In this opening chapter, we give an informal and qualitative overview – a pep talk – to help you appreciate why sustained nonequilibrium systems are so interesting and worthy of study.

We begin in Section 1.1 by discussing the big picture of how the Universe is filled with nonequilibrium systems of many different kinds, a consequence of the fact that the Universe had a beginning and has not yet stopped evolving. A profound and important question is then to understand how the observed richness of structure in the Universe arises from the property of not being in thermodynamic equilibrium. In Section 1.2, a particularly well studied nonequilibrium system, Rayleigh–Bénard convection, is introduced to establish some vocabulary and insight regarding what is a nonequilibrium system. Next, in Section 1.3, we extend our discussion to representative examples of nonequilibrium patterns in nature and in the laboratory, to illustrate the great diversity of such patterns and to provide some concrete examples to think about. These examples serve to motivate some of the central questions that are discussed throughout the book, e.g. spatially dependent instabilities, wave number selection, pattern formation, and spatiotemporal chaos. The humble desktop-sized experiments discussed in this section, together with theory and simulations relating to them, can also be regarded as the real current battleground for understanding nonequilibrium systems since there is a chance to compare theory with experiment quantitatively.

Perhaps the most magical moment in a pattern-forming system is when a pattern first appears out of nothing, the genesis of structure. The “nothing” that one starts with is not empty space but some spatially uniform system. From our study of equilibrium systems, we are used to expecting that if the external conditions are constant in space and time, then, perhaps after some transient dynamics, the system will relax to a state that is time independent and spatially uniform on macroscopic scales. However, as the system is driven further and further out of equilibrium by turning some experimental knob in small successive steps, it is often the case that a point is eventually reached such that a spatial structure spontaneously appears. This is the beginning of pattern formation. In many examples of interest, the novel state with spatial structure develops because the spatially uniform state becomes unstable toward the growth of small perturbations. Analyzing this linear instability provides a first approach to understand the beginnings of pattern formation.

In this chapter, we discuss linear instability: how to predict when a spatially uniform time-independent state becomes unstable to tiny perturbations. We also discuss some details of the growing spatial structure such as its characteristic length and time scales, and how the structure depends on symmetries of the system.

In this chapter, we discuss model evolution equations that serve as a bridge between the previous chapter on the qualitative properties of nonlinear saturated states and later chapters on the amplitude equation and phase diffusion equation, which provide a way to understand many quantitative details of pattern formation. Model equations are a natural next step after the previous chapter because they demonstrate a nontrivial insight, that many details of experimental pattern formation can be understood without having to work with the quantitatively accurate but often difficult evolution equations that describe pattern formation in say a liquid crystal or some reaction–diffusion system. Analytical and numerical calculations show that if a simplified model contains certain symmetries (say rotational, translational, and inversion), has structure at a preferred length scale (generated say by a type-I instability), and has a nonlinearity that saturates exponentially growing modes, the model is often able to reproduce qualitative features, and in some cases quantitative details, observed in experiments.

The same insight that mathematically simplified models can have a rich pattern formation is useful for later chapters because model equations provide a more efficient way to carry out and test theoretical formalisms than would be the case for quantitatively accurate evolution equations. We have already used the Swift–Hohenberg model in this way, for example to calculate how the growth rate σq of an unstable mode depends on the perturbation wave number q (Eq. (2.11)), or to calculate approximate stationary nonlinear stripe solutions near onset (Eqs. (4.21) and (4.24)). Model equations can also usually be studied numerically more thoroughly than the fully quantitative equations.

We have approached the formation of patterns in nonequilibrium systems through the notion of states that develop via a supercritical linear instability and so saturate at small amplitudes near the threshold of the instability. The resulting patterns retain to some degree the features of the linearly growing mode and this allows many aspects of the pattern formation to be analyzed in a tractable way via the amplitude equation formalism. In nature, however, most nonequilibrium systems are not close to any threshold, and the amplitudes of their corresponding states cannot be considered small. Even near the threshold of a linear instability, structure can emerge via a subcritical bifurcation such that the exponential growth saturates with a large amplitude. What can be said about these strongly nonlinear patterns that are far from the linearly growing mode?

Experiments and simulations indicate that patterns far from threshold can be divided into two classes. One class qualitatively resembles patterns that, at least locally, take the form of stripes or lattices. The other class of patterns far from threshold involves novel states that do not correspond locally to lattice structures.

Far from threshold less can be said about stripe and lattice states with any generality. One question that can be addressed generally is the slow variation of the properties of the stripes or lattices over large distances in a sufficiently big domain. As we explain in Section 9.1.1 these slow dynamics are connected with symmetries of the system.

One of the fundamental ways that a stationary dynamical system can become time dependent as some parameter is varied is via a Hopf bifurcation (see Appendix 1). In the supercritical case, a fixed point becomes unstable at the same time that a stable periodic orbit grows smoothly out of the fixed point. In this chapter, we use amplitude equations and comparisons of calculations with experiments to discuss the universal dynamics that arise near the onset of a Hopf bifurcation in a spatially extended homogeneous nonequilibrium medium. Although many of the concepts and issues are similar to those already discussed in Chapters 6–8 for the type I-s instability (e.g. amplitude equations, stability balloons, defects, phase equations), a new feature of oscillatory media is the appearance of propagating waves. For media with one extended direction, there are typically right- and left-propagating waves that interact in a nonlinear way with each other, and these waves can also interact nonlinearly with waves generated by reflection from a lateral boundary. An intriguing one-dimensional example that we discuss later in this chapter is the blinking state, which can be observed when a binary fluid (e.g. a mixture of water and alcohol) convects in a narrow rectangular domain, see Fig. 10.11. In a two dimensional oscillatory medium, the propagating waves most often take the form of rotating spirals (see Figures 1.9, 1.18(a), 10.3, and 10.4).

Propagating waves and spiral structures also are observed in so-called excitable media.

Introduction

Three kinds of mathematical problems have appeared frequently earlier in the book: the time evolution of a pattern-forming system, the identification of stationary states (e.g. a uniform state or a periodic hexagonal lattice), and the calculation of growth rates σq (eigenvalues) for small-amplitude perturbations of a stationary state. Except for simplified mathematical models that often cannot be compared quantitatively with experiment, and except for rather narrow parameter regimes such as just beyond the onset of a supercritical bifurcation, these three classes of problems cannot be solved analytically. It can then be helpful to use numerical methods on a digital computer.

In this chapter, we discuss some numerical ideas and algorithms to solve the first two of these three kinds of problems. The discussion will be useful in several ways. First, many difficult concepts associated with pattern formation such as spatiotemporal chaos can often first be conveniently studied using a numerical method since the alternatives of experiments or analytics can be more time consuming, expensive, or difficult. Second, the great power of current computers and of modern numerical algorithms increasingly allow the investigation of evolution equations that describe a nonequilibrium system quantitatively and sometimes provide the only way to obtain information about a system. Simulations thus complement theory and experiment as an important third way of exploring and understanding nonequilibrium phenomena. Third, the following discussion should help you to understand the assumptions that underlie some of the numerical methods used to study pattern-forming systems and so give you a sense of when you can trust the simulations.

Chapter 4 discussed the evolution of infinitesimal perturbations of a uniform state into saturated, stationary, spatially periodic solutions. By restricting attention to such solutions, we were able to study the effects of the nonlinearities, using analytical methods near threshold and numerical methods further from threshold. However, most realistic geometries do not permit spatially periodic solutions since these solutions are usually not compatible with the boundary conditions at the lateral walls. Even if periodic solutions are consistent with some finite domain, they do not exhaust all the possible patterns. As we have seen in Section 4.4, typically patterns have the ideal form (stripes, hexagons, etc.) only over small regions and these ideal forms are distorted over long length scales or disrupted in localized regions by defects. In addition, the distortions and defects are often time-dependent.

In this chapter, we introduce the amplitude equation formalism which provides a powerful and broadly useful method to study spatial and temporal distortions of ideal patterns. The formalism represents a substantial conceptual and technical simplification in that, near onset and for slowly varying distortions of periodic patterns, the evolution of the many fields u(x, t) that describe some physical system (e.g. temperature, velocity, and concentration fields) can be described quantitatively in terms of the evolution of a single scalar complex-valued field A(x, t) called the amplitude. The evolution equation for the amplitude is called the amplitude equation and is typically a partial differential equation (pde). Amplitude equations capture three basic ingredients of pattern formation: the growth of a perturbation about the spatially uniform state, the saturation of the growth by nonlinearity, and what we will loosely call dispersion, namely the effect of spatial distortions.

This book is intended as a text for undergraduates in the atmospheric sciences. The students are expected to have some calculus, general chemistry and classical physics background although we provide a number of refreshers for those who might have less experience or need reminders. Our students have also had a survey of the atmospheric sciences in a qualitative course at freshman level. The primary aim of the book is to prepare the student for the synoptic and dynamics courses that follow. We intend that the student gain some understanding of thermodynamics as it applies to the elementary systems of interest in the atmospheric sciences. A major goal is for the students to gain some facility in making straightforward calculations. We have taught the material in a semester course, but in a shorter course some material can be omitted without regrets later in the book. The book ends with two chapters that are independent of one another: Chapter 8 on thermochemistry and Chapter 9 on the thermodynamic equation.

This book is the result of teaching an introductory atmospheric thermodynamics course to sophomores and juniors at Texas A&M University. Several colleagues have taught the course using earlier versions of the notes and we gratefully acknowledge Professors R. L. Panetta, Ping Yang, and Don Collins as well as the students for their many helpful comments. In addition, we have received useful comments on the chemistry chapter from Professors Sarah Brooks, Gunnar Schade, and Renyi Zhang. We also thank Professor Kenneth Bowman for many fruitful discussions. We are grateful for financial support provided by the Harold J. Haynes Endowed Chair in Geosciences.

Often in meteorology we deal with a fixed mass of a gas whose volume and other characteristics may change as the air mass moves about. The particular mass of gas may be thought of as a small parcel of matter that is transported through the environment by natural forces acting upon it. We could also imagine moving it virtually via an abstract thought experiment, for example to determine its stability under small perturbations. As an air parcel rises for whatever reason in the real atmosphere, it will almost instantaneously adapt its internal pressure to the external pressure exerted by the local surroundings, but the temperature and composition adjust more slowly. In convection, entrainment of neighboring air also speeds up the process of equalizing the temperature between inside and outside air. Still there is a huge separation of equalization times between pressure and temperature and/or trace gas concentrations. This time scale separation has made the parcel concept a useful and even powerful tool in the atmospheric sciences. We will return to it often.

Thermodynamics is concerned with the state of a system (an example of which is the parcel alluded to above now treated as an approximate thermodynamic system) and the changes that occur in its state when certain processes occur (such as its being lifted). In the case of a parcel composed of an ideal gas, the state is completely described by the state variables p, V, M and T (actually in equilibrium only three variables need to be specified, since the equation of state in the form of the Ideal Gas Law can be used to calculate the fourth from knowledge of the other three).

In nature water presents itself in solid, liquid and gaseous phases. Energy transfers during transformations among these phases have important consequences in weather and climate. The system of redistribution of water on the planet constitutes the hydrological cycle which is central to weather and climate research and operations. Water is also an important solvent in the oceans, soils and in cloud droplets. The presence of tiny particles in the air can influence the formation of cloud drops and thereby change the Earth's radiation balance between absorbed and emitted and/or reflected radiation. These and other effects lead us into the fascinating role of water in the environment. Of course, thermodynamics is an indispensable tool in unraveling this very challenging puzzle.

Vapor pressure

We start with a discussion of the equilibrium gas pressure in a chamber in diathermal contact with a reservoir at a fixed temperature, T0. The chamber is to have a volume that is adjustable, as shown in Figure 5.1. In the following let the chamber have no air present – only the gas from evaporation of the liquid. We are to choose a volume V such that there is some liquid present at the bottom of the chamber (we say here the bottom of the chamber, but otherwise we ignore gravity). There are gas molecules constantly striking the liquid from above and sticking. […]

Ideal gas basics

Gases are a form of matter in which the individual molecules are free to move independently of one another except for occasional collisions. Most of the time the individual molecules are in free flight out of the range of influence of their neighbors. Gases differ from liquids and solids in that the force between neighbors (on the average over time) is very weak, since the intermolecular force is of short range compared to the typical intermolecular distances for the individual gas molecules.

If an imaginary plate is held vertically in a gas as shown in Figure 2.1, there will be a force exerted on the thin plate from each side. The forces on opposite sides of the inserted plane are equal; otherwise, if forces on the opposing sides were unbalanced, the plate would experience an acceleration. The force on the left side of the plate is caused by the reflection of molecules as they hit the left face of the plate and rebound. These impulsive forces are so frequent that the resulting macroscale force is effectively steady. The force is perpendicular to the face and has the same value no matter how the face is oriented. This can be seen by considering the collisions with the wall and the tendency for no momentum to be transferred parallel to the plane surface. The perpendicular component of the force per unit area on the plane is called the pressure. Tangential components of the force cancel out (when averaged over many collisions with the wall) and therefore vanish when averages are taken over a large number of collisions with the surface.

Chapter 1

1. 1.1 26 428 ft for H = 8 km.

2. 1.2 0.0013.

3. 1.3 819 hPa (using H = 8 km, 1 mile = 1.6 km).

4. 1.4 (a) 40200 km, (b) 111.7 km deg−1, (c) 96.7 km deg−1.

5. 1.5 50000 km2. 43000 km2.

6. 1.8 11.2 km s−1.

7. 1.9 2.

Chapter 2

1. 2.1 1.29, 1.25, 1.2 kg m−3.

2. 2.2 0.80 kg m−3.

3. 2.3 212 hPa.

4. 2.4 12.7 N.

5. 2.5 2.69×1025 molecules m−3.

6. 2.6 Rd = 2.87 hPa K−1 m3 kg−1.

7. 2.7 Differentiate P(v) with respect to v and set it to zero. See Table 2.2.

8. 2.8 462, 493, 413, 1846 ms−1.

9. 2.9 630 Pa. 1 kg m−1. Pressure with moisture = 78646 Pa. Density of moist mix = 1 kg m−3. Density of dry air at same temperature and pressure: 1.003 kg m−3.

10. 2.10 1.6×108 J.

11. 2.11 4.3×107 J.

12. 2.12 n0 = 2.69 ×1025 molecules m−3, H = 8000 m, N =2.15×1029 molecules.

13. 2.13 zλ=H = H ln(n0σcH).

14. 2.14 (gp0/RT0) ∫∞0ze−z/H dz = gp0H2/RT0.

15. 2.15 ½ Nm0v2 =32; NkBT, see Problem 2.12 for N.

16. 2.16 fyellow = c/λ = 6.0 × 1014 s−1. fcoll = n0σc≈ 6.5 × 109 s−1. An excited atom might suffer tens of collisions before it relaxes to the lower energy level.