1. 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

2. Isostasy: the Airy hypothesis (application of Archimedes’ principal)u s h1 r3 d
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:
Ocean basin depth, h2, is underlain by an anti-root of thickness:

3. 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:
Ocean basin whose depth is h2 is underlain by a high density material, 2, that is given by:

4. Isostasy
Questions:
Which is the correct hypothesis?
Does isostatic equilibrium apply everywhere?

5. Isostasy
Is the person resting on top of a spring-matress in a state of isostatic equilibrium?

6. 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.

7. Isostasy: elastic flexure
The response of the lithosphere to a vertical load depends on the lithosphere elastic properties as follows:
where D is the flexural rigidity, that is given by:
with:
E being Young Modulus
h being the plate thickness
 being Poisson’s ratio

8. Isostasy: elastic flexure
The figure below show the solution for the simplest case for:
Figure from Fowler
Note the flexural bulge on either side of the depression.
Of course in reality things are more complex…

9. Isostasy: example from the Hawaii chain
bathymetry
free-air
Two effects:
Elastic flexure due to island load.
A swell due to mantle upwelling.
Figure from Fowler

10. Isostasy: example from the Mariana subduction zone
Fluxural bulge
deflection [km]
distance [km]
Figure from Fowler
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.

11. Isostasy example from the Tonga subduction zone
deflection [km]
distance [km]
Figure from Fowler
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.

12. 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.

13. 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.

14. Isostasy
Questions:
Isostatic equilibrium means no excess mass. Does this mean no gravitational anomaly.
Can we distinguish compensated from uncompensated topographies?

15. 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

16. 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

17. Question: is the MAR in isostatic equilibrium?
Isostasy: gravity
bathymetry
But the ambiguity is always there. Here’s an example from a Mid-Atlantic 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.
Bouguer
anomaly
3 shallow
modles
Question: is the MAR in isostatic equilibrium?
Figure from Fowler

18. 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

19. 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

20. 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.