## MAA Short Course on Conceptual Climate Models

### ***THIS COURSE IS SOLD OUT***Contact the MMSB if you would like to be placed on a waiting list.

### Organizers

**Anna Barry**, Institute for Mathematics and Its Applications

**Hans Kaper**, Georgetown University

**Richard McGehee**, University of Minnesota

**Samantha Oestreicher**, University of Minnsota

**James Walsh**, Oberlin College

**Esther Widiasih**, University of Arizona

**Mary Lou Zeeman**, Bowdoin College

### Registration

This two-day course, cosponsored by the Mathematics and Climate Research Network, will take place on Monday and Tuesday, January 7 and 8, in Room 5B, Upper Level, San Diego Convention Center, before the annual meeting begins.

Advance registration fees are: member of the AMS or MAA, US$156; nonmembers are US$225; students, unemployed, or emeritus are US$78. These fees are in effect until **December 17**. If you choose to register at the meeting, the fees are US$166 for members of the AMS or MAA, US$235 for nonmembers, and US$88 for students, unemployed, or emeritus.

Advance registration will begin on **September 1, 2012**. Onsite registration will take place on Monday, **January 7, 2013**, 8:00 a.m. - noon, outside rooms 4, Upper Level, San Diego Convention Center.

**Enrollment for this course is limited to 40.**

### Overview

In this two-day short course, the presenters will introduce various conceptual models of the Earth's climate system. The first day will be devoted to Energy Balance Models (EBMs)—differential equations which express the physical law of energy conservation in mathematical terms. It will be shown how the models can be modified to include the effects of greenhouse gases and the ice-albedo feedback mechanism. The second day will be devoted to paleoclimate studies. It will be shown how observational data from the paleoclimate record and computational data from simulations of the Earth's orbit during the Pliocene and Pleistocene can be incorporated into EBMs.

During the two-day course, participants will have the opportunity to conduct hands-on simulations with models to explore the interplay between energy balance, ice-albedo feedback, Milankovitch cycles in Earth's orbit, and other feedback mechanisms. This will build insight into which features of the paleoclimate record can be explained by the dynamics of low-dimensional conceptual models. Modules for bringing the material into a range of core undergraduate mathematics classes will be provided.

### Background

Earth's climate is a fascinating complex system of multiple components, including the atmosphere, oceans, biosphere, cryosphere, and more. The components interact and influence each other while the system evolves on many different spatial and temporal scales. The traditional approach of studying a physical or biological system by combining carefully designed lab experiments with mathematical models is not available, for obvious reasons: We have only one Earth, we are living in it, and our life span is too short compared to the climate time scale. Instead, our only data come from current observations and from Earth's climate history. Mathematical and computational models are the primary experimental tools in our quest to understand Earth's climate, and mathematicians have a fundamental role to play. We need to develop a hierarchy of models in which models at each level inform models at other levels, and we need to help train our students and future researchers to develop and analyze those models.

The simplest climate models are conceptual low-order dynamical models which are based on our understanding of the physics, chemistry, geology, biology and other sciences pertaining to climate. These models often allow for mathematical analysis using established techniques from dynamical systems theory and other fields. Conceptual models can thus be used to provide intuition for the simulated behavior of vastly more complicated IPCC-level computational models. In the words of the eminent climate scientist Isaac Held: "It is useful to complement the results of global climate models with understanding gained from low-order models." The goal of the proposed short course is to build familiarity with the first level of low-order models.

The material in the proposed Short Course is accessible to a broad range of faculty and scholars, whether they are interested in teaching at the high-school, undergraduate or graduate level, or whether they are interested in understanding the basic conceptual models from which research at a variety of levels of model complexity can grow. We start the the first session with an introduction to Energy Balance Models (EBMs) and the associated science of solar radiation, black-body radiation and greenhouse effects, and a discussion of "Snowball Earth." This introduction will be followed by a hands-on session of numerical simulations of climate of other planets--for example, Mars, Venus and some extrasolar planets. In the afternoon, we will introduce so-called one-dimensional EBMs which allow for variations of the temperature with latitude. The additional dimension leads to partial differential equations which, in turn, lead to infinite-dimensional dynamical systems. Approximation by Legendre polynomials then leads to a low dimensional system of ODEs. One-dimensional EBMs can be used to study the ice-albedo feedback mechanism, which seems to have played an important role in the occurrence of ice ages. The subsequent hands-on lab will explore this mechanism through numerical simulations. On the following day, we start with a discussion of paleoclimate observational data obtained from ice cores and ocean sediment, and Milankovitch cycles in Earth's orbit. We extend the energy balance models to include a dynamic ice line and Milankovitch cycle forcing. In the subsequent hands-on lab, we compare the model simulations with the paleoclimate record. In the afternoon, we will discuss ways to incorporate carbon dioxide as a greenhouse gas into an energy balance model. The discussion will be extended in the hands-on lab session with more explorations and its applications to paleoclimate (glaciation cycles, snowball earth). Each topic in this course lends itself to classroom use. We will provide participants with modules designed for core math classes at a variety of levels. We hope this short course opens the door for more mathematicians and educators to become involved in the exciting area of climate mathematics.

### Structure of the Course

There will be eight sessions over the two-day period, four lectures and a hands-on lab session following each lecture. Each session will be approximately 90 minutes. The 90-minute lecture slots will leave ample time for questions and discussion. The 90-minute hands-on slots will allow for a more in-depth exploration to the materials provided; the session leaders will circulate among the participants as they work through the materials in small groups. Participants will be advised (in advertising materials and when registering for the course) to bring laptops equipped with MATLAB. In addition there will be optional activities that use the software XPPAUTO to explore various bifurcations.

#### Day 1. Energy Balance Models: A Primer

**Session 1**: *Zero-dimensional Energy Balance Models*, **Hans Kaper**, Georgetown University. In this lecture we will introduce the concept of a zero-dimensional energy balance model of climate. Topics will include model derivation, solar energy spectrum, black-body radiation, composition of the atmosphere, and inclusion of greenhouse gases. We will discuss the concept of a “Snowball Earth” and give a brief introduction to bifurcation phenomena.

**Session 2**: *Hands-On: Dynamics of Energy Balance Models*, **Anna Barry**, Institute for Mathematics and its Applications, and **Samantha Oestreicher**, University of Minnesota. Participants will use hands-on numerical simulation, to explore the dynamics of the energy balance models introduced in the first session, applied to Earth, and possibly Mars, Venus and some extra solar planet.

**Session 3**: *One-dimensional Energy Balance Models*, **Hans Kaper**, Georgetown University. In this lecture we will introduce the concept of a one-dimensional energy balance model of climate. These are partial differential equations. Using the method of eigenfunction expansions we will show how these models lead to infinite-dimensional dynamical systems, which then need to be truncated to yield models that can be analyzed.

**Session 4**: *Hands-On: Dynamics of Energy Balance Models*, **Anna Barry**, Institute for Mathematics and its Applications, and **Samantha Oestreicher**, University of Minnesota. We continue the discussion of one-dimensional energy balance model and explore how they are applied to Earth, Mars, Venus, and extra-solar planets by adjusting some parameters.

#### Day 2. Applications of Energy Balance Models

**Session 5**: *PaleoClimate Data, Milankovitch Cycles, and Extending Energy Balance Models*, **Richard McGehee**, University of Minnesota. In this lecture, we will discuss paleoclimate data from ice cores and ocean sediments, and Milankovitch cycles in Earth’s orbit. To then explore how well glaciations and other features of the paleoclimate record may be explained by an energy balance model, we extend the 1-dimensional model to include a dynamic iceline and Milankovitch cycle forcing.

**Session 6**: *Hands-On: Comparing Energy Balance Models with the PaleoClimate Record*, **Richard McGehee**, University of Minnesota, and **Esther Widiasih**, University of Arizona. In this hands-on session, we will compare simulations of the extended energy balance models with the paleoclimate record and discuss the importance of the green house gas effect in amplifiying the effect of orbital variations in Earth climate.

**Session 7**: *The Greenhouse Effect in Energy Balance Models*, **Jim Walsh**, Oberlin College. In this lecture, we introduce ways to incorporate the effect of carbon dioxide into the energy balance model to illustrate the snowball earth and possibly the glacial/interglacial cycles. The discussion will lead to interesting dynamical systems.

**Session 8**: *Hands-On: Green House Gas Effect Explorations*, **Anna Barry**, Institute for Mathematics and its Applications, and **Esther Widiasih**, University of Minnesota. Participants will use hands-on simulation to explore the incorporation of the greenhouse gas effect in EBM's. Participants will also have the option to use the software XPP-AUTO to explore bifurcations of various parameters.