The average elevation of continental regions is about 875 m above sea level. Ocean depths average ab

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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.
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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.
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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.
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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.
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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.
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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.
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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?
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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?
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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!
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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.).
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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?
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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?
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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?
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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
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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?