JEE 661 ATMOSPHERIC BOUNDARY LAYER SCIENCE

Lecture 3: Radiation in the Atmosphere

Contents

1. Solar Radiation Quantities
2. Apparent Position of the Sun
3. Solar Radiation Outside the Atmosphere
4. Effects of the Atmosphere and the Earth on Solar Radiation
5. Terrestrial Radiation from the Earth's Surface
6. Terrestrial Radiation from Clouds
7. Longwave Radiation from the Atmosphere
8. Radiation Fluxes at the Earth's Surface

Solar Radiation Quantities

Beam Irradiance

Beam irradiance G_b (direct solar radiation) is the flux from the sun's disc on a surface perpendicular to the beam. It is about 0.9kW/m² in bright sunlight.

Diffuse Irradiance

Diffuse irradiance G_d (diffuse solar radiation) is the solar radiation from the sky, omitting the sun's disc, on a horizontal surface. It is about 0.1kW/m² under a clear sky, and 0.3kW/m² to 0.6kW/m² under cloudy skies.

Global Irradiance

Global irradiance G (global solar radiation) is the sum of the direct and diffuse irradiances on a horizontal surface (facing upwards). It is given by:

G = G_bcos theta_z + G_d,

where theta_z is the zenith angle of the sun (see below).

Apparent Position of the Sun

The zenith angle theta_z is the angle between the zenith directly above the site, and the direction of the sun (see Fig. 6).

The azimuth alpha is the angle between the south point on the horizon and the foot of the perpendicular from the sun to the horizon, positive towards the west (see Fig. 6).

Fig. 6. The zenith angle theta_z, and the azimuth alpha of the sun.

The zenith angle and azimuth of the sun can be calculated from the time of day and day of the year, and the latitude and longitude of the site.

Zone Mean Time (ZMT) is the mean time on the standard meridian. Clocks are set to ZMT.

Local Mean Time (LMT) is the mean time on the meridian of the site. To get LMT, subtract from ZMT 4min per degree of longitude the site is west of the standard meridian.

Example: The standard meridian in Thailand is 105°E. Bangkok has longitude 100.5°E. Therefore LMT in Bangkok is 18min behind ZMT.

Apparent Solar Time (AST) is such that the sun crosses the meridian of the site at exactly 12:00 noon AST. The ellipticity of the Earth's orbit around the sun, and the inclination of the Earth's polar axis to the plane of the orbit, cause the difference between AST and LMT to vary.

The Equation of Time E is the difference:

E = AST - LMT.

Let n_d be the day of the year (Jan 1 = 1, ... Dec 31 = 365). Let theta_d = 360°(n_d - 80)/365. Then:

E = 0.013 + 7.342cos(theta_d - 194.9°) + 9.939cos(2.theta_d - 93.1°) minutes.

The Hour Angle omega of the sun is the angle between the meridian and the plane containing the Earth's axis and the sun, positive towards the west. Then:

omega = 15(AST - 12) degrees,

where AST is in hours.

The Declination delta of the sun is the angluar distance of the sun north of the equator. It can be calculated each day of the year using the formula:

delta = 0.386 + 23.272cos(theta_d - 92°) + 0.4cos(2.theta_d - 19.2°) degrees.

The zenith angle theta_z and azimuth alpha of the sun can be found from:

cos theta_z = cos phi.cos delta.cos omega + sin phi.sin delta
sin alpha = (cos delta.sin omega)/sin theta_z
cos alpha = (sin phi.cos delta.cos omega - cos phi.sin delta)/sin theta_z,

where phi is the latitude of the site. At Bangkok phi is 13.5°.

Refraction by the atmosphere decreases the zenith angle theta_z by 0.5° when the sun is on the horizon. The decrease by refraction is less than 0.1° when theta_z < 80°.

Solar Radiation Outside the Atmosphere

The spectrum of solar radiation outside the atmosphere is that of a blackbody at temperature 5900K. The energy distribution in the spectrum is: ultra-violet 8%, visible 44%, near infra-red 48%. The mean flux density is:

I₀ = 1.37 ± 0.02kW/m².

Variations in the flux density due to the ellipticity of the Earth's orbit are given by:

I = I₀[1 - 0.0335sin(360°(n_d - 94)/365)], where n_d is the day in the year.

Effects of the Atmosphere and the Earth on Solar Radiation

Scattering by air molecules, water vapour molecules, water droplets, and dust particles causes 6% of solar radiation to be returned to space; 20% of solar radiation reaches the surface of the Earth as diffuse radiation.

Air molecules scatter blue light the most and red light the least. This makes the sky blue and the setting sun red.

Scattering by large particles (water droplets and dust particles) is independent of wavelength, so mist and haze look white.

Above 40km absorption is mainly by ozone (O₃); this absorbs 3% of solar radiation. At lower levels water vapour absorbs about 14%, mainly in the near infrared.

Clouds absorb very little solar radiation, which is why clouds do not evaporate. Their effect on solar radiation is by scattering and reflection.

The albedo of the Earth's surface is the fraction of solar radiation reflected. It depends on the colour of the surface. Examples of albedos with the sun overhead are given in Table 4.

Note: When the sun is low in the sky the albedo of water is much greater than 0.1.

Solar Radiation Under Clear Skies

Solar radiation from a clear sky depends on the zenith angle of the sun, the water content of the atmosphere, and the turbidity (dust and smoke). Values of the global solar irradiance G and the direct solar irradiance perpendicular to the beam G_b for low water vapour content (20mm precipitable water) and low turbidity are given in Table 5.

Effects of Cloud

Direct solar irradiance perpendicular to the beam G_b is reduced or eliminated by cloud. Diffuse solar irradiance G_d may be greater than under a clear sky, or less. Thin layers of cloud increase G_d, and thick layers reduce G_d.

Global solar irradiance G is usually reduced by cloud. But if the sun is in a clear part of the sky, and there are bright clouds nearby, then G may be greater than under a totally clear sky.

Shortwave Radiation Balance at the Earth's Surface

Fig. 7. Shortwave radiation flux balance at the Earth's surface.

The shortwave radiation fluxes at the surface of the Earth are shown in Fig. 7, where G is the global solar irradiance. If the reflectance or albedo of the surface is rho, and the absorptance of the surface is alpha, then the radiation balance is given by the following equations:

G = rho.G + alpha.G,   rho + alpha = 1.

Terrestrial Radiation from the Earth's Surface

This is upward longwave (far infrared) blackbody radiation at the temperature of the surface. The radiant flux density phi is given by the Stefan-Boltzmann Law:

phi_surface = sigma.T⁴,   where sigma = 5.67×10^-8W/m²K⁴.

Terrestrial Radiation from Clouds

The downward longwave radiation flux density from clouds phi_cloud is given by:

phi_cloud = epsilon_cloud.sigma.T⁴,

where epsilon_cloud is the emittance of the cloud (See Table 6).

The relation between the emittance and absorptance of clouds (and other gases in the atmosphere) is given by Kirchoff's Radiation Law, which states that

At a given wavelength lambda the emittance epsilon_lambda and the absorptance alpha_lambda are equal.

Longwave Radiation from the Atmosphere

In clear air longwave radiation is emitted and absorbed by the following gases in the following wavelength bands:

• Water vapour: 5µm to 8µm, and 18µm to 30µm.
• Carbon dioxide: 3µm to 5µm, and 13µm to 17µm.
• Ozone: 9µm to 10µm.

In the atmospheric window from 8µm to 13µm the atmosphere is almost transparent. (The effect of ozone is very small.)

Downward atmospheric irradiance on a horizontal surface is a function of surface air temperature T. From a clear sky the Idso-Jackson formula can be used:

L₀/(sigma.T⁴) = 1 - 0.261exp[-0.000777(T - 273)²],

where T is in kelvins.

The effective sky temperature T_e is the temperature of a blackbody giving the same irradiance as the sky (see Table 7 for clear skies).

Two corrections to the values of the downward atmospheric irradiance in Table 7 are needed to obtain good estimates of the actual downward atmospheric radiation.

Corrections for diurnal variations of air temperature lapse rate near the surface:

In the afternoon with high temperature at the surface and cooler air above:

L_actual approx= L₀ - 20W/m².

At dawn with low temperature at the surface and warmer air above:

L_actual approx= L₀ + 15W/m².

Corrections for longwave radiation by clouds:

L_actual = L₀ + (sigma.T⁴ - L₀)kn^1.4,

where n is the cloud amount (for a clear sky: n = 0, for an overcast sky: n = 1), and k depends on the type of cloud (for low cloud: k = 0.86, for middle cloud: k = 0.50, for high cloud: k = 0.17).

Radiation Fluxes at the Earth's Surface

The radiation fluxes at the Earth's surface are shown in Fig. 8. All the downward atmospheric radiation is absorbed by the surface. The upward longwave radiation emitted by the surface is blackbody radiation at the surface temperature.

Fig. 8. Shortwave (solar) and longwave radiation fluxes at the Earth's surface.
T₀ is the temperature of the Earth's surface.

By R. H. B. Exell, 2001. King Mongkut's University of Technology Thonburi.
Back to Home Page