Introduction

When the velocities of particles change slowly with time the geostrophic approximation to the horizontal wind becomes more accurate. While at first sight this seems a good thing, because it makes the equations simpler, but it is also worrying, for the momentum equations lose some of their ability to be predictive. Thus the term in Dυ/Dt is a small residual between two large terms. If horizontal gradients of pressure are largely balanced by the Coriolis force, how are we to find what is left over to make the momentum evolve? This dilemma can be tackled by eliminating the pressure term from the momentum equations. This gives, of course, the vertical component of the vorticity equation.

Scale analysis

We use the equations of motion to establish a set of order-of-magnitude relations between variables. This process is analogous to solving the equations, except that it has the lesser aim of seeing that the numbers involved could possibly represent a solution. While we aim to be deductive, we find that it is easier to justify some approximations only after they have been made. Thus our end point is really one of plausible consistency.

Many weather systems have a much longer transverse than longitudinal scale. This is consistent with the notion that they are there in order to transfer properties like heat in the transverse direction.

This volume is the condensation of lectures given to students at Imperial College Department of Meteorology, while it existed, and to students of Environmental Science at the University of East Anglia when the department at Imperial College was closed down. Any good sense it might contain is almost certainly attributable to colleagues. Eric Eady opened my eyes to many fascinating phenomena, though when I taught students I began to realise that such communication is perhaps more of a two-way process than I imagined at the time. It is one thing to speculate on an interpretation, but another when this stimulates a response. Recalling a technical difficulty to a non-technical partner is perhaps a simple illustration of the suggested interaction. There is also a lot of Frank Ludlam here. He asked for explanations of mathematical theories that he could understand without having to go through the detailed mathematics. I remember an early encounter when he described his method of evaluating integrals. Simple, he said, ‘you just move the variables, one by one, through the integral sign ’till you are left with an integral that you can do’. Aspects of this collaboration can be seen in his book on Clouds and Storms, which has some of me in it, but which for political reasons was forgone; Frank died before his wonderful book was published. The philosophy here is almost all that of Professor Peter Sheppard, who tormented, teased, threatened, sometimes disillusioned, many generations of students, for his criticisms were accurate and acid.

Perturbations of inconstant shape: the missing baroclinic wave

A major forecast problem is concerned with forecasting development; intensification of more-or-less observable existing systems. While exponentially amplifying waves are a useful description of some aspects of wave generation, they are not the whole story, and there are some paradoxical cases. For example, consider the Eady system, in which the baroclinity is independent of height, with no variation of inertial density, and constant stratification, but with β not necessarily zero, as illustrated in Figure 10.1. With β zero there is a short-wave cut-off to instability at k = 2.4 nearly, and one pair of non-amplifying waves with steering levels above and below the middle level for shorter waves. But with β = +0.01 there is no short-wave cut-off, but only a pair of short waves with steering level near the lower boundary, one amplifying weakly, the other diminishing weakly. The wave that had its steering level near the upper boundary has vanished. Conversely, if β were small and negative, then the lower-level wave would disappear. We argue that these disappearing waves are part of a continuous spectrum of solutions that occasionally attain the property of being of constant shape.

Transfer

We are concerned here with the transfer of properties from one place to another. This transfer may be by bodily contact as when momentum is transferred from one airstream to another by the action of pressure forces. Transfer may be by pure advection with the bulk motion of the fluid, as when water vapour is carried along at constant mixing ratio together with the dry air. Such simple advection may be complicated by the relative motion of a different phase as when water droplets fall relative to the air. Molecular diffusion may be important, as when warm air is brought close to cold air then brought back to where it started, but cooler. To form a cloud droplet, water vapour diffuses towards a cloud particle and the latent heat released is conducted away from a hot particle. Electromagnetic radiation transfers energy from one absorber (emitter) to another, and sound waves carry pressure information.

Here we concentrate on the advection by fluid motion, but find that we must be aware of these other processes at the same time. For example, when the fluid motion is inexactly known, the statistical fluctuations round the known state may resemble diffusion in some respects, but not in others. It is necessary at least to be aware of the possible differences. The transfer of a variety of properties by fluctuating unresolved, but not necessarily random, fluid motion, is at the heart of the turbulence problem.

Definition of mesoscale

There is no universally accepted definition of mesoscale. Here we use it to denote motion that is not dominated by the mechanics of gravity waves, nor by the near geostrophy of Chapter 9, but is aware of, and compromises between, both. We have in mind the motion in frontal zones, the organisation of several cumulonimbus to make a severe storm, squall line, or even a hurricane, and the longer-term evolution of sea breezes. Sometimes the word mesoscale is used to indicate motion whose scale is of order 100 km but this geometrical definition includes a number of distinctly different physical processes. For example, motion on the scale of 100 km in the ocean is very similar in mechanism to the scale of 1000 km in the troposphere in that baroclinity, static stability, and variation of Coriolis parameter with latitude play similar roles. We prefer to use a criterion based on the dominant physical processes rather than on the geometry. There are disadvantages, and we find that the meridional overturning, of global scale, comes within our new definition of mesoscale, which is at first sight astonishing. There is often no very obvious energy source for mesoscale motion defined this way. Rather the effect is to organise the energy already available.

Nature of approximation

to thine own self be true

The smallness of a term in an equation is a hint that the term might be omitted, but we must be careful because a quantity that is negligible in one equation, may not be in another, perhaps because it appears there multiplied by a large factor. This is because it represents a different process in the new context. Micawber commented that even a small excess of expenditure over income eventually led to disaster.

Mathematicians like to expand in powers of a small number ∊ say, and keep only the lowest orders. This often presents difficulties in physical interpretation because the terms in ∊2 often represent very different physics to those in ∊. For example a wave of small amplitude ∊ advects wave properties at a rate proportional to ∊2, and often it is the transfer we are really interested in, not just the existence of the wave. Thus we find the shape of the motion pattern, represented by the correlation between different properties, interesting, as well as the amplitude.

The longer-term evolution of the amplifying waves found in unstable linearised systems, depends on the non-linear terms, because it is only these that prevent the unlimited increase in amplitude. In contrast, the amplitude of forced waves is often a more incidental property. It might be argued that this is untrue of breaking waves where non-linearity is an essential ingredient.

Philosophy

An important aspect of scientific study is crystallised by the idea of a model. Having thought about a problem and gathered together all sorts of useful data, we begin to think we can see what is going on. In general the picture will be horrifyingly complicated, with many physical processes involved. For example, no model of large-scale atmospheric motion is likely ever to be able to reproduce the behaviour of individual cumulus clouds, but every model must include the vertical transport of heat, and possibly water vapour, by motion on that scale if it is to model properly the energy input to the free atmosphere.

Simulation

It may be that what we really want to do is to imitate the real world. This would be so if we wanted to sell our prediction to a user for example. We might then try to describe as many of the relevant processes as we could, usually in the form of equations, and using the best estimates of their parameters. This set could then be solved for appropriate initial conditions. Such a task would require the application of numerical methods and a large computer; hence the usual name of numerical model. When this simulation is executed using observed initial data, deficiencies become apparent.

Introduction

We usually notice phenomena; events isolated in space, but most of this book will rely on a wave formalism. This chapter explores some of the relations between the two forms of description. We are going to describe a great variety of motion systems, with broad classes, but with each member of each class different. Moreover, the definition of the class depends, to some extent, on the reason we are trying to classify the phenomenon. A cumulus cloud is fairly well defined, in the sense that two independent observers will (usually) agree that a specific cloud should be put in that broad category. There will be some debate as to whether this specimen is young or old, becoming congested, and such subtleties are important as indicators of the future development of the convection. The particular one we see now is an individual, and studied for a specific purpose; it might make a gust that will disturb my boat, cover up the sun, precipitate soon, carry aphids/momentum/water vapour, into the higher levels of the troposphere; a whole host of things that will affect the way I look at it.

We might see the objective of understanding meteorology, as getting a feel for why the planet has its observed mean temperature, which sets the field for water vapour transformations, clouds, latent heat, and life. A second objective might well be the assesment of fluctuations; the global and seasonal variation of winds and temperatures and humidities, which is what we are about to engage apon. Further objectives might well begin with persistent anomalies from the climate thereby established.

Definition of general circulation

Some set of mean overall properties of the atmosphere that change slowly with time are called the general circulation. Perhaps the most obvious of these sets, though not necessarily the easiest to explain, is the zonal mean of wind, temperature, humidity, etc. as functions of height. We find that some average variables are related to other average variables rather simply. For example it is difficult to justify a large imbalance between the zonal mean of the zonal component of the wind and the pole–equator temperature gradient, as constrained by the thermal wind. Other quantities depend almost entirely on the transport of various properties by eddy motion which may occur on a variety of scales. Thus the non-zero value of the zonally averaged surface wind depends almost entirely on the transport of momentum across latitude belts by eddies on the scale of weather systems.