Problems
8. A way to get insight into the physics of subduction is to use a
classic result from fluid mechanics, called Stokes’ problem, which
describes the terminal velocity v at which a sphere of radius a and
density ρ sinks due to gravity in a fluid with viscosity η and lower
density ρ′. The result is v = 2ga
2 (ρ − ρ′)/9η. Estimate the subduction
velocity of a slab assuming the slab is a sphere with radius equal to
half its thickness. To do this, estimate the density contrast from the
thermal model (Eqn 5.4.14) and a coefficient of thermal expansion
α = 3 × 10
−5 °C
−1 . Use a mantle viscosity from Section 5.5.3.
Because this is a back-of-the-envelope calculation, there is no correct answer, but you should be able to come up with something
reasonable (within an order of magnitude or two of reality).
9. The result that a subducting slab that reaches the core should still
be thermally distinct (Fig. 5.4-5) may seem surprising. For another
estimate, use the one-dimensional cooling equation in Section
5.3.2 to estimate how long a slab should need to warm up to 90%
of the ambient lowermost mantle temperatures, assuming that
it were immediately transported to the base of the mantle and that
κ = 10
−6 m
2 s
−1 .
10. Using the definition of the slab pull force (Eqn 5.4.15):
(a) Write the force in terms of the age of the subducting plate.
(b) Explain whether this force would be greater or smaller, and
why, for increased values of subducting plate age, coefficient
of thermal expansion, and thermal diffusivity.
11. Assume that in a subducted slab the depth of the spinel–perovskite
phase transition deepens from its usual 660 km outside the slab
to 700 km, and that the core of the slab is 800° colder than the
surrounding mantle. What is the Clapeyron slope of the phase
change?
12. The surface of Venus is much hotter (450°C) than that of Earth. If
Venus had plate tectonics and the rocks were similar, so that the
temperature gradient in old lithosphere there were the same as on
Earth, how would the thickness of the “oceanic” lithosphere
differ? How would the slab pull and ridge push forces differ? What
other differences might you expect?
13. Express the ratio of the slab pull (Eqn 5.4.15) and ridge push
(Eqn 5.5.6) forces. Explain why this ratio depends on thermal
diffusivity. Estimate this ratio near a trench where old oceanic
lithosphere is subducting, assuming that κ = 10
−6 m
2 s
−1 .
14. To see if momentum can be responsible for the Indian plate’s
northward motion long after its collision with Asia began, estimate
the momentum of the Indian plate and that of an ocean liner, and
compare the two.
15. Use Mohr’s circle to show why
(a) Rocks at depth do not fracture under lithostatic pressure
alone.
(b) The deviatoric stress needed for fracture increases at greater
depth.
16. Suppose that a rock is stressed close to its brittle limit. Show
graphically which will make the rock fracture sooner: (a) increasing σ 1 or (b) decreasing σ 2 by the same amount (assume a twodimensional case where σ 1 and σ 2 are both negative, and internal
friction exists).
17. Suppose that the fracture line for a particular rock is τ = 80 − 0.5σ,
where stresses are in MPa. What angle would the normal to a fracture plane make with σ 1 ? If σ 1 is 400 MPa at failure, what is σ 2 ?
18. For the slider block earthquake model in Section 5.7.5:
(a) Derive an expression for the time between successive slip
events.
1. Assume that Pacific–North America plate motion along the San
Andreas fault occurs at 35 mm/yr.
(a) If all this motion occurs seismically in earthquakes about
22 years apart, which is a typical recurrence interval for the
Parkfield fault segment, how much slip would you expect in
the earthquakes? From Fig. 4.6-7, estimate likely fault
lengths and magnitudes for such earthquakes.
(b) Give similar estimates if the earthquakes occur about
132 years apart, as at Pallett Creek.
2. Assume that all the earthquakes in the Pallett Creek sequence (Fig.
1.2-15) involved 4 m of seismic slip. Using the time interval from
the present to the 1857 earthquake, calculate the seismic slip rate
on this portion of the San Andreas fault. Next, do so by averaging the recurrence intervals for the past two earthquakes (1857
and 1812), the past three, and so on for the entire earthquake
history. What are the implications of this simple experiment for
seismic slip estimates? What other sources of uncertainty should
also be considered, and how might they affect this estimate?
3. (a) Use Table 5.2-1 to find the rate that the Juan de Fuca plate
subducts beneath North America at 46°N, 125°W.
(b) If all this motion occurs in large earthquakes, how often would
you expect an earthquake if the slip in each were 5 m? How
would this estimate change if the slip were 10 or 20 m?
(c) How would the answers to (b) change if only 25% or 50% of
the plate motion occurred by seismic slip?
(d) Paleoseismic observations and historic records of a tsunami
imply that this subduction zone has had very large earthquakes
approximately 500 years apart. Suggest some possibilities in
view of parts (a)–(c). How might you attempt to distinguish
between them?
(e) The crust subducting at this trench is about 10 million years
old. Given the convergence rate and the observations from
other trenches in Fig. 5.4-30, what might you infer about the
moment magnitude of the largest earthquake expected here?
Find the corresponding seismic moment and suggest a plausible fault geometry and amount of slip that would also be consistent with the paleoseismic and plate motion observations.
4. For rigid plates, Eqn 5.2.10 shows that we can find the angular
velocity vector of one plate from the sum of two others. Show that
at a point we can also do this for the linear velocity vectors.
5. The news media sometimes ask “How large would the largest
possible earthquake be?” Estimate the seismic moment and
moment magnitude by assuming that all the trenches in the world
(48,000 km) slip at the same time, that 10 m of slip occurs, and the
fault width is 250 km.
6. Estimate the thermal Reynolds number R defined in Eqns 5.3.19
and 5.4.3, assuming that κ = 10
−6 m
2 s
−1 . What does this estimate
imply about the processes of plate cooling and subduction?
7. Assume that oceanic lithosphere has a thermal conductivity of
3.1 Wm
−1
°C
−1 .
(a) Find the heat flow for old oceanic lithosphere, assuming a
linear temperature gradient (Fig. 5.3-8), a basal temperature
of 1450°C, and a plate thickness of 95 km.
(b) How would this value change for a basal temperature of
1350°C and plate thickness 125 km?
(c) If the lithosphere under a midplate region were thinned to
50 km while the basal temperature remained 1350°, what
would the heat flow be, assuming a linear temperature
gradient?
Problems 367
8. A way to get insight into the physics of subduction is to use a
classic result from fluid mechanics, called Stokes’ problem, which
describes the terminal velocity v at which a sphere of radius a and
density ρ sinks due to gravity in a fluid with viscosity η and lower
density ρ′. The result is v = 2ga
2 (ρ − ρ′)/9η. Estimate the subduction
velocity of a slab assuming the slab is a sphere with radius equal to
half its thickness. To do this, estimate the density contrast from the
thermal model (Eqn 5.4.14) and a coefficient of thermal expansion
α = 3 × 10
−5 °C
−1 . Use a mantle viscosity from Section 5.5.3.
Because this is a back-of-the-envelope calculation, there is no correct answer, but you should be able to come up with something
reasonable (within an order of magnitude or two of reality).
9. The result that a subducting slab that reaches the core should still
be thermally distinct (Fig. 5.4-5) may seem surprising. For another
estimate, use the one-dimensional cooling equation in Section
5.3.2 to estimate how long a slab should need to warm up to 90%
of the ambient lowermost mantle temperatures, assuming that
it were immediately transported to the base of the mantle and that
κ = 10
−6 m
2 s
−1 .
10. Using the definition of the slab pull force (Eqn 5.4.15):
(a) Write the force in terms of the age of the subducting plate.
(b) Explain whether this force would be greater or smaller, and
why, for increased values of subducting plate age, coefficient
of thermal expansion, and thermal diffusivity.
11. Assume that in a subducted slab the depth of the spinel–perovskite
phase transition deepens from its usual 660 km outside the slab
to 700 km, and that the core of the slab is 800° colder than the
surrounding mantle. What is the Clapeyron slope of the phase
change?
12. The surface of Venus is much hotter (450°C) than that of Earth. If
Venus had plate tectonics and the rocks were similar, so that the
temperature gradient in old lithosphere there were the same as on
Earth, how would the thickness of the “oceanic” lithosphere
differ? How would the slab pull and ridge push forces differ? What
other differences might you expect?
13. Express the ratio of the slab pull (Eqn 5.4.15) and ridge push
(Eqn 5.5.6) forces. Explain why this ratio depends on thermal
diffusivity. Estimate this ratio near a trench where old oceanic
lithosphere is subducting, assuming that κ = 10
−6 m
2 s
−1 .
14. To see if momentum can be responsible for the Indian plate’s
northward motion long after its collision with Asia began, estimate
the momentum of the Indian plate and that of an ocean liner, and
compare the two.
15. Use Mohr’s circle to show why
(a) Rocks at depth do not fracture under lithostatic pressure
alone.
(b) The deviatoric stress needed for fracture increases at greater
depth.
16. Suppose that a rock is stressed close to its brittle limit. Show
graphically which will make the rock fracture sooner: (a) increasing σ 1 or (b) decreasing σ 2 by the same amount (assume a twodimensional case where σ 1 and σ 2 are both negative, and internal
friction exists).
17. Suppose that the fracture line for a particular rock is τ = 80 − 0.5σ,
where stresses are in MPa. What angle would the normal to a fracture plane make with σ 1 ? If σ 1 is 400 MPa at failure, what is σ 2 ?
18. For the slider block earthquake model in Section 5.7.5:
(a) Derive an expression for the time between successive slip
events.
1. Assume that Pacific–North America plate motion along the San
Andreas fault occurs at 35 mm/yr.
(a) If all this motion occurs seismically in earthquakes about
22 years apart, which is a typical recurrence interval for the
Parkfield fault segment, how much slip would you expect in
the earthquakes? From Fig. 4.6-7, estimate likely fault
lengths and magnitudes for such earthquakes.
(b) Give similar estimates if the earthquakes occur about
132 years apart, as at Pallett Creek.
2. Assume that all the earthquakes in the Pallett Creek sequence (Fig.
1.2-15) involved 4 m of seismic slip. Using the time interval from
the present to the 1857 earthquake, calculate the seismic slip rate
on this portion of the San Andreas fault. Next, do so by averaging the recurrence intervals for the past two earthquakes (1857
and 1812), the past three, and so on for the entire earthquake
history. What are the implications of this simple experiment for
seismic slip estimates? What other sources of uncertainty should
also be considered, and how might they affect this estimate?
3. (a) Use Table 5.2-1 to find the rate that the Juan de Fuca plate
subducts beneath North America at 46°N, 125°W.
(b) If all this motion occurs in large earthquakes, how often would
you expect an earthquake if the slip in each were 5 m? How
would this estimate change if the slip were 10 or 20 m?
(c) How would the answers to (b) change if only 25% or 50% of
the plate motion occurred by seismic slip?
(d) Paleoseismic observations and historic records of a tsunami
imply that this subduction zone has had very large earthquakes
approximately 500 years apart. Suggest some possibilities in
view of parts (a)–(c). How might you attempt to distinguish
between them?
(e) The crust subducting at this trench is about 10 million years
old. Given the convergence rate and the observations from
other trenches in Fig. 5.4-30, what might you infer about the
moment magnitude of the largest earthquake expected here?
Find the corresponding seismic moment and suggest a plausible fault geometry and amount of slip that would also be consistent with the paleoseismic and plate motion observations.
4. For rigid plates, Eqn 5.2.10 shows that we can find the angular
velocity vector of one plate from the sum of two others. Show that
at a point we can also do this for the linear velocity vectors.
5. The news media sometimes ask “How large would the largest
possible earthquake be?” Estimate the seismic moment and
moment magnitude by assuming that all the trenches in the world
(48,000 km) slip at the same time, that 10 m of slip occurs, and the
fault width is 250 km.
6. Estimate the thermal Reynolds number R defined in Eqns 5.3.19
and 5.4.3, assuming that κ = 10
−6 m
2 s
−1 . What does this estimate
imply about the processes of plate cooling and subduction?
7. Assume that oceanic lithosphere has a thermal conductivity of
3.1 Wm
−1
°C
−1 .
(a) Find the heat flow for old oceanic lithosphere, assuming a
linear temperature gradient (Fig. 5.3-8), a basal temperature
of 1450°C, and a plate thickness of 95 km.
(b) How would this value change for a basal temperature of
1350°C and plate thickness 125 km?
(c) If the lithosphere under a midplate region were thinned to
50 km while the basal temperature remained 1350°, what
would the heat flow be, assuming a linear temperature
gradient?
Problems 367