This book has been written specifically for the AIMS Library Series, so its intended audience is students who are attending, have attended, or have backgrounds that would make them eligible to attend the post-graduate programs offered at the African Institute for Mathematical Sciences. The contents of this book could easily be delivered as one of the AIMS postgraduate courses, though it is primarily intended as a self study introductory guide to mathematical modelling in the atmospheric sciences. It has been prepared so that readers with a fairly thorough applied mathematics or physics background can easily, and with little additional reading, understand the main approaches, theoretical and observational underpinnings, intellectual history and challenges of the subject. It is neither a broad introduction to atmospheric science (there exist many such books which serve a very different audience than that intended here), nor is it a review of current research (since that will not serve my intended audience). This book has four distinct, but linked objectives:

1. • introduce the beauty and wonder of atmospheric phenomena by examining a representative selection;

2. • explain the importance of scale analysis and scaling arguments in studies of atmospheric phenomena;

3. • emphasize the power of mathematics in developing an understanding of these phenomena;

4. • demonstrate how a combination of mathematical modelling, numerical modelling and observations are needed to achieve the understanding.

I start with two rather lengthy introductory chapters designed to introduce the governing equations, their analytical difficulties, and how scale analysis is conducted. The substantive content of this book is organized according to the conventional scale analysis of atmospheric phenomena, and within each scale-specific section I will cover in some detail theoretical (analytical) modelling approaches. Wherever possible and appropriate, I will refer to numerical modelling and observations of the phenomena being discussed. This will be done in order to emphasize the richness of method that characterizes atmospheric science as an academic and professional discipline, but will not constitute a full discussion of atmospheric numerical modelling, or observational meteorology.