Unit 5: Growth and Decay of Ice Sheets
Large continental ice sheets, such as the Laurentide Ice Sheet from the last glaciation, as well as Antarctica and Greenland of today, are some of the most important features of the global climate system — they exert a major control on sea level and they represent an important feedback mechanism on the global temperature via their very high albedo.
In 1976, Johannes Weertman created an elegant, simple mathematical model of a large ice sheet and showed how the ice sheet would respond to insolation forcing related to orbital changes — the Milankovitch cycles. This paper, in conjunction with the famous "Pacemaker" paper by Hays et al. in the same year — which showed how the marine oxygen isotope record matched the fundamental frequencies of orbital variations — helped to establish the importance of the Milankovitch astronomical climate theory.
In this module, we first review the importance of these ice sheets and how the model emerges from some simple assumptions about ice flow. We then review some basic things about orbital variations and the resulting changes in insolation, and how those insolation changes are likely to affect something like an ice sheet. Students are then guided through the construction of the model and they carry out a series of experiments with the model to learn how the ice sheet responds to changes. The experiments illustrate the nature of the system's steady state, its response time, and the existence of a threshold that separates growth from rapid melting. Oxygen isotope data from the oceans, which give an indication of ice volume changes, are used to compare the model's response to orbital changes with the real world.
On completing this module, students are expected to be able to:
- Create a model of ice sheet growth and decay based on Weertman's (1976) classic model.
- Experiment with the model to understand the importance of initial conditions.
- Calculate the response time of the ice sheet and compare that to the frequency of Milankovitch climate cycles to predict how the ice sheet will be influenced by these orbital changes.
- Discover through experimentation the importance of thresholds that separate growth and stability from total collapse.
- Use the model to predict what should be happening now in terms of climate change, and what the next 20 kyr (kiloyear) should see in terms of ice buildup.
This exercise addresses several of the guiding principles of the InTeGrate program. In particular, it helps add some sophistication to their systems thinking toolbox, develops students' abilities to use numerical modeling to generate and test geoscientific hypotheses, uses astronomical data to drive the model, compares the model output with oxygen isotope data from marine sediments, and addresses a grand challenge facing society, the potential danger of thresholds or tipping points in the climate system.
Context for Use
We intend this module to be used in a three- to four-hour class period that meets once a week (or two shorter periods in the same week). It can be used as part of this modeling course or it can be adapted as a lab exercise for a course in paleoclimatology or in some other course dealing with the cryosphere. We assume that the students will have a basic understanding of mathematics, which essentially provide the recipe for making this model — they should understand the concepts of integration and differential equations, but they do not need to solve these problems on their own. For this module, students should come to class prepared to take a short quiz on the assigned reading. Thereafter they will be led through a series of prompts designed to help them create and experiment with a number of simple models using the iconographic box modeling software STELLA (see https://www.iseesystems.com/store/products/ for different options for purchasing student or computer lab licenses of STELLA or for downloading a trial version). Students should also have access to Microsoft Excel or similar spreadsheet software.
For those learning to use STELLA, we suggest the online "play-along" tutorials from isee systems. You can find them here: isee Systems Tutorials.
Description and Teaching Materials
In preparation for the exercise, students should read the following: Unit 5 Student Reading.
For advanced courses, instructors may also wish to have students read and present the papers by Weertman and Hays et al. cited below.
Students should take the following quiz prior to coming to class to ensure they have done the assigned reading: Ice Sheet Pre-Lab Quiz (Microsoft Word 2007 (.docx) 48kB Aug11 16). The instructor's key to the quiz is here:
Before class, students should be provided with the exercise found here: Ice Sheet Modeling Activity (Microsoft Word 2007 (.docx) 131kB Dec3 16). They should be required to study the first part of this document, which leads them through the process of creating the model and making a paper and pencil version of the model design, showing all the relevant reservoirs, converters, and connector arrows — these sketch models should be submitted to the instructor several days before the class session. When they actually begin to make the STELLA model, they also need to download a STELLA file consisting of two converters that contain the data for mean summer insolation and the SPECMAP oxygen isotope data: Ice Sheet Template STELLA model (Stella Model (v10 .stmx) 11kB Aug11 16). The insolation data are used to force the model, while the SPECMAP data are included for comparison with the model output.
An answer key for the exercise can be found here:Ice sheet model w/o interface (Stella Model (v10 .stmx) 22kB Aug11 16) . The model was created using STELLA Professional. If you are using an earlier version of STELLA, the complete model graphic and equations can be found in the answer key so that you can reconstruct the model yourself.
Teaching Notes and Tips
We generally post the readings and assignments for students to an LMS site (e.g. Moodle, Blackboard, Canvas). Students can open the assignment in Microsoft Word on the same computer they are using to construct the STELLA model and then answer the questions by typing directly into the document. Students can either print a paper copy to hand in to the instructor or email their modified file to the instructor. It is straightforward to copy graphs and model graphics out of STELLA and to paste them into Word. Simply select the items to be copied, hit copy in STELLA, and paste into Word. There is no need to export graphics to jpg.
We think the course is best taught in a three- to four-hour block once a week because we have found that models require a lot of uninterrupted time to construct. If students have a 50- or 75-minute class period several times a week, they spend time trying to figure out where they left off, making this inefficient. However, we also know that sustaining attention for this length of time can be difficult. We therefore recommend allowing students the freedom to take breaks throughout the modeling session to get snacks or coffee.
Because the students are asked to make their own models from scratch, and the model is a bit complex, they should be required to make some initial attempt to create the model in the form of a sketch with paper and pencil before the class meets — this will prepare them to make real progress during the class period.
If people have trouble creating their models (and they almost always do), it often turns out that there are problems involving parentheses in equations, which can mess up the order of operations. For instance, one might intend to write A divided by the product of B and C and type A/B*C, but you really have to type A/(B*C). If a student is really struggling and they have made a good effort to prepare a version of the model before coming to class, we recommend giving them the pre-made version of the model and have them compare the two and figure out where they went wrong, and then proceed with the pre-made version. There is little to be gained by struggling endlessly with troubleshooting, but there is something to be gained by comparing their version with the pre-made one.
A typical four-hour class session might be broken up into the following sections:
- 30-minute discussion of the reading to ensure all the students are familiar with the mathematics behind the model and the relationship between the differential equations and the system components. If you have students make a pencil sketch of their model designs, you could use this time to comment on them.
- 1+ to hour to build the model — this presumes that the students have made some initial attempt to create a pencil and paper version of the model, following the directions in the exercise.
- 2+ hours to conduct experiments
For instructors who have more limited contact hours with their students, we suggest that the model construction parts of this exercise be assigned as a pre-lab to be handed in several days before class as a completed STELLA model. This would allow the instructor to determine whether students' models are working correctly and to provide feedback to address errors in construction that might lead to spurious model behavior. Class time could then be devoted to running experiments and analyzing the results. If access to STELLA outside of class time is impossible due to computer lab scheduling or to financial constraints that prevent students from purchasing their own STELLA licenses, students could be asked to create a detailed pencil and paper sketch of what their model should look like, annotated with equations, and send it to the instructor several days before class for feedback. This should facilitate a faster model completion time during the limited class hours.
References and Resources
Weertman, J., 1976, "Milankovitch solar radiation variations and ice age sizes of ice sheets," Nature, v. 261, p. 17–20.
Hays, J.D., Imbrie, J., and Shackleton, N.J., 1976, "Variations in Earth's orbit: Pacemaker of the Ice Ages," Science, v. 194, p. 1121–1131.
Imbrie, J., and Imbrie, K.P., 1979, Ice Ages: Solving the Mystery. Macmillan Press Ltd., London, 224 pp.
Milankovitch, M.M., 1941. Canon of Insolation and the Ice Age Problem. Königlich Serbische Academie, Belgrade. English translation by the Israel Program for Scientific Translations, United States Department of Commerce and the National Science Foundation, Washington D.C. [Note: Milankovitch's first published works on this topic appeared in the 1920s]