Download Homework 6: Heat Transfer (Lowrie Chapter 4.2)

GG 304: Physics of the Earth and Planets
Homework 6: Heat Transfer (Lowrie Chapter 4.2)
Due Thursday March 17, 2011
1. Surface heat flow at a given location on the Colorado Plateau is measured to be q0 = 90 mW/m2.
Various lines of evidence indicate that at a depth of z = 35 km (the base of the crust), the
temperature of the crust was recently T(z=35) ~ 800°C. The thermal conductivity is 3
W/(m°C), and the surface temperature averages to be 0°C.
(a) Calculate and roughly plot the equilibrium geotherm (i.e., T versus depth, z) assuming that
the Colorado Plateau is in thermal steady state.
(b) Estimate the average rate of heat production H in the crust beneath the above location.
(c) At what rate is heat flowing from the underlying mantle into the crust?, i.e., what is Q(z=35)
km? Briefly explain why the difference between q0 and q(z=35) does (or does not) make
sense given your estimate of H.
2) Recently, a new eruption site developed at Volcano National Park on the Big Island. Once
lava is in place and begins to cool, it start to solidify to form crust from the bottom up as well
as from the top down. The thermal diffusivity of the lava is ~10-6 m2/s. Answer the
following questions, explaining your reasoning and simplifying assumptions.
(a) From what you have learned about time and length scales of thermal diffusion, roughly how
thick is the crust after 1 day? How about 1 week and 1 month?
(b) If the lava flow is 1 meter thick, how long should you wait before you can safely hike over
the flow? (remember that it is cooling from both the top and bottom)
3) Consider an ocean basin that is 4000 km wide with a spreading ridge in its center. Assume
that this is Earth’s only ocean basin and its width is not changing.
(a) The ridge been spreading 2 cm/yr for over 200 Myr. What the average seafloor depth?
(b) If the spreading rate of the central ridge doubles to 4 cm/yr and remains that way for another
200 Myr, how does sea level change? (assume that the volume of seawater is constant).
4) Consider the following parameters that describe convection in the Earth’s mantle:
density: ρ = 3300 kg/m3
mantle depth: b = 2880 km gravity: g = 10 m/s2
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expansivity: α = 3×10 K
diffusivity: κ = 10-6 m2/s
Temp. Drop: ΔT = 3000 K
Average Mantle Viscosity: η = 1022 Pa s
(a) By calculating the Rayleigh number for the mantle, estimate the surface plate velocity that
is produced by whole mantle convection.
(b) However, the lower mantle is thought to be about 100 times more viscous than the upper
mantle. Now consider the following viscosity layering of the mantle:
Layer
Thickness
Viscosity
Temperature Drop
Upper Mantle
bum = 670 km
ηum = 1021 Pa s
900 K
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Lower Mantle
blm = 2210 km
ηlm = 10 Pa s
2100 K
What is the Rayleigh number for the upper and lower mantles if they are convecting
separately? Which is convecting more vigorously? Consider the heat balance of the top
layer – is this layer heating up or cooling down (and why)?
(c) Compare the expected surface plate velocities for whole mantle flow (from a) with those
for upper mantle flow (as in b)? Which flow pattern seems more reasonable given
observed plate speeds?
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