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Unit 5: Growth and Decay of Ice Sheets

David Bice, Department of Geosciences, Penn State
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These materials have been reviewed for their alignment with the Next Generation Science Standards as detailed below. Visit InTeGrate and the NGSS to learn more.


In this unit, students develop a model to experiment with ice sheet dynamics, testing the influence of Earth's orbital cycles and human-induced climate change.

Science and Engineering Practices

Using Mathematics and Computational Thinking: Use simple limit cases to test mathematical expressions, computer programs, algorithms, or simulations of a process or system to see if a model “makes sense” by comparing the outcomes with what is known about the real world. HS-P5.4:

Obtaining, Evaluating, and Communicating Information: Critically read scientific literature adapted for classroom use to determine the central ideas or conclusions and/or to obtain scientific and/or technical information to summarize complex evidence, concepts, processes, or information presented in a text by paraphrasing them in simpler but still accurate terms. HS-P8.1:

Developing and Using Models: Develop, revise, and/or use a model based on evidence to illustrate and/or predict the relationships between systems or between components of a system HS-P2.3:

Developing and Using Models: Develop and/or use a model (including mathematical and computational) to generate data to support explanations, predict phenomena, analyze systems, and/or solve problems. HS-P2.6:

Constructing Explanations and Designing Solutions: Construct and revise an explanation based on valid and reliable evidence obtained from a variety of sources (including students’ own investigations, models, theories, simulations, peer review) and the assumption that theories and laws that describe the natural world operate today as they did in the past and will continue to do so in the future. HS-P6.2:

Cross Cutting Concepts

Stability and Change: Stability might be disturbed either by sudden events or gradual changes that accumulate over time. MS-C7.3:

Systems and System Models: When investigating or describing a system, the boundaries and initial conditions of the system need to be defined and their inputs and outputs analyzed and described using models. HS-C4.2:

Systems and System Models: Models can be used to predict the behavior of a system, but these predictions have limited precision and reliability due to the assumptions and approximations inherent in models. HS-C4.4:

Systems and System Models: Models (e.g., physical, mathematical, computer models) can be used to simulate systems and interactions—including energy, matter, and information flows—within and between systems at different scales. HS-C4.3:

Stability and Change: Change and rates of change can be quantified and modeled over very short or very long periods of time. Some system changes are irreversible. HS-C7.2:

Cause and effect: Changes in systems may have various causes that may not have equal effects. HS-C2.4:

Disciplinary Core Ideas

Weather and Climate: The foundation for Earth’s global climate systems is the electromagnetic radiation from the sun, as well as its reflection, absorption, storage, and redistribution among the atmosphere, ocean, and land systems, and this energy’s re-radiation into space. HS-ESS2.D1:

Weather and Climate: Current models predict that, although future regional climate changes will be complex and varied, average global temperatures will continue to rise. The outcomes predicted by global climate models strongly depend on the amounts of human-generated greenhouse gases added to the atmosphere each year and by the ways in which these gases are absorbed by the ocean and biosphere. HS-ESS2.D4:

Earth and the Solar System: Cyclical changes in the shape of Earth’s orbit around the sun, together with changes in the tilt of the planet’s axis of rotation, both occurring over hundreds of thousands of years, have altered the intensity and distribution of sunlight falling on the earth. These phenomena cause a cycle of ice ages and other gradual climate changes. HS-ESS1.B2:

Earth Materials and Systems: Earth’s systems, being dynamic and interacting, cause feedback effects that can increase or decrease the original changes. HS-ESS2.A1:

Performance Expectations

Earth's Systems: Use a model to describe how variations in the flow of energy into and out of Earth’s systems result in changes in climate. HS-ESS2-4:

This material was developed and reviewed through the InTeGrate curricular materials development process. This rigorous, structured process includes:

  • team-based development to ensure materials are appropriate across multiple educational settings.
  • multiple iterative reviews and feedback cycles through the course of material development with input to the authoring team from both project editors and an external assessment team.
  • real in-class testing of materials in at least 3 institutions with external review of student assessment data.
  • multiple reviews to ensure the materials meet the InTeGrate materials rubric which codifies best practices in curricular development, student assessment and pedagogic techniques.
  • review by external experts for accuracy of the science content.

This page first made public: Sep 15, 2017


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.

Learning Goals

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 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:

Ice Sheet Pre-Lab Quiz key

This file is only accessible to verified educators. If you are a teacher or faculty member and would like access to this file please enter your email address to be verified as belonging to an educator.


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:

Key to Ice Sheet Modeling exercise

This file is only accessible to verified educators. If you are a teacher or faculty member and would like access to this file please enter your email address to be verified as belonging to an educator.

. It contains not only answers to the different questions but also strategies instructors can use to guide students through the exercise and information on typical stumbling blocks.

Instructors can download a version of the STELLA ice sheet model by clicking on this link: 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.


Answers to exercise questions are located in the answer key for this unit (see Description and Teaching Materials section above). Instructors may download an assessment rubric for the modeling exercise here: Assessment rubric (Microsoft Word 2007 (.docx) 121kB Jan8 15). Rather than assign a point value to every question in the exercise, we employ a holistic approach that determines the extent to which a student has correctly built the model, supplied appropriate documentation of equations and units, thoroughly answered questions throughout the assignment, and provided appropriately labeled graphs and figures in answering questions.

References and Resources

Additional Reading:

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]

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These materials are part of a collection of classroom-tested modules and courses developed by InTeGrate. The materials engage students in understanding the earth system as it intertwines with key societal issues. The collection is freely available and ready to be adapted by undergraduate educators across a range of courses including: general education or majors courses in Earth-focused disciplines such as geoscience or environmental science, social science, engineering, and other sciences, as well as courses for interdisciplinary programs.
Explore the Collection »