In Chapter 7 we allowed patterns to deviate slightly from a regular lattice by permitting modulations on long scales. However, in a large-aspect-ratio system that can accommodate a large number of pattern wavelengths in all directions, the size and orientation of the pattern will typically change slowly in space and time. In spiral defect chaos, for example, you tend to see patches of rolls that look quite regular locally, but in fact are curved with a large radius of curvature (Figure 10.17). Fingerprints, though stationary, also look like stripes that vary slowly in orientation over a large domain. The ridges in fingerprints are believed to form through the buckling of the lower layer of the skin; recently Kuecken (2004) has derived roll-hexagon amplitude equations from a buckling model of fingerprint formation, suggesting that the analysis of fingerprints as a pattern-forming system may be valid.

Obviously we can't describe patterns in a large-aspect-ratio system by assuming that they lie almost on a lattice, since they clearly don't. However, far from onset in the fully nonlinear regime, we can use the slowness with which the patterns evolve in time and space to develop an asymptotic description of them. The full nonlinearity is a requirement of the theory, so we will lose the small parameter measuring the distance from onset that we used previously to derive amplitude equations, but the slow rates of change will give us a new small parameter to work with.

The theory presented in this chapter was originally developed by Cross and Newell (1984) and later expanded by Passot and Newell (1994). Here we follow their treatment of the problem quite closely.

In Chapter 2 we saw that a system with discrete eigenmodes can be reduced to its evolution on a centre manifold. By requiring patterns to be periodic with respect to a lattice in Chapter 5 we could distinguish between critical and decaying modes and apply the centre manifold theorem to extract amplitude equations for the critical modes. However, if the pattern is not exactly periodic our analysis must allow for the possibility that modes arbitrarily close to the critical modes in Fourier space contribute to the pattern. Then the distinction between stable and unstable modes becomes a little blurred: a mode with growth rate infinitesimally greater than zero will grow, but infinitely slowly, whereas a mode with growth rate infinitesimally less than zero will decay, but again infinitely slowly. In this case, we cannot perform a centre manifold reduction, since we cannot separate the growing and decaying modes well enough. Specifically, we cannot find an appropriate δ in equation (2.30) of Chapter 2. In cases such as these we must use an alternative method of analysis. This chapter describes how envelope equations can be used to describe the evolution of patterns that fit almost, but not exactly, onto a lattice.

Envelope equations for specific models

As explained in Chapter 5 pattern-forming systems can often be described adequately by a set of partial differential equations for a marker quantity, such as the density or temperature perturbation in a convecting fluid, together with appropriate boundary conditions. In this form, the problem is amenable to analysis using envelope equations. To explain the method, we will look at a specific example.

This book is about patterns: stripes on tigers, whorls in your fingerprints, ripples in sandy deserts, and hexagons you can cook in your own kitchen. More precisely it will be concerned with fairly regular spatial or spatiotemporal patterns that are seen in natural systems – deserts, fingertips, animal coats, stars – and in laboratory or kitchen experiments. These are structures you can pick out by eye as being special in some way, typically periodic in space (Figure 1.1), at least locally. The most common are stripes, squares and hexagons – periodic patterns that tesselate the plane – and rotating spirals or pulsating targets. Quasipatterns with twelvefold rotational symmetry (Figure 1.2) never repeat in any direction, but they look regular at a casual glance, while spiral defect chaos (Figure 1.3) is disordered on a large scale, but locally its constituent moving spirals and patches of stripes are spatially periodic.

Similar patterns are seen in wildly different natural contexts: for example, zebra stripes, desert sand ripples, granular segregation patterns and convection rolls all look stripy, and they even share the same dislocation defects, where two stripes merge into one (Figure 1.4). Rotating spirals appear in a dish of reacting chemicals and in an arrhythmic human heart. Squares crop up in convection and in a layer of vibrated sand. It turns out to be common for a given pattern to show up in several different systems, and for many aspects of its behaviour to be independent of the small details of its environment.

Introduction

Experience shows that in all thermodynamic systems the state variables temperature, internal energy and pressure assume a prominent role. Many experiments have been performed whose description require the introduction of temperature. This fact is expressed in the so-called zeroth law of thermodynamics which simply states that every thermodynamic system is associated with a variable of state T, called temperature. If two systems are in thermal equilibrium, the temperature of both systems must be identical.

Before discussing the all-important first law of thermodynamics it will be necessary to describe briefly the concepts of heat and work. If two isolated systems are brought in thermal contact by a conducting wall, heat will flow from the warmer to the colder system as long as a temperature difference exists. Heat is energy in transition between the systems. It is not a property of the physical system, therefore, it is not a variable of state. Thus, it makes no sense to speak of the heat of a particular system. The amount of heat that is transferred depends on the “path” for how it is added to the system. If, for example, the temperature of unit mass of air is to be increased by a certain number of degrees, it makes a significant difference if the heat is added at constant pressure or at constant volume.

The task ahead is the formulation of prognostic and diagnostic equations to describe the future development of the atmospheric thermodynamic state. The variables describing this state are also called the state variables of the thermodynamic system. These quantities may be the pressure, temperature and density of the air as well as the concentrations of the water substance in its different phases, that is water vapor, liquid water, and ice. As will be later seen, other choices of state variables are also possible.

The laws of atmospheric motion, as far as they are required in this book, are considered to be known. If necessary, additional details will be given at the appropriate places. A suitable reference book for our purposes is Dynamics of the Atmosphere by the present authors, which from now on will be abbreviated by DA. Of course, any other suitable textbook on atmospheric dynamics might be just as satisfactory.

Description of the atmospheric thermodynamic state

The physical object to be investigated is called the thermodynamic system, anything interacting with the system is defined as the surroundings. The thermodynamic state defines all properties of the system that have to be determined from measurements or from calculations.

The purpose of this chapter is to derive expressions for the specific partial quantities ψk = vk, Sk, ek, hk, fk, μk, k = 2, 3 for the condensed phases of water vapor. These quantities are needed to describe the thermodynamics of cloud air. In our treatment we assume that the liquid water and ice occur unmixed and neither liquid water nor ice contain foreign materials. Due to this assumption we cannot describe the formation of a water droplet. Such a droplet forms when water vapor condenses on a suitable aerosol particle so that the resulting droplet cannot be viewed as a pure substance. With this in mind, we may consider the ψk as the pure phase, i.e.. For simplicity we leave out the superscript ∘ denoting the pure phase and also drop the suffix k since confusion is unlikely. In the final section we will add the suffix k for completeness and accuracy.

The material coefficients

First of all, we need to define the coefficients of the isothermal compressibility (K) and the adiabatic compressibility (?S). They are defined by

Similarly, we define the isochoric pressure coefficient β and the isobaric expansion coefficient v* by

The quantities ?, β and v* are not independent. We recognize this by writing the state equation according to the Appendix of Chapter 6 in the form

Expansion of (9.5) gives

This expression is also valid if we choose any of the independent variables T, v and p as constant.