Introduction

Any numerical model of a physical process necessarily represents that process by a set of mathematical relations that approximate the underlying physical laws. The level of approximation is determined principally by two factors: (1) knowledge of the physical processes; (2) computational limitations. The latter forces one to view the problem of the evolution of waves on an oceanic scale as a statistical one. The process of global wave evolution encompasses scales as large as the basin (107 metres) involving ocean currents, topography and wind variations, and as small as the smallest waves (10-3 metres) that modify the wind stress – it could be argued that even smaller scales associated with turbulent interfacial couplings play a role in the process. The statistical approach essentially treats the bottom 4/5 of this physical range of ten orders of magnitude as though those scales respond to well-defined physical laws imposed at the nodes of a numerical grid of typical size of one degree of latitude. The task of defining appropriate mathematical expressions that reflect the essential physics of the process is one of synthesizing the results of theoretical calculations and observational programs. As discussed in chapter I, the mathematical framework is based on a statistical description of waves having a range of scales of about one metre to one kilometre. These waves evolve in response to an action balance equation in which the ‘physics’ is embodied in a set of source functions. In this chapter, we first discuss the source functions individually and then examine the observational evidence of spectral characteristics and wave growth in fetch-limited and in shallow water situations and directional adjustment to turning winds.

Introduction

The principles of wave prediction were already well known at the beginning of the sixties (§ I.I). Yet, none of the wave models developed in the 1960s and 1970s computed the wave spectrum from the full energy balance equation. Additional ad hoc assumptions have always been introduced to ensure that the wave spectrum complies with some preconceived notions of wave development that were in some cases not consistent with the source functions. Reasons for introducing simplifications in the energy balance equation were twofold. On the one hand, the important role of the wave–wave interactions in wave evolution was not recognized. On the other hand, the limited computer power in those days precluded the use of the nonlinear transfer in the energy balance equation.

The first wave models, which were developed in the 1960s and 1970s, assumed that the wave components suddenly stopped growing as soon as they reached a universal saturation level (Phillips, 1958). The saturation spectrum, represented by Phillips' one-dimensional f-5 frequency spectrum and an empirical equilibrium directional distribution, was prescribed. Nowadays it is generally recognized that a universal high-frequency spectrum (in the region between 1.5 and 3 times the peak frequency) does not exist because the high-frequency region of the spectrum not only depends on whitecapping but also on wind input and on the low-frequency regions of the spectrum through nonlinear transfer. Furthermore, from the physics point of view it has now become clear that these so-called first generation wave models exhibit basic shortcomings by overestimating the wind input and disregarding nonlinear transfer.

In the previous chapters we have been dealing with theoretical, physical and numerical aspects of wave modelling, and the WAM model in particular. It is now time to face practical problems and to turn to applications. The WAM model is one of the most widely used wave models in the world, both for forecasting and hindcasting on global and regional scales. There is therefore a wealth of applications and experience which can be used to discuss and to judge the behaviour of the model.

The purpose of applications

Model applications serve two purposes: a practical one and a scientific one. In practical applications the model is accepted as a reliable tool. As such it can be used either in real-time, to forecast sea conditions for ship routing, offshore operations and for coastal protection, or in a hindcasting mode for computation of the sea state during a particular event and to determine wave climatology and extremal statistics. For scientific purposes, once properly verified, the model is, within the limits derived from its formulation, a representation of the real world. It can therefore be used for numerical experiments to simulate field experiments under conditions that would be difficult to find or expensive to organize. The model is a tool for the better understanding of the relevant phenomena and how they interact. It can also be used to study the sensitivity of the results to a change in the input conditions.

General features of wave data assimilation and inverse modelling

Background

In previous chapters we have discussed the use of a mathematical model, the third generation wave prediction model, to compute the state of the sea. We can distinguish between hindcasts, nowcasts and forecasts, the difference being the time for which the sea state is computed relative to the clock time. A forecast field can be computed only by a model. However, the model estimation of hindcast and nowcast fields can be improved using observations, which have been considered in the previous chapters of this book only for validation of the underlying physics or for verification of model results. This combination of using model results and observations to create an optimal estimate of the sea state is called data assimilation or analysis. The word analysis originates in early meteorological applications, for which meteorologists would subjectively draw isobaric patterns on the basis of isolated pressure observations. They analysed the weather. Later, this work was carried out with numerical models.

Observed data can also be used to validate and improve models. When the model improvement is carried out using numerical automatic model fitting techniques, one speaks of inverse modelling: instead of using a given model to compute data, which are then compared with observations, the observations are used in an inverse mode to construct an optimally fitted model.

The wind blows where it wills; you hear the sound of it, but you do not know where it comes from, or where it is going. (John, 3, 8)

In the ancient world, the question of where the wind came from and where it was going to was one of life's unanswerable puzzles. Indeed, so unpredictable and unknowable was the wind, that the evangelist used it as an elaborate pun on the inscrutable purposes of God's Spirit (in the Greek of the New Testament, the same word means ‘wind’ and ‘spirit’). Meteorology of any sort must have been a most frustrating study. Hints of regular patterns emerged, only to vanish on closer inspection. Wise saws about the weather were wrong as often as they helped. And if you were a farmer, a sailor or a campaigning soldier, guessing the winds or the weather wrongly could lead to disaster. Truly, through the weather and the winds, the gods played with man, teased and tormented him, and confirmed their authority.

And here matters more or less rested until the Newtonian revolution of the seventeenth century. By then, many different aspects of the natural world had been reduced to reproducible laws. Kepler showed how the motions of the planets were governed by strict rules, although he did not quite succeed in explaining why his laws of planetary motion had the forms they did.

Low frequency transients

In the preceding chapter, Fig. 7.14 compared the eddy correlation tensor for the higher frequency eddies, whose periods are less than around ten days, with the lower frequency transients. The high frequency eddy statistics have a well-defined structure in the midlatitudes, with maxima in the storm track regions. The high pass filter used in Chapter 7 served to isolate a specific family of dynamical processes, namely, those associated with baroclinic instability and the subsequent evolution of baroclinically unstable waves. The low frequency eddy kinetic energy is much less clearly structured. Figure 7.14 shows some evidence of maxima downstream of the high frequency maxima, as well as some correlation between the jet centres and maxima of low frequency variability. But none of these patterns is especially marked.

One reason for this is that the low frequency band covers a very wide range of frequencies. There are disturbances whose periods are very little longer than those of the baroclinic disturbances; indeed, the maxima downstream of the storm track centres are at least partly due to the occlusion and decay of midlatitude cyclones, which become slower moving as they fill. But there are also transients of significant amplitude with very much longer periods. Indeed, a spectral analysis of any long sequence of atmospheric data reveals that variability is observed on as long a period as one cares to specify.

Observations of steady eddies

Despite the eddy–zonal flow partitioning which we have employed in preceding chapters, the seasonal mean flow is very far from being zonally symmetric. Such departures from symmetry are important in accounting for regional variations of climate. They also modify the global patterns of heat and momentum transport, especially in the northern hemisphere winter. In this chapter, we will discuss some observations of the steady wave pattern, and show how rather simple theories based on linear wave propagation can account for some of the gross features of these observations.

The steady waves are most pronounced in the northern hemisphere winter, and have their largest amplitudes in the upper troposphere. In some circumstances, they also become very important at high levels in the winter stratosphere, a point that we will return to in Chapter 9. Figure 6.1 shows the winter mean geopotential height field at 25 kPa in both hemispheres. The characteristic features of the northern hemisphere picture are the pronounced troughs over Canada and Japan, with ridges over the eastern side of the two ocean basins. One's subjective impression is of a predominantly zonal wavenumber 2 pattern. This general pattern is very persistent and can be seen in individual seasons with only relatively small variations. The corresponding picture for the southern hemisphere looks, at first sight, much more axisymmetric.

Major influences on planetary circulations

Until recently, the study of global circulations has been confined to the circulation of a single system, namely that of the Earth. Throughout the earlier part of this book, we, too, have concentrated upon the Earth, showing how the poleward and upward transports of heat generate the kinetic energy associated with observed atmospheric circulations. We have described some of the forms which these heat fluxes can take, including the essentially axisymmetric circulations of the Hadley cells in low latitudes and the wavelike baroclinically unstable waves of the midlatitudes. These principles need not be restricted to the Earth's system alone. In this chapter, we will enquire how general are the particular heat transporting circulations observed in the Earth's atmosphere, and how they might be modified in different circumstances.

Such a discussion has become much more informed in the last 20 years or so, as the study of planetary atmospheres has advanced considerably. Spacecraft have now paid at least fleeting visits to every planet with a substantial atmosphere in the solar system, with the exception of Pluto, which may possess an atmosphere. In the case of Venus and Mars, direct in situ measurements of meteorological parameters have been made in addition to the more usual remotely sensed data. In the coming years, plans are under way for entry probes and direct measurements of other atmospheres, including those of Jupiter and Titan.

Observational basis

The large scale structure of the atmospheric flow varies most rapidly in the vertical direction, and least rapidly in the zonal direction. Zonal averaging therefore makes the important vertical and meridional variations plain, and has been employed for many years as a compact way of studying the global circulation. Indeed, for many writers, the global circulation is simply the pattern of flow projected on to the meridional plane. In this book, we will take a broader view by attempting to summarize our current understanding of the full, evolving three-dimensional pattern of winds and temperature in the atmosphere. But the traditional zonal mean view is a useful starting point which we will explore in this chapter.

The zonal mean wind and vectors of the mean meridional wind are illustrated in Fig. 4.1, based on ECMWF analyses. Rising motion is seen in the tropics, with the maximum vertical velocity in the summer hemisphere. Strongest descent is at latitudes of around 25 – 30° in the winter hemisphere, with flow towards the equator near the surface and away from the tropics in the upper troposphere, as is required by continuity. Such an axisymmetric circulation is the most obvious response of the atmospheric flow to the net heating excess in the tropics and the deficit at high latitudes discussed in the preceding chapter.

The aim of this chapter is to introduce the basic physical laws which govern the circulation of the atmosphere and to express them in convenient mathematical forms. No attempt is made at either completeness or rigour beyond the requirements of the later chapters. Those who wish for a more detailed discussion are referred to one of the many excellent texts on dynamical meteorology which are now available. Those by Holton (1992) and by Gill (1982) are particularly recommended.

The first law of thermodynamics

The first law may be stated simply in its qualitative form: heat is a form of energy. The transformation of heat energy into various forms of mechanical energy is the process which drives the global circulation of the atmosphere and which is responsible for the formation of the weather systems whose cumulative effects define the climate of a particular region. These transformations will be discussed in more detail in Chapter 3. In this section, the first law will be expressed in mathematical terms. But, first, it will be necessary to consider the thermodynamic properties of the air which makes up the atmosphere.

The ‘thermodynamic state’ of a parcel of air is defined by specifying its composition, pressure, density, temperature, and so on. In fact, these properties are not independent of one another, but are related through the ‘equation of state’ of the air.

The seasonal cycle of the stratospheric circulation

Up until this point, we have concentrated almost exclusively upon the troposphere, which is characterized by a relatively weak stratification, with a temperature lapse rate of around 6–7 K km–1. At the tropopause, the lapse rate becomes close to zero; the lower stratosphere is nearly isothermal. The corresponding change in stratification, as measured by the Brunt-Väisälä frequency, is by a factor of around two, from values of 10–2 s–1 in the troposphere to values of 2 × 10–2 s–1 in the lower stratosphere. In the upper stratosphere, from heights of 30 km to around 50 km, the temperature actually increases with height. The transition to stably stratified conditions is called the tropopause, which is extremely sharp in the tropics and midlatitudes. It is rather more gradual in polar latitudes, especially in winter when there is no incoming sunlight. The abrupt increase of stratification at the tropopause means that the stratosphere is dynamically very different from the underlying troposphere. Baroclinic instability is virtually suppressed and disturbances are mainly forced from below. The stratification acts as a filter, removing the smaller scale disturbances and allowing only the longest waves to propagate out of the troposphere to great heights in the stratosphere. Shorter wavelength disturbances are thereby trapped in the troposphere, which behaves as a waveguide, the upper boundary of which is the tropopause.

Zonal variations in the tropics

Up to this point, we have followed a traditional exposition of the global circulation by concentrating upon the zonal mean circulation and upon the zonal mean fields of eddy quantities. But the global circulation is far from zonally symmetric. Tropical heating has distinct maxima at particular longitudes. In the midlatitudes, the transient eddies are not distributed uniformly around the latitude circles, but are concentrated into isolated ‘storm tracks’, especially in the northern hemisphere. This chapter will be devoted to a description of such zonal asymmetries and their consequences.

The various diagnostics of the steady and transient eddy activity which we have considered in earlier chapters become small in the tropics. Eddy kinetic energy is much smaller in the tropics than in the midlatitudes. Similarly, eddy temperature and momentum fluxes, both steady and transient, are much smaller in the tropics. Thus a picture emerges in which heat and momentum are transported, essentially by axisymmetric motions in the tropics, with eddies taking over in the subtropics and midlatitudes.

There is some truth in this picture. But it can also be misleading. First, consider the heating fields shown in Fig. 3.8. The forcing of the circulation is certainly not axisymmetric, especially in the tropics. Rather, there are a small number of centres of intense heating.

Timescales of atmospheric motion

The circulation of the atmosphere is intrinsically unsteady; fluctuations on all timescales are observed. In the last chapter, it was shown that the fluxes of temperature, momentum, and so on, carried by such transients play an important part in determining the time mean circulation of the atmosphere. Our task in this chapter will be to describe the transients on various timescales, and to discuss the mechanisms which can give rise to transient behaviour. Just as the atmosphere contains a wide range of spatial scales, from the molecular to the global, so the atmospheric circulation exhibits a wide range of timescales, ranging from timescales of just a few seconds associated with the overturning of small turbulent eddies, to geological timescales for major climate changes.

Some of the frequencies observed are directly related to the frequencies of periodic forcings. For example, diurnal and semi-diurnal variations of temperature and wind are associated with the diurnal variation of solar heat input. These ‘thermal tides’ are important at high levels in the atmosphere and can be detected in the lower atmosphere. More importantly for our purposes, the annual cycle of radiative forcing has a profound effect on large scale atmospheric circulation. This seasonal cycle of meteorological quantities affects nearly all parts of the globe.

But in addition to these externally imposed periods, the atmospheric flow itself generates all kinds of timescales internally.