Absorption by Atmospheric Gases

Digital archives offer the investigator an up-to-date analysis of an extensive and specialized literature. Periodic revisions are reported in the open literature and it seems unlikely that future investigators will attempt to use any other source where archives can provide the required data. For this reason, we shall confine our comments on permitted vibration-rotation transitions to describing the AFGL tape contents, but we shall add two areas not contained in it: first, electronic bands, and second, the related topics of forbidden transitions, collision-induced transitions, and polymer spectra. The AFGL tape lists data on one important set of electronic transitions, those giving rise to the near-infrared atmospheric bands of molecular oxygen. These bands behave in the same way as vibration rotation bands, except for the frequency displacement caused by the change in electronic energy and the symmetry conditions imposed by the electronic wave functions. Other electronic transitions usually involve larger differences between energy levels and cannot be understood as completely as the lower energy, vibrational and rotational transitions. Fortunately, visible and ultraviolet bands of importance for atmospheric problems are less complicated than vibration—rotation bands and they are usually less affected by state parameters. Atmospheric absorption calculations in the visible and ultraviolet spectrum are commonly made on the basis of empirical data without requiring the level of understanding developed in Chapters 3 and 4 for vibration-rotation bands. The altitude of unit optical depth for ultraviolet atmospheric bands is illustrated in Fig. 5.1. The intensity of solar radiation falls off rapidly with decreasing wavelength in the spectral region shown (the irradiance at 2000 Å compared to that at 3000 Å is 10-2 whereas at 1000 Å it is 10-5, see Appendix 9). For heating rate calculations at altitudes less than 100km, only O2 and O3 are important, except under special conditions when the atmosphere contains large amounts of volcanic aerosols, or polar stratospheric clouds at high latitudes. All of the absorptions shown in Fig. 5.1 are important for other reasons that do not directly concern us here.

Theory of Radiative Transfer

In common with astrophysical usage the word intensity will denote specific intensity of radiation, i.e., the flux of energy in a given direction per second per unit frequency (or wavelength) range per unit solid angle per unit area perpendicular to the given direction. In Fig. 2.1 the point P is surrounded by a small element of area dπs, perpendicular to the direction of the unit vector s. From each point on dπs a cone of solid angle dωs is drawn about the s vector. The bundle of rays, originating on dπs, and contained within dωs, transports in time dt and in the frequency range v to v + dv, the energy . . . Ev = Iv(P,S) dπs dωs dv dt, (2.1). . . where Iv(P, s) is the specific intensity at the point P in the s-direction. If Iv is not a function of direction the intensity field is said to be isotropic ; if Iv is not a function of position the field is said to be homogeneous.

Evolution of a Thermal Disturbance

The thermodynamic equation for an ideal gas is where ρcv6 is the internal energy per unit volume and hR is the radiative heating rate. For the sake of clarity we omit diabatic terms additional to the radiative heating. We may expand the left-hand side of (10.1) and write it in the form of an enthalpy equation, and cp is the specific heat at constant pressure. The left-hand side of includes both internal and potential energy, d is the dynamic heating. If the vertical coordinate is pressure, ∂p/∂t = 0 and all terms in d tend to zero as the velocities tend to zero. Solutions to (10.2) in conjunction with the equations of motion, the equation of continuity, and the gas law are the matter of dynamic meteorology. In this chapter, we look at a single aspect, namely the coupling between radiation and dynamics as expressed by the thermodynamic equation, (10.2). We shall use the methods of perturbation theory. Assume the existence of a basic, steady-state (suffix 0) for which This basic state could be a state of radiative equilibrium or a state dominated by dynamic transports.

Radiative Transfer in a Scattering Atmosphere

The source function for scattering, (2.32), is more complicated than a thermal source function on two accounts: it is not a function of local conditions alone, but involves conditions throughout the atmosphere, through the local radiation field, and the phase function, Pij(s, d), may be an extremely complex function of the directions, s and d, and the states of polarization, i and j. The general solution, (2.87), is still valid, but it is now an integral equation, involving the intensity both on the left-hand side and under the integral on the right-hand side. Successive approximations, starting with the first-order scattering term [third term on the right-hand side of (2.116)], are an obvious approach, and would lead to a solution, but there are more efficient and more accurate ways to solve the problem. Many methods are available because their fundamental theory has proved to be mathematically interesting and because there are important applications in neutron diffusion theory and astrophysics. These motivations are extraneous to atmospheric science, but the availability of the methodology has led to its adoption and extension to atmospheric problems. Many methods are available because their fundamental theory has proved to be mathematically interesting and because there are important applications in neutron diffusion theory and astrophysics. These motivations are extraneous to atmospheric science, but the availability of the methodology has led to its adoption and extension to atmospheric problems. Solutions to scattering problems can be elaborate and mathematically elegant; they can also be numerically onerous but, with access to modern computers, “exact” solutions are feasible, given the input parameters τv, av (=sv/ev), and Pi j. For monochromatic calculations with simple phase functions, numerical solutions present few difficulties. Nevertheless, the combination of unfamiliar formalism with inaccessible and undocumented algorithms can be daunting for those with only a peripheral interest in radiation calculations. It is, therefore, relevant to note that available data are imprecise and virtually never require the accuracy available from exact methods. Easily visualized two-stream approximations, combined with similarity relations to handle complex phase functions (see §§8.4.4 and 8.5.6), are often more than adequate, and some angular information can be added, if required, from the use of Eddington’s second approximation (§ 2.4.5).

Extinction by Molecules and Droplets

The formal theory developed in Chapter 2 assumed the Stokes parameters to be additive. The sufficient condition for additivity is that the radiation fluxes in the atmosphere shall have no phase coherence. Thermal emission from independently excited molecules is necessarily incoherent with respect to phase. Atmospheric scattering centers are widely and randomly spaced, and they can be treated as independent and incoherent scatterers. The situation differs, however, when we consider details of the scattering process within a single particle, and in order to derive the extinction coefficient and the scattering matrix (see § 2.1.3) we must make use of a theoretical framework that involves the phase explicitly. The problem of the interaction between an electromagnetic wave and a dielectric particle can be precisely formulated using Maxwell’s equations. For a plane wave and a spherical particle, Mie’s theory provides a complete solution (see §7.6). But the general problem is complicated and our understanding is rendered more difficult by preconceptions based on the approximations of elementary optics. This chapter provides a brief survey of the important results and the underlying concepts. The geometry of the problem is illustrated in Fig. 7.1. An isolated particle is irradiated by an incident, plane electromagnetic wave. The plane wave preserves its character only if it propagates through a homogeneous medium; the presence of the scattering particle, with electric and magnetic properties differing from those of the surrounding medium, distorts the wave front. The disturbance has two aspects: first, the plane wave is diminished in amplitude; second, at distances from the particle that are large compared with the wavelength and particle size, there is an additional, outward-traveling spherical wave. The energy carried by this spherical wave is the scattered energy; the total energy lost by the plane wave corresponds to extinction; the difference is the absorption. The properties of the spherical wave in one particular direction (the line of sight) will be considered. This direction can be specified by the scattering angle 6 (see Fig. 7.1) in a plane containing both the incident and scattered wave normals (the plane of reference), and the azimuth angle ϕ) between the plane of reference and a plane fixed in space.

Vibration-Rotation Spectra of Gaseous Molecules

In this chapter we discuss the characteristics of absorption by gaseous constituents of the earth’s atmosphere. This is a complex topic and atmospheric investigators may be disturbed by the idea that weather and climate might be affected by details of the kind we shall discuss. But, as yet, we lack criteria as to what is important and what is not, leaving little alternative to developing a general understanding of the field. A full description of the atmospheric absorption spectrum involves the intensities, state dependence, and detailed line profiles of 105 to 106 lines of 20 or more different chemical species. Given the capabilities of modern computers, it is possible to store, retrieve, and manipulate such data and this is the method of choice for purposes such as the identification of lines in high-resolution spectra. One of a number of current attempts to assemble an up-to-date archive of molecular data is the Air Force Geophysics Laboratory (AFGL) magnetic tape. Not only does this tape provide an economical means of access to the best data from a vast literature, but it also provides a convenient international standard atmosphere. Two numerical climate models, both using the AFGL data, cannot attribute their differences to the radiation data employed. We shall, therefore, address the subject of molecular spectroscopy in the general context of the AFGL tape and many of our illustrations are composed from the tape in preference to seeking out observed spectra. As will be apparent by the end of this chapter, it may sometimes take an expert to distinguish between the two. Figure 3.1 offers an overview of the atmospheric absorption spectrum. The six gases considered are the most important radiators, although climate studies often involve more and rarer species. All six gases are minor species (and therefore in dilute mixtures with nitrogen and oxygen) and are very simple molecules (methane is the most complex). Figure 3.1 shows no visible or ultraviolet spectra. The missing features are mainly electronic bands of oxygen and ozone; they will not be treated in this chapter since they are more complex theoretically but easier to handle empirically than the bands shown in Fig. 3.1.

Introduction

Earth, like the other inner planets, receives virtually all of its energy from space in the form of solar electromagnetic radiation. Its total heat content does not vary significantly with time, indicating a close overall balance between absorbed solar radiation and the diffuse stream of low-temperature, thermal radiation emitted by the planet. The transformation of the incident solar radiation into scattered and thermal radiation, and the thermodynamic consequences for the earth’s gaseous envelope, are the subjects of this book. The scope must be narrowed, however, because in its broadest interpretation our title could include atmospheric photochemistry and many other topics usually treated in books dealing with the upper atmosphere. By restricting attention to the thermodynamic aspects, this problem of selection usually resolves itself. For example, the absorption of energy accompanying photodissociation or photoionization will be considered if the energy involved is comparable to that of other sources or sinks, but not otherwise. Similarly, the oxygen airglow has some thermodynamic consequences in the upper atmosphere, but the important topic of the airglow will be mentioned only in this limited context. The irradiance at mean solar distance—the solar constant—is slightly less than 1400 Wm-2, giving an average flux of solar energy per unit area of the earth’s surface equal to 350 W m-2 (the factor 4 is the ratio of surface area to cross section for a sphere). Of this energy, approximately 31% is scattered back into space, 43% is absorbed at the earth's surface, and 26% is absorbed by the atmosphere. The ratio of outward to inward flux of solar radiation is known as the albedo. We may speak of the albedo of the entire earth or of individual surfaces with reference either to monochromatic radiation or to a weighted average whole is about 0.31, and an average of 224 Wm-2 is available for heating, directly and indirectly, the earth and its atmosphere. The redistribution of this absorbed solar energy by dynamic and radiative processes and its ultimate return to space as low-temperature planetary or terrestrial radiation are the most important topics of this book.

Atmospheres in Radiative Equilibrium

In this chapter we discuss radiative equilibrium models of the earth’s atmosphere and the closely related radiative—convective models, for which small-scale convection is included in a highly parameterized form. In both cases, heat transports by planetary-scale motions are neglected. Despite their limitations, radiative equilibrium and radiativeconvective studies have provided stimuli for many of the fundamental ideas discussed in this book. Their value is principally heuristic. The radiative equilibrium state is one conceivable state of a planetary atmosphere that may be analyzed so that the implications of parameteric changes can be understood in simple terms (e.g., changes in atmospheric composition, earth orbital elements, solar emission, etc.). The same cannot yet be said of any dynamic model. While numerical solutions are available from general circulation models, their behavior is often no easier to interpret than that of the atmosphere itself. For studies that are not based on the existence of day-to-day observations, radiative equilibrium considerations provide the irreplaceable first step in a number of fields: the atmospheres of other planets, stellar atmospheres, the earth’s primitive atmosphere; and much of the progress in studies of climate change has been based on the simplest energy balance models. In addition to their value in examining general principles, there is a recurrent, although disputed theme that radiative equilibrium has direct relevance to the observed atmospheric structure. This proposition embraces a number of instructive ideas but, before examining them, we consider some of the observational evidence that motivates them. From the earliest days following the discovery of the stratosphere, theoretical workers assumed that the stratosphere, unlike the troposphere, was in radiative equilibrium. The reasoning was that no forms of heat transport, other than radiative, could be important in a highly stable atmosphere. Since nothing was known about planetary-scale motions at that time, this conclusion was premature. If we turn to modern data, Fig. 9.1a presents the observed climatological temperatures in the middle atmosphere, to be compared with radiative equilibrium calculations shown in Fig. 9.1b. Agreement between theory and observation is fairly good except in the region of the polar winter.

Radiation Calculations in a Clear Atmosphere

This chapter is concerned with the requirements of numerical weather prediction and general circulation models. These numerical models always assume a stratified atmosphere and utilize a limited number of grid points in the vertical direction. Computations are repeated at many horizontal grid points and at frequent time intervals; a premium is placed on computational economy. The nested integrals involved in radiative flux and heating calculations, particularly the frequency integration, can create an unacceptable computational burden unless approximated. In this chapter we limit attention to clear-sky conditions, i.e., to absorbing constituents and a thermal source function (§2.2). For a Planck function, the formal solution, (2.86), is a definite integral involving measurable quantities, temperatures, and gaseous densities. Scattering problems, on the other hand, involve the intensity in the source function and cannot be solved by a single application of this integral. Scattering calculations will be discussed further in Chapter 8; it will be shown that scattering can be neglected if the volume scattering coefficient is not very much larger than the volume absorption coefficient. This is usually the case for aerosols in the thermal region of the spectrum. As regards boundary conditions, it is usual for clear-sky calculations to assume that the earth’s surface and the upper and lower surfaces of clouds can be treated as black surfaces in the thermal spectrum. Equations (2.86) and (2.87) are stated in terms of general boundary conditions. In the flux and heating integrals, (2.106) and (2.110), these conditions are specialized to a black surface at ground level, but they can be generalized without difficulty to include a black surface at any level or partial reflection from these surfaces, if appropriate. The equations for which efficient algorithms are required are the flux equations, (2.107) and (2.108), the heating equations, (2.110) or (2.111), and the solar flux equations, (2.115). The nested integrals are 1. the vertical integral, (2.92), for the optical depth; 2. the integral, (2.86), along the optical path; 3. the angular integral, (2.102); 4. an integral over all frequencies. We may introduce the issues by considering a restricted example, that of the intensity recorded outside the atmosphere by a downward pointing satellite spectrometer.

Band Models

Radiative heating calculations in the atmosphere involve four distinguishable scales of frequency. First, there is the comparatively slow variation with frequency of the Planck function and its derivative with respect to temperature. About one-half of the radiation from a black body at terrestrial temperatures lies in a wave number range of 500 cm-1. The second scale is that of the unresolved contour of a band. For atmospheric molecules other than water vapor, the Planck function is effectively constant over a single band; water vapor bands must be divided into sections of the order of 50 cm-1 wide before this is so. For a rotating molecule, the next relevant scale of frequency is that of the spacing between rotation lines, approximately 1-5 cm-1. Finally, there is the monochromatic scale on which the absorption coefficient may be treated as a constant, and for which Lambert’s absorption law is obeyed: of the order of one-fifth of a line width ≃ 2 x 10-2 cm-1 for a gas at atmospheric pressure, down to 2 x 10-4cm-1 for a Doppler line in the middle atmosphere. This step takes us to a division of the frequency scale that, when taken together with other features of the calculation, presents a formidable computation task. Calculations can, of course, be made and are made at this limiting spectral resolution (line-by-line calculations) but, despite the fact that they are technically feasible with modern computers, such calculations are rare and are usually performed to provide a few reference cases. The great majority of investigations make use of averages over many lines, embracing spectral ranges that are small compared to a band contour (narrow-band models), or over complete bands (wide-band models), or over the entire thermal spectrum (emissivity models.) There are a number of reasons for working with spectral averages. Practical considerations are that important classes of laboratory measurements, and most atmospheric observations (e.g., satellite radiometry) are made with some spectral averaging, often comparable to that of narrow-band models.