uncompensated
topographies?
Isostasy: gravity
100% compensated
A rule of thumb: A region is in
isostatic equilibrium if the
Bouguer anomaly is a mirror
image of the topography.
Figure from Fowler
Isostasy: gravity
Uncompensated
A rule of thumb: A region is NOT
in isostatic equilibrium if the
Bouguer anomaly remains flat
under topographic highs and
lows.
Figure from Fowler
Isostasy: gravity
bathymetry
But the ambiguity is always there.
Here’s an example from a MidAtlantic Ridge (MAR).
free-air
deep model
The observed anomaly may
be explained equally well
with deep models with
small density contrast or
shallow models with greater
density contrast.
Question: is the MAR in
isostatic equilibrium?
Bouguer
anomaly
3 shallow
modles
Figure from Fowler
Isostasy: isostatic rebound
The rate of isostatic rebound depends on
the elastic properties of the lithosphere
(including its thickness) as well as the
mantle viscosity.
Isostatic rebound can be observed if a
large enough load has been added or
removed fast enough.
Figure from Fowler
Isostasy: isostatic rebound
Small loads, say ~100 km diameter, can tell us about the viscosity
of the asthenosphere.
shoreline
Lake Bonneville, Utha:
• A lake 300 m deep dried up
10,000 years ago.
• Lake center has risen by
65 m.
Images from: academic.emporia.edu/aberjame/histgeol/gilbert/gilbert.htm
Isostasy: isostatic rebound
Large loads, say ~1000 km diameter tell us about the upper and
lower mantle viscosity.
Fennoscandia:
• Removal of 2.5 km thick
ice at the end of the last ice
age 10,000 years ago.
• Current peak uplift rate is 9
mm/yr.
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