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Final examination, Geol.

2

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Answer 12 of the 1

4

questions. All questions have equal value. There is a

page of useful formulas and constants on the last page. Submit your com-

pleted exam by email to sam.butler@usask.ca by midnight, Saskatoon time,

on April 2

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th. Calculators and reference materials are permitted. If you

answer more than 12 questions, clearly indicate which ones you would like

marked. It is up to all of us to maintain the integrity of course eval-

uations during the pandemic. You are not allowed to communicate

with anyone about the exam or ask for help other than from the

course instructor for clarification.

1) The IGRF coefficients for 2020 include g01 = −29405 nT, g11 = −1451

nT, and h11 = 4

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52 nT. These coefficients describe the best fitting dipole at

the centre of the Earth.

a) Which coefficient describes an axial dipole (a dipole aligned with Earth’s

rotation axis)?

b) Calculate the current dipole moment of Earth’s magnetic field (m =

4πR

3

µ0

√

(g01)

2 + (g11)

2 + (h11)

2). Give your answer in A m2

c) Calculate the longitude of Earth’s geomagnetic pole. (φ = arctan[h11/g

1

1]).

Express your answer in degrees.

2) The Nubian, Antarctic and Australian plates meet at a triple junction

at 23◦ South,

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1◦ East. The boundaries between the plates are as follows

(Antarctic-Nubian, ridge), (Nubian-Australian, ridge), (Australian-Antarctic,

ridge). The following velocities are known:

ANvNU16.48 mm/yr at 348.61

◦ or (16.15 mm/yr North, -3.25 mm/yr East)

AUvAN = 50.59 mm/yr at 229.95

◦ or (-32.55 mm/yr North, -38.72 mm/yr

East)

a) Calculate NUvAU. Give your answer either as a magnitude and an angle

measured clockwise from north or as north and east components.

b) Determine the strike of each of the ridges as an angle clockwise from north.

3) a) Using the data from question 2, draw the velocity diagram for the

Nubia-Antarctica-Australia triple junction.

b) Show that this triple junction is stable.

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4) The solar constant (the solar power per unit area at Earth) is 1370 W/m2.

Calculate the solar power per unit area at Mars. The distance from the Sun

to Mars is 2.1822 ×1011m. Explain your reasoning.

5) This question refers to figure 1. As you can see, there is a trend of

increasing earthquake depth as you move to the north-west. Note also that

the deepest earthquakes are very deep (> 300 km) and that there are vol-

canoes (marked in yellow). Is there likely to be a subduction zone, ridge or

transform fault here? Roughly what is the orientation of the fault? If it is

a subduction zone, in which direction does the slab dip? If it is a transform

fault, explain the sense of the slip (dextral or sinistral). Explain your rea-

soning in a paragraph.

Figure 1: Map of Earthquakes in Indonesia (question 5)

6) Give two examples of human activities that can lead to earthquakes.

Briefly describe (in a few sentences) the mechanism by which the earth-

quakes are triggered.

7) Referring to figure 2, we see a series of seamounts on the ocean floor

off of the east coast of North America. These seamounts have been inter-

preted to be a hot-spot track.

a) Knowing that spreading is occurring in the middle of the atlantic, would

you expect the oldest rocks to be found on the seamounts to the north-west,

in the middle, or to the south-east?

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b) Given the length of the seamount track and the half-spreading rate of the

mid-atlantic ridge of 1cm/yr, what would you expect to be the range of ages

or rocks on the seamount track?

Figure 2: Seamounts off the east coast of North America (question 7)

8) Referring to figure 3, assume that the continental crust is support iso-

statically.

a) Derive an expression relating H, the thickness of the continental crust to

h, the height of the continental shelf in terms of ρm, ρw and ρc.

b) Calculate a numerical value for H assuming h = 3 km, ρw = 1000 kg/m

3,

ρm = 3300 kg/m

3, ρc = 2700 kg/m

3.

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Figure 3: Schematic of a passive continental margin (question 8)

9) If a planet is modeled as consisting of a core of radius rc and density

ρc as well as a mantle of density ρm and the total radius of the planet is re,

show that the moment of inertia can be written

I =

8π

15

(ρcr

5

c + ρm(r

5

e − r5c ).

You can start from the fact that the moment of inertia of a sphere with con-

stant density is I = 2

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Mr2 where M is the total mass and r is the radius of

the planet.

10) Figure 4 shows elevation profiles across two mid-ocean ridges. One is

the mid-Atlantic ridge and one is the East Pacific rise.

a) Which of the profiles (upper or lower) corresponds to a ridge that is spread-

ing faster?

b) Which one is the East-Pacific rise and which one is the mid-Atlantic ridge?

11) The Rayleigh number, Ra, is a dimensionless number that describes the

degree of vigor of thermal convection. If you run computer simulations of

convection with different Rayleigh numbers, describe what the temperature

contours will look like when

a) Ra < Racrit where Racrit is the critical Rayleigh number.
b) Ra > Racrit but where convection is steady.

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Figure 4: Two profiles across mid-ocean ridges (question 10).

c) Ra is large enough that convection is unsteady.

12) Consider a planet that rotates on its own axis with exactly the same

period as its orbital period about its star.

a) What is the length of a sidereal day measured in units of years on that

planet?

b) What is the length of a solar day measured in units of years on that

planet?

c) Does the star rise in the west or east on that planet?

13) The Geoid represents a surface of constant gravitational potential that is

coincident with mean sea level and the measured Geiod is shown in figure 5.

There is a signficant geoid low (roughly 100 m) just south of the southern

tip of India.

i) If a shape sails through the geoid low, does it get closer to the center of

the Earth?

ii) Do the ship’s engines have to work harder coming out of the geoid low

than going in?

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iii) Would the amount of sky that is visible change if you are at the base of

the geoid low compared with if you are outside of it?

Figure 5: The Geoid (question 13).

14) The needle of a magnetic compass aligns itself with the horizontal com-

ponent of the local magnetic field.

a) Why are magnetic compasses useless for navigation at the magnetic poles?

b) Define magnetic declination.

c) Why is knowledge of the local magnetic declination important if you are

navigating in the woods.

Useful numbers:

Gravitational constant: G = 6.67× 10−11 m3

kgs2

1AU (distance from Earth to Sun)= 1.4959789× 1011m

radius of the Earth: re = 6.37× 106m

radius of the core: rc = 3.48× 106m

Earth’s polar moment of inertia: C = 8.02× 1037kgm2

Mass of Earth: 5.976× 1024kg

Mass of the core: 1.95× 1024kg

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Mass of the Moon: 7.348× 1022kg

Distance from the Earth to the moon: 3.84× 108m

Radius of Moon 1.738× 106 m

Formulas

Newton’s law of gravitation

g =

GMe

r2

(1)

Centrifugal acceleration

ac = ω

2r (2)

Hydrostatic equation

dp

dz

= −ρg (3)

Polar coordinates

θ = arctan(

√

x2 + y2

z

) (4)

φ = arctan(

y

x

) (5)

Cosine Law

a2 = b2 + c2 − 2bc cos(θ) (6)

Sine Law

sin(a)/A = sin(b)/B (7)

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