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Wavelength (I0-6 m)

Figure 2.36. Spectral absorption efficiency of selected gaseous constituents of the atmosphere, and for the atmosphere as a whole (bottom) (from Fleagle and Businger, An Introduction to Atmospheric Physics, Academic Press, 1963).

The concentrations of nitrogen oxides in the stratosphere may be increased by a number of human activities. The ozone concentrations vary between years (Almquist, 1974), but anthropogenic emissions of fluorocar-bons are believed to dominate the picture, as they have been demonstrated to cause increased ozone to accumulate near the poles (cf. Fig. 2.35).

While the ultraviolet part of the solar spectrum is capable of exciting electronic states and molecular vibrational-rotational levels, little visible light is absorbed in the atmosphere. The energy is insufficient for most electronic excitations, and few molecular bands lie in this frequency region. Of course, harmonics of lower fundamental bands can be excited, but the cross section for such absorption processes is very low.

In the infrared region, at wavelengths above 7 X 10-7 m, the fundamental vibrational and rotational bands of several molecules are situated. They give rise to clearly identifiable signatures of a large number of molecules, including H2O, CO2, N2O, CH4, CO, SO2, H2S, NO, NO2 and NH3. A few of these absorption spectra are shown in Fig. 2.36. Owing to the variations in water content, as well as the seasonal changes in the concentration of those molecules formed in biological cycles, the combined absorption spectrum is far from invariant.

Models for the description of scattered radiation

The cross section for scattering processes in the atmosphere is large enough to make multiple scattering important. The flux reaching a given scattering centre is thus composed of a unidirectional part from the direction of the Sun, plus a distribution of intensity from other directions. On the basis of scattering of radiation on atoms and molecules much smaller than the wavelength (the so-called Rayleigh scattering, described above), the intensity distribution over the sky may be calculated, including the direct (unscat-tered) part as well as the simply and multiply scattered parts. Figure 2.37 gives the result of such a calculation.

Figure 2.37. Luminance distribution over the sky (as a function of the zenith angle and azimuth relative to that of the Sun, for the direction of observation): (a) for pure Rayleigh atmosphere (multiple Rayleigh scattering on gases only), (b) for Mie atmosphere (including multiple Mie scattering on particles typical of continental Europe for a cloudless sky) and (c) measured on a clear day in the Swiss Alps. The three luminance distributions have been arbitrarily normalised to 15 at the zenith. The Sun's position is indicated by a circle (based on Möller, 1957).

Figure 2.37. Luminance distribution over the sky (as a function of the zenith angle and azimuth relative to that of the Sun, for the direction of observation): (a) for pure Rayleigh atmosphere (multiple Rayleigh scattering on gases only), (b) for Mie atmosphere (including multiple Mie scattering on particles typical of continental Europe for a cloudless sky) and (c) measured on a clear day in the Swiss Alps. The three luminance distributions have been arbitrarily normalised to 15 at the zenith. The Sun's position is indicated by a circle (based on Möller, 1957).

Assuming the solar radiation at the top of the atmosphere to be unpolar-ised, the Rayleigh scattered light will be linearly polarised, and the distribution of light reaching the Earth's surface will possess a component with a finite degree of polarisation, generally increasing as one looks away from the direction of the Sun, until a maximum is reached 90° away from the Sun (Fig. 2.38).

The Rayleigh distribution of luminance (intensity) over the sky does not correspond to observations, even on seemingly clear days without clouds or visible haze (see Fig. 2.37). The reason for this is the neglect of scattering on particulate matter. It follows from the particle size distribution given in Fig. 2.31 that particles of dimensions similar to the wavelength of light in the solar spectrum are abundant, so that a different approach to the problem must be taken.

The theory of scattering of radiation on particles, taking into account reflection and refraction at the surface of the particle, as well as diffraction due to the lattice structure of the atoms forming a particle, was developed by Mie (1908). The cross section for scattering on a spherical particle of radius r is expressed as a= n r2K(y, n), where the correction factor K multiplying the geometrical cross section depends on the ratio of the particle radius and the wavelength of the radiation through the parameter y = 2k r/X and on the refraction index (relative to air) of the material forming the particle, n.

Figure 2.38. Distribution of polarisation over the sky (in %) for a typical clear sky atmosphere (based on Sekera, 1957).

Numerical calculations of the function K (Penndorf, 1959) are shown in Fig. 2.39 for fixed refraction index. The behaviour is very oscillatory, and the absorption may exceed the geometrical cross section by as much as a factor of four.

Angular distributions of Mie scattered light are shown in Fig. 2.40 for water particles (droplets) of two different size distributions representative for clouds and haze. The Rayleigh scattering distribution is shown for comparison. The larger the particles, the more forward peaked is the angular distribution, and the higher the forward to backward ratio.

In Fig. 2.37b, the results of a Mie calculation of the luminance distribution over the sky are presented. Here, a particle size distribution proportional to r"4 has been assumed, and the mean refraction index of the atmosphere has been fixed at n = 1.33 (Möller, 1957). Multiple scattering has been included as in the case of pure Rayleigh scattering, shown in Fig. 2.37a. For comparison, Fig. 2.37 also shows the results of measurements of luminance performed under the "clean air" conditions of a high-altitude Alpine site. The calculation using the Mie model is in qualitative agreement with the measured distribution, while the Rayleigh calculation is in obvious disagreement with the measurements.

Figure 2.39. Cross-section function K(y, n) for Mie scattering, as function of y = 2n r/X for fixed n = 1.50 (cf. text) (based on Penndorf, 1959).

Figure 2.39. Cross-section function K(y, n) for Mie scattering, as function of y = 2n r/X for fixed n = 1.50 (cf. text) (based on Penndorf, 1959).

Figure 2.40. Angular distributions of radiation for simple (not multiple) scattering on water particles with different size distributions. Mie scattering theory was used in connection with the size distributions of clouds (comprising very large particles or droplets) and haze (smaller particle sizes but still comparable to or larger than the wavelength of solar radiation). For comparison, the angular distribution for Rayleigh scattering (particle sizes small compared to wavelength) is also shown. The intensity is averaged over different polarisation directions (based on Hansen, 1975).

Figure 2.40. Angular distributions of radiation for simple (not multiple) scattering on water particles with different size distributions. Mie scattering theory was used in connection with the size distributions of clouds (comprising very large particles or droplets) and haze (smaller particle sizes but still comparable to or larger than the wavelength of solar radiation). For comparison, the angular distribution for Rayleigh scattering (particle sizes small compared to wavelength) is also shown. The intensity is averaged over different polarisation directions (based on Hansen, 1975).

The mean free path in the atmosphere, averaged over frequencies in the visible part of the spectrum, gives a measure of the visibility, giving rise to a clarity classification of the condition of the atmosphere (this concept can be applied to traffic control as well as astronomical observation, the main interest being on horizontal and vertical visibility, respectively). The atmospheric particle content is sometimes referred to as the turbidity of the air, with the small particles being called aerosols and the larger ones dust.

The types of stored energy

The net radiation fluxes discussed in section 2.2 showed the existence of energy transport processes other than radiation. Energy in the form of heat is transferred from the land or ocean surface by evaporation or conduction and from the atmosphere to the surface by precipitation, by friction and again by conduction in small amounts. These processes exchange sensible* and latent heat between the atmosphere and the oceans and continents. The exchange processes in the atmosphere include condensation, evaporation and a small amount of conduction. In addition, energy is added or removed by transport processes, such as convection and advection. The turbulent motion of the convection processes is often described in terms of overlaying eddies of various characteristic sizes. The advective motion is the result of more or less laminar flow. All motion in the atmosphere is associated with friction (viscosity), so that kinetic energy is constantly transformed into heat. Thus, the general circulation has to be sustained by renewed input of energy. The same processes are involved in the circulation in the oceans, but the quantitative relations between the different processes are different, owing to the different physical structure of air and water (density, viscosity, etc.). As mentioned earlier, the source of energy for the transport processes is the latitude variation of the net radiation flux. Additional transport processes may take place on the continents, including river and surface run-off as well as ground water flow. Direct heat transport in dry soil is very small, because of the smallness of the heat conductivity.

The amount of energy stored in a given volume of air, water or soil may be written

where Wot is the geopotential energy, Wkin is the kinetic energy of external motion (flow), Wsens is the amount of sensible heat stored (internal kinetic motion) and Wlat is the amount of latent heat, such as the energies involved in phase changes (solid to quid, fluid to gaseous, or other chemical rearrangement)**. The zero point of stored energy is, of course, arbitrary. It may

* That is, due to heat capacity

The sum of sensible and latent energy constitutes the total thermodynamic internal enerqy (cf. section 4.1).

be taken as the energy of a state where all atoms have been separated and moved to infinity, but in practice it is customary to use some convenient average state to define the zero of energy. The geopotential energy may be written

where p is the density (of air, water or soil) and g is the gravitational constant at the distance r from the Earth's centre (eventually r is replaced by z = r - rs measured from the surface rs). Both p and g further have a weak dependence on the geographical location (0, X). The kinetic energy Wkm receives contributions both from the mean flow velocity and from turbulent (eddy) velocities.

The sensible heat is

where the heat capacity cP (at constant pressure) depends on the composition, which again may be a function of height and position. All quantities may also depend on time. The latent heat of a given constituent such as water may be written

where Lv (2.27 J kg-1 for water) is the latent heat of vaporisation and Lm (0.33 J kg-1 for ice) is the latent heat of melting. mv is the mixing ratio (fractional content by volume) of water vapour and mw is the mixing ratio of liquid water (applicable in oceans and soil as well as in the atmosphere). For other constituents it may be necessary to specify the chemical compound in which they enter in order to express the content of latent energy.

Total energy fluxes

As in the case of radiation, all relevant energy fluxes must now be specified, and the sum of radiation and non-radiation fluxes (Emomr) gives the net total energy flux,

The sign convention introduced for radiative fluxes should still be applied.

Since many energy conversion processes (e.g. melting of snow and ice, run-off) take place at the surface of the Earth, the fluxes through strictly two-dimensional planes may vary rapidly for small vertical displacements of the defining plane and hence may be inconvenient quantities to work with. For this reason, the net total energy flux through such a plane will, in general, be different from zero, except possibly for year averages. For the upward boundary surface between Earth (continent or ocean) and atmosphere, the total net energy flux may be written

Figure 2.41a,b. Flux of sensible heat from the Earth's surface to the atmosphere (—Es ) for January (above) and July (below) 1997, in W m-2 (based on NCEP-NCAR, 1998).

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