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SSAChaz.TCJ.3(trad)
Oceanic Lithosphere: Sink or Swim
The fate of oceanic plates depends on their
density—how does it change?
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Core Quantitative Issue
Weighted in
average
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Geoscience
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Plate tectonics, lithosphere
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Thomas
Juster
Department of Geology, University of South Florida
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1
The paradox of oceanic lithosphere
Oceanic lithosphere plays two very important roles in plate tectonics. First, it is what the oceanic
plates are made of, and thus underlies 70% of Earth’s surface. Second, it is thought to provide the
most important driving force for the motion of the plates as it sinks into the asthenosphere at
subduction zones, dragging the rest of the plate along with it (a process called “slab pull”).
Figure from USGS web site
So how can the density of the
lithosphere be both greater
and less than the density of
the asthenosphere? This is
the paradox of the oceanic
lithosphere.
Here the oceanic lithosphere is less
dense than the asthenosphere, causing
it float. This is a good thing, because
foundering of the oceanic plate would
destroy the oceans and all life on Earth!
(For a review of density, see Endnote 1)
Here in the subduction zone the oceanic
lithosphere is denser than the
asthenosphere, causing it to sink. This
tugging force drags the entire plate
along, causing it to move on the surface.
2
Objectives of this module
Upon completion of this module you should be able to:
• Explain what the weighted average is, and compare it to the simple non-weighted average;
• Compute the weighted average;
• Compute the density of the oceanic lithosphere given different proportions of mantle and
crustal rock;
• Explain how oceanic lithosphere thickens as it ages, and how its density changes during this
process
• Explain how oceanic lithosphere can be both less dense than the underlying asthenosphere
(in the ocean basins) and more dense than the underlying asthenosphere (in subduction
zones)
First, extract the embedded Excel
spreadsheet where you will do your
homework. Remember to immediately
save it under a new, unique name.
Q1. Quick review: what kinds of geologic hazards
are commonly at subduction zone plate
boundaries, like the boundary shown in the
diagram to the left? Go directly to End-of-Module Questions
3
Math concept: what is a weighted average?
But here’s the key point: the weights don’t have to all be equal. And If they’re not
all equal, then some terms will get more weight than others in computing the
average.
4
Weighted average, con’t.
When would the weights be unequal? When we’re taking the average of numbers that don’t
represent individual values, but groups of values. For example, suppose we wanted to calculate the
average age of students at a college given these data:
Class
Number of
students
% of college
population
Average age
of class (yrs)
Freshman
135
33.75%
18.25
Sophomore
107
26.75%
19.37
Junior
85
21.25%
20.83
Senior
73
18.25%
22.09
Whole college
400
Q2. If we didn’t use a
weighted average,
what would the
weights equal for a
simple average?
(HINT: we would be
simply averaging four
numbers)
Go directly to End-of-Module Questions
A simple average wouldn’t be appropriate because it would weigh each class equally in the average,
but the classes aren’t equal—there are more freshman, for example, than seniors. To calculate an
accurate average we need to weigh the averages for each class by the fractions of each class in the
whole population, so that classes—like the freshman class—which contain a higher percentage of the
college’s students, contribute more to the average.
5
An example: what is the average asking price of a house in Tampa?
Real estate brokers list homes, and you can use these data to compute the average asking
price. In many cases the data are broken down by the number of bedrooms (groups of values).
For example, here are the data for home listings in Tampa, Florida, in December 2010:
Looking at the data, you can see the obvious—larger houses in
general cost more than small ones, and the largest houses—
those that have five or more bedrooms—can cost millions.
Data from trulia.com
You could compute an average of these prices, but what does
this mean? The average is heavily influenced by the cost of the
largest houses, but there were only 500 of them.
Instead you
calculate a
weighted
average, with
the weights
equal to the
proportion of
listings.
Multiply the
weight (W) times
the average price
for each size
The sum of these values is the
weighed average asking price.
Total of all listings using
the SUM() function
Weight (W) for each size house computed as
the number of listings divided by the total of
all listings. The weights should sum to one.
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An example: what is the average asking price of a house in Tampa?
Here’s what the Excel cell formulas look like. Study them so you can create your own
spreadsheet to calculate a weighted average. This table is also found on the embedded
spreadsheet file.
Q3. What does the reference
$E$16 mean (it’s found in
the formula for cell G11)
=E11/$E$16
Go directly to End-of-Module Questions
=F11*G11
Copy and paste
the formula in
cell H11 into
these cells
=SUM(E11:E15)
Copy and paste
the formula in
cell G11 into
these cells
=SUM(H11:H15)
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Another example: what is the average tuition paid by USF students?
Here’s another example: what is the average tuition
paid by USF students? If you look it up you will find
that the tuition depends on whether a student is instate or not, and it’s a big difference!
In state tuition: $5,124
Out of state tuition: $15,933
However, there are far more in-state students than
out-of-staters, so we need to use the weighted
average to compute the average tuition. Here are
the complete data:
Q4. Fill in the rest of the this table,
which computes the weighted average
of tuition for USF students, both instate and out-of-state. Note that some
of the computed values are revealed
so you can check your formulas.
Q5. Here is the weighted average.
Is it closer to the in-state or out-ofstate tuition? Why?
Go directly to End-of-Module Questions
8
Review: what is lithosphere?
Crust: Intermediate or
mafic rock covered by
sediment, 7-30km thick
Mantle:
Ultramafic rock
Core: Iron (Fe)
metal
Layering based
on composition
Plates: Lithosphere: Rigid rock
100-250 km, 1,300°C
Asthenosphere:
Solid rock that can
flow
Liquid Metal
Layering based on
style of deformation
Recall that the lithosphere is
the relatively cool, rigid outer
layer of Earth, and is underlain
by the asthenosphere, solid
rock that is hot enough to flow
like a fluid.
The lithosphere is not the same
thing as the crust, which is the
outermost layer of rock on
Earth defined on the basis of its
chemical composition.
The lithosphere consists of two
compositional layers: the crust
and the uppermost part of the
mantle. The transition from
lithosphere to asthenosphere
occurs at ≈ 1,300°C, the
temperature at which mantle
rock begins to flow.
9
Review: what is oceanic lithosphere?
140
20
70
8 m.y.
m.y.
m.y.
MidOcean
Ridge
Crust
Mantle
1,300°
C
Click here to see how the oceanic
lithosphere thickens as it moves
away from the ridge and ages
The oceanic crust is formed at mid-ocean ridges, and consists primarily of mafic volcanic rocks
(basalt). Underlying the crust is a small piece of mantle rock cool enough to be rigid, and these
two components form the oceanic lithosphere.
Typically, about 7 km of volcanic rocks can accumulate at the mid-ocean ridge to form the
oceanic crust before the plate moves away from the source of heat and magma.
As the lithosphere moves away from the mid-ocean ridge and its source of heat, it cools. As it
cools more and more of the mantle rock becomes rigid, and the mantle component of the
oceanic lithosphere thickens. Notice that as it thickens, the crust becomes a smaller and
smaller proportion of the lithosphere.
10
Average density of oceanic lithosphere
The average density of the mafic crustal rocks in the oceanic lithosphere is 2,800 kg/m3. The
average density of the ultramafic mantle rock in the oceanic lithosphere is 3,400 kg/m 3.
Because it consists of both crust and mantle, the average density of the lithosphere will
therefore be a weighted average of the densities of the crustal and mantle components.
When the oceanic lithosphere is approximately 8 million years old it consists of about 13 km of
mantle overlain by 7 km of crust.
Q6a. Fill in the orange cells in this
table in Excel that will calculate the
average density of the oceanic
lithosphere when it is about 8 million
years old, and consists of 7 km of
crust overlying 13 km of rigid mantle.
Q6b. The density of the asthenosphere below the lithosphere is about 3,350 kg/m3.
Based on the density of the lithosphere you just calculated, will the lithosphere float
in this asthenosphere or sink through it? Enter “float” or “sink” in this cell.
Go directly to End-of-Module Questions
11
Average density of oceanic lithosphere as a function of age
As shown on Slide 10, the oceanic lithosphere gets thicker with age, as the mantle component
grows. Now that you know how to compute the average density of the lithosphere using the
weighted mean, you can investigate how the density changes with time.
Q7a. Fill in the orange cells in this Excel table
that will calculate the average density of the
oceanic lithosphere at 9 different ages, from 8
to 140 million years old. Note that the
arrangement of the table is a little different
from the ones you’ve done before, but the
equations are all the same—just make sure
you enter the cell references properly. I’ve
revealed the density for 25 m.y. old lithosphere
so you can check you’re doing it right.
Q7b. In column N
calculate the density
difference between
the lithosphere and
the asthenosphere
[ρ(asth), shown in
column M].
Q7c. Once again,
decide whether the
lithosphere will
float or sink in the
asthenosphere,
and enter “float” or
“sink” in column
O.
Go directly to End-of-Module Questions
12
Density of oceanic lithosphere under subduction zones
Most geologists think that the motion of
the plates is driven by the sinking of
oceanic lithosphere at subduction zones
(the force is indicated with the arrow).
What is different about the oceanic
lithosphere here as opposed to on the
surface?
Basalt
Eclogite
Q8. Calculate the density of the subducting oceanic
lithosphere, and decide whether it floats or sinks.
Go directly to End-of-Module Questions
Here’s the big difference: basalt,
which forms near the surface and is
stable there, transforms at depth into
a new rock called eclogite. Eclogite is
much denser than basalt. Endnote 2
Notice that all the densities are larger because of the
greater pressure at 150 km. The greater pressure
compresses the minerals so they occupy less volume.
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End-of-Module Assignment
Answer all questions in the spaces provided in the embedded spreadsheet (Slide 3), which you
should have saved with a different name (e.g., “YourName-density.xls”).
1.Answer questions 1-8 on Slides 3, 5, 7, 8, 11, 12, and 13.
9.How does the density of the subducted oceanic lithosphere change as it warms up? How
would this change the “slab pull” driving force for plate tectonics?
10.Continental crust and lithosphere is much thicker than oceanic lithosphere. The average
thickness of the continental crust is 30km, and the average thickness of the continental
lithosphere is 200km. Calculate the average density of the continental crust assuming that the
crustal rocks have a density of 2,700 kg/m3. Show all your work in the spreadsheet.
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Endnotes
1. Density is a measure of the amount of mass per volume. The modern metric unit of density
is kilograms-per-square-meter, kg/m3, though many people are more familiar with the older
grams-per-cubic-centimeter (g/cm3). Water at normal surface conditions has a density of
1,000 kg/m3 or 1.00 g/cm3. Return to Slide 2.
2. The difference between basalt and eclogite is the mineral composition. Basalt consists
primarily of three minerals: olivine, plagioclase, and pyroxene. When exposed to high
pressures, the olivine and plagioclase transform into garnet in the rock eclogite. Here are
the densities of the pertinent minerals:
Mineral
Density (kg/m3)
Olivine
3,300
Plagioclase
2,700
Pyroxene
3,400
Garnet
3,500
You can see that a rock made of pyroxene + garnet (eclogite) will be denser than a rock
made of olivine, plagioclase, and pyroxene (basalt). Return to Slide 13.
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