Mathematics and Climate

Goal

The purpose of this Graduate Student Summer Program is to introduce graduate students to interactions between mathematics and climate science and build toward current research problems. The program will focus on both models and data and culminates in using data to parameterize models using data assimilation techniques. The program will last three weeks with both lectures and project and will have an interdisciplinary focus.

Description

The use of mathematical models to understand climate phenomena is an important and growing area, most recently fueled by the looming climate disruption caused by the burning of fossil fuels and the consequent release of greenhouse gasses. Earth’s climate is a large, complex system evolving with time and subject to external forces, such as variation in the Sun’s intensity and cycles in the elements of the Earth’s orbit. Accurately modeling and predicting the climate can only be attempted through the use of large computer models. However, some of the basic processes can be understood from fundamental laws of physics and with the use of conceptual models taking into account only macro-­scale variables.

The full climate system is, of necessity, understood through the use of coupled models that attempt to capture the detail necessary to understand the dynamics of the climate sub-systems. No model can, however, capture everything relevant as there will always be processes occurring below the grid scale of the model’s resolution and physical effects whose processes are not fully modeled. The standard strategy for dealing with this situation is to parameterize these processes by tuning the various parameters impacted by them in the larger model in order to account for their effect. A typical parameterization will work offline by building a more refined model for the missing process and using the relevant observational data. The process by which these data are used is known broadly as data assimilation. With data assimilation, the information from computational models and observations are combined to produce an optimal estimate of the system state and/or prediction. In its sequential, or filtering, form the evolutionary nature of the system is at the fore and it can be cast as a problem in dynamical systems. The issue is two‐fold: how should the system state be re-initialized to account for observational data and how does the system dynamics evolve the uncertainty of both the model and the data-based re-adjustments. In a full Bayesian framework, this can be achieved, in theory, by invoking the dynamical system’s Fokker-Planck equation. But, in practice, this is not computationally feasible and various approximations are made. Many of these, including the celebrated Ensemble Kalman Filter, are motivated by dynamical considerations, such as Lyapunov exponents.

Basic Prerequistes

This summer graduate program is geared toward graduate students who have completed a year of graduate school (some background in analysis, differential equations, numerical analysis, and/or statistics is desirable). Preference will be given to students at the early stages of graduate study.

Program Outline

Research topics include modeling paradigms of climate systems, methods for analysis of ODEs, PDEs and dynamical systems, numerical methods for ODEs and PDEs, high performance computing, optimization techniques, spatial statistics, uncertainty quantification, and data assimilation.

Week 1: Basic models (both conceptual and physical) and how to analyze these models using techniques from PDEs, ODEs, and dynamical systems

Week 2: Data and how data are acquired and techniques for data analysis

Week 3: How to use models and data together and how data assimilation can be used to parameterize models.