Overview

Warm clouds typically form in the lower troposphere when ice is not important to the microphysics. The development of warm clouds depends on condensation to activate aerosol particles and grow liquid droplets initially. Subsequent processes include drop–drop collisions, coalescence, and disruption during the formation of rain in mature clouds. Such processes may also occur in colder, mixed-phase clouds as long as the presence of ice does not interfere significantly with any of the warm-cloud microphysical processes.

The focus of this chapter is the spectral evolution of drop populations, how small (~ 10 μm radius) cloud droplets grow to large (~ 1 mm radius) raindrops. Many of the individual processes responsible for particle formation and growth have already been treated, so we now look at how these processes work collectively to form rain.

A perspective of how liquid-phase microphysics leads to rain in a warm cloud can be obtained by viewing the different processes as discrete boxes operating simultaneously inside the cloud. As shown in Fig. 11.1, one process leads to another, from initial activation of the aerosol entering the cloud in its updraft to rain falling out through its base. Overall, rain formation requires several broad categories of spectral evolution: condensation, coalescence initiation, and continuous collection. As supersaturation develops by adiabatic cooling of the rising air, the aerosol particles grow first as haze droplets, then as cloud droplets by condensation.

Transformations, changes in the structure or composition of something, are common in our everyday world. Lakes freeze in the winter; ponds dry up on hot summer days. Salt melts ice and dissolves in water. Such examples may be classified as either physical or chemical, if you wish, but many natural phenomena represent blends of both disciplines. Of primary interest is the notion that the entity in question (here, water or salt) has undergone a change of one sort or another. Learning how transformations proceed in nature is a fundamental goal of science, so we are often concerned with “processes” and “mechanisms”, the sequence of discrete events that leads to a particular outcome. Clouds owe their existence to particular transformations: Water vapor changes into liquid droplets, and those droplets later freeze when the temperature becomes low enough. Clouds would never form (or dissipate) were water not to change, be transformed from one state to another. We simply cannot understand much about clouds without dealing with transformations in detail. In fact, it is often useful to view clouds as processes, rather than as objects or entities. This part of the book lays out the guiding principles upon which to build our deeper understanding of atmospheric processes leading to clouds.

Natural transformations take place when a system deviates in some way from its most balanced or equilibrium state.

There are two protagonists in this story: inertia and friction. One meets them first in the mechanics of particles and solids where their interplay is not very complicated: inertia tries to keep the motion while friction tries to stop it. Going from a finite to an infinite number of degrees of freedom is always a game-changer. We will see in this book how an infinitesimal viscous friction makes fluid motion infinitely more complicated than inertia alone ever could. Without friction, most incompressible flows would stay potential, i.e. essentially trivial. At solid surfaces, friction produces vorticity, which is carried away by inertia and changes the flow in the bulk. Instabilities then bring about turbulence, and statistics emerges from dynamics. Vorticity penetrating the bulk makes life interesting in ideal fluids though in a way different from superfluids and superconductors.

On the other hand, compressibility makes even potential flows non-trivial as it allows inertia to develop a finite-time singularity (shock), which friction manages to stop. It is only in a wave motion that inertia is able to have an interesting life in the absence of friction, when it is instead partnered with medium anisotropy or inhomogeneity, which cause the dispersion of waves. The soliton is a happy child of that partnership.

In this chapter, we consider systems that support small-amplitude waves whose speed depends on wavelength. This is in distinction from acoustic waves (or light in the vacuum) that all move with the same speed so that a small-amplitude one-dimensional perturbation propagates without changing its shape. When the speeds of different Fourier harmonics are different, the shape of a perturbation generally changes as it propagates. In particular, initially localized perturbation spreads. That is, dispersion of wave speed leads to packet dispersion in space. This is why such waves are called dispersive. Since different harmonics move with different speeds, then they separate with time and can subsequently be found in different places. As a result, for quite arbitrary excitation mechanisms one often finds locally sinusoidal perturbation, the property well known to everybody who has observed waves on water surface. Surface waves form the main subject of analysis in this section but the ideas and results apply equally well to numerous other dispersive waves that exist in bulk fluids, plasma and solids (where dispersion usually results from some anisotropy or inhomogeneity of the medium). We shall try to keep our description universal when we turn to a consideration of non-linear dispersive waves having finite amplitudes. We shall consider weak non-linearity, assuming amplitudes to be small, and weak dispersion, which is possible in two distinct cases: (i) when the dispersion relation is close to acoustic and (ii) when waves are excited in a narrow spectral interval.

Fluid flows can be kept steady only for very low Reynolds numbers and for velocities much less than the velocity of sound. Otherwise, either flow experiences instability and becomes turbulent or sound and shock waves are excited. Both sets of phenomena are described in this chapter.

A formal reason for instability is non-linearity of the equations of fluid mechanics. For incompressible flows, the only non-linearity is due to fluid inertia. We shall see how a perturbation of a steady flow can grow due to inertia, thus causing an instability. For large Reynolds numbers, the development of instabilities leads to a strongly fluctuating state of turbulence.

An account of compressibility, on the other hand, leads to another type of unsteady phenomena: sound waves. When density perturbation is small, velocity perturbation is much less than the speed of sound and the waves can be treated within the framework of linear acoustics. We first consider linear acoustics and discover what phenomena appear as long as one accounts for a finiteness of the speed of sound. We then consider non-linear acoustic phenomena, the creation of shocks and acoustic turbulence.

Instabilities

At large Re most of the steady solutions of the Navier–Stokes equation are unstable and generate an unsteady flow called turbulence.

Kelvin–Helmholtz instability

Apart from a uniform flow in the whole space, the simplest steady flow of an ideal fluid is a uniform flow in a semi-infinite domain with the velocity parallel to the boundary.

In this chapter, we define the subject, derive the equations of motion and describe their fundamental symmetries. We start from hydrostatics where all forces are normal. We then try to consider flows this way as well, neglecting friction. This allows us to understand some features of inertia, most importantly induced mass, but the overall result is a failure to describe a fluid flow past a body. We are then forced to introduce friction and learn how it interacts with inertia, producing real flows. We briefly consider an Aristotelean world where friction dominates. In an opposite limit, we discover that the world with a little friction is very much different from the world with no friction at all.

Definitions and basic equations

Here we define the notions of fluids and their continuous motion. These definitions are induced by empirically established facts rather than deduced from a set of axioms.

Definitions

We deal with continuous media where matter may be treated as homogeneous in structure down to the smallest portions. The term fluid embraces both liquids and gases and relates to the fact that even though any fluid may resist deformations, that resistance cannot prevent deformation from happening. This is because the resisting force vanishes with the rate of deformation. Whether one treats the matter as a fluid or a solid may depend on the time available for observation. As the prophetess Deborah sang, ‘The mountains flowed before the Lord’ (Judges 5:5).

Why study fluid mechanics? The primary reason is not even technical, it is cultural: a physicist is defined as one who looks around and understands at least part of the material world. One of the goals of this book is to let you understand how the wind blows and the water flows so that flying or swimming you may appreciate what is actually going on. The secondary reason is to do with applications: whether you are to engage with astrophysics or biophysics theory or build an apparatus for condensed matter research, you need the ability to make correct fluid-mechanics estimates; some of the art of doing this will be taught in the book. Yet another reason is conceptual: mechanics is the basis of the whole of physics in terms of intuition and mathematical methods. Concepts introduced in the mechanics of particles were subsequently applied to optics, electromagnetism, quantum mechanics, etc.; here you will see the ideas and methods developed for the mechanics of fluids, which are used to analyze other systems with many degrees of freedom in statistical physics and quantum field theory. And last but not least: at present, fluid mechanics is one of the most actively developing fields of physics, mathematics and engineering, so you may wish to participate in this exciting development.

Even for physicists who are not using fluid mechanics in their work, taking a one-semester course on the subject would be well worth the effort.

Now that we have learnt basic mechanisms and elementary interplay between non-linearity, dissipation and dispersion in fluid mechanics, where can we go from here? It is important to recognize that this book describes only a few basic types of flow and leaves whole sets of physical phenomena outside of its scope. It is impossible to fit all of fluid mechanics into the format of a single story with a few memorable protagonists. Here is a brief guide to further reading, more details can be found in the endnotes.

A comparable elementary textbook (which is about twice as big) is that of Acheson [2]; it provides extra material and some alternative explanations on the subjects described in Chapters 1 and 2. On the subjects of Chapter 3, a timeless classic is the book by Lighthill [15]. For a deep and comprehensive study of fluid mechanics as a branch of theoretical physics one cannot do better than use another timeless classic, volume VI of the Landau–Lifshitz course [14]. Apart from a more detailed treatment of the subjects covered here, it contains a variety of different flows, a detailed presentation of the boundary layer theory, the theory of diffusion and thermal conductivity in fluids, relativistic and superfluid hydrodynamics, etc. In addition to reading about fluids, it is worth looking at flows, which is as appealing aesthetically as it is instructive and helpful in developing a physicist's intuition.