Global Warming: A Zonal Energy Balance Model

Victor Padron
Normandale Community College,
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This is a teaching module, directed to undergraduate students in applied mathematics, that presents a Zonal Energy Balance Model to describe the evolution of the latitudinal distribution of Earth's surface temperature subject to incremental levels of cumulative carbon emissions in the atmosphere. A strategy to avert "dangerous levels" of global warming is imbedded in the model. Students working with the module will write a computer code, using a software such as MATLAB or Mathematica, to obtain numerical solutions of the model and simulate strategies that guarantee controlled levels of global warming.

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Learning Goals

The primary goals of this activity are twofold.

1. To increase student's motivation to learn mathematics by making it relevant to their lives; connecting mathematics learning to the goals and interests that students bring to college; and showing how mathematics relates to other disciplines, important civic questions, and technological challenges.

2. To develop basic skills in programming and scientific computing by writing their own computer code with graphical representation of the solutions.

Context for Use

This module is directed to undergraduate students in applied mathematics, as in a course of mathematical modeling or calculus with differential equations. I have used versions of this material as an end of course group project, but it could also be used as individual project.

Description and Teaching Materials

The Global Warming Handout begins with a self-contained derivation of a spatially discrete version of Budyko-Sellers zonal energy balance model that describes the evolution of the latitudinal distribution of Earth's surface temperature.

The concept of equilibrium climate sensitivity is incorporated in the model, using Arrhenius' logarithmic greenhouse law, to account for the warming impact of the incremental levels of cumulative carbon emissions in the atmosphere.

The presentation continues with a discussion of anthropogenic interference with the climate system and the estimation of a dangerous threshold value beyond which the increase of global temperature becomes catastrophic. A strategic reduction of the annual growth rate of carbon emissions that averts this scenario is incorporated into the model.

The last section includes the description of two assignments. The first one is to develop a computer code to obtain a numerical solution to the model. In the second assignment students apply the model with their own selection of values for the equilibrium climate sensitivity and the dangerous threshold to develop a strategy of reduction of the annual rate of carbon emissions, and verify that future global mean surface temperature remains below the chosen dangerous levels of global warming.

A few exercises are included along the presentation.

The Global Warming Evolution (private instructor-only file) is a MATLAB script that uses the ODE solver ode45 to obtain numerical solutions of the system of differential equations of the model, with corresponding graphical outputs, under the assumption of constant annual growth rate of carbon emissions. It calls the MATLAB function Global Warming Vector Field (private instructor-only file) to compute the vector field of the ODE system. A Mathematica version for these computation is included in the Global Warming Mathematica Notebook (private instructor-only file.)

Teaching Notes and Tips

Although the module includes a self-contained description of the basics of global warming and derivation of the model, I usually dedicate a few lectures during the semester to discuss the material. The references provided below are a good supplement to the material discussed in the module.

Choosing their own values for the equilibrium climate sensitivity and dangerous threshold provides a good opportunity for interesting students' discussions within their groups. The handout contains some references to support these discussions, but much more is available on the web.

I use the exercises as individual submissions to further engage students with the subject, and to have record of the work of each student apart from their group-work.

Basic skills in programming are expected to complete the project. These skills could be developed during the semester by integrating a software such as MATLAB or Mathematica as the main simulation tool within the course. Depending on the level of expertise required, instructors may or may not choose to provide the included private instructors-only MATLAB and Mathematica files to their students.


I use SENCER SALG pre- and post-assessment to assess the changes in student's perception of mathematics, its interaction with other disciplines, and its role in addressing relevant social issues. The SENCER SALG,, is an online free course-evaluation tool that allows college-level instructors to gather learning-focused feedback from students. Instructors are guided through a wizard, using an adaptable template, to design their own instrument.

References and Resources

  1. H. Kaper & H. Engler, Mathematics & Climate, SIAM (2013).
  2. K. K. Tung, Topics in Mathematical Modeling, Princeton University Press, Princeton, New Jersey, USA, 2007