Isostasy answers (?) these questions•Is the deep earth rigid as a ‘steel ball’ or is it able to flow viscously like a ‘sea of lava’ ?

•What processes create the low standing oceans and the high standing continents ?

•What forces hold-up the mountains or conversely holds-down basins ?

•What is the earth’s figure best represented as: a sphere or a flattened ellipsoid ?

•What causes the stranded shorelines Scandanavia (uplift evidence) ?

•How does measured gravity relate to the mode of compensation of the earth surface?

•How strong in compression/tension/shear is the earth’s crust and lithosphere ?

•What are the rheologic laws that govern the deformation (stress/strain) of the shallow and deep portions of the planet ?

Mis-conceptions about crust, lithosphere, asthenosphere*

(1) Crust (compositional). The crust is the residue from melting the mantle. Crustal thicknesses is 10 km (oceans) and 30-80 km (continents).

(2) Mantle (compositional). The mantle +crust+core = chondrite meteorites.

(3) Core (compositional). Made of liquid/solid iron mostly.

(4) Lithosphere (strength). Also, called a plate. The strong layer that slides over the asthensophere. The lithosphere is strong because it is colder.

(5) Asthenosphere (strength). The mantle just below the lithosphere is often weaker than the deeper mantle due to pressure and temperature effects.

Pressure gradient and gravity

Early evidence and thoughts on Isostasy

Hypsometry (distribution of elevations) of planet earth

How would gravity vary over small and large mountains ?

Remember, a fundamental property of mass is that it has gravity (and inertia). So, the mass of the two mountains (above) above the surface will increase the gravity recorded by the gravimeter in the balloon. But, what about the lower density root ? What is important is two questions: 1) is the crust strong enough to support the weight of a mountain? 2) if the crust ‘breaks’, then what forces support the weight of the mountains? Note that mountain (A) has no change in density beneath the level, but mountain (C) has a ‘lower density root’ under it.

In general, we’ll find that there are two end-member processes that support or compensate the weight of the mountains: the strength of the crust for small mountains, and a bouyant low density root beneath large mountains.

10 km 100 km

Earth model: an elastic lithosphere (crust), that does not flow (but does break), over a ‘fluid’ asthenosphere/mantle

Historically, whether the earth viscously flowed has been vigorously debated from 1500 until 1930-1960. Some people said the earth’s interior was as strong as a steel ball. They were wrong! Other said, the earth’s interior was liquid. They were right (outer core) and wrong (mantle). The truth is that the mantle is 99.9% solid AND it does flow at 1-10 cm/yr (10-100 km/Ma) rates. This is called convection. It might take millions of years to move things around, but the earth is 4,500 million years old!

Local versus regional compensation

The Lithosphere (crust) is strong enough to support the load (weight) of the mountain. But, the lithospheric strength is finite and the surface of the lithosphere is bowed down ‘regionally’ to support the load.

Gravity effect •Positive free air anomaly over load•Near zero Bouguer anomaly over load

The Lithosphere (crust) is NOT strong enough to support the load (weight) of the mountain. In fact, in the limit, the lithosphere is broken on either side of the load and has a near zero strength. The load is supported by the hydrostatic pressure of the asthenosphere pushing on the bottom of the loaded block.

Gravity effect •Zero free air anomaly over load•Negative Bouguer anomaly over load

Regional compensation Plate flexure examples

Regional compensation of Hawaiian Island Chain

Note the strong correlation between the bathymetry and the free air gravity profile. This is because this topography is NOT isostatically compensated but is ‘regional’ compensated by the strength of the lithosphere. Also, note the downwarps to either side of the big island caused by the load that down-flexes the lithosphere (see Fig. 9.10).

Oceanic island volcanic loads compared to seismic evidence

Modelling gravity over Hawaiian Chain: finding best elastic plate thickness of 30 km

4 km deep trenches on Venus, is this subduction ?

What is a buoyancy force ?A buoyancy force arises when a solid block (boat, mountain) is placed into an (ideal)-liquid.The buoyancy force is specified by Archimede’s Principle which states: the decrease in weight of the block equals the weight of the liquid displaced by the submerged portion of the body.

An ideal-liquid is simple in that the only force it transmits is pressure which means that an ideal-liquid CANNOT support any shear stresses. The pressure in a liquid is: P= depth * liquid-density * little-g (Passcals or N/m2 ).

The sign of the density contrast between the block and the liquid determines whether the block floats (block density less than liquid) or sinks (block density greater than liquid).Note below that the weight of the block (a) decreases as the block is lower into the fluid (b) and when the block is floating (c) the effective weight becomes zero as the buoyancy force supports the weight of the block!.

Examples of bouyancy

An object can change it density to change buoyancy force and move up or down in gravity field.

Layered floating blocks and gravity

Figure 9.3 shows two blocks with the same density that are floating in a liquid. Note that when a few layers of block (a) are removed, hence reducing the blocks height (mass), then the block floats at a new level to maintain isostatic equilibrium. Also, note that block (b) extends less above the liquid surface because LESS fluid volume displaced by the small weight of the shorter block. One could also view this plot as adding layers (mass) to block (b) which causes block (a) to float with a higher surface and deeper root.

Measuring gravity at points A,B,C finds no change in gravity! This is because the blocks are all in isostatic equilibrium which means that the total mass of the block and liquid beneath each blocks is the same! (I am missing a detail here, what is that?) Remember that for the blocks to float (not sink), the liquid must be denser than the blocks. Note the higher block’s roots extends deeper into the fluid.

Deriving the isostatic mass and height equations

To calculate the isostatic balance between two blocks (A and B), the height and weight (force) equations are derived. To do this, first the upper-level and lower-level must be defined. We assume that the atmosphere is always at the top of each block and that its weight, hence density, is negligible (assume zero). One must always choose the upper-level to be ABOVE the top of either block! For the lower-level, one must always choose this depth to be at the base of the deepest block. We also assume that the asthenospheric ‘fluid’ that the blocks float in is at the base of each block.

Specifying the height and weight equations

1 1 2 2 3 3 . 1 1 2 2 .

1 1 2 2 3 3 . 1 1 2 2 .

1 2 3 .

: ( ) ( )

: ( ) ( )

: ( ) (

air air asth asth blk A air air asth asth blk B

asth asth

asth

asth blk A asth blk B

blk Aair

Weight g h h h h h g h h h h

which simplifes to h h h h h

Height h h h h h

h h

h

1 2 .)bls Bth kaair hh h

The Weight and Height equations are defined as:

We now have two equations and many unknowns that cannot be solved algebraically. But, if one knows densities and block layer thickness, we have 2 equations and 1 unknown and can uniquely solve the algebraic system of equations.

Example: add 2 km ice to top of block

. .(0* 2*3 2.7*30 3.1*70 3.2* ) (0.9*2 2*3 2.7*30 3.1*70)

0.9*23.2* 0.9*2 0.56 : 2

3.2

air asth blk A blk B

asth asth asth air

h h

Notethis equationcanbe greatly simplied by removing common terms

h h km Height Eqn h h km

To the right block B, two km of ice are added. This added mass causes the bottom of the block to sink into asthenosphere by ha (km) as shown by the right side block.

The addition of 2 km of ice causes block B to sink 0.56 km. Thus, using the height equation, the former air/sediment surface sank by 2 - 0.56 = 1.44 km.

Block A Block B

Isostasy: filling a lake with sedimentNotice there are two unknowns in this problem: the thickness of the new sediments (hs ) and the amount the block sinks into the asthenosphere (ha ).

Amazing! Due to isostasy the 2 km deep lake actually filled with 3.14 km of sediment.

: 1.0* 2 3.2 1.8* 3.2* 1.8* 2

3.2* 1.8* (2 ) 2 1.14

: 2 2 1.14 3.14

aa s

a

s

a a

a s s

Weight h h

h h h km

Height h h h km

h h

One equation and two unknowns!

Back substitute hs

Height equation

Airy and Pratt end-member isostasy models

This was a big debate in 1855 after a British gravity/geodetic survey in India.

Mr. Pratt suggested that mountains do NOT have roots but instead the topography is compensated by a less dense (hence lighter) block. And, that all topography was compensated at the same depth.

Mr. Airy suggested that mountains had thick low density roots supporting mountains and that the depth of compensation was NOT constant, but the density of the blocks was constant.

TIME TO TEST THE HYPOTHESES!

Who was correct ?Airy was mostly correct about what supports large (wide) mountains, but it took until the 1970’s to prove this with seismic work that measured the thickness of the crust and lithosphere beneath mountains.

Pratt was correct in that the difference between the low standing ocean basins and the high standing continents is partially due to the fact that oceans have dense gabbroic composition crust whereas continents have lighter less dense ‘Andesitic’ composition crust.

Test: what are density variations between blocks

Free-air/Bouguer/Airy-isostatic gravity with and without compensation

Extra mass problem Free-air Bouguer Airy isostatic

Airy comp. (0) w/ edge effect (-) 0

No comp. (regional) (+) (0) (+) with edge effect

Important is to know WHY the gravity effects of the two mass anomalies makes the above table true. What happens if I remove the area (mass) above the red line from the blocks ?

Viscous mantle (asthenosphere) response to loading

Glacial isostatic adjustmentU

M v

isco

sity

Gravity modelling: mid-ocean ridge and passive margin

Gravity over Amazon delta

Continental yield strength and earthquake depths

The strength (flow stress) of the lithosphere is given by Byerlee’s brittle failure law and measured flow law for different indicated minerals for assumed geotherm for 100 Ma lithosphere.

Earthquake in continents occur mostly at <25 km depth, but <50 km in oceans. Difference in earthquake depths manifests variation in strength with depth in oceanic and continental crust/lithosphere

Changing strength of oceanic lithosphere in time

(a) Note that as the lithosphere gets older (right to left ‘flag plots’), it gets stronger due to thermal cooling of lithosphere.

(b) Integrated strength of lithosphere (gray area under (a) curves) increase in time.

Water height in oceans is proportional to gravity fieldBy radar mapping the ocean surface, the gravity field is measured (also current pressures).

This is because the liquid ocean surface is perpendicular to the gravity field.

Figure of the earthTerms to know:

•Spherical radius•Equatorial radius•Polar radius•Reference ellipsoid•Geoid (Gravitational Potential)

Non-spherical shape and centrifugal force makes the gravity vary as IGF equation.

Also, the earth rotates once a day around its spin axis to make forces.

What force makes the Earth a flattened ellipsoid with a smaller polar radius ?

Measured figure of earth

Isostasy quotes (1)That part of the surface of any heavy body will become more distant from the centre of its gravity which becomes of greater lightness. The earth therefore, the element by which the rivers carry away the slopes of mountains and bear them to the sea, is the place from which gravity is removed: it will make itself lighter……The summits of the mountains in course of time rise continually (da Vinci, 1505).

The mountains, I think, are to be explained chiefly as due to thermal expansion of material at depth, whereby the rock layers near the surface are lifted up. The uplifting does not mean the inflow or addition of material at depth, the void within the mountain compensates for the overlying mass (Boscovich, 1755).

The state of the Earth’s crust lying upon the lava may be compared with perfect correctness to the state of a raft of timber floating upon water; in which, if we remark one log whose surface floats much higher than the upper surfaces of the others, we are certain that its lower surface lies deeper in the water than the lower surfaces of the others (Airy 1855).

The amount of matter in any vertical column drawn from the surface to a level surface below the crust is now and ever has been, approximately the same in every part of the Earth (Pratt, 1861).

Isostasy quotes (2)The crust must be in a condition of approximate hydrostatical equilibrium, such that any considerable addition of load will cause any region to sink, or any considerable amount deduced off an area will cause it to rise……..the crust is analogous to the case of a broken-up area of ice, refrozen and floating upon water. (Fisher, 1881).

The hypothesis (interior contraction by secular cooling) is nothing but a delusion and a snare, and the quicker it is thrown aside and abandoned, the better it will be for geological science (Dutton, 1882).

In an unpublished paper I have used the terms isostatic and isostasy to express that condition of the terrestrial surface which follow from the floatation of the crust upon a liquid or highly plastic substratum…….isobaric would have been a preferable term, but it is preoccupied in hypsometry………….For this condition of equilibrium of figure, to which gravitation tends to reduce a planetary body, irrespective of whether it is homogeneous or not, I propose the name isostasy (Dutton 1882).

An Airy theory is untenable……The Pratt theory is the only one so far that is sound……..It seems safe to assert that the teory of isostasy has been proven……….. (Bowie 1927).

Convenience of computation, and perhaps tradition, rather than any physical probability, has been the chief reasons for the attention given to Pratt’s hypothesis, instead of Airy’s (Jeffreys 1926).

Isostasy quotes (3)Various geological observations and deductions, which the geologists regard as established facts (horizontal shortening), seems inconsistent with the isostatic theory (Chamberlin 1932).

Geologists often ask too much of the principle of isostasy. When they find that it will not explain all earth movements, they think it is not a true principle (Reid 1922).

Mountains, mountain ranges, and valleys of magnitude equivalent to mountains, exist generally in view of the rigidity of the Earth’s crust; continents and plateaus and oceanic basins exist in virtue of isostatic equilibrium in a crust heterogeneous wrt density (Gilbert 1889).

The success of the isostatic hypothesis in reducing gravity anomalies is to show that isostatic adjustment in the earth’s crust is nearly perfect (Gilbert 1913).

The excesses and deficits of mass……..will be a more accurate measure of the capacity of the rigid crust to carry without viscous yielding loads which have borne through geological time, hidden loads whose magnitudes in many regions appear to mask by contrast the present relief between mountains and valleys……….The deep zone is the hydraulic agent which converts the gravity of the excess matter in the heavy column into a force acting upwards against the lighter column….By this means even the continental interiors are kept in isostatic equilibrium with the distant ocean basins (Barrell 1914).

No question that a mountain’s load is distributed beyond the area of its base (Putnam 1935).

Comparison of astronomic and geodetic determined latitude

The keynote of isostasy is a working towards equilibrium. Isostasy is not a process that disturbs equilibrium, but one that restores equilibrium.

Gravity deficient and the Himalaya’s

Airy’s floating table-land hypothesis (1855)

block liquid

liquid

block

B b

b

B

Global elastic thickness

Free air gravity

Global free air gravity from satellite

Crustal thickness: Airy, Pratt or both or

NOT?

Crustal thickness in western US

from Earthscope array data

Free-air gravity through Hawaiian Islands