Homework Problem Set for Week 5, due Friday, February 5.
Name_______________________
In this problem set, we will use an Excel spreadsheet to calculate the VARIABILITY in
solar insolation due to earth's orbital parameters: precession (P), eccentricity (e), and tilt (T).
There are some simplifying assumptions used in the formulation below, but the principles are
correct. If you are confused about the terms used, re-read Chapter 8 of the text.
Assume that P, e, and T vary as
P = P0 sin Pt
period = 23,000 years
P0 = 4 w/m2
T = T0 sin Tt
period = 41,000 years
T0 = 5 w/m2
e = e0 sin et
period = 100,000 years
e0 = 12 (no units)
remember that for a sine function, the period is = 2 
1a. Assume that the Long Term Average solar insolation is 340 w/m2. Plot the variability of the
solar insolation (Isolar) over 500,000 years when only the variation due to precession is included;
i.e. Isolar = 340 + P w/m2. Identify the maximum and minimum values of this curve in an Excel
plot and calculate the change in temperature of a 'black body' earth due to these values. Give your
values of [Imax, Imin, TEMPmax, TEMPmin] where TEMP is the difference between the global
black body temperature at equilibrium with the Long Term Average of 340 w/m2 and those at
equilibrium with Imax or Imin. Turn in the plot for 500,000 years.
1b. For this part, do the same thing for the variability due to tilt alone; i.e., Isolar = 340 + T w/m2.
Identify the maximum and minimum values (list these in your solution) and calculate the change
in temperature of a 'black body' earth due to these values. Give values for [Imax, Imin, TEMPmax,
TEMPmin] and turn in the plot for 500,000 years.
1c. Plot eccentricity as a function of time [e = e0 sin et], just to get a feel for the 100,000 year
variation. Turn in the plot for 500,000 years.
1d. Calculate the variability of total solar insolation using all 3 parameters (T, P, and e), assuming
that precession and tilt is additive and eccentricity modulates precession in the form
Isolar = 340 + (P * e + T) w/m2
Give [Imax, Imin, TEMPmax, TEMPmin] and include your plot of Isolar as a function of time for
500,000 years.
Total things to turn in for problem set: Your four Excel plots and a table with 3 sets of values
for [Imax, Imin, TEMPmax, TEMPmin].
SEE EXAMPLE PLOTS AT END OF SOLUTIONS.
2a. Which parameter, tilt or precession, most influenced the magnitude of insolation 41 kya ago?
300 kya?
41 kya: Precession dominates—tilt was at 0 amplitude, precession at maximum negative. 300
kya: Tilt dominates—precession was at 0 amplitude, tilt at maximum positive.
2b. Which parameter dominates the overall shape of the graph? Which parameter dominates the
overall magnitude of the plot?
Eccentricity has the longest period, and therefore determines the basic period of the system. Also
acceptable to say tilt and eccentricity since both go into determining positions of highs and lows
at similar time scales. Eccentricity dominates the overall magnitude of the system since it has the
largest amplitude.
2c. Mars has a much more variable eccentricity than Earth (greater amplitude), and has an
eccentricity period of 2.2 My. How would you expect Mars’ insolation graph to appear relative
to Earth’s, assuming it’s precession and tilt periods and amplitudes are of similar magnitude to
that on Earth?
Longer dominant cycle (eccentricity), producing a longer overall period between highest highs
and lowest lows. Larger eccentricity amplitude means higher highs and lower lows - assuming
eccentricity amplitude is still larger than relative to tilt and precession amplitudes.
3. Chapter 10 in the text discusses the deformation of continental bed rock due the presence of
massively heavy ice sheets during the Last Glacial Maximum (see Figure 10-10).
3a. Using your own resources (i.e., the web) define the term ‘Pro-Glacial Lakes’.
Freshwater lakes of melt water that form in the depression adjacent to ice sheets.
3b. Give an example of a pro-glacial lake that formed adjacent to these ice sheets in North
America during the Last Glacial Maximum
(i) Laurentide Ice Sheet __________Agassiz_________________
(ii) Cordilleran Ice Sheeet_________Lake Missoula____________
3c. The Puget Lobe was the ice sheet over Puget Sound that was the southern extent of of the
Cordilleran Ice Sheet, and was responsible for most of the hills (glacial moraines) that you see
around Seattle. It was about 1000 meters thick over Seattle at the maximum at 18,000 years BP.
Assume that the present shore line of Puget Sound is at sea level. Use the relationship between
ice sheet total thickness and the amount of bedrock depression shown in Figure 10-10 to estimate
how far below sea level the Puget Lobe ice sheet depressed the Puget Sound area at the Last
Glacial Maximum.
(1/3.3) * (1000 meters) = 300 meters below PRESENT sea level.
3d. During the period of the Puget Lobe Advance during the Last Glacial Maximum, enough ice
had formed on continents to lower global sea level 130 meters. Use your answer in part 3c to
calculate how deep the Puget Sound shoreline would have been under water at 18,000 years BP.
300 (from depression of ice sheet) – 130 (sea level drop) = 170 meters
3e. Given the elastic response time of bed rock rise/fall of 3000 years due to loading and unloading of the bedrock due to ice sheets (see Chapter 10 in text), and assuming that the Puget
Lobe melted abruptly at 18,000 years BP, calculate the residual depression of the Puget Sound
shoreline that is still present due to this ice sheet.
18,000 years = six 3000 year half-lives. Divide 300 meters by 2 six times = 4.5 meters of
depression left. Or write exponential expression to get same answer.
3f. The North Atlantic is a region of bottom water formation (down-welling of the surface waters
due to temperature and salinity). If the pro-glacial lake adjacent to the Laurentide Ice Sheet
drained abruptly into the North Atlantic at the end of the Last Glacial Maximum, how would that
impact the formation of bottom water in the North Atlantic?
The freshwater from the lake would make the surface waters of the North Atlantic less dense, and
bottom water formation (down-welling) would be less likely to occur.
eccentricity
15
10
5
0
eccentricity
0
100
200
300
400
500
600
-5
-10
-15
I +precession alone
I=340+precession
I=340+precession
346
344
342
340
338
336
334
0
100
200
300
400
500
600
years, 1000s
Precession, tilt and eccentricity
Insolation in W/m2
400
380
360
340
320
300
280
0
100
200
300
thousands of years
400
500