Homework Problem Set, Week 2.
Due Friday, January 22:
Please show your calculations.
Name________________________
1. The total amount of carbon on the earth’s surface depends on a balance between the input
of new carbon – from the CO2 derived from volcanism at mid-ocean ridge spreading centers,
and the sink of carbon-rich sediments being carried back down into the earth’s mantle at
subduction zones. Most variations in global climate are determined by the partitioning of
this carbon into the various reservoirs on the earth’s surface (air, oceans, sediments, biology),
and there is a general balance between input of new carbon and burial in the deep mantle of
old carbon.
a. Assume that the total length of all seafloor spreading centers (globally) is 59,200 km and
that the average spreading rate is 55.7 ± 3.4 mm/year. Also assume that oceanic crust is
about 6 km thick, with the Upper Crust is made up of a 1 km thickness of extrusive
(volcanic) basalts and upper feeder dikes, and the Lower Crust is a 5 km thick layer of lower
dikes and gabbros (a frozen magma chamber). Calculate the total volume of oceanic crust
produced each year (including both upper and lower crust) in km3/year.
(6km)(5.92x104 km)(5.57x10-5km yr-1)=19.78 km3 yr-1
b. Assume the Upper and Lower Crust has a density of 3.0 gm/cm3. Upper Oceanic Crust
(the basalts and upper dikes) are about 0.04 % CO2 by weight. Assume that all of the CO2 in
the Upper Crust is removed immediately after formation (relative to the spreading rate) – the
gas is ejected by the rocks as they solidify and carried by hydrothermal circulation into
bottom seawater at the ridge axis. Calculate the amount of CO2 (in kilograms/year) that is
injected into seawater each year from the formation of Upper Crust.
3g cm-3=3x1012 kg km-3, total crust=(3x1012 kg km-3)(19.78km3 yr-1)=5.93x1013 kg yr-1
CO2 upper crust=(1/6)(5.93x1013 kg yr-1)(4x10-4)=3.95x109 kg yr-1
c. The Lower Oceanic Crust (the 5 km thickness of lower dikes and gabbros) has a much
smaller CO2 content (0.006 % by weight) than the Upper Crust. There is considerable debate
if this CO2 is ‘extracted‘ from the lower crust immediately after formation, or is vented in
older, off-axis vents. We will assume that it is all vented immediately after formation at the
ridge axis for ease of calculation. Compute the total input of CO2 from the formation of lower
crust in kg/year.
CO2 lower crust=(5/6)(5.93x1013 kg yr-1)(6x10-5)=2.96x109 kg yr-1
d. Total CO2 emitted from crustal formation (under the above assumptions) is (b) plus (c).
Assume that the total carbon inventory in the ocean is stable over geological time and the
hydrothermal input above is balanced by the transport of carbon-rich sediments into the
mantle at subduction zones. The carbon (not CO2) inventory dissolved in seawater is 38 x
10+15 kg. What is the residence time of carbon in the ocean, with respect to seafloor
spreading? Assume that all carbon in the ocean starts as CO2 at a spreading ridge.
Total C = (massC/massCO2)(1b+1c)=(12/44)(6.912x109 kg yr-1)= 1.885x109 kg yr-1
Residence time= inventory/flux=38x1015kg/1.885x109 kg yr-1= 20x106 yr
e. During the Cretaceous period, when seafloor spreading increased dramatically, the balance
between input and burial may have been temporarily tipped in favor of ‘input’. If seafloor
spreading increased by a factor of 2.4 times during the early Cretaceous, what would you
expect to happen to the rate of seafloor consumption at subduction zones?
Conservation of mass requires subduction rates also increase by 2.4 times
f. Assume that the CO2 emitted by volcanism during seafloor spreading ends up 50% in the
oceans (dissolved as carbonate, bicarbonate and dissolved CO2) and 50% in the atmosphere.
Describe in the simplest terms how carbon in each of these two reservoirs ends up back in the
mantle.
Atmospheric CO2: Weathering => rivers => oceans => biology => sediments => subduction
Dissolved CO2: ocean => biology => sediments => subduction
g. What volcanic and tectonic factors discussed in lecture OTHER than CO2 derived from
seafloor spreading volcanism could be responsible for the extreme warmth of the Cretaceous
compared to today?
1. isolation of Antarctica at high latitude
2. continental positions and ocean gateways
3. formation of large igneous provinces
4. break-up of super-continent Pangea
2. At steady state, the rate of CO2 addition to the surface of the earth should be balanced by the
rate of removal by weathering of continental rocks and eventual subduction of carbonate rich
sediment deposits. In this problem you will estimate the amount of continental rock consumed
by chemical weathering from the input of CO2 from seafloor spreading. We will ignore any
changes in carbon storage in surface reservoirs (rocks, seawater, air) and assume that all CO2
from seafloor volcanism is consumed by weathering immediately.
a. The weathering of silicate minerals (the majority of the exposed continents) and their
deposition as carbonate minerals is chemically expressed as:
(equation 1)
CaSiO3+H2O+CO2 CaCO3+SiO2*H2O
The molar mass of CaSiO3 (a generic silicate mineral) is 116.16 g/mol. Ignoring anthropogenic
impacts and assuming the carbon cycle is at steady state, and further assuming that all weathering
and deposition on the modern Earth is of silicate rocks with the above chemical formula,
calculate the total mass of silicate consumed by chemical weathering in one year. (HINT: See
solution to problem 1b and c)
Total CO2 flux from 1b+1c =6.912x109 kg yr-1
(6.912x109 kg yr-1CO2)(116.16kg CaSiO3/ 44kg CO2)=1.825x1010 kg yr-1 CaSiO3
b. The increase in atmospheric CO2 during the Cretaceous has been estimated at 3 to 9 times
the present pre-industrial concentration. It has also been estimated that the global average
temperatures during this period was between 8 and 14°C higher than today. How would
these changes have impacted weathering rates as a sink for atmospheric CO2?
Higher temperatures and CO2 concentrations would have dramatically increased weathering
rates; CO2 drawdown would have been high.
c. Mountain building associated with subduction almost always occurs at the edges of
continents (i.e., Andes, Cascades, Aleutian volcanoes). How would an increase in subduction
rates in a warmer climate have impacted weathering during the Cretaceous?
Higher atmospheric temperatures would have led to more ocean evaporation and therefore more
rainfall on the continental edges. If subduction-related mountains that are built at the continental
edges expose more bare rock to this rainfall, weathering (and therefore the atmospheric CO2 sink)
would increase.
3. In the simplest possible form, the radiative balance of the earth (from Problem Set #1) is:
(Equation 2)
S = σ • T4
Where S is the effective solar insolation (accounting for albedo) of 240 W/m2, σ is the StefanBoltzman constant [5.67 x 10-8 W/(m2 °K4)] and T is absolute temperature (°K). Underline
implies ‘global average value’. An alternative method used for including the impact of
Greenhouse gases on earth’s surface temperature modifies Eqn 2 as:
(Equation 3)
S + G = σ • T4
Where G is the greenhouse gas warming term in units of W/m2.
a. If the present average temperature at the surface of the earth is now +15°C, calculate the value
of G; the amount of heating due to all greenhouse gases in W/m2.
240 = (5.67x10-8) * T4  T=255 K w/o greenhouse gases Present T=15+273=288 K
240+G = (5.67x10-8)(288)4 = 150 W m-2
b. To modify this expression to allow for variations of average global temperature due to
changes in CO2 greenhouse gas forcing, we can write Gnew (the new GHG heating) as:
(Equation 4)
Gnew = G + A • ln [ C / Co ]
Where G is the average value calculated above, C is the new concentration of CO2, Co = the preindustrial value of CO2 in the atmosphere of 280 ppm, A (in W/m2) represents the ‘sensitivity’ of
greenhouse gas warming to CO2 concentrations, and ln is the natural log. Based on the
examination of paleoclimate data, a doubling of CO2 concentrations (2 X 280 ppm) in the past
has produced a 2.6°C temperature increase, which is equivalent to A = 20.5 W/m2. Use this
information and Eqn 3 to calculate the difference in temperature from present conditions you
would expect for the Cretaceous, when the CO2 levels were ~10 times the pre-industrial value.
Gnew = 150 + (20.5) * ln(10) = 197.2 W m-2  CrT: (240+197.2) = (5.67x10-8)(T)4 
T = 296°K = 23°C
Change from present T= 23°C – 15°C = +8°C
4. Sea level is roughly proportional to land coverage—as the ocean rises, less land is exposed,
and vise versa. The albedo of water is very different than that of continental materials, and
changes in the proportion of exposed land mass can alter the effective solar insolation. In this
problem you will estimate the impact on global temperature by changing sea-level.
a. The figure below shows ocean height above the present sea-level. When was sea-level at it’s
highest in the Cretaceous, and at what height relative to the present? You will need an accurate
answer to finish the problem set, so answer to the nearest 5 m.
Sea level vs age
300
Sea Level (meters)
250
200
150
100
50
0
0
20
40
60
80
100
120
140
age (My)
95My, 250 m
b. What phenomena would have produced the rapid drop in sea level at 30 and 5 My?
1. Ice sheets forming in Antarctica at 30 My
2. Ice sheets increasing in Antarctica and now in North America (and Eurasia).
160
c. The percentage of Earth’s surface that is land can be expressed as:
(Equation 5)
Po - P = S/25
Where Po is the present percentage of land (30%) and S is the sea-level in meters relative to the
present. Using your answer to part a, find the percentage of land.
30 - P = 250/25  P = 20%
d. The table below shows the albedo of a variety of substances. Assume the average albedo of
land in the Cretaceous, including rock, clouds, and vegetation, is 30%, and that of water is 10%.
Calculate the average global albedo. How does the compare to the modern value of 39.1%, and
what two major differences that might cause this?
(0.2x0.3)+(0.8x0.1)=0.14 or 14%. Roughly one third as large as modern albedo. Less land, no
ice/snow in Cretaceous. Difference in vegetation also valid if point out Cretaceous had more
long grasses/coniferous trees than modern Earth.
d. Using the first equation in the problem set from week 1 and your answer to part d (above),
determine the expected difference in steady state temperature of the Earth during the Cretaceous
compared to the present (+15°C). Assume all other constants are unchanged from problem set 1,
and that f=61%.
342(1-0.12) = 0.61(5.67x10-8) T4  303°K = 30°C
30° - 15° = +15°C
e. How does your value from 4e compare with that from question 3b? Identify potential causes
of any discrepancies.
Much higher, 3b answer was +8°C: Albedo plays large role in increasing effective insolation,
while question 3 only considered the heat trapping effects of greenhouse gases. Also, f number
for Cretaceous may not be same as modern as we assumed.
5. During the Cretaceous, much of the global temperature increase occurred at high latitudes
(Arctic and Antarctic) compared to recent times. What impact would these high latitude high
temperatures have on ocean circulation?
Reduced meridional temperature gradients between equatorial and high latitudes would have
reduced ocean currents, and reduced heat transfer from tropics to poles. Bottom water formation
would be extremely reduced, allowing the formation of anoxic black shales on a global scale.