SOLAR RADIATION AND PLANETARY TEMPERATURE

The temperature of a planet irradiated by solar radiation can be estimated by balancing the amount of radiation absorbed (Ra) against the amount of outgoing radiation (Ro). The Ra will be the product of:

(i) Solar irradiance (I)

(ii) Area of planet. Area of planet relevant for such calculations is the area of the planet as seen by incoming radiation which is given by r² where r = radius of the planet.

(iii) Absorbed fraction of radiation. The fraction of radiation that is absorbed is given by (1 – A where A = albedo of planet. This albedo represents the fraction of radiation that is reflected back from the planet.

Figure. 1. Energy balance of the Earth. (Components values in kcal/cm²/year).

Thus the energy absorbed by the planet will be:

Ra = I ^ (1 – A)

Intensity of outgoing radiation of a body is given by Stefan-Boltzmann Law i.e.

Io =  T⁴

where = Stefan-Boltzmann Constant = 5.6 x 10^-8 Wm^-2 K^-4. The total energy radiated by the planet will be the product of the intensity of outgoing radiation (Io) and the area of the whole planetary surface giving out radiation (4  r ²). Thus, the outgoing radiation (Ro) from the planet is given by:

Ro = 4r² T⁴

Effective planetary temperature

Since Ra = Ro i.e. system is assumed to be in steady-state where radiation absorbed and outgoing radiation are equal, an expression for the effective planetary temperature (Te) can be obtained from the above equation and it may be given by:

Te = [I – (1 – A)/4 ]^0.25

In this expression of effective planetary temperature, effect of atmosphere has not been taken into account. For Earth, solar irradiance (I) at the top of atmosphere is about 1.4 x 10³/m²/s and albedo of Earth as a whole is about 0.33. From these values, the calculated equilibrium temperature of Earth comes to be 254 K. However, the actual observed average ground level temperature of Earth is about 288 K. This higher effective temperature of Earth from the calculated value is due to the greenhouse-effect of atmosphere.

The black-body spectrum of Earth at 288 K shows that radiation from Earth is of much longer wavelength and is at much lower intensity than radiation from Sun. The absorption spectrum of Earth’s atmosphere overlaps fairly well with the solar emission spectrum. Except for a very narrow window in the absorption bands, much of the long-wave radiations from Earth correspond with the region of absorption in the atmosphere. This means that much of the incoming radiation reaches the Earth’s surface while the outgoing thermal radiation is largely absorbed by the atmosphere rather than being lost to space. Thus, the effect of atmosphere is to trap the outgoing thermal radiation. This effect is termed green-house effect.. The thermal radiation i.e. the heat trapped by the atmosphere due to green-house effect is responsible for the effective temperature of Earth being higher than the temperature calculated without taking into account the effect of atmosphere. In general, absorption of re-emitted long-wave radiation and vertical mixing processes determine the temperature profile of the lower part of atmosphere (troposphere) which in turn determine the Earth’s temperature.

Optical depth of atmosphere and Earth’s surface temperature

The atmosphere is not transparent to the outgoing long-wave radiation and much of this radiation is absorbed in the lower part of the atmosphere, which is warmer than the upper parts. Simple radiative equilibrium models have been developed for Earth and to account for this effect, these models divide the atmosphere into layers that are just thick enough to absorb the outgoing radiation. These atmospheric layers are said to be optically thick and the atmosphere is discussed in terms of its optical depth based on the number of these atmospheric layers of different optical thickness. Earth’s atmosphere is sometimes said to have two layers while that of planet Venus has almost 70 layers which are largely due to enormous amount of CO₂ in the atmosphere of Venus. The radiation equilibrium model indicates that the effective planetary temperature (Te) is thus related to ground-level planetary temperature (Tg) by the equation:

Tg⁴ = (1 – )Te⁴ (where  = optical depth of atmosphere)

The optical depth of atmosphere increases with increase in atmospheric concentrations of carbon dioxide and water vapor because both these are principal atmospheric absorbers of outgoing long-wave radiation. With increasing concentrations of CO₂ in lower layers of atmosphere, other such gases that are responsible for radiating heat to outer space are pushed to slightly higher and colder levels of atmosphere. The radiating gases will radiate heat less efficiently because they are colder at higher altitudes. Thus, the atmosphere becomes less efficient radiator of heat and this results in rise of atmospheric temperature. This rise in atmospheric temperature, in turn, leads to more evaporation and increase in atmospheric water vapor, which is a greenhouse gas and further increases the absorption of outgoing long-wave thermal radiation. This positive feedback results in further increase in atmospheric temperature. The model also suggests that increase in atmospheric CO₂ is associated with decrease in temperatures of upper (stratospheric).

Vertical heat transport and Earth’s surface temperature

Simple models of radiation balance of atmosphere do not take into account various other processes that transport heat vertically in the atmosphere and, therefore, overestimate the surface temperature of Earth. Convection is major process of vertical heat transport and is very important in lowering the surface temperature. Convection occurs because warm air is lighter than cool air and so rises upwards carrying heat from Earth’s surface to the upper atmosphere. As warm air rises up, it expands due to fall in pressure and work done in expansion causes it to cool adiabatically. Thus,

C_v T = – P V (where Cv = molar heat capacity at constant volume)

Ideal gas equation PV = RT takes the differential form P dV + V dP = R dt which may be rearranged in incremental form as:

– P V + R  = V P

This equation may be combined with equation C_v T = – P V using the fact that C_p – C_v = R, where C_p = molar heat capacity at constant pressure = 29.05 J/mol/K. This results in following equation:

C_p T = (C_v + R) T = – P V + R T = V P = (RT/P) P ………….(a)

It can be shown that P/P = – M_mg z /RT where M_m = mean molecular weight of air = 0.028966 kg/mol; g = acceleration due to gravity = 9.8065 m/s/s; z = altitude. This gives:

RT/P = – Mmg z/ P……………………………………………….(b)

Substitution of the above equation (b) in equation (a) gives:

C_p T = – M_mgz

or, T/ z = – (M_mg/Cp)

For Earth’s atmosphere, the lapse rate ( T/ z) works out to be -9.8 k/km for dry air. However, the air is usually wet and as it rises up, it releases latent heat so the measured lapse rate is -6.5 K/km.

If atmospheric temperature falls much less slowly with height than the lapse rate (or even rises with height) then inversion conditions exist and air is very stable with respect to vertical convective mixing. Conversely, if temperature falls very rapidly with height, at a rate greater than lapse rate, then the atmosphere is unstable and convective mixing will be active.

Short-wave radiation and temperature

The discussion till now has assumed total transparency of atmosphere to incoming solar radiation. Though it is true for visible range of radiation, it is not true for ultra-violet region of the solar spectrum. Though the amount of such short-wave radiation is very small, it has important consequences for the temperature of Earth-atmosphere system.

Various ultra-violet wavelengths are absorbed in the atmosphere at different heights. At just over 40 km, absorption of ultra-violet radiation by ozone results in considerable warming of stratosphere and in this zone, temperature rises with altitude. Average temperature of stratosphere is 250 K. Considering it to be a black-body radiator, maximum power radiation would be expected at 11.5 µm. This value is very close to absorption band of carbon dioxide which means that this gas also plays important role in stratospheric temperature. Increase in concentration of carbon dioxide in stratosphere might allow more effective radiation from stratosphere and, therefore, its cooling. This effect is quite opposite to that noted for troposphere.

Further, at the altitude of thermosphere, atmosphere is very thin. In this zone, molecules are exposed to unattenuated solar radiation of extremely short wavelength i.e. of high energy. This radiation arises from the outer region of Sun. At wavelengths below 50 nm, effective emission temperature exceeds 10,000 K. High-energy solar protons of such wavelengths are absorbed by gas molecules giving them high transitional energies i.e. high temperatures. The energies may be large enough to dissociate oxygen and nitrogen. Temperatures in thermosphere undergo wide variations depending upon the state of Sun. During solar disturbances, output of high-energy protons is very much enhanced that results in very high atmospheric temperatures. Temperature in this zone may further be increased by another mechanism. The temperature is normally defined in terms of transitional energy but absorption and emission of radiation occur through vibrational and rotational changes. In upper atmosphere, the frequency of molecular collisions is relatively low and so exchange of translational, vibrational and rotational energies is infrequent. Hence the cooling of thermosphere by re-radiation is very inefficient. The temperature of thermosphere increases with height so it is also stable against convection. Heat can be lost only by very inefficient diffusion processes and as a result, thermospheric temperatures are extremely high.

SOLAR RADIATION ON EARTH

The Sun of planet Earth emits radiation at a temperature of about 6000 degrees K. Average radiation emitted by Sun at its surface (Sun’s radiant emittance) is 73×10⁶ W per square meter. Spectrum of this solar radiation shows distinct emission lines indicating that it comprises of radiation of different wavelengths. The intensities and thus the magnitudes of the radiation of different wavelengths are also different. The intensity of far ultra-violet radiation is very low due to absorption of radiation by outer photosphere of the Sun. Further, extreme ultra-violet and X-ray parts of solar radiation are emitted from the chromosphere and corona regions of the Sun. These regions have temperatures as high as 1 million K. The solar radiation falling at the upper boundary of Earth’s atmosphere is termed incident solar radiation. Its average magnitude over the Earth is given by (Solar constant x r²)/4r², where r = radius of Earth.

The Solar constant is the irradiance on an area at right angle to solar beam and outside the Earth. Its value is 1353 W per square meter. The average incident solar radiation at upper boundary of Earth’s atmosphere is approximately 11 x 10⁹ J/m² /yr. Important factors that affect this solar radiation received by Earth are:

1. Spherical shape of Earth: Earth is a sphere and, therefore, the angle at which incoming solar radiation strikes the upper boundary of atmosphere is not same at all points. The radiation strikes Earth at right angle in the center but the angle gradually becomes more acute towards periphery. As a result, the amount of solar radiation reflected back from the upper surface of atmosphere is zero at the center and increases gradually towards periphery. Thus, the amount of radiation penetrating atmosphere and entering Earth-atmosphere system is maximum in the center and gradually decreases towards periphery.

2. Orbit of Earth: The orbit of Earth around Sun is not perfectly circular but is slightly elliptic. Sun occupies one focus of this elliptic orbit. The mean distance between Sun and Earth is about 150 million kilometers. However, the orbit of Earth is elliptical and so the distance changes during different times in the year. Earth is closest (about 91.5 million miles) to Sun on about January 3, at which time it is said to be in perihelion. It is at greatest distance (about 94.5 million miles) from Sun on about July 4 when it is said to be in aphelion. These differences in distance also cause some difference in amount of solar radiation received by Earth. However, the ellipticity of orbit is not the reason of seasons on Earth. An important fact related to the Earth’s orbit around Sun is that the geometry of Earth’s orbit is not constant. The orbit at present is very slightly elliptical being nearly circular but the shape of Earth’s orbit changes cyclically from almost circular to markedly elliptical and back with a periodicity of 100,000 years. The solar constant is the mean solar irradiance on an area at upper boundary of Earth’s atmosphere perpendicular to incoming solar beam, which is about 1353 W per square meter. In the present state of Earth’s orbit being nearly circular, the difference between perihelion and aphelion is about 3.5% and the difference in solar constant at these two points is about 6.66%. In the state of most elliptic orbit, difference in solar constant at perihelion and aphelion may be as large as 30%.

3. Inclination of Earth’s axis of rotation: The axis of rotation of Earth is not perpendicular to the plane of ecliptic i.e. the plane in which Earth’s orbit and Sun lie. Earth’s axis of rotation makes an angle of about 66.5 degrees with the plane of ecliptic and is tilted 23.5 degrees from the line perpendicular to plane of ecliptic. The Earth’s axis although always makes an angle of 66.5 degrees with plane of ecliptic, also maintains a fixed orientation with respect to stars. The Earth’s axis continues to point to the same spot in the heavens as it makes its yearly circuit around Sun. This inclination of Earth’s rotational axis alongwith its fixed orientation throughout the whole orbit around Sun causes different seasons on Earth. Between September 23 and March 21, North Pole of Earth’s axis is tilted towards the Sun and South Pole is away from the Sun. During this period, Northern Hemisphere has summers and Southern Hemisphere has winters. In this period, daylength and, therefore solar radiation received increases towards North Pole and decreases towards South Pole. From March 21 to September 23, South Pole is tilted towards Sun and North Pole is tilted away from it, North Hemisphere has winters and South Hemisphere has summers during this period. In this period, daylength and, therefore, solar radiation received increases towards South Pole and decreases towards North Pole. Maximum tilts of North Pole and South Pole towards Sun occur on June 21 and December 22 respectively and these dates are termed summer solstice and winter solstice respectively. Midway between the dates of solstices, twice the Earth’s axis is at right angle to the line drawn from Sun to Earth and neither pole is tilted towards Sun. This condition occurs on March 21 or 22 (vernal equinox) and on September 22 or 23 (autumn equinox). Two important cyclic changes related with inclination of Earth’s axis of rotation have been noted. First is the wobbling of Earth’s axis of rotation with a periodicity of 21,000 years. This causes continuous and cyclic hemispheric variation in the solar constant. Second change is cyclic variation in the angle of inclination of Earth’s axis of rotation within the range of 21.8^o and 24.4^o (23.45 degrees at present) with a periodicity of about 40,000 years. Therefore, the distribution of solar irradiance at Earth’s two hemispheres varies continuously with this 40,000 years cyclic periodicity.

1TRANSFORMATIONS OF SOLAR RADIATION

The general fate of the incident solar radiation in Earth-atmosphere system is as below:

(a) Absorbed in stratosphere (mainly by Ozone) = 3%

(b) Absorbed in troposphere by:

(i) Carbon dioxide = 1%

(ii) Water vapor = 12%

(iii) Dust = 2%

(iv) Water droplets in clouds = 3%

(c) Reflected from clouds = 21%

(d) Scattering back into atmosphere 6%

(e) Reflected back from Earth’s surface = 4%

(f) Received at Earth’s surface as:

(i) Direct radiation = 27%

(ii) Diffused radiation via clouds or downward scattering = 21%

Solar radiation received by Earth i.e. incident solar radiation undergoes various transformations after entering the uppermost boundary of atmosphere. The solar radiation is absorbed by atmosphere, hydrosphere, lithosphere and biosphere. Some part of solar radiation absorbed in a component provides energy for the dynamic functions of that component. The remaining part of absorbed radiation is re-emitted from the component as long-wave radiation. Two important features in the study of the transformation of solar radiation are albedo and effective radiation, which are discussed below.

1.1Albedo

The fraction of solar radiation received by a body that is returned back from it forms the albedo of that body. In general, the radiation reflected back from clouds (21%), scattered back into atmosphere (6%) and reflected back from the Earth’s surface (4%) together constitute Earth’s albedo. The average value of albedo of Earth as a whole comes to about 33% or 0.33 (represented as fraction of unity).

The albedo of Earth as a whole has two components:

(a) Albedo of Earth’s surface: It is that fraction of radiation received at the Earth’s surface, which is returned back from the surface. Its value varies depending on the extent of snow cover, vegetation and the soil characteristics. Average albedo values for the snow range from 0.7 to 0.8 and may be as low as 0.4 to 0.5 in case of wet or dirty snow. In the deserts that have light sandy soil and are without vegetation, surface albedo values are typically 0.4 to 0.5. Albedo of damp soil is usually less than that of the corresponding dry soil. In case of damp chernozem soils the albedo values may be as low as 0.05. Albedo of natural Earth surface covered with thick vegetation cover generally ranges from 0.1 to 0.25. Areas covered with coniferous forests have lower albedo than those covered with meadows.

The height of sun in the sky determines the absorption of radiation in the water bodies, mainly the oceans. When sun is relatively high, radiation reaches water surface at high angle. A large part of the incoming radiation penetrates upper layers of water body and is absorbed. When sun is low, the radiation reaches the water surface at low angles and most of it is reflected. Thus it does not penetrate much and the albedo value of water surface increases sharply at low sun. However, in case of diffused radiation, albedo of water surface is much less variable and is about 0.1.

(b) Albedo of Earth-atmosphere system: It is more complex in nature than that of Earth’ surface. Its value is largely determined by the presence-absence, nature and thickness of clouds. In the absence of clouds in the sky, albedo of Earth-atmosphere system depends largely on the albedo of Earth’s surface. If clouds are present, a large portion of solar radiation reaching atmosphere is reflected back from the upper surface of clouds and albedo value of system increases. Albedo of clouds is usually 0.4 to 0.5. In the presence of clouds, albedo of Earth-atmosphere system is usually greater than that of Earth’s surface, except where surface is covered with relatively clean snow.

1.2Effective radiation

It is the difference between the amount of Earth’s radiation from Earth’s surface and the amount of long-wave counter-radiation from atmosphere. Most important factor connected with effective radiation is ‘green-house effect’ due to presence of atmosphere. The atmosphere has various gases viz. carbon dioxide, methane, water vapor etc. which selectively absorb long-wave radiation. Due to this Earth’s atmosphere is relatively more transparent to short-wave radiation than to long-wave radiation. Since long-wave radiation from the Earth’s surface is trapped in the atmosphere, average effective radiation from Earth’s surface as a whole is much lower than the short-wave radiation absorbed at the surface.

The effective radiation of Earth’s surface largely depends on the temperature at Earth’s surface, atmospheric humidity and clouds. Experimental data has shown that radiation of Earth’s natural surfaces is generally quite close to the radiation of black body at corresponding temperatures. Further, a significant part of long-wave radiation lost from Earth’s surface is compensated by long-wave counter-radiation from the atmosphere. This counter-radiation mainly depends on the amount of atmospheric water vapor i.e. air humidity and clouds and so these factors affect the effective radiation of Earth’s surface.