Download Natural Hazards - Geology 209 Homework assignment #3

Due Date:
Natural Hazards - Geology 209
Homework assignment #3 - Tectonics
February, 14th
Name____________
Plate Motion Calculator
1. Listed in the table below are absolute plate velocities for the listed locations calculated from the
following web page (http://ofgs.ori.u-tokyo.ac.jp/~okino/platecalc_new.html).
Location
Philadelphia, PA
Las Vegas, Nevada
Los Angeles, California
Salt Lake City, Utah
London England
Shanghai, China
Perth Australia
Bombay, India
Velocity (cm/year) Direction (degrees from north)
2.8
247
3.6
241
6.0
296
2.5
240
1
231
1
310
8.4
17
5.4
14
Use these values to plot plate velocity vectors on the world map attached to this problem set. To do this
you will need a protractor - if you need to borrow one please come see me, but I am sure you can get one
from a friend. Velocity vectors represent the direction and magnitude of plate motion. To draw them draw
an arrow that begins at the point of interested (e.g., Philadelphia, PA) and points in the direction of plate
motion (i.e. the Direction given above = 247o). An approximate location of the 8 sites is sufficient for this
exercise. The velocity direction of plate motion is given as an azimuth with respect to north, which is a
compass direction between 0o and 360o, where 360o or 0o is north, 90o is east, 180o is south and 270o is west
(thus, Philadelphia's plate motion is almost due west, or pointed to the left on your maps). The length of the
arrow should be drawn proportional to its velocity - for this exercise use a scale such that 1 inch equals a
4.0 cm/year velocity. This means if the plate motion calculator gives a plate velocity of 3 cm/year toward
270o, you should draw a 0.75-inch long arrow pointing to the west (or to the left on your map).
a) Why do you suppose the velocities change as you move across North America?
b) What does the plate motion profile across North America tell us about the rigid nature of plates in terms
of the plate tectonic paradigm?
c) Are we moving closer or further away from England? Does this answer make since knowing what you
know about the direction of seafloor spreading in the middle of the Atlantic Ocean (see relative plate
velocities in your book - Figure 2.6 on page 34)?
d) Is Australia going to eventually collide with Asia?
e) Do you think there is or is not still a high probability for geophysical hazards in the Himalayas (between
India and Asia)? Why or why not?
f) Does the motion of Europe and North America seem paradoxical given that the Atlantic Ocean is getting
bigger and bigger with time?
0
20
4 mm/year
2. Using the attached figure of the Emperor-Hawaiian Island seamount chain, determine the rate and
direction for the Pacific plate for the past 80 Ma. To do this use Excel to plot the measured distance
between the island of interest and the main island of Hawaii versus there determined age. To measure the
distances, select the center of the island of interest and measure, with a ruler, the distance between the
island and the active Hawaiian Hotspot (make sure that you measure your distance along the axis of
the island chain - not as the crow flies). Remember to change your measured distance into km using the
scale provided - I measured 6 inches between Suiko and the hotspot, which converts to about 6000
kilometers - i.e. a half inch is 500 kilometers). Once you are finished with your plot answer the following:
a) Discuss a method geologists could have used to determine the age of the islands.
b) What is the approximate velocity of the Pacific plate in i) cm/year, ii) mi/hr and iii) km/Ma?
c) What do you think happened 40 Ma ago to cause the large kink in the Hawaiian island chain?
d) What is the difference between relative plate motion and absolute plate motion? Is your
calculated velocity an absolute or relative rate? Why?
500 km
3. Paleomagnetism
Under the proper conditions many rocks preserve a record of the orientation of the Earth's
magnetic field at the time that the rock was formed. The study of this ancient fossilized
magnetism is called paleomagnetism - what Professor Weil does. Paleomagnetic studies were
instrumental in leading to the widespread acceptance of plate tectonics.
The Earth's magnetic field behaves very much like a bar magnet - meaning it has two poles, one
positive and one negative. The Earth's bar magnetic is located in the center of our planet and is
oriented along the spin axis. Consequently, the north and south magnetic poles are very similar to
the north and south geographic poles. Thus, knowing the magnetic pole position provides a very
useful reference frame for ancient continental reconstructions.
At any point on the Earth's surface, the magnetic field can be expressed in terms of two
components: declination and inclination. The declination is the angle between the geographic
North Pole and the magnetic North pole. The inclination is simply the angle the magnetic field
lines make with the Earth's surface (see figure 17.16 below). By convention, a downward
inclination is positive. Note that there is a systematic relationship between latitude and inclination.
Thus, if a paleomagnetist can determine the ancient inclination of a continent, then he/she can
determine the relative latitudinal position of that continent at the time of magnetization acquisition.
Note, however that paleomagnetic data cannot provide any information regarding paleolongitude.
Paleomagnetic studies can therefore document plate motion in the north-south direction (changes
in latitude), but not in the east-west direction (changes in longitude). The mathematical
relationship between latitude and inclination is:
tan I = 2 tan λ
where I is the magnetic inclination and λ is the latitude relative to the magnetic north pole (i.e.,
+90o = the north pole).
Consider the following situations. If a plate were
to remain fixed in position or move only in an eastwest direction, rocks of all ages from that plate
would show the exact same magnetic inclination.
Simply - all rock samples would define the same
magnetic pole (see figure 17.18 below). If a plate
moves towards the equator with time, younger
rocks will show progressively more gently
plunging magnetic inclinations. From the point of
view of a fixed plate, the calculated paleomagnetic
poles from progressively younger samples would
appear to move away from the plate (see figure
17.19 below). This apparent motion of the
paleomagnetic pole is termed apparent polar
wander. Conversely, if a plate moves poleward
with time, younger rocks will show progressively
steeper magnetic inclinations through time.
By collecting paleomagnetic data from samples of different ages from different continents, we can
track the motion of their respective plates relative to one another.
On the class web page is an Excel file of paleomagnetic inclination data from four continental
blocks involved in the formation of the Himalayan Mountains.
1. Convert inclination values for each block into paleo-latitude using the equation given
above. You can use your calculator to solve for the individual results, or better yet, you
can use Excel to do your calculations for you. However, it is important to note that Excel
does its trigonometric calculations using radians not degrees. Consequently, you must
first change your input degrees (in this case inclination) into radians. To do this use the
Excel equation:
=Radians(angle you want changed)
Also, note that your answer will be in radians - and you want degrees latitude. So, to
change back to degrees, use the Excel equation:
=Degrees(radians you want changed)
2. Make a plot of latitude versus time that includes data from all 4 continents. Use a different
plot symbol for each continental block. In geology, or any science that looks at temporal
evolution, time is always plotted in reverse, which means that the oldest age is on the left
and the youngest on the right.
3. Paleomagnetic data has an intrinsic error of plus or minus 10 degrees. Plot error bars for all
of your data. To do this double click on your data in the graph and a format data series
window will appear. Choose the tab for the appropriate axis (X or Y error bars) for
adding your error bars. Within that menu choose fixed value in error amount and type in
10. In Display type select the option so that error bars are plotted on both sides of your
data point. Click OK and you should see a small bar that now bounds each of your data
points.
4. What can you infer about the tectonic history - especially the continental drift history - of
these four blocks. Think about what it means of the continents are at different, ort similar
latitudes at the same time.
5. Approximately when did India collide with Asia to form the final stages of Himalayan
evolution?
4. Crustal Residence Time
Background: The basaltic crust that forms the upper 7 km of oceanic lithosphere is created by
partial melting of uprising mantle at mid-ocean ridges and represents the most abundant igneous
rock type in the Earth's crust. Oceanic crust rafts along with the rest of the oceanic lithosphere,
eventually to be subducted and recycled back into the mantle. The oldest oceanic crust surviving
today is 180 Ma (million years) old, suggesting that the cycle of birth at a ridge, transport with the
plate and subduction into the mantle typically takes less than 180 Ma. By comparison, continental
crust is typically much, much older, suggesting that it is recycled very slowly (if at all). In this
problem set, you will determine the present-day volume of oceanic crust, determine its
production/destruction rate, and examine several implications for the Earth's mantle and crust.
An important concept we will need is residence time. The matter comprising Earth's crust and
mantle is part of the rock cycle. Mass is transferred back and forth between the crust and mantle.
As a result, rock doesn't stay forever in either layer. Residence time refers to the average length of
time mass spends in a layer. To compute residence time, we assume that the sum total of material
transferred out of a layer (crust or mantle) exactly equals the sum total of material transferred into
the layer, such that the mass of the layer is constant through time. This is the assumption of steady
state: output equals input (or constant layer mass - i.e., the Earth surface is not growing or
shrinking).
Residence time is the average time spent by mass in a particular layer and is equal to the mass of
the layer divided by the flux of mass into (or out of) the layer. In mathematical terms, we say that
RT =
M
M
=
Fin Fout
where M is the mass of the layer, Fin is the flux of mass into the box and Fout is the flux of mass
out of the box. Residence time (RT) has units of time, whereas Fin and Fout have units of mass per
time. For example, to find the residence time of mass in the oceanic crust we simply divide the
mass of the crust by the rate at which oceanic crustal mass is subducted back into the mantle (Fout).
a. Surface area of the oceans: At present, oceanic crust covers 71% of the surface. Use this to
compute the area of the globe covered by oceanic crust. Express your answer in km2. You will
need to know that the radius of the Earth is 6,371 km and that the surface area of a sphere is:
Surface area = 4 πr2
where r is the radius of the sphere.
b. Volume of oceanic crust: On average, oceanic crust is 7 km thick. Using the surface area of
oceanic crust, compute the total volume of oceanic crust. Express your result in km3. For this
problem, it is acceptable to approximate the volume as the product of surface area and thickness.
c. Mass of oceanic crust: To compute the mass of oceanic crust, we simply multiple volume by
average density. The latter is: 3000 kg/m3. Use this value to compute the mass of oceanic crust
in kg. Note: this will require conversion of units!
d. Mean age (residence time) of oceanic crust: Tabulated below are the fractions of oceanic crust
that fall into each of 8 age intervals. Use this data to compute the average age of the Earth's
oceanic crust. To do this, begin by computing an average age for each of the 8 age intervals. Then
multiply each of these by the fraction of crust of that age and sum your results. Your calculated
sum represents the mean age, or residence time (RT), in Ma (millions of years).
Age Interval (Ma)
0-15
15-30
30-45
45-60
60-90
90-120
120-150
150-180
Fraction of Total
0.10
0.15
0.15
0.18
0.18
0.15
0.07
0.02
e. Production/Subduction rate: We now know the mass of oceanic crust and its mean age or
residence time. We can use these numbers to compute the rate at which new crust is created by
mid-ocean ridge processes. Under the assumption of steady state, this rate in must equal the rate
out.
Fin = Fout =
M
RT
(i) Use this equation to estimate the flux of material in (or out) of the oceanic crustal
layer. Express your result in kg/yr.
(ii) Use the average density of oceanic crust (3000 kg/m3) to convert your result in the
previous problem (part (i) of question e. ) to km3/yr.
f. Mean spreading rate: Given that the total length of mid-ocean ridges is 60,000 km, what yearly
spreading rate (length/time) is needed to produce the volume of oceanic crust created in a
year (from your answer for e.ii)? Express your result in cm/yr, which is the unit for spreading
rate (conversions will be required). Assume that the crust produced is 7 km thick. Hint: Imagine a
rectangular box that is 60,000 km long and 7 km tall. How wide must it be to have a volume of
41.2 km3? The width of the box divided by 1 year is the yearly spreading rate.
g. Mantle volume and mass: We start with the equation for the volume of a sphere:
4
V = πr 3
3
where r is radius. The mantle is not a sphere, but rather a spherical shell. To compute its volume,
we use the above equation to compute the volume of the sphere formed by the combined mantle
and core and subtract from that the volume of the core (this way both volumes, mantle and core,
and core, are spheres). The outer radius of the mantle is 6355 km; the outer radius of the core is
3485 km.
(i) Use these values to compute the volume of the mantle in km3.
(ii) Use the average density of the mantle (4000 kg/m3) to compute the mass of the mantle.
Note: this will require conversion of units!
h. Mantle residence time: We now know the mass of the mantle (from question g.ii) and the flux of
mass from the mantle (from question e.i) to oceanic crust (and vice versa). Use the definition of
residence time to compute the residence time of rock in the Earth's mantle (note, we are
ignoring interactions with the continental crust, but these are relatively unimportant). Express your
result in years.
i. The result is much larger than the age of the Earth, what does this imply about mantle rocks?