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Module 4-1

The Floating Lithosphere 1 Isostasy

The average elevation of continental regions is

about 875 m above sea level. Ocean depths

average about 3730 m below sea level.

Continental regions, therefore, are about 4.6 km

higher than oceanic regions, on average. If we

could view the Earth from space and see through

the ocean at the same time, this would be the

most striking topographic feature that we would

see. Why is it so that continents are higher than

ocean basins?

Quantitative Concepts and Skills Manipulating

equations Deriving equations Numerical vs.

analytical solutions Role of conceptual

models Unit conversion Logic function

The reason is isostasy, a state of gravitational

equilibrium continental regions float higher

than oceanic regions.

Where we are going in these modules

This classic diagram is a collection of seismic

sections across North America. The vertical axis

is depth in km. The numbers on the sections are

seismic velocities (P-wave) in km/sec. The large

white thicknesses with 6 km/sec are continental

crust, which is underlain by mantle (8 km/sec)

at the Mohorovicic Discontinuity. The shallow

units with smaller velocities are sediments and

sedimentary rocks. In Module 4-3, we will

calculate the pressure along a surface at 40-km

depth to test the notion of isostasy. In modules

4-1 and 4-2 we will do some preliminary

calculations to explore the concept.

PREVIEW

The parent of isostasy is hydrostatics. The

hydro in hydrostatics refers to water.

Isostasy refers to the Earths crust and upper

mantle the fluid is the asthenosphere. The

physical principles are the same (First

Principles). Slide 4 introduces the

concept of depth of compensation in the context

of hydrostatics (a block of ice in water).

Pressure along the depth of compensation is key

to isostatic calculations. Slides 5-8 ask you to

develop a spreadsheet that calculates the

hydrostatic pressure along the depth of

compensation below a block of ice totally or

partially immersed in water. You will use the

spreadsheet, together with some logic statements,

to find the percentage of the ice that sticks up

above the water. Slide 9 solves the same problem

using algebra. Slides 10-12 ask you to

develop a spreadsheet that applies the algebraic

solution from Slide 9 to find the difference in

elevation of the top of two floating blocks. In

Slides 13 and 14 you will adapt this spreadsheet

to two different conceptual models that try to

duplicate the 4.6-km difference in floating

levels of continental and oceanic crust.

Hydrostatics

This block of ice (density 0.917 g/cm3) is

floating in the water (density 1.00 g/cm3).

This block is not in an equilibrium position. PB

PA. Why?

Water will flow from B to A and the block will

rise because PB PA. This will continue until

PB PA.

The block is in this vertical position, rather

than some other, because the pressure along the

dashed horizontal line (or any other horizontal

line below the block) is constant.

In the language of isostasy, the dashed

horizontal line along which pressure is the same

is known as the depth of compensation.

cell containing a number.

Building the spreadsheet

cell containing an equation.

Suppose the block is 3 m thick and submerged so

that its surface is right at the water level as

shown. What are pressures PA and PB? By how

many kPa is PB larger than PA ?

The mks unit of pressure is Pa (Pascals), or

N/m2. 1 kPa 1000 Pa.

P r g z, where P is pressure, r is density of

the overlying material, and z is its thickness.

In this case, there is only one material above

each point (ice at A and water at B). Each P is

a one-layer calculation.

The Target Problem How much does the 3-m block of

the previous slide have to rise to come into

equilibrium?

Building the spreadsheet, 2

Step 1, Revise your spreadsheet (add a row) so

that it splits the total thickness of the ice

into two parts one above the water level and the

other below the water level. You will also need

to revise the formula for Pressure at A to take

into account the two portions of ice.

Building the spreadsheet, 3

Step 2. Revise your spreadsheet further so that

it calculates the fraction of the blocks

thickness that is under water (Row 19) and

contains a logic statement announcing whether or

not the block is in equilibrium (Row 21). Make

the logic function return YES if the absolute

value of the difference is less than 0.05 Pa.

The idea is that you can add bits of ice above

the water level until the logic statement

produces a YES, and thereby find the answer by

trial and error.

But what if you add too much ice and go past the

range where the logic function returns YES?

Building the spreadsheet, 4

So, finally, revise the logic function to return

three possible answers YES, needs more ice

above water, needs less ice above water. You

can do this with nested logic functions.

Now you are in a position to start with zero ice

above the water level, then incrementally add

ice, and know if you have overshot the mark. For

example, 0.1 m of ice is not enough.

What answer do you get? That is, what value for

the thickness of ice above water produces YES?

Is there more than one possible answer? Do you

get a different value if you use 0.005 Pa rather

than 0.05 Pa in your logic function?

An alternative approach

Let ZT total thickness. Let ZU the upper part

(above water level). Let ZL the lower part

(below water level).

ZU

ZT

ZL

Then -- ZT ZU ZL PA ?ice g ZT PB ?water

g ZL PA PB So ZL / ZT ?ice / ?water and

ZU / ZT (?water ?ice ) / ?water

This is the fraction of the block that lies below

the water level. Does it agree with what you

found in your spreadsheet?

This says that pressure is the same along the

depth of compensation

What is the sum of the last two equations? Does

it make sense?

Sometimes the easiest thing to do is to use a

little algebra!

Two blocks of ice floating in a bucket of water

Use the equations of the previous slide to make a

spreadsheet that calculates the difference in

elevation of the top of two blocks of floating

ice, one 5 m thick and the other 7 m thick.

Not to scale

This is a preview of Airy isostasy mountains

are higher than plains because they have roots.

Thus, continents are higher than ocean basins

because continental crust is thicker than oceanic

crust.

(Note, although the depth of compensation is

drawn in the figure, it isnt used in the

calculation in this spreadsheet.).

Two blocks of metal floating in a bucket of

mercury

Change the numbers in your spreadsheet to

calculate the difference in elevation between the

top of a 5-m-thick block of copper and a

5-m-thick block of lead, both floating in liquid

mercury. The densities are 8.94 g/cm3, 11.37

g/cm3, and 13.6 g/cm3 for the Cu, Pb, and Hg,

respectively.

Not to scale

This is a variation of Pratt isostasy mountains

are higher than plains because they are less

dense. Thus, continents are higher than ocean

basins because continental crust is less dense

than oceanic crust.

Actually Pratt envisioned the blocks ending at

the same depth. How would you modify the

spreadsheet to handle that conceptual model?

Two blocks of crust floating in a bucket of

asthenosphere

Change the numbers in your spreadsheet of Slide

10 to calculate the difference in elevation

between the top of a 28-km-thick block of granite

(density 2.67 g/cm3) and a 7.9-km-thick block of

gabbro (density 2.99 g/cm3) both floating in a

fluid with density 3.3 g/cm3.

Not to scale

Compare your calculated topographic difference to

the difference between continents and ocean basin

averages cited in the first slide. What do you

think of this conceptual model as an explanation

for the topographic difference between continents

and ocean basin?

An alternative picture Together, the blocks

cover the entire surface so that the fluid

(asthenosphere), which the blocks are floating

on, does not reach the surface.

Isostasy

Rethink your spreadsheet. Recall depth of

compensation.

Not to scale

ZC thickness cont. crust ZO thickness oceanic

crust ZM thickness upper mantle DH difference

in elevation.

From ?O ZO ?M ZM ?C ZC

From ZO ZM DHd ZC

You should have gotten this result in the

previous spreadsheet.

Do you still like the conceptual model?

Isostasy, 2

You shouldnt, because (Oops!) we forgot the

ocean!!

Also, the depth of compensation actually lies

within the lithosphere, above the asthenosphere.

The floating blocks contain subcrustal

lithosphere (mantle material) as well as the

crust.

So, add a second layer (seawater) to the

ocean-basin block.

Not to scale

Bottom line is same as in previous spreadsheet.

Note changes in parameters

End of Module Assignments

- The spreadsheets of Slides 5-8 are an example of

a numerical solution to a problem. The algebra

of Slide 9 produces an analytical solution to the

same problem. With these examples in mind,

discuss the differences between a numerical and

an analytical solution. Include comment about

your answer to the question in Slide 8. Would

you say that the solution to the problem posed in

Slide 14 is numerical or analytical? - The language of isostasy includes the colorful

terms root and antiroot. Define these terms

based on the figures in the module. - Let the antiroot density be 3.2 g/cm3 in Slide

14. What must the other densities be to produce

the same 4.6-km structural difference between

continents and ocean basins? Is your answer

unique? What densities affect the bottom line

the most? In answering this last question you

are doing a sensistivity analysis. What is a

sensitivity analysis? - From the figure in Slide 14, what is continental

freeboard? What would the continental freeboard

be if the continent in Slide 14 was covered with

a 3-km thick ice sheet? - Mathematical modeling of the type done in this

module depends on (a) First Principles, (b) the

conceptual model, (c) the method of solution, and

(d) the numerical values of parameters. Discuss

the role of each in getting the right answer. - Find out who Pratt and Airy were, and why they

thought up what has come to be known as isostasy.

Was this before or after Mohorovicic discovered

the crust?

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