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.