I will first give a few example applications of "IPCC class" climate models - coupled atmosphere/ocean/ice/land models with sufficient realism to be used in the Intergovernmental Panel on Climate Change (IPCC) 4'th assessment report. One example is tangentially related to the celestial mechanics theme of the previous lectures in this seminar: estimating the threat from climate change using the same methodology as used to to compute the threat of a cataclysmic Earth/asteroid impact (Harris, Nature 2008, Boslough & Harris, AGU 2008).

A recent paper "Target Atmospheric CO₂: Where Should Humanity Aim?" by James Hansen et al will be discussed.
Reference: James Hansen, et al, Target Atmospheric CO₂: Where Should Humanity Aim? The Open Atmospheric Science Journal 2 (2008) 217-231. doi:10.2174/1874282300802010217. Cached copy.

Reference: W. Buermann et al, The changing carbon cycle at Mauna Loa Observatory, PNAS 104 (2007), 4249–4254. doi:10.1073/pnas.0611224104. Cached copy.

State estimation techniques are used in weather and climate prediction, hydrogeology, seismology, as a way to blend model output and real data in order to improve on predictions from the exclusive use of the model or the data alone. Techniques that are based upon least-squares ideas, such as the family of Kalman Filter/Smoothers, or Variational Data Assimilation, are optimal in linear/Gaussian problems. However, they often fail in problems in which nonlinearities are important and/or when Gaussianity in the statistics cannot be assumed. Even linearization may fail, and so do ensemble techniques that make nonlinear predictions but rely on linear analyses. These comprise the practical state of the art, at least in weather forecasting and in hydrogeology. I will describe these as well as how failures arise in these methods.

We have created a number of nonlinear/non-Gaussian data assimilation techniques. Our present efforts are to make them computationally practical as well as to use of these to do problems that are otherwise intractable using conventional means.

One such application is in Lagrangian data assimilation: here we tackle the problem of blending data that has been sampled along paths, which when blended in traditional ways on Eulerian grids will lead to loss of critical features even though the estimates may be variance-minimizing.

The ocean is about 2 billion years old. Why has the ocean's temperature been relatively constant (aside from some 10 degree deviations), if during that time it has been subjected to about 1.7X10¹⁷ W from the sun? Why do we worry about CO₂ and greenhouse effects in the atmosphere, if 86% of the Earth's carbon is in the oceans? Why do we worry about Greenland ice melt?

This is a decidedly non-mathematical and introductory talk on Earth's heat engine. In addition to answering the questions just posed, we will describe why the quest for an understanding of climate is tied to understanding ocean dynamics.

The 1-dimensional energy balance model was first introduced in late sixties independently by Mihail Budyko and William Sellers as a differential equation that governs the evolution of the one hemispheric temperature distribution, by taking into account the ice albedo feedback.

In this paper we introduce an equation that induces ice line dynamics, then we analyze the dynamics of the Budyko's equation coupled with this ice line equation. We found that the coupled temperature-ice line system has a one dimensional center stable manifold. Furthermore, the model suggests an optimistic view that the ice free earth is unstable.