Mathematics in climate research is often thought to be mainly a provider of techniques for solving the continuum mechanical equations for the flows of the atmosphere and oceans, for the motion and evolution of Earth’s ice masses, and the like. Three examples will elucidate that there is a much wider range of opportunities.

Climate modellers often employ reduced forms of the continuum mechanical equations to efficiently address their research questions of interest. The first example discusses how mathematical analysis can provide systematic guidelines for the regime of applicability of such reduced model equations.

Meteorologists define climate, in a narrow sense, as the statistical description in terms of the mean and variability of relevant quantities over a period of time (World Meteorological Society; see the website for a broader sense definition). Now, climate researchers are most interested in changes of the climate over time, and yet there is no unique, well-defined notion of time dependent statistics. In fact, there are restrictive conditions which data from time series need to satisfy for classical statistical methods to be applicable. The second example describes recent developments of analysis techniques for time series with non-trivial temporal trends.

Modern climate research has joined forces with economy and the social sciences to generate a scientific basis for informed political decisions in the face of global climate change. One major type of problems hampering progress of the related interdisciplinary research consists of often subtle language barriers. The third example describes how mathematical formalization of the notion of vulnerability has helped structuring related interdisciplinary research efforts.

Flows of the Earth’s atmosphere cover a very wide range of scales, from micrometer-sized raindrops to planetary-scale climate phenomena. The range of relevant time scales is equally broad. A central task of theoretical meteorology is to identify specific weather- or climate-related flow phenomena that are associated with particular length and time scales, and to construct simplified models describing these phenomena in terms of some reduced set of effective degrees of freedom.

Textbooks on theoretical meteorology offer derivations of such reduced models from more comprehensive fluid dynamical governing equations through scale analysis, which involves often ingenious combinations of physical intuition and skillful mathematical derivations. Nevertheless, keeping track of how these models relate to each other and what were the underlying assumptions in their derivations is a formidable task. If one is interested, however, in how phenomena associated with different length and/or time scales interact, then knowing how to consistently couple or synchronize the individual reduced models becomes crucial.

In the first part of the lecture I will introduce a systematization of reduced model equations of theoretical meteorology that is based on the principles of dimensional and asymptotic analysis. This approach allows one to re-derive a large number of reduced models of theoretical meteorology in a unified fashion from the full compressible flow equations via classical single-scale asymptotics.More importantly, this unified approach lends itself naturally to multiple scales analyses, i.e., to studies of how scale-dependent phenomena described by different reduced model equations are coupled across the scales. The second part of this lecture will cover examples of such multiscale interaction theories selecting from an asymptotic model for tropical storm intensification, the derivation and justification of sound-proof atmospheric flow models involving a non-standard asymptotic three