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.

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-bibPicture 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:

• Ocean basin depth, h2, is underlain by an anti-root of thickness:

€

r1 =h1ρ uρ s − ρ u

.

€

r3 =d(ρ u − ρw )

ρ s − ρ u .

r1

u

s

h1

r3

d

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

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:

€

1 = ρ uD

h1 + D .

€

d =ρ uD− ρwd

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:

€

Dd4w

dx 4=V (x) ,

where D is the flexural rigidity, that is given by:

with:E being Young Modulush being the plate thickness being Poisson’s ratio

€

D =Eh3

12(1−ν ) ,

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 .

Note the flexural bulge on either side of the depression.

Of course in reality things are more complex…

Figure from Fowler

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

Isostasy: example from the Mariana subduction zone

defle

ctio

n [k

m]

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.

Fluxural bulge

Figure from Fowler

Isostasy example from the Tonga subduction zone

defle

ctio

n [k

m]

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.

Figure from Fowler

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

But the ambiguity is always there. Here’s an example from a Mid-Atlantic Ridge (MAR).

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?

Isostasy: gravity

bathymetry

free-air

deep model

Bougueranomaly

3 shallowmodles

Figure from Fowler

Isostasy: isostatic rebound

Figure from Fowler

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.

Small loads, say ~100 km diameter, can tell us about the viscosity of the asthenosphere.

Isostasy: isostatic rebound

Lake Bonneville, Utha:• A lake 300 m deep dried up 10,000 years ago.• Lake center has risen by 65 m.

shoreline

Images from: academic.emporia.edu/aberjame/histgeol/gilbert/gilbert.htm

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.

Large loads, say ~1000 km diameter tell us about the upper and lower mantle viscosity.

Isostasy: isostatic rebound