Relevant Subjects

Atmospheric composition played an important part in the development of chemistry, following the work of Priestley, Lavoisier, and Dalton. Since air is a mixture of gases, many of them chemically reactive, see for example Finlayson-Pitts and Pitts (2000) and Graedel et al. (1986), which is subject to solar photons, absorbs and emits infrared photons, experiences temperatures ranging from −100 to 40° C, is exposed to the ocean, encompasses phase changes of water and sustains turbulent flow, it involves significant parts of physical chemistry. Pedagogically, the three-volume set by Berry, Rice, and Ross (2002a, b, c) covers the basic physicochemical material clearly and thoroughly, particularly Chapters 19, 20, 27, 28, 30, and 31. In addition to kinetic molecular theory, chemical kinetics, spectroscopy, and equilibrium statistical mechanics, there are other branches of physical science which are applicable to the atmosphere; in our context they include of course meteorology and turbulence theory. It ought to be recognized that the atmosphere has high complexity arising from a vast number of degrees of freedom, several anisotropies, and morphologically complicated boundaries extending over 15 orders of magnitude in scale from the molecular mean free path to the Earth’s circumference; these factors and the concomitant non-linearities make the application of non-equilibrium statistical mechanics a daunting prospect, but nevertheless one which should be attempted, for the reason that the energy distributions and their transformations in the atmosphere need to be accurately described, particularly in the representation and prognosis of the climatic state. We will also show that vorticity is the fundamental variable, since vortices are generated from molecular populations subjected to an anisotropy, on very short space scales and fast time scales. In this Chapter we will give a skeletal survey connecting these basic subjects, with references to more comprehensive, individual sources. The simplest possible molecular model for a gas is a collection of spherical ‘billiard balls’—the intermolecular potential consists of an infinite repulsive force on contact. This approach, pioneered by Waterston, Maxwell, and Boltzmann, is successful for air as a first approximation. The idea is that collisions are completely elastic, with no interaction between potential collidant molecules until physical contact occurs, whereupon an infinite repulsive potential operates.

Initial Survey of Observations

The observations are our starting point in this book, having been obtained from research aircraft in the last two decades. Justification for this approach can be found in Section 1.3 and by noting that there are no known analytical solutions to the Navier–Stokes equation, preventing the possibility of a priori prediction of the atmosphere’s turbulent structure. We note the pioneering power spectral analysis of wind, temperature, and ozone from commercial Boeing 747 aircraft (Nastrom and Gage 1985) and the more recent data from Airbus 340 aircraft under the aegis of the MOZAIC programme (Marenco et al. 1998). Multifractal analysis was first applied to observations from an IL-12 aircraft in the tropics (Chigirinskaya et al. 1994) and has been applied to a large body of observations taken from ER- 2, WB57F, DC-8, and G4 aircraft, with dropsondes from the last of these; Chapters 2, 4 and 5 are largely devoted to the results. Many of these data were obtained in the lower stratosphere from the ER-2 in the course of investigating ozone loss in both Arctic and Antarctic regions, where there exists a reasonably well-defined, durable circulation system offering clear dynamical, chemical, and radiative signatures. A more climate-driven imperative exists to investigate the tropical upper troposphere and lower stratosphere, largely pursued with the WB57F. The recent G4 and dropsonde data were acquired in the troposphere over the eastern Pacific Ocean, in the course of investigating northern hemisphere winter storms there. The utility of balloons and then, 120 years later, from 1903, of powered aircraft for exploring atmospheric properties, were immediately obvious. The Second World War saw aircraft attaining stratospheric altitudes, revealing a very dry lower stratosphere with westerly winds in winter and easterlies in summer, with accumulation of high ozone abundances in polar regions (Brewer 1944; Dobson et al. 1945; Brewer et al. 1948; Brewer 1949; Murgatroyd and Clews 1949; Bannon et al. 1952).

Introduction

The atmosphere consists of molecules in motion, yet it is often hard to find any mention of the fact in meteorological texts. This absence is also true of substantial areas of physics and chemistry which have evolved to provide quantitative descriptions of the behaviour of atoms and molecules in the gas phase: in particular, non-equilibrium statistical mechanics and molecular dynamics have had less overlap with the theory and observation of turbulence than perhaps might have been expected. Meteorology of course has had fluid mechanics at front and centre for over a century and has had to face issues in turbulence for over half that time. The purpose of this book is to show that atmospheric turbulence is an emergent property arising from the anisotropic environment of populations of gas molecules, linking molecular dynamics with fluid mechanics through the generation of vorticity. The anisotropies arise from gravity, planetary rotation, the solar beam, and the nature of the topography, the sea and ice surfaces, and the vegetative cover. We shall see that analysis of high resolution data of adequate quality, as yet available largely from only a few aircraft, leads to the emergence of a correlation of the multifractal, turbulent scaling at the smaller scales with some characteristics of the larger scale meteorological flow, such as the intensity and depth of jet streams. Lest the reader should think that the formulation of events at the microscopic scale has little or nothing to do with the central concerns of modern meteorology, we note that climate is determined through the absorption and emission of photons by molecules in the atmosphere and at the surface. The nature and distribution of these molecules is determined by photochemical kinetics acting in the presence of turbulent transport and biogeochemical fluxes from the surface. Quantitative calculation of the rates of these processes must necessarily account for both the skewed probability distributions of molecular velocities maintained interactively by vorticity structures and the effect of the scale invariant turbulent structures, on such large volumes of chemical reaction as the stratospheric polar vortex.

Summary, Quo Vadimus? and Quotations

In this chapter, we offer a summary of the book’s results and conclusions, ask what future developments might be contemplated, both theoretical and experimental, and provide some scientific quotations which seemed relevant. The quotations are collected here rather than dispersed through the text, because some of them apply at several junctures and one or two apply to the whole book. It is hoped that they will underline some important points in a memorable and even entertaining manner. Application of generalized scale invariance to large amounts of research quality in situ airborne observations of the free troposphere and lower stratosphere has shown that the atmosphere behaves as a random, non-Gaussian, Lévy stable process. The scaling exponents describing the resultant statistical multifractality are the conservation H1, the intermittency C1 and the departure from monofractality α, the Lévy exponent. They had average values of 0.55, 0.05, and 1.6 respectively as deduced from airborne time series of wind speed and temperature. Certain regimes, such as jet streams, however showed correlation within the mean; the value of H1(s) for horizontal wind speed s was positively correlated with the magnitude of the horizontal speed shear and the value of H1(T ) for temperature was positively correlated with the meridional (equator-to-pole) temperature gradient. The value of H1(s) in the vertical showed clear correlation with vertical measures of jet stream strength, such as depth and maximum speed. The vertical scaling of temperature showed the paramount influence of gravity, having H1 close to unity, while horizontal wind speed and relative humidity were about 0.75. These results show that large scale ordered flow can be interpreted as emerging from less ordered smaller scale motions. At the same time, the smaller scale motions are never truly random in the atmosphere and the larger scale motions are never perfectly correlated, smooth flow. Ozone and water, while occasionally behaving as passive scalars, that is to say as tracers, more often showed the presence of sources and sinks: a numerical model-independent demonstration of the operation of photochemistry and precipitation respectively.

Non-Equilibrium Statistical Mechanics

The Earth’s atmosphere is far from equilibrium; it is constantly in motion from the combined effects of gravity and planetary rotation, is constantly absorbing and emitting radiation, and hosts ongoing chemical reactions which are ultimately fuelled by solar photons. It has fluxes of material and energy across its boundaries with the planetary surface, both terrestrial and marine, and also emits a continual outward flux of infrared photons to space. The gaseous atmosphere is manifestly a kinetic system, meaning that its evolution must be described by time dependent differential equations. The equations doing this under the continuum fluid approximation are the Navier–Stokes equations, which are not analytically solvable and which support many non-linear instabilities. We have also seen that the generation of turbulence is a fundamentally difficult yet central feature of air motion, originating on the molecular scale. Non-equilibrium statistical mechanics may offer insight into which steady states a system far from equilibrium as a result of fluxes and anisotropies may migrate, without the need for detailed solution of the explicit path between the states. However, it does not seem possible to demonstrate mathematically that such steady states exist for the atmosphere. A physical view of the planet’s past and probable future suggests that the past and future evolution of the sun and its outgoing fluxes of energy may mean that the air-water-earth system may never have been or will ever be in a rigorously defined steady state. Also, to the human population, the detailed, time-dependent evolution is what matters in many respects. Nevertheless, non-equilibrium statistical mechanics is a discipline which should be applicable in principle to yield information about approximate steady states. These steady states may as a practical matter be definable from the observational record, for example the ice ages and the intervening periods evident in the geological record, or between states with two differing global average abundances of a radiatively active gas such as carbon dioxide. There has been great progress recently in non-equilibrium statistical mechanics, stemming from recent work on the concept of the maximization of entropy production.

Radiative and Chemical Kinetic Implications

The laws governing the dynamical behaviour of atoms and molecules are quantum mechanical, and specify that their internal energy states are discrete, with only definite photon energies inducing transitions between them, subject to selection rules. These energy levels appear as spectra in different regions of the electromagnetic spectrum: pure rotational lines in the microwave or far infrared, ‘rovibrational’ (rotation + vibration) lines in the middle and near infrared, while electronic transitions, sometimes with associated rotational and vibrational structure (‘rovibronic’) occur from the near infrared through the visible to the ultraviolet. An important feature of these spectra in the atmosphere is that they do not appear as single sharp lines, but are collisionally broadened about the central energy into ‘line shapes’ which frequently overlap with other transitions, both from the same molecule and from others. One of the primary dynamical quantities involved in the processes broadening these line shapes is the relative velocity of the molecules with which the photon absorbing and emitting molecules are colliding. These are primarily N2 and O2 in the atmosphere; if they have an overpopulation of fast moving molecules relative to a Maxwell–Boltzmann distribution, as we have suggested, the line shapes will be affected. Molecules such as carbon dioxide, water vapour, and ozone are all active in the infrared via rovibrational transitions, with water vapour being light enough and so having sufficiently rapid rotation that it has rotational bands appearing in the far infrared rather than the microwave. Nitrous oxide, N2O, and methane, CH4, are also active, but make smaller contributions because of their lower abundances. Molecular nitrogen and molecular oxygen, because they are homonuclear diatomic molecules, do not absorb or emit via electric dipole allowed transitions in the atmospherically important regions of the electromagnetic spectrum. Molecular oxygen, having a triplet ground state, does have weak forbidden and magnetic dipole transitions which, however, play only a very small role in the radiative balance. It should be noted that the translational energy of molecules in a large system like the atmosphere is effectively continuous rather than quantized.

Temperature Intermittency and Ozone Photodissociation

During the last two missions performed by the ER-2 in the Arctic lower stratosphere, POLARIS in the summer of 1997 and SOLVE during the winter of 1999–2000, an unexpected correlation emerged when the data were subjected to analysis by generalized scale invariance. It was between the intermittency of temperature, a number which can be determined for each segment of analysable flight from the temperature measurements, and the average over the flight segment of the photodissociation rate of ozone, which was calculable as a time series along the flight segment by taking the product of the 1Hz measurements of the local ozone concentration and the 1Hz measurements of the ozone photodissociation coefficient. In searching for a physical explanation of this correlation, it was realized that the common link between the quantities was that ozone photodissociation produces photofragments of atomic and molecular oxygen that recoil very fast, while temperature itself is the integral of the translational energy of all air molecules. The next step therefore was to ask if the intermittency of temperature was correlated with the average of the temperature itself over the flight segment: it was. One might think that because ozone is present at about 20km altitude in mixing ratios of about 2−3×10−6, the rapid quenching of the translational energies of the recoiling photofragments by molecular nitrogen and molecular oxygen would prevent any possible effects from showing up in the bulk, observed temperature. However, during the POLARIS mission, it was possible to fly the ER-2 near the terminator, the boundary between day and night, because at Arctic latitudes the planet was rotating slowly enough that it could fly legs in the same, stagnant air mass in both sunlight and darkness. These flights showed that the heating rate was significant, about 0.2Kper hour, and since heating in the stratosphere arises from the absorption of solar radiation by ozone, which leads to photodissociation, there is a prima facie case for considering non-local thermodynamic equilibrium effects from the recoiling fast photofragments. Two arguments may be deployed at this point, both from the theoretical literature; there are as yet no experiments on the translational speed distributions of atmospheric molecules.

Generalized Scale Invariance

Probability distributions plotted to date from large volumes of high quality atmospheric observations invariably have ‘long tails’: relatively rare but intense events make significant contributions to the mean. Atmospheric fields are intermittent. Gaussian distributions, which are assumed to accompany second moment statistics and power spectra, are not seen. An inherently stochastic approach, that of statistical multifractals, was developed as generalized scale invariance by Schertzer and Lovejoy (1985, 1987, 1991); it incorporates intermittency and anisotropy in a way Kolmogorov theory does not. Generalized scale invariance demands in application to the atmosphere large volumes of high quality data, obtained in simple and representative coordinate systems in a way that is as extensive as possible in both space and time. In theory, these could be obtained for the whole globe by satellites from orbit, but in practice their high velocities and low spatial resolution have to date restricted them to an insufficient range of scales, particularly if averaging over scale height-like depths in the vertical is to be avoided; analysis has been successfully performed on cloud images, however (Lovejoy et al. 2001). Some suitable data were obtained as an accidental by-product of the systematic exploration of the rapid (1–4% per day) ozone loss in the Antarctic and Arctic lower stratospheric vortices during winter and spring by the high-flying ER-2 research aircraft in the late 1980s through to 2000. Data initially at 1Hz and later at 5Hz allowed horizontal resolution of wind speed, temperature, and pressure at approximately 200 m and later at 40 m, with ozone available at 1 Hz, over the long, direct flight tracks necessitated by the distances involved between the airfield and the vortex. Some later flights also had data from other chemical instruments, such as nitrous oxide, N2O, reactive nitrogen, NOy, and chlorine monoxide, ClO, which could sustain at least an analysis for H1, the most robust of the three scaling exponents. Better than four decades of horizontal scale were available for 1Hz and 5Hz data. Since then, a lesser volume of adequate data has been obtained away from the polar regions by the WB57F.