Download Please be neat and organized in what you hand in. Once you have

Geodynamics 456-556
Problem set #2. Heat problems
(Due in class on April 15th)
Please be neat and organized in what you hand in. Once you have found the answer, rewrite it in
an orderly fashion so that I can follow your steps. You are welcome to do the calculations and
plots by hand, or use programs like Excel or Matlab. If you use programs, be sure to include
your coding and/or excel spreadsheets.
Equations that you’ll need:
___________________
The heat flow equation for conduction
T −T
q = −K * 1 2
x1 − x 2
Where q is the heat flow in W/m2
K is thermal conductivity (~2.5 W/m°C for typical rocks)
______________________
The 1-D, equilibrium heat-conduction equations for the continental lithosphere, assuming
uniform distribution of heat-producing elements in the crust, above a layer of sub-crustal
lithospheric mantle (with no heat-producing elements). Available as a matlab script,
1-D_Geotherm.m, on the g456 web-site, under problem set 2.
T = T0 −
A
A * h⎞
⎛q
* z2 + m +
z
⎝K
2K
K ⎠
Temperature within the crust 0 ≤ z < h
A 2
⎛ qm ⎞
Temperature within the lithospheric mantle: h ≤ z < L
z+
h
⎝K⎠
2K
Where T0 is the temperature at the surface of the Earth in °C (typically assumed to be 0°C)
h is the depth to the base of the crust in km,
L is the depth to the base of the lithosphere in km,
A is the heat production rate in µW/m3 and
qm = heat flow into the base of the lithosphere
T = T0 +
This set of equations is available as a matlab script, on the course web-site, under
problem set 2.
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Problems
1. Heat flow in the Basin and Range. In the Basin and Range province of the western US the heat
flow at the earth's surface is considerably enhanced. In a 300 m deep borehole in Owens
Lake in the eastern California portion of the Basin and Range, the temperature increased
by 18°C from the surface to the base of the borehole. The conductivity of the materials is
-1
-1
4 W m °C . What is the heat flux (also sometimes called the heat flow) at the earth
2
surface (in W/m ).
2. Thermal profile in and beneath an ice sheet. Consider a portion of the Antarctic ice sheet that
is 2 km thick (in places it is significantly more than this). The mean annual surface
-2
temperature is -50°C. The surface heat flow out of the top of the ice is 54 mW m , which
-1
-1
is a decent average for Antarctica. The conductivity of ice is 2.2 W m °C , and that of
-1
-1
the underlying bedrock is 3.5 W m °C .
a) Calculate and then plot the steady state geotherm for this location, taking the
temperature profile down into the underlying rock by 2 km. Assume no heat
production takes place. Hint: since we are assuming steady-state conditions, with no
heat production, then heat flow out of the top of the ice is equal to heat flow out of
the top of the bedrock. You can use the heat flow equation for conduction to calculate
the temperature at the bottom of the ice (using K for ice). Now you know the
temperature at the top of the bedrock, and you can use the heat flow equation for
conduction to determine the temperature within the bedrock (using K for bedrock).
b) In many locations in Antarctica the ice is thick enough to melt at the base forming subglacial lakes (if your interested, google Lake Vostok for info on one of the largest and
best studied sub-glacial lakes). Calculate how thick the ice has to be to melt using the
same values as above.
3. Contribution of radioactive heat sources to surface heat flow. Consider the case of uniformly
2
distributed radioactive heat sources. The surface heat flow is measured to be 54 mW m ,
and the measured heat production rate due to radioactive heat sources, A, is 1.25
µW/m3. The lithosphere is 125 km thick, and the crust is 30 km thick
a) Estimate the mantle heat flow in this location.
b) What fraction of the surface heat flow in this location comes from the mantle, and
what fraction comes from the decay of radioactive elements?
4. Internal heat of Venus. On missions to Venus, the surface temperature was measured to be
470°C, and at heat producing elements were measured as follows:
Heat generation (10-10 W/kg) of
elements at site Venera 8
K
0.17
U
0.60
Th
0.97
The measured density of the Venusian crust is 2.8 x 103 kg/m3.
a) calculate the heat generation in µW/m3.
b) Assuming a uniform distribution of the heat-producing elements through out the top
50 km, use the one-dimensional equilibrium heat-conduction equation to calculate and
plot the Venus geotherms (Aphroditotherms) down to 50km depth. Assume that
conductivity is 2.5 W/(m °C); that the heat flow from the mantle and deep lithopshere
of Venus is 21 mW/m2.
c) Discuss what these Aphroditotherms suggest about the internal structure of the
planet.
5. Archean geotherms: Early in the Earth’s history there was a much higher concentration of
the heat producing isotopes of U, Th, and K. It is estimated that in the Archean (~3500 Ma)
heat generation was about 3.5x what it is today. Today, a “typical” value of crustal heat
production is 1.25 µW/m3. Assuming an Archean mantle heat flux of 63 mW/m2, and a
uniform distribution of heat producing elements in the crust, and an Archean crustal
thickness of 20 km, determine the thickness of Archean continental lithosphere assuming that
the temperature at the base of the lithosphere is 1100°C (the temperature where mantle
behaves elastically, and can be considered part of the plate)