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.

This introduction to atmospheric modelling covered a wide range of topics in a sadly short space. In spite of the brevity, I hope to have conveyed some of the most important ideas and approaches employed in mathematical analysis of atmospheric dynamics. In my opinion, the salient ideas are scale based analysis built on an underlying set of equations whose intractability and breadth conspire to prevent simple analysis. That progress can be made using relatively simple mathematics is remarkable. That progress is made is a tribute to the rigour and persistence of a cohort of extraordinary scientists, only some of whom have been named in this book. Ultimately, the subject has an enormously practical expression, that is the societal need for meteorological forecasts. The numerical methods that lie behind modern weather forecasts have as essential foundations the equations I have presented, and the methods I have sketched.

The compact nature of this book, and the specific focus on mathematical modelling have had a less than desirable consequence. The focus on dynamics means that I have not been able to deal with many fascinating and awe inspiring phenomena encountered in the atmosphere. I have not touched on any atmospheric optical phenomena, so I have deprived readers of the wonders and beauty of rainbows, arcs and haloes. I have dealt with neither clouds in all their variety, nor precipitation in its many forms. This means readers will have to look elsewhere to discover the mystery and beauty of a cumulonimbus or a funnel cloud, and will have to broaden their reading to understand the distinction between graupel and hail. I could not ignore atmospheric thermodynamics, but had to give it only enough attention to help in the understanding of atmospheric dynamics. Similarly, radiation in the atmosphere was treated only to the extent that it served an understanding of dynamics. I hope readers will be inspired by what I have presented to undertake their own reading of these omitted topics, and many others.

Equations 1.9a, 1.9b and 1.9c (more commonly presented without rotational effects) are known as the Navier–Stokes equations, and are a notoriously difficult mathematical problem. Demonstrating that they have a smooth, physically reasonable solution is one of the Clay Mathematics Institute's Millennium Problems. There are two major difficulties with these equations. In a strange sense, the equations are too complete! They contain as solutions, and cannot discriminate between, the multiplicity of diverse processes that make up the wide spectrum of atmospheric phenomena. Furthermore, there are strong non-linearities embodied in the advective parts (terms of the form u∂u/∂x) of the material derivative. This non-linearity results in broad-spectrum or multi-scale solutions, and underlies the existence of chaotic behaviour. Our task is to find a way of simplifying the equations through identification of approximations that may render the equations at least partially tractable. As will be seen, the approximate forms will be applicable only to a limited range of scales, which conveniently helps limit the set of phenomena that must be dealt with. In effect the approximations act as a band-pass filter, thereby narrowing our field of view. As will be demonstrated, some of the approximations can be identified with the three scales illustrated in Figure 1.1. While there exist no usefully applicable solutions to the full equations, it is nonetheless possible to make enormous strides in understanding atmospheric phenomena by a careful consideration of scales of phenomena involved, and an analysis of approximate forms of the full equations. In a very real sense, this is an admission that while we do have a complete, unified theory of atmospheric motion, it is not a very helpful theory because of the intractability of the full equations. Fortunately, the equations are enormously helpful, if we consider phenomena according to their scale.

Order of magnitude analysis

The first approach is to investigate the possibility that some of the terms (hopefully the troublesome ones) in Equations 1.9a, 1.9b and 1.9c can be neglected because of their size relative to the remaining terms. This procedure is conducted by an order of magnitude analysis.

It is common to observe patterns in thin, semi-continuous layers of cirrus clouds that are highly suggestive of wave-like motion in the middle and upper troposphere. Astute observers may even have seen clouds that appear to mark a sequence of breaking waves near a large mountain or mountain range. These are indeed indications that waves can exist in the atmosphere, but are a misleading indication because these waves are only evident because they are made visible by clouds whose presence may not be directly related to the waves. There is the possibility that waves could exist in clear air, and also that waves could exist on such a large scale that they are not evident to observers looking up from Earth's surface. An example of a wave-filled atmosphere is shown in Figure 5.1. Studies of atmospheric waves have uncovered a wide range of wave types, all having different dynamical bases, different characteristics, and existing under different atmospheric conditions. All waves have a simple dynamical basis in common – they are driven by restoring forces that act in opposition to a displacement from an equilibrium position. The elasticity of air gives rise to sound waves. If the restoring force is gravity, the atmosphere will support gravity waves. If the restoring force is both gravity and the Coriolis force, the atmosphere will support inertia-gravity waves, while the Coriolis force alone gives rise to inertial waves. If the variation of Coriolis force with latitude provides the restoring force, Rossby waves will result. These waves are all particular solutions of the governing equations. We will explore only two simple wave types in order to illustrate the approaches needed for their analysis.

Sound waves are unlike other atmospheric waves in that they are longitudinal waves, in which the oscillation is in the direction of propagation. By contrast, transverse waves have an oscillation perpendicular to their direction of propagation.

Earth's atmosphere is a shallow fluid held by gravity to the surface of a spinning sphere whose surface is heated by electromagnetic radiation from the sun. Roughly two thirds of the sphere is covered by water, which continuously undergoes evaporation, condensation, freezing, thawing and sublimation. There is continuous turbulent transport of water vapour and heat between atmosphere and surface. At global scale, the atmosphere is in continuous motion, driven by a relative excess of heating in equatorial regions relative to higher latitudes. The net effect of this motion is a latitudinal redistribution of heat, either directly or by a net transport of moist air from the tropics to higher latitudes where it condenses and falls as precipitation.

Atmospheric large scale motion results in a cascade of energy to smaller scales, producing a complex palimpsest of motion of various types, and at a wide range of scales from global (tens of thousands of kilometres) to a microscale on the order of millimetres. In spatial terms, these motions include some that are quasi two-dimensional, some that are fully three-dimensional, some that are strongly wave-like, and some that are appropriately described as chaotic. Temporally, the motions have time scales of variability that range from astronomically forced variations over tens of thousands of years to turbulent fluctuations of a few seconds in duration, and more recently, decade scale temporal trends driven by human industrial activities. In addition to the purely dynamical phenomena I have just described, the atmosphere includes phenomena whose dynamics are powerfully influenced by thermodynamic processes (such as cloud and precipitation processes) and a wide range of fascinating atmospheric optical phenomena (such as rainbows and circumsolar haloes). This book will primarily concentrate on atmospheric dynamical phenomena, whose time and space scales are graphically shown in Figure 1.1. These phenomena are conventionally grouped into micro, meso and macro scales, and most analyses of atmospheric phenomena focus only on one of the ‘scales’. This narrowing of focus has become so sharp that most atmospheric scientists will label themselves according to the ‘scale’ they study.

This is a book for students who have some familiarity with general physics and the calculus to learn what thermodynamics and statistical mechanics is all about, starting with the fact that you don't have to know what a system is doing, just how many things it could be doing, to get precise values of its thermodynamic variables, particularly its temperature and pressure. That insight is discussed and explained in the first chapter, along with some simple observations about the peculiar behavior of very large numbers. It uses the perfect gas to illustrate its points.

Chapter 2 looks into the behavior of thermodynamic variables and gives a lesson in partial derivatives, the essential language of thermodynamics. It also considers the various ways of expanding an ideal gas. Chapter 3 deals with the states of matter and the phase transitions between them, and also covers temperature scales, the ideal gas equation of state and the behavior of the atmosphere.

Chapter 4 considers the origin and meaning of the first and second laws of thermodynamics and goes on to discuss Carnot engines and refrigerators and the notion of reversibility. Chapter 5 addresses the probability of finding a system in a particular state and goes on to deal with the partition function and the density of states.

From atoms to thermodynamics

Imagine a box, a cube 10 cm on each edge, with 1022 atoms of helium gas in it. The atoms share among them some total energy U; say, 2 × 106 ergs, which cannot change because the box is isolated from the rest of the world. Inside the box the atoms fly around, banging into each other or the walls, exchanging energy and momentum. If there is only one atom in the box, and we know how it started out, we might imagine being able to calculate its precise trajectory for a while, predicting just where it would end up at some later time. If there are twenty atoms, the same job becomes horribly more complicated. With 1022 atoms it is obviously hopeless. Moreover, according to the laws of quantum mechanics, it would not be possible even in principle. If we knew precisely where the atoms were at some time, we could have no idea of how fast they were moving, according to the uncertainty principle. Obviously, a very short time after we start things off, there is not much we can say about what's going on inside the box.

Nevertheless, it is possible to make some very precise statements about the properties of the gas in the box, especially if we allow some time to pass after we start it off. For example, the gas will have some pressure, P, and some temperature, T, and, given the information we already have, these can be predicted with extreme accuracy and confidence. Temperature and pressure are macroscopic or thermodynamic quantities. The problem before us in this section is to describe the connection between these (predictable) thermodynamic quantities and the (unpredictable) microscopic quantities that somehow give rise to them.