Unit 7: Heat Flow in Permafrost
In developing their numerical model of heat flow in permafrost, students learn about the physics of heat conduction and the factors that influence the geothermal gradient. They use trigonometry to impose a thermal oscillation at Earth's surface and explore the effects of different periods of oscillation on the depth of penetration of the temperature disturbance.
On completing this module, students are expected to be able to:
- Create a model of heat flow through the outer kilometer of Earth's crust using Fourier's law of heat conduction.
- Experiment with different thermal conductivities and heat capacities to see the impacts of changes on the crust's geothermal gradient.
- Experiment with a step increase in surface temperature to explore the impact of anthropogenic warming on the geothermal gradient.
- Explain how borehole temperature data in permafrost can be used to show that the Arctic has experienced recent warming.
- Evaluate the impact of different frequencies of surface temperature oscillation on the geothermal gradient over time.
- Compare model results to published borehole temperature data.
This exercise addresses several of the guiding principles of the InTeGrate program. In particular, it requires the use of systems thinking, develops students' abilities to use numerical modeling to generate and test geoscientific hypotheses, makes use of authentic borehole temperature data, and addresses a grand challenge facing society, anthropogenic warming.
Context for Use
This unit is intended to be used in a three- to four-hour class period that meets once a week. It can be used as part of this modeling course or it can be adapted as a lab exercise for a course in geophysics. 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 have access to Microsoft Excel or similar spreadsheet software to allow them to graph temperature profiles as a function of depth as STELLA will only display time series of temperature change for an individual level in the crust.
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 7 Student Reading.
For advanced courses, instructors may also wish to have students read and present Lachenbruch, A.H., and Marshall, B.V., 1986, "Changing Climate: Geothermal Evidence from Permafrost in the Alaskan Arctic," Science, v. 234, p. 689–696, and Davis, M.G., Harris, R.N., and Chapman, D.S., 2010, "Repeat temperature measurements in boreholes from northwestern Utah link ground and air temperatures at the decadal time scale," Journal of Geophysical Research, v. 115, B05203, doi: 10.1029/2009JB006875.
Students should take the following quiz prior to coming to class to ensure they have done the assigned reading: Heat flow in permafrost reading quiz (Microsoft Word 2007 (.docx) 137kB Dec3 16). An answer key for the quiz can be found here:
In class, students should be provided with the exercise found here: Permafrost exercise for students (Microsoft Word 2007 (.docx) 310kB Dec3 16).
An answer key for the exercise can be found here:
Instructors can download a version of the STELLA heat flow in permafrost model by clicking on this link: Heat flow in permafrost STELLA model (Stella Model (v10 .stmx) 45kB Aug11 16). The model is "spun up," containing values in the reservoirs that are non-zero. See the answer key for explanation. The model was created using STELLA Professional and should open on any subsequent version of STELLA. 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 teach the course 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 at least 20 minutes of subsequent class periods trying to figure out where they were in the exercise at the beginning of the week. This is not a good use of time, hence the recommended three- to four-hour class session once per week. 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.
A typical 4-hour class session might be broken up into the following sections:
- 20-minute discussion of the reading to ensure all the students are familiar with the mathematics behind the model, specifically Fourier's law of heat conduction.
- 1.5 to 2 hours to build the model
- 1.5 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 a day or two before class along with the completed STELLA model itself. This would allow the instructor to determine whether students' models are working correctly and to provide feedback to address errors in construction, omissions in documentation, problems with unit conversions, and inappropriately sized time steps 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 pencil and paper sketch of what their model should look like, annotated with equations and then sent to the instructor in advance of class for feedback. This should facilitate a faster model construction time during the limited class hours.
References and Resources
This exercise is based on the following references:
Lachenbruch, A.H., and Marshall, B.V., 1986, "Changing Climate: Geothermal Evidence from Permafrost in the Alaskan Arctic," Science, v. 234, p. 689–696.
Turcotte, D.L., and Schubert, G., 2000, Geodynamics, 2nd ed., Cambridge, U.K.: Cambridge University Press, pp. 132–143, 150–152.
Another reading that can be given to advanced students is:
Davis, M.G., Harris, R.N. Chapman, D.S., 2010, "Repeat temperature measurements in boreholes from northwestern Utah link ground and air temperatures at the decadal time scale," Journal of Geophysical Research, v. 115, B05203, doi: 10.1029/2009JB006875.
Additional web resources include:
- Data and Maps from the Permafrost Laboratory
- U.S. Geological Survey's Borehole Temperature Logs from Arctic Alaska, pre-1989