Modeling the Earth

Kirsten Menking, Vassar College

The class in which my students acquire the most hands-on experience with complex systems is my senior seminar on numerical modeling, entitled Modeling the Earth. This course introduces students to finite difference modeling through a series of geological and environmental science problems. Each week students read articles that form the basis for the week's project. They then construct models using the STELLA software and run a number of experiments. The projects include:

• the global phosphorus cycle
  
• the U-Pb Concordia/Discordia dating method
  
• flow of ice in glaciers
  
• impact of changes in runoff and evaporation on the volume of water contained within a chain of lakes in eastern California
  
• heat flow in permafrost
  
• scarp retreat
  
• Earth's energy balance with the sun and resulting temperature
  
• impact of biological organisms with different albedos and temperature-dependent growth functions on planetary temperature under conditions of increasing solar luminosity (Daisyworld)
  

I have chosen these topics to expose students to a variety of system behaviors (e.g. steady state, linear growth or decay, exponential growth or decay, oscillatory) and types (open vs. closed) as well as to introduce positive and negative feedback loops, boundary conditions, initial conditions, and response and residence times. I find that the act of creating models and experimenting with them is a powerful way of learning about and understanding complex systems. Model construction requires students to identify the different components of a system and how they are related to one another physically and mathematically. Modeling also engages students' critical thinking skills as they compare their outputs to empirical data and try to explain system behavior.

The STELLA software is well suited for introducing undergraduates to model construction. It is icon-based and represents reservoirs as boxes and flows between reservoirs as arrows. Additional tools include circles that hold values of constants or equations and linking arrows that are used to show dependencies between variables. A drop down menu specifies run time parameters and model time step from which the software automatically constructs the do loop architecture necessary to execute each iteration. Double clicking on reservoirs and flow arrows allows the specification of initial and boundary conditions, and a graphing window shows the values of variables over time. The visual nature of the software allows students to quickly develop working models without having to learn a programming language such as Fortran or C++, and is therefore less intimidating for math phobic students. That being said, the STELLA software includes a menu item that allows students to see the first order differential equations the software is solving.

Response of students to the course has been highly positive. Many have commented on the value of using models to understand complex systems composed of numerous interacting parts. Many have said that they felt empowered by learning a new skill and that they enjoyed the ability to develop hypotheses, run experiments, and receive confirmation or negation of those hypotheses in real time. Students also remarked that the exercises gave them newfound appreciation for mathematics. One of their favorite aspects of the course was the end of semester project, in which they worked on a problem of personal interest. These projects have been quite diverse in reflection of students' majors or minors and have included eutrophication of lakes, the flow of traffic on city streets, the wage-fund doctrine economic model, groundwater flow, and the production of tidal power.

Though the course has been successful and well received by students, it has also had some challenges. First, most of the times I've taught the course, there has been a student who has found the process of modeling to be highly frustrating. This student typically has difficulty sustaining the patience required to find the one misplaced parenthesis or exponent that is making his or her model behave incorrectly. This student often stews in silence while his or her classmates are asking each other or me for help and then storms out of the classroom in tears or a fit of anger. In recent years I have addressed this problem at the beginning of the semester by telling students that they should expect to be frustrated, angry, and in tears at times, and that they need to take a deep breath and ask for help before they get so frustrated that they are no longer capable of carrying out the assignment.

Another challenge of this course is that students occasionally forget to use their intuition and critical thinking skills, or show that they have never fully developed these skills. When presented with odd model behavior that is caused by a misplaced parenthesis or exponent, students may try to explain away the behavior by invoking variables that aren't included in the model. For example, they may attribute odd behavior in a lake model that incorporates runoff, overflow, and lake surface evaporation to changes in temperature over time, even though temperature appears nowhere in the model. These students seem to have difficulty understanding the old "garbage in, garbage out" mantra and think that because their model runs, it must be behaving correctly. This is the same sort of student who makes conversion errors and seems incapable of spotting those errors even when their results are clearly ridiculous. They might, for example, calculate a discharge of 1 million m³/s for a campus stream that flows at 0.1 m³/s, having incorrectly converted from centimeters to meters. I have not yet found a way to assist these students to my satisfaction and hope to learn strategies from my colleagues at the Complex Systems workshop.

A final challenge of teaching a course such as this is the fact that modeling can be difficult, small errors are hard to find, and there is only one of me to help debug the models of a class of 8-10 students. As a result, students spend a lot of time waiting for my assistance. Unfortunately, we teach this course on an alternating year schedule, primarily to juniors and seniors, so it's impossible to have a student who has already taken the course act as a teaching assistant. Having a TA would be very helpful in a course such as this.